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ISATIS 2011

Oil & Gas Case Studies

Published, sold and distributed by GEOVARIANCES49 bis Av. Franklin Roosevelt, BP 91, 77212 Avon Cedex, France

Web: http://www.geovariances.com

Isatis Release 2011, February 2011

Contributing authors:

Catherine Bleinès

Matthieu Bourges

Jacques Deraisme

François Geffroy

Nicolas Jeannée

Ophélie Lemarchand

Sébastien Perseval

Jérôme Poisson

Frédéric Rambert

Didier Renard

Yves Touffait

Laurent Wagner

All Rights Reserved

© 1993-2011 GEOVARIANCES

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means including photocopying, recording or by any information storage and retrieval sys-

tem, without written permission from the copyright owner.

"... There is no probability in itself. There are only probabilistic mod-els. The only question that really matters, in each particular case, is

whether this or that probabilistic model, in relation to this or that real phenomenon, has or has not an objective meaning..."

G. Matheron Estimating and Choosing - An Essay on Probability in Practice

(Springer Berlin, 1989)

1

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 About This Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 Property Mapping & Risk Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . .161

4.1 Presentation of the Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1624.2 Estimation of the Porosity From Wells Alone. . . . . . . . . . . . . . . . . .1644.3 Fitting a Variogram Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1684.4 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1704.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1734.6 Estimation with External Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1774.7 Cokriging With Isotopic Neighborhood . . . . . . . . . . . . . . . . . . . . . .1804.8 Collocated Cokriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1864.9 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192

5 Non Stationary & Volumetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1995.1 Presentation of the Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2005.2 Creating the Output Grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2025.3 Estimation With Wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2045.4 Estimation With Wells and Seismic . . . . . . . . . . . . . . . . . . . . . . . . .2105.5 Assessing the Variability of the Reservoir Top . . . . . . . . . . . . . . . . .2215.6 Volumetric Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227

6 Plurigaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2496.1 Presentation of the Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2506.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2586.3 Creating the Structural Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2596.4 Creating the Working Grid for the Upper Unit . . . . . . . . . . . . . . . . .2606.5 Computing the Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2696.6 Lithotype Rule and Gaussian Functions . . . . . . . . . . . . . . . . . . . . . .2856.7 Conditional Plurigaussian Simulation. . . . . . . . . . . . . . . . . . . . . . . .2986.8 Simulating the Lithofacies in the Lower Unit . . . . . . . . . . . . . . . . . .3016.9 Merging the Upper and Lower Units . . . . . . . . . . . . . . . . . . . . . . . .313

7 Oil Shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3177.1 Presentation of the Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3187.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3227.3 Fitting a Variogram Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3267.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3287.5 Displaying Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331

8 Multi-layer Depth Conversion With Isatoil . . . . . . . . . . . . . . . . . . . . .335

2

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3368.2 Field Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3368.3 Loading the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.4 Master File Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468.5 Building the Reservoir Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 3618.6 Filling the Units With Petrophysics. . . . . . . . . . . . . . . . . . . . . . . . . 3748.7 Volumetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3808.8 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

9 Geostatistical Simulations for Reservoir Characterization . . . . . . . . 4159.9 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4179.10 General Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4189.11 Data Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4209.12 Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4239.13 2D Petrophysical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.14 Modeling 3D Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4539.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

5

Introduction

6

1 About This Manual

Note - The present document only contains case studies related to a specific field of application. The full Case Studies Manual can be downloaded on Geovariances web site.

A set of case studies is developed in this manual. It is mainly designed:

• for new users to get familiar with the software and gives some leading lines to carry a study through,

• for all users to improve their geostatistical knowledge by following detailed geostatistical workflows.

Basically, each case study describes how to carry out some specific calculations in Isatis as precisely as possi-ble. The data sets are located on your disk in a sub-directory, called Datasets, of the Isatis installation directory.

You may follow the work flow proposed in the manual (all the main parameters are described) and then com-pare the results and figures given in the manual with the ones you get from your test.

Most case studies are dedicated to a given field (Mining, Oil & Gas, Environment, Methodology) and therefore grouped together in appropriate sections. However, new users are advised to run a maximum of case studies, whatever their field of application. Indeed, each case study describes different functions of the package which are not necessarily exclusive to one application field but could be useful for other ones.

Several case studies, namely In Situ 3D Resources Estimation (Mining), Property Mapping (Oil & Gas) and Pollution (Environment) almost cover entire classic geostatistical workflows: exploratory data analysis, data selections and variography, monovariate or multivariate estimation, simulations.

The other Case Studies are more specific and mainly deal with particular Isatis facilities, as described below:

• Non Linear: anamorphosis (with and without information effect), indicator kriging, disjunctive kriging, uniform conditioning, service variables and simulations.

• Non Stationary & Volumetrics: non stationary modeling, external drift kriging and simulations, volume-tric calculations, spill point calculation, variable editor.

• Plurigaussian: an innovative facies simulation technique.

• Oil Shale: fault editor.

• Isatoil: multi-layer depth conversion with the Isatoil advanced module.

8 Case Studies

• Young Fish Survey, Acoustic Fish Survey: polygons editor, global estimation.

• Image Filtering: image filtering, grid or line smoothing, grid operator.

• Boolean: boolean conditional simulations.

Note - All case studies are not necessarily updated for each Isatis release. Therefore, the last update and the corresponding Isatis version are systematically given in the introduction.

Property Mapping & Risk Analysis 237

4 Property Mapping & Risk Analysis

This case study is based on a real data set kindly provided by AMOCO for teaching purposes, and that has been used in the AAPG publication Stochastic Modeling and Geostatistics, edited by Jeffrey M. Yarus and Richard L. Chambers. It demonstrates several capabilities offered by Isatis to cope with two variables whose coverage of the field are different, typically a few wells on one hand and a complete 3D seismic on the other hand. The study covers the use of estimation and simulations, from Kriging to Cokriging, External Drift and Collocated Cokriging. Last update: Isatis version 2012

238

4.1 Presentation of the Dataset

First, create a new study using the Study / Create facility of the File / Data File Manager window.

(snap. 4.1-1)

Then, set the Preferences / Study Environment / Units:

m default input-output length unit in foot,

m X, Y and Z graphical axis in foot.

The datasets are located in two separate ASCII files (in the Isatis installation directory, under the Datasets/Petroleum sub-directory):

m The file petroleum_wells.hd contains the data collected at 55 wells. In addition to the coordi-nates, the file contains the target variable (Porosity) and the selection (Sampling) which concerns the 12 initial appraisal wells,

m The file petroleum_seismic.hd contains a regular grid where one seismic attribute has been measured: the normalized acoustic impedance (Norm AI). The grid is composed of 260 by 130 nodes at 40ft x 80ft.

Both files are loaded using the File / Import / ASCII facility in the same directory (Petroleum), in files respectively called Wells and Seismic.

Property Mapping & Risk Analysis 239

(snap. 4.1-2)

Using the File / Data File Manager, you can check that both files cover the same area of 10400ft by 10400ft. You can also check the basic statistics about the two variables of interest.

At this stage, no correlation coefficient between the two variables can be derived, as they are not defined at the same locations.

In this case study, the structural analysis will be performed using the whole set of 55 wells, whereas any estimation or simulation procedure will be based on only the 12 appraisal wells, in order to produce stronger differences in the results of various techniques.

Variable Porosity (from Wells) Norm AI (from Seismic)

Number of samples 55 33800

Minimum 6.1 -1

Maximum 11.8 0.

Mean 8.2 -0.551

Std Deviation 1.4 0.155

240

4.2 Estimation of the Porosity From Wells Alone

The first part of this case study is dedicated to the mapping of the porosity from wells alone. In other words, we simply ignore the seismic information. This step is designed to provide a compari-son basis, although it would probably be skipped in an industrial study. The spatial correlation of the Porosity variable is studied through the Statistics / Exploratory Data Analysis procedure. The following figures are displayed: a base map where the porosity variable is represented with propor-tional symbols, an histogram and the omnidirectional variogram calculated for 10 lags of 1000ft. In the Application / Graphic Specific Parameters of the Variogram window, the Number of Pairsoption is switched ON.

(fig. 4.2-1)

The area of interest is homogeneously covered by the wells. The Report Global Statistics item from the Menu bar of the variogram graphic window produces the following printout where the vario-

0 5000 10000 X (ft)

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6 7 8 9 10 11 12 Porosity

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Nb Samples: 55Minimum: 6.1Maximum: 11.8Mean: 8.2Std. Dev.: 1.4

73

94

199

217194

160203

142 99

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Property Mapping & Risk Analysis 241

gram details can be checked. The number of pairs is reasonably stable (above 70) up to 9000ft: this is consistent with the regular sampling of the area by the wells.

Variable : Porosity Mean of variable = 8.2 Variance of variable = 1.862460 Rank Number Average Value of pairs distance 1 73 1301.15 1.143562 2 94 1911.80 1.460053 3 199 2906.72 1.863894 4 217 4054.00 2.068571 5 194 5092.86 1.987912 6 160 5882.27 1.817500 7 203 6895.25 1.909532 8 142 8014.89 2.118310 9 99 8937.23 2.070556

Coming back to the variogram Application / Calculation Parameters, ask to calculate the vario-gram cloud. Highlight pairs corresponding to small distances (around 1000ft) and a high variability on the variogram cloud: these pairs are represented by asterisks on the variogram cloud; the corre-sponding data are highlighted on the base map and joined by a segment. No point in particular can be designated as responsible for these pairs (outlier): as usually, they simply involve the samples corresponding to high porosity values.

242

(fig. 4.2-2)

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Property Mapping & Risk Analysis 243

(fig. 4.2-3)

To save this experimental variogram in a Parameter File in order to fit a variogram model on it, click on Application / Save in Parameter File and call it Porosity.

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244

4.3 Fitting a Variogram Model

Within the procedure Statistics / Variogram Fitting, define the Parameter File containing the exper-imental variogram (Porosity) and the one which will contain the model. The latter may also be called Porosity; indeed, although these two Parameter Files have the same name, there will be no confusion as their type is different. Visualize the experimental variogram and the fitted model using any of the graphic windows; as there is only one variable and one omnidirectional variogram, the global and the fitting windows are similar. From the Model Initialization frame, select Sphericaland Add Nugget. These are the structures that will be fitted on the experimental.

The model can be fitted using the Automatic Fitting tab by pressing Fit.

(snap. 4.3-1)

Pressing the Print button in this panel produces the following printout where we can check that the model is the nesting of a short range spherical and a nugget effect.

Property Mapping & Risk Analysis 245

(snap. 4.3-2)

The corresponding graphic representation is presented in the next figure.

(fig. 4.3-1)

A final Run(Save) saves this model in the Parameter File Porosity.

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246

4.4 Cross-Validation

The cross-validation technique (Statistics/Modeling/Cross-validation) enables you to evaluate the consistency between your data and the chosen variogram model. It consists in removing in turn one data point and re-estimating it (by kriging) from its neighbors using the model previously fitted.

An essential parameter of this phase is the neighborhood, which tells the system which data points, located close enough to the target, will be used during the estimation. In this case study, because of the small number of points, a Unique neighborhood is used; this choice means that any information will systematically be used for the estimation of any target point in the field. Therefore, for the cross-validation, each data point is estimated from all other data.

This neighborhood also has to be saved in a Parameter File that will be called Porosity.

(snap. 4.4-1)

When a point is considered, the kriging technique provides the estimated value Z* that can be com-

pared to the initial known value Z, and the standard deviation of the estimation * which depends on the model and the location of the neighboring information. The experimental error between the estimated and the true values (Z - Z*) can be scaled by the predicted standard deviation of the esti-

mation ( *) to produce the standardized error. This quantity, which should be a normal variable,

σ

σ

Property Mapping & Risk Analysis 247

characterizes the ability of the variogram model to re-estimate correctly the data values from their neighboring information only. If the value lies outside a given interval, the point requires some attention: defining for instance the interval as [-2.5 ; 2.5] (that is to say, setting the threshold to 2.5), enables to focus on the 1% extreme values of a normal distribution. Such a point may arbitrarily be called an "outlier".

The procedure provides the statistics (mean and variance) of the estimation raw and standardized errors, based on the 55 data points. The same statistics are also calculated when the outliers have been removed: the remaining data are called the robust data.

Statistics based on 55 test data Mean Variance Error -0.00533 1.18778 Std. Error -0.00257 1.02851 Statistics based on 53 robust data Mean Variance Error 0.10776 0.88043 Std. Error 0.10193 0.76465 A data is robust when its Standardized Error lies between -2.500000 and 2.500000

Note - The key values of this printout are the mean error, which should be close to zero, and the variance of the standardized error which should be close to 1. It is not recommended to pay too much attention to the variance of the results obtained on the «robust data» alone, as the model has been fitted taking this outlier into account.

The procedure also provides four standard displays which reflect the consistency between the data, the neighborhood and the model: each sample is represented with a + sign whose dimension is pro-portional to the variable, whereas the outliers are figured using a l symbol. They consist of:

m the base map of the variable,

m the histogram of the standardized errors,

m the scatter plot of the true value versus the estimated value,

m the scatter plot of the standardized error of estimation versus the estimated value.

248

(fig. 4.4-1)

A last feature of this cross-validation is the possibility of using this variance of standardized error (score) to rescale the model. As a matter of fact, the kriging estimate and therefore the estimation error does not depend on the sill of the model, whereas the variance of estimation is directly propor-tional to this sill. Multiplying the sill by the score ensures that the cross-validation performed with this new model, all other parameters remaining unchanged, provides a variance of standardized error of estimation exactly equal to 1.

This last possibility must be manipulated with caution, especially if the score is far from 1 as one can hardly imagine that the only imperfection in the model could be its sill. Instead, it is recom-mended to check the outliers first and possibly re-run the whole procedure (structural analysis and cross-validation).

In the following, the Porosity variogram model is considered to be the best possible one.

Property Mapping & Risk Analysis 249

4.5 Estimation

The task is to estimate by kriging the value of the porosity based on the 12 appraisal wells at the nodes of the imported seismic grid, using the fitted model and the unique neighborhood.

The kriging operation is performed using the Interpolate / Estimation / (Co-)Kriging procedure. It is compulsory to define:

l the variable of interest (Porosity) in the Input File (Wells). As discussed earlier, the estimation operations will be performed using the 12 appraisal wells only. This is the reason why the Sam-pling selection is specified,

l the names of the output variables for the estimation and the corresponding standard deviation,

l the Parameter File containing the Model: Porosity,

l the Parameter File containing the Neighborhood: Porosity.

(snap. 4.5-1)

250

The Test button can be used to visualize the weight attached to each data point for the estimation of one target grid node. It can also be used to check the impact of a change in the Model or the Neigh-borhood parameters on the Kriging weights.

The 33800 grid nodes are estimated with values ranging from 6.6 to 11.3. These statistics can inter-estingly be compared with the ones from the original porosity variable, which lies between 6.1 and 11.8. The difference reflects the smoothing effect of kriging.

The kriging results are now visualized using several combinations of the display capabilities. You are going to create a new Display template, that consists in an overlay of a grid raster and porosity data locations. All the Display facilities are explained in detail in the "Displaying & Editing Graph-ics" chapter of the Isatis Beginner's Guide.

Click on Display / New Page in the Isatis main window. A blank graphic page is popped up, together with a Contents window. You have to specify in this window the contents of your graphic. To achieve that:

l Firstly, give a name to the template you are creating: Phi. This will allow you to easily display again this template later.

l In the Contents list, double click on the Raster item. A new window appears, in order to let you specify which variable you want to display and with which color scale:

m In the Data area, in the Petroleum / Seismic file select the variable Kriging (Porosity),

m Specify the title that will be given to the Raster part of the legend, for instance Phi,

m In the Graphic Parameters area, specify the Color Scale you want to use for the raster dis-play. You may use an automatic default color scale, or create a new one specifically dedi-cated to the Porosity variable. To create a new color scale: click on the Color Scale button, double-click on New Color Scale and enter a name: Porosity, and press OK. Click on the Edit button. In the Color Scale Definition window:

- In the Bounds Definition, choose User Defined Classes.

- Click on the Bounds button, enter 14 as the New Number of Classes, 6 and 13 as the Min-imum and Maximum values. Press OK.

- In the Colors area, click on Color Sampling to choose regularly the 25 colors in the 32 colors palette. This will improve the contrast in the resulting display.

- Switch on the Invert Color Order toggle in order to affect the red colors to the large Phi values.

- Click on the Undefined Values button and select for instance Transparent.

- In the Legend area, switch off the Automatic Spacing between Tick Marks button, enter 10 as the reference tickmark and 1 as the step between the tickmarks. Then, specify that you do not want your final color scale to exceed 6 cm. Switch off the Automatic Formattoggle, and enter 0 as the number of digits. Switch off the Display Undefined Values tog-gle.

- Click on OK.

Property Mapping & Risk Analysis 251

m In the Item contents for: Raster window, click on Display current item to display the result.

m Click on OK.

(snap. 4.5-2)

l Back in the Contents list, double-click on the Basemap item to represent the Porosity variable with symbols proportional to the variable value. A new Item contents window appears. In the Data area, select Wells / Porosity variable as the Proportional Variable and activate the Sam-pling selection. Leave the other parameters unchanged; by default, black crosses will be dis-played with a size proportional to the Porosity value. Click on Display Current Item to check your parameters, then on Display to see all the previously defined components of your graphic.

Click on OK to close the Item contents panel.

252

l In the Item list, you can select any item and decide whether or not you want to display its leg-end. Use the Up and Down arrows to modify the order of the items in the final Display.

l Close the contents window. Your final graphic window should be similar to the one displayed hereafter.

(fig. 4.5-1)

The label position may be modified using the Management / View Label / Move unconstrained

The * and [Not saved] symbols in the name of the graphic page indicate that some recent modifica-tions have not been stored in the Phi graphic template, and that this template has never been saved. Click on Application / Store Page to save them. You can now close your window.

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Property Mapping & Risk Analysis 253

4.6 Estimation with External Drift

Actually, two types of data are available:

l one scarce data set containing few samples of good quality (this usually corresponds to the well information),

l one data set containing a large amount of samples covering the whole field but with poor accu-racy (this usually corresponds to the seismic information).

In this case, one well-known method consists in integrating these two sources of information using the Kriging with External Drift technique. It consists in performing the standard kriging algorithm, based on the variable measured at the wells, considering that the drift (overall shape) is locally rep-resented by the seismic information. This requires such information (or background) to be known everywhere in the field or at least to be informed densely enough so that the value at any point (well location, for instance) can be obtained using a quick local interpolation.

As in any kriging procedure a model is required about the spatial correlation. In the External Drift case, this model has to be inferred knowing that the seismic information serves as a local drift: this refers to the Non-stationary Structural Analysis.

The application Interpolate / Estimation / Bundled External Drift Kriging provides all these steps in a single procedure which assumes that:

l the seismic background is defined on a regular grid. It is interpolated at the well locations from the target nodes using a quick bilinear interpolator.

l the model of the target variable (measured at the wells) taking the seismic information into account as a drift can be either provided by the user interactively or automatically calculated in the scope of the Intrinsic Random Functions of order k theory, using polynomial isotropic gen-eralized covariances. For more information about the structural analysis in IRF-k, the user should refer to the "Non stationary modeling" technical reference (available from the On-Line documentation). The only choice using the automatic calculation is whether to allow a nugget effect as a possible component of the final model or not. To impose the estimation to honor the well information and avoid misties, a quite common practice is to forbid this nugget effect com-ponent.

Still using the Sampling selection and the unique neighborhood (Porosity), the procedure first determines the optimal structure forbidding any nugget effect component and then performs the estimation.

The results are stored in the output grid file (called seismic) with the following names:

254

l ED Kriging (Porosity) for the estimation,

l ED Kriging St. Dev. (Porosity) for its standard deviation.

(snap. 4.6-1)

The printout generated by this procedure details the contents of the optimal model that has been used for the estimation:

====================================================================== | Structure Identification | ====================================================================== .../... Drift Identification ==================== The drift trials are sorted by increasing Mean Rank The one with the smallest Mean Rank is preferred Please also pay attention to the Mean Squared Error criterion T1 : 1 f1 T2 : 1 x y f1 Mean Mean Sq. Mean Trial Error Error Rank T2 9.194e-03 5.547e-01 1.417 T1 1.370e-02 6.223e-01 1.583 Results are based on 12 measures Covariance Identification ========================= The models are sorted according to the scores (closest to 1. first) When the Score is not calculated (N/A), the model is not valid as the coefficient (sill) of one basic structure, at least, is negative S1 : Order-1 G.C. - Scale = 1462.30ft S2 : Spline G.C. - Scale = 1462.30ft S3 : Order-3 G.C. - Scale = 1462.30ft

Property Mapping & Risk Analysis 255

Score S1 S2 S3 0.869 1.099e-01 2.141e-02 0.000e+00 1.192 0.000e+00 6.281e-02 0.000e+00 0.771 1.871e-01 0.000e+00 0.000e+00 1.869 0.000e+00 0.000e+00 3.409e-02 Successfully processed = 12 CPU Time = 0:00:00 (0 sec.) Elapsed Time = 0:00:00 (0 sec.)

The 33800 grid nodes are estimated with values ranging from 5.1 to 13.3 and should be compared to the one of the data information where the porosity varies from 6.1 to 11.8.

To display the ED Kriging result, you can easily use the previously saved display called Phi. Click on Display / Phi in the main Isatis window. You just need to modify the variable defined in the Grid Raster contents: replace the previous Kriging (Porosity) by ED Kriging (Porosity) and click on Display.

(fig. 4.6-1)

The impact of the seismic information used as the external drift is clear, although both estimations have been carried out using the same amount of data (hard) information, namely the 12 appraisal wells.

The External Drift method can be seen as a linear regression of the variable on the drift information. In other words, the result is a combination of the drift (scaled and shifted) and the residuals. The usual drawbacks of this method are that:

l the final map resembles the drift map as soon as the two variables are highly correlated (at the well locations) and tends to ignore the drift map in the opposite case.

l the drift information is used as a deterministic function, not as a random function and the esti-mation error does not take into account the variability of this drift.

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4.7 Cokriging With Isotopic Neighborhood

One drawback of the previous method is the lack of control on the quality of the correlation between the variable measured at the wells and the seismic information. This paragraph will focus on this aspect.

Cokriging is the traditional technique for integrating several variables in the estimation process: the estimation of one variable at a target point consists of a linear combination of all the variables avail-able at the neighboring points. This method is obviously more demanding than the kriging algo-rithm as it requires a consistent multivariate model.

When all variables are not known at the same locations and particularly when an auxiliary variable (here seismic) is densely sampled, one problem is the choice of the neighborhood. Here seismic will be used only where the porosity is known (isotopic neighborhood).

4.7.1 Structural Analysis

To derive a multivariate model, some of the information on both variables has to be defined at the same points. This is not directly possible as the porosity is defined at 55 wells and the normalized acoustic impedance is measured on the output grid: the two variables are in two different files. Therefore, the preliminary task consists in "getting" the values of the seismic information at the well locations. Due to the high density of the seismic grid, all quick local interpolation techniques will give similar results. The simplest one is offered by the Tools / Migrate / Grid to Point proce-dure which gives to a well the value of the closest grid node. This is how we define the new variable in the Wells data file, called Impedance at wells.

(snap. 4.7-1)

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The Statistics / Exploratory Data Analysis application is used to check the correlation between the two variables: on the basis of the 55 wells, the correlation coefficient is 0.826 and is visualized in the following scatter diagram where the linear regression line of the impedance versus the porosity has been plotted. The two simple variograms and the cross-variogram are also calculated for 10 lags of 1000ft each, regardless of the direction (omnidirectional).

(fig. 4.7-1)

Note - The variance of the acoustic impedance variable sampled at the 55 well locations (0.027) is close from the variance of the variable calculated on the entire data set (0.024).

The calculation parameters being similar to the previous (monovariate) structural analysis, the sim-ple variogram of Porosity has obviously not changed. This set of experimental variograms is saved in a Parameter File called Porosity & Impedance.

The Statistics / Variogram Fitting procedure is used to derive a model which should match the three experimental variograms simultaneously. To fit a model in a multivariate case, in the framework of the Linear Coregionalization Model, the principle is to define a set of basic structures by clicking the Edit button. Any simple or cross variogram will be expressed as a linear combination of these structures. The two basic structures that will compose the final model are:

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l a nugget effect,

l a spherical variogram with a range of 4000ft.

Once you have entered the two structures the use of the Automatic Sill Fitting option ensures that the cokriging matrix is positive definite.

(snap. 4.7-2)

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(fig. 4.7-2)

Pressing the Print button in Model Definition panel produces the following printout. This model is finally saved in a new Parameter File called Porosity & Impedance.

Model : Covariance part ======================= Number of variables = 2 - Variable 1 : Porosity - Variable 2 : Impedance at wells .../...Number of basic structures = 2 S1 : Nugget effect Variance-Covariance matrix : Variable 1 Variable 2 Variable 1 0.0039 0.0111 Variable 2 0.0111 0.3162 .../... S2 : Spherical - Range = 4000.00ft Variance-Covariance matrix : Variable 1 Variable 2 Variable 1 0.0258 0.1915 Variable 2 0.1915 1.6755 .../...

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4.7.2 Cross-Validation

The Statistics / Cross-Validation procedure checks the consistency of the model with respect to the data. When performing the cross-validation, in the multivariate case, for each target point, it is pos-sible to choose in the Special Kriging Options:

l to suppress all the variables relative to this point,

l to suppress only the target variable at this point.

Note - The latter possibility is automatically selected in the Unique Neighborhood case. In order to try the first solution, the user should use the Moving Neighborhood instead, which can be extended by increasing the radius (20000ft) and the optimum count of points (54) for the neighborhood search.

(snap. 4.7-3)

The cross-validation results are slightly better than in the monovariate case. This is due to the fact that the seismic information (correlated to the porosity) is used even at the target point where the porosity value is removed.

====================================================================== | Cross-validation | ====================================================================== Statistics based on 55 test data Mean Variance Error -0.00427 0.52898

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Std. Error -0.00293 1.01547 Statistics based on 55 robust data Mean Variance Error -0.00427 0.52898 Std. Error -0.00293 1.01547 A data is robust when its Standardized Error lies between -2.500000 and 2.50000

4.7.3 Estimation

The estimation is performed using the Cokriging technique where, at each target grid node, the porosity result is obtained as a linear combination of the porosity and the acoustic impedance mea-sured at the 12 appraisal wells only (isotopic neighborhood). The Interpolate / Estimation / (Co-)Kriging panel requires the definition of the two variables of interest in the input file (Wells), the model (Porosity & Impedance) and the neighborhood (Porosity). It also requires the definition of the variables in the output grid file (Seismic) which will receive the result of the estimation:Cokriging (Porosity) for the estimation of the porosity and Cokriging St. Dev. (Porosity) for its standard deviation.

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(snap. 4.7-4)

It is obviously useless to compute the estimation of the acoustic impedance obtained by cokriging based on the 12 appraisal wells only.

The 33800 grid nodes are estimated with values ranging from 6.8 to 11.2. The cokriging estimate is displayed using the same parameters as before.

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(fig. 4.7-3)

This map is very similar to the one obtained with the porosity variable alone: the few differences are only linked to the auxiliary variable (seismic information) and to the choice of the multivariate model.

Obviously, a large amount of information is lost when reducing the seismic information to its value at the well locations only.

The next part of the study deals with the Collocated Cokriging technique, which aims at integrating through a cokriging approach the whole auxiliary information provided by the Norm AI variable,exhaustively known on the seismic grid.

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4.8 Collocated Cokriging

The idea of this technique is to enhance the cokriging process by adding, for each target grid node, the value of the acoustic impedance at this location.

The system resembles the traditional cokriging technique where one additional fictitious sample is added which coincides with the target grid node and for which only the acoustic impedance value is provided. This is therefore an heterotopic case, as both variables are not only informed at the same locations.

The multivariate model defined for the standard cokriging procedure (Porosity & Impedance) is still used here. Concerning the neighborhood (Porosity), the term Unique may be misleading: all the samples of the data file (Wells) are taken into account, but one fictitious sample is added at the target grid node.

Compared to the previous cokriging, it is only compulsory to define, in the Standard (Co-)Krigingwindow:

l a new name for the variable to be created, for instance Collocated Cokriging (Porosity),

l the collocated variable in the Output File variable list: this refers to the seismic information called Norm AI.,

l the collocated cokriging as a Special Kriging Option in the main window; the collocated vari-able in the Input File should be indicated: this refers to the variable carrying the seismic infor-mation called Impedance at Wells (which is defined as target variable #2 in the Input File)

(snap. 4.8-1)

The 33800 grid nodes are estimated with the values ranging from 5.6 to 12.5.

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Note - The kriging matrix systematically involves one extra point whose location varies with the target grid node. Therefore, the Unique Neighborhood computer trick which consists in inverting the kriging matrix only once cannot be exploited anymore. A partial inversion is used instead, but the computing time is significantly longer than for the traditional cokriging.

(fig. 4.8-1)

Compared to the External Drift technique, the link between the two variables is introduced through the structural model rather than via a global correlation: this allows more flexibility as this correla-tion may vary with the distance. This is why it is essential to be cautious when performing the struc-tural analysis.

Collocated Cokriging with Markov-Bayes assumption:

The idea in this paragraph is to take the full advantage of the seismic information, especially during the structural analysis, by choosing a simplified multivariate model based on the seismic informa-tion. This may be useful when the number of wells is not large enough to allow a proper variogram calculation.

The next graphic shows a variogram map obtained from the Exploratory Data Analysis window (last statistical representation at the right) for the Norm AI variable defined on the grid, using 50 lags of 120 ft for the calculation parameters. This tool allows to easily investigate potential anisotropies. In this case, a direction of better continuity N10E can be quite clearly identified: just click on one of the small tickmarks corresponding to directions, on the mouse right button and finally on Activate Direction.

Note - This calculation can be quite time demanding when it is applied to large grids. In such cases, a Sampling selection can be preliminary performed to subsample the grid information; the variogram map calculation is then performed only on this selection.

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(snap. 4.8-2)

It is advised to cautiously analyze this apparent anisotropy. Actually, in the present case, this anisot-ropy is not intrinsic to the impedance behavior over the area of interest; it is more likely due to the presence of a North-South low impedance band around X equal to 2000 to 4000ft. It is therefore ignored and a standard experimental variogram is computed. By default, the grid organization is used, as it allows a more efficient computation of the variogram, for instance along the main grid axes. Switch off the Use the Grid Organization toggle to compute an omnidirectional variogram of the NormAI variable on the grid. Compute 50 lags of 120ft and save (Application / Save in Param-eter File menu of the graphic page) the experimental variogram under a new Parameter File called Norm AI.

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The Statistics / Variogram Fitting procedure is used to fit a model to the acoustic impedance exper-imental variogram. A possible model is obtained by nesting, in Automatic Sill Fitting mode:

l a Generalized Cauchy structure with a range of 1750ft (third parameter equal to 1),

l a spherical variogram with a range equal to 6000 ft.

The following figure presents the resulting model.

(fig. 4.8-2)

To run a Bundled Collocated Cokriging procedure, it is still compulsory to define a completely con-sistent multivariate model for porosity and acoustic impedance.

The idea of the Markov-Bayes assumption is simply to derive the cross-variogram and the vario-gram of the porosity by simply rescaling the acoustic impedance variogram. The scaling factors are obtained by dividing the experimental variances of the two pieces of data using; Var Norm AI (0.0313) / Var Porosity (3.24) = (0.00966) and using the correlation coefficient at wells (0.915) between the 2 variables Porosity and Impedance at wells that can be obtained from the scatter dia-gram of the Exploratory Data Analysis.

Note - This correlation coefficient corresponds to the porosity values within the Samplingselection and the Norm AI background variable after migration from grid to wells location (Grid to point option).

The cokriging process, by construction, operates within the scope of the model of intrinsic correla-tion. In this case, kriging and cokriging lead to the same result for isotopic data sets (all variables informed at all data points). In the collocated cokriging case, an additional acoustic impedance sam-ple, located at the target grid node, is introduced in the estimation process.

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To perform Collocated Cokriging with Markov hypothesis, select the window Interpolate / Estima-tion / Bundled Collocated Cokriging. The results of this bundled Collocated Cokriging process are stored in variables called:

l CB Kriging (Porosity) for the estimation,

l CB Kriging St.dev. (Porosity) for its standard deviation.

(snap. 4.8-3)

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(fig. 4.8-3)

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4.9 Simulations

As a matter of fact, linear estimation techniques, such as kriging or cokriging, do not provide a cor-rect answer if the user is interested in estimating the probability that the porosity overcomes a given threshold. Applying a cutoff operator (selecting every grid node above the threshold) on any of the previous maps would lead to a two-color map (each value is either above or below the thresh-old); this cannot be used as a probability map and it can be demonstrated that this result is biased. At least a simple proof consists in noticing that the standard deviation of the estimation (which proves that this estimated value is not the truth) is not used in the cutoff operation. Drawing a value of the error at random within an interval calibrated on a multiple of this standard deviation, centered on the estimation would correct this fact on a one-grid node basis. But drawing this correction at random for two consecutive nodes does not take into consideration that the estimation (and there-fore its related standard deviation) should be consistent with the spatial correlation model.

A correct solution is to randomly draw several simulations, which reflect the variability of the model, and to transform each one of them into a two-color map by applying the cutoff operator. Then, on a grid node basis, it is possible to count the number of times the simulated value passes the threshold and normalize it by the total number of simulations: this provides an unbiased probability estimate. The accuracy of this probability is ensured when a lot of simulations are drawn assuming that they are all uncorrelated (up to the fact that they share the same model and the same condition-ing data points).

As implemented in Isatis, the simulation technique is based on a random number generator which ensures this independence. Any series of random numbers is related to the value of a seed which is defined by the user. Therefore, in order to draw several series of independent simulations, it suffices to change this seed.

Several simulation techniques are available in Isatis. The one which presents a reasonable trade-off between quality and computing time is the Turning Bands Method which will be used for all tech-niques described in this paragraph. The principle of this technique is to produce a non-conditional simulation first (this is a map which reflects the variogram but does not honor the data) and then to correct this map by adding the map obtained by interpolating the experimental error between the data and the non-conditional simulated value at the data point: this is called conditioning. This last interpolation is performed by kriging (in the broad sense) using the input model. The final map is called conditional simulation. The only parameter of this method is the count of bands that will be fixed to 200 in the rest of this section. For more information on the simulation techniques, the user should refer to the On-Line documentation.

Each conditional simulation is supposed to be similar to the unknown reality. It honors the few wells and reproduces the input variogram (calculated from these few data).

An additional constraint is to reproduce the histogram. Actually, most simulation techniques assume (multi)gaussian distributions. It is therefore usually recommended to transform the original data prior to using them in a simulation process, unless:

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l the experimental histogram is not a good representation of a meaningless theoretical histogram model: this is the case when the data variable is not stationary.

l the variable measured at the data points is already normally distributed.

This can be checked on both variables: the porosity from the well data file and the acoustic imped-ance from the seismic grid file. In Statistics / Exploratory Data Analysis, the Quantile-Quantile plot graphically compares any experimental histogram to a set of theoretical distributions, for instance gaussian in the present case.

(fig. 4.9-1)

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(fig. 4.9-2)

Visual comparison shows that the hypothesis that the distribution is normal does not really hold. Nevertheless, for simplicity, it is decided to perform the simulations directly on the raw variables, bypassing the gaussian anamorphosis operation. Hence, each spatial correlation model used in the estimation section can be used directly.

Note - An example of gaussian transformation (called «anamorphosis») can be found in the Non Stationary & Volumetrics case study, for the thickness variable.

These simulations are illustrated in the next paragraphs in the univariate case and for the external drift technique. Similarly, cosimulations and collocated cosimulations (bundled or real) could be performed using the same model than for the estimation step.

4.9.1 Univariate Simulations

The menu Interpolate / Conditional Simulations / Turning Bands used with the target variable (Porosity) in the Data file (Wells) and with Sampling selection performs sequentially the non-con-ditional simulation and the conditioning with kriging.

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For instance, perform ten simulations using 200 turning bands, storing their results in one Macro Variable called Simu Porosity.

Should you wish to generate several batches of simulations (say 10 at one time), you have to mod-ify the seed for each run, as discussed earlier. You also have to increase the index given to the first simulation by 10 if you want the indices in the Macro Variable to be consecutive.

Finally, specify the model (Porosity) and the neighborhood (Porosity) to be used during the condi-tioning kriging step, based on the 12 appraisal wells only. Two simulation results are displayed below.

(snap. 4.9-1)

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(fig. 4.9-3)

The Tools / Simulation Post Processing facility is used to compute the probability that the porosity is greater than one threshold (9. in this case).

Among various possibilities, define in the Iso-Cutoff Maps one new macro-variable that will con-tain the probability that the variable remains above the threshold 9; the resulting map will be stored under the name Proba Porosity (kriging) {9.000000}. The resulting map is displayed with a new color scale for the probability map (in raster mode); this color scale is derived from the «Red Yel-low» palette. The porosity at the 12 appraisal wells is overlaid using the Symbols type of represen-tation, with + symbols for porosity values above 9 and o symbols for porosity values below.

(fig. 4.9-4)

The noisy aspect of the result is due to the small number of simulations.

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4.9.2 External Drift Simulations

The External Drift Kriging introduces the seismic information as the background shape which con-ditions the local drift (hence its name). The same formalism can be transposed in the simulation domain. The Interpolate / Conditional Simulation / External Drift (bundled) menu offers this possi-bility by nesting the following phases:

l local interpolation of the seismic at the data points,

l determination of the optimal model inferred taking into account the seismic information at the data point as an external drift: as for kriging, we forbid any nugget effect component,

l the conditional simulations.

The resulting Macro variable name is set to Simu Porosity ED.

(snap. 4.9-2)

The next graphic shows two output realizations. Due to the control brought by the seismic informa-tion, the variability between the simulations is much smaller than for the univariate simulations.

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The corresponding probability map (below 9.) is finally displayed.

(fig. 4.9-5)

(fig. 4.9-6)

This probability map reflects the ambiguity of the status of the auxiliary seismic variable used as an external drift: this quantity is assumed to be a known function. Hence this drift component does not introduce any randomness in the simulation process. Moreover, the scaling and shifting factors which are automatically derived by the kriging system remain constant from one simulation to the next one, and even more, they are the same all over the field due to the Unique Neighborhood. Therefore, because of the high level of the correlation between the acoustic impedance and the porosity, the seismic variable controls almost completely the estimation of the probability to exceed a threshold.

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5 Non Stationary & Volu-metrics

This case study is based on a real 2D data set kindly provided by Gaz de France. Its objective is twofold:

• to illustrate the application of geostatistics for non-stationary phe-nomena in the scope of the theory of Intrinsic Random Function of order k (IRF-k) and the use of kriging with external drift,

• to illustrate how volumetric calculations are derived from condi-tional simulations within Isatis. Last update: Isatis version 2012

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5.1 Presentation of the Dataset

The information consists in the depth of an horizon top. It is composed of:

l A few wells in the ASCII file gdf_wells.hd, containing depth measurements in meters corre-sponding to the top of a reservoir and its respective thickness values.

l 2D seismic survey, containing depth measurements in meters corresponding to the same top structure (after velocity analysis), in the ASCII file gdf_seismic.hd.

A new study has first to be created. Then, both data sets are imported using the File / Import / ASCIIprocedure in a new Directory Non Stationary; the Files are called Wells and Seismic.

(snap. 5.1-1)

The Data File Manager can be used to derive the following statistics on:

l the depth measured at the wells Directory Name : Non Stationary File Name : Wells Variable Name : depth at wells .../... Printing Format : Decimal, Length = 10, Digits = 2 MINI= 2197.00 Q.25= 2208.00 Q.50= 2214.50 ST.D/MEAN= 0.0187787 Q.75= 2284.00 MAXI= 2343.00 Defined Samples= 87 / 87 MEAN= 2241.17 ST.D= 42.09

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l the seismic depth Directory Name : Non Stationary File Name : Seismic Variable Name : seismic depth .../... Printing Format : Decimal, Length = 10, Digits = 2 MINI= 2147.00 Q.25= 2190.00 Q.50= 2215.00 ST.D/MEAN= 0.0164406 Q.75= 2235.00 MAXI= 2345.00 Defined Samples= 1351 / 1351 MEAN= 2215.60 ST.D= 36.43

The next figure illustrates a basemap of both data sets: seismic data (black crosses) and the well data (red squares), using two basemap representations in a new Display page. The area covered by the seismic data is much larger than the area drilled by wells.

(fig. 5.1-1)

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5.2 Creating the Output Grid

All the resulting maps will be produced on the same output grid covering the entire seismic infor-mation, even if a lot of extrapolation is then required when working with the well data alone. The procedure File / Create Grid File is used to create the grid file Grid containing 3420 grid nodes, with the following parameters:

l X origin: 327500m, Y origin: 10500m,

l X and Y mesh: 250m,

l X nodes number: 60, Y nodes number: 57.

(snap. 5.2-1)

For comparison purpose a quick estimation of depth at wells is performed, using standard kriging using Interpolate / Quick Interpolation in a Linear Model Kriging mode using all samples for each grid estimation (Unique neighborhood).

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(snap. 5.2-2)

The estimated depth, Depth from wells (quick stat), is displayed below with several types of rep-resentation:

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l a Raster display, using a new color scale ranging from 2200 to 2520 by steps of 10m,

l an Isolines display of the estimated depth, with isolines defined from 2200 to 2500m with a 100m step (thin black lines) and between 2219.5 and 2220.5 with a 1m step to illustrate in bold the 2220m value,

l a Basemap of the wells.

(snap. 5.2-3)

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5.3 Estimation With Wells

The purpose of this section is to perform an estimation of the depth using the information given by the 87 wells alone in a non-stationary framework.

5.3.1 Exploratory Data Analysis

The spatial structure of the depth at wells variable is analyzed through the Statistics / Exploratory Data Analysis procedure. The omnidirectional variogram is computed for 12 lags of 500m, with a tolerance on distance equal to one half of the lag (all pairs are used).

(fig. 5.3-1)

The variogram reaches the dispersion variance (1772) around 3km and keeps rising with a parabolic behavior: this could lead to modeling issues, as the smoothest theoretical variogram model pre-cisely has a parabolic behavior. This is actually a strong indication that the variable is not stationary at the scale of a few kilometers.

5.3.2 Non-stationary Model Fitting

A first solution consists in fitting a generalized covariance of order k (rather than a variogram) where k designates the degree of the polynomial drift which suits the representation of the global behavior of the variable.

The utility Statistics / Non Stationary Modeling performs the structural inference in the scope of the theory of the Intrinsic Random Functions of order k (IRF-k) in the following two steps:

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l Determination of the optimal polynomial drift (among the possible drift trials specified by the user). The default drift trials is selected by pressing the button Automatic (no ext. drift). Once determined, this polynomial drift is substracted from the raw variable to derive residuals.

l Determination of the best combination of generalized covariances, from a list of basic structures that could be modified by the user.

(snap. 5.3-1)

The Parameter File where the Model will be ultimately stored is called Wells. Edit it in order to open the Model Definition panel and ask for the Default Model; it is composed of:

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l A nugget effect,

l A linear generalized covariance term (similar to a linear variogram),

l A spline generalized covariance,

l A third order generalized covariance.

The scale factor of all the basic structures is automatically calculated, being equal to 10% of the field diagonal. The value of these parameters has no consequence on the model, and is just kept by consistency with the variogram definition.

It also requires the definition of the Neighborhood used during this structural inference. Because of the small amount of data, we keep the Unique neighborhood previously defined.

The structural inference in Unique Neighborhood produces the following results:

l The optimal drift is quadratic: this makes sense when trying to capture the dome-shape of the global reservoir.

l The corresponding optimal generalized covariance is composed only of a nugget effect (struc-ture 1), with a sill coefficient of 336.74.====================================================================== | Structure Identification | ====================================================================== Data File Information: Directory = Non Stationary File = Wells Target File Information: Directory = Non Stationary File = Wells Seed File Information: Directory = Non Stationary File = Wells Variable(s) = depth at wells Type = POINT (87 points) Model Name = Wells Neighborhood Name = unique - UNIQUE .../... Drift Identification ==================== The drift trials are sorted by increasing Mean Rank The one with the smallest Mean Rank is preferred Please also pay attention to the Mean Squared Error criterion T1 : No Drift T2 : 1 x y T3 : 1 x y x2 xy y2 Mean Mean Sq. Mean Trial Error Error Rank T3 -3.833e-02 3.996e+02 1.276 T2 -7.317e-01 1.156e+03 2.103 T1 -3.136e-14 1.813e+03 2.621 Results are based on 87 measures Covariance Identification ========================= The models are sorted according to the scores (closest to 1. first)

286

When the Score is not calculated (N/A), the model is not valid as the coefficient (sill) of one basic structure, at least, is negative S1 : Nugget effectS2 : Order-1 G.C. - Scale = 1400.000mS3 : Spline G.C. - Scale = 1400.000mS4 : Order-3 G.C. - Scale = 1400.000m

Score S1 S2 S3 S4 1.042 3.367e+02 0.000e+00 0.000e+00 0.000e+00 1.171 0.000e+00 0.000e+00 2.857e+02 0.000e+00 0.806 0.000e+00 2.007e+02 0.000e+00 0.000e+00 1.447 0.000e+00 1.023e+01 0.000e+00 7.715e+02 1.471 4.710e-01 0.000e+00 0.000e+00 8.652e+02 1.644 0.000e+00 0.000e+00 0.000e+00 1.071e+03 N/A 0.000e+00 0.000e+00 3.004e+02 -2.867e+01 N/A -1.015e+00 3.292e+01 0.000e+00 5.998e+02 N/A -6.779e-01 0.000e+00 3.486e+02 -7.207e+01 N/A 0.000e+00 -2.837e+01 4.476e+02 -1.221e+02 N/A 0.000e+00 -2.307e+01 3.716e+02 0.000e+00 N/A -6.011e-01 0.000e+00 3.095e+02 0.000e+00

Successfully processed = 87CPU Time = 0:00:01 (1 sec.)Elapsed Time = 0:00:02 (2 sec.)

The Model Parameter File (Wells) has been updated

It is frequently observed that after the drift identification process, the resulting residuals present an erratic (non structured) behavior; consequently, a covariance structure composed only of a nugget effect should not be a surprise. In some cases it is advised to force the model to be structured by removing the nugget effect from the list of basic structures. To achieve this, click on Default Modelin the Model Definition window and remove the Nugget Effect from the list. Click on OK and then on Run. This new structural inference using the same Unique Neighborhood produces the follow-ing results for the Covariance Identification Step (the drift identification results obviously remain the same):

.../... Covariance Identification=========================The models are sorted according to the scores (closest to 1. first)When the Score is not calculated (N/A), the model is not validas the coefficient (sill) of one basic structure, at least, is negative

S1 : Order-1 G.C. - Scale = 1400.000mS2 : Spline G.C. - Scale = 1400.000mS3 : Order-3 G.C. - Scale = 1400.000m

Score S1 S2 S3 1.171 0.000e+00 2.857e+02 0.000e+00 0.806 2.007e+02 0.000e+00 0.000e+00 1.447 1.023e+01 0.000e+00 7.715e+02 1.644 0.000e+00 0.000e+00 1.071e+03 N/A -2.307e+01 3.716e+02 0.000e+00 N/A 0.000e+00 3.004e+02 -2.867e+01 N/A -2.837e+01 4.476e+02 -1.221e+02

Successfully processed = 87CPU Time = 0:00:00 (0 sec.)Elapsed Time = 0:00:01 (1 sec.)

The Model Parameter File (Wells) has been updated

Non Stationary & Volumetrics 287

The optimal generalized covariance is composed only of a Spline (structure 2), with a sill coeffi-cient of 285,7.

5.3.3 Estimation

The estimation by kriging can now be performed using the standard procedure Interpolate / Estima-tion / (Co-)Kriging. The target variable depth at wells has to be defined in the Input File Wells, and the names of the resulting variables in the Output File Grid:

l Depth from Wells (Non-Stat) for the estimate,

l Depth from Wells (St. Dev.) for the corresponding standard deviation.

The estimation is performed with the non-stationary model Wells, and the unique neighborhood. The results are visualized in the following figure, where the estimated depth is represented in the same way as for the previously quick estimation. The only difference is that the color scale is mod-ified in order to avoid expanding anymore the values greater than 2520; these values are set to blank.

Additionally, a red isoline is displayed for a standard deviation value equal to 15m. The value, which indicates a rather poor precision, is exceeded almost on the entire field, excepted close to the wells.

288

(snap. 5.3-2)

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5.4 Estimation With Wells and Seismic

The information provided by seismic data is of prime interest in making the depth estimation map more reliable especially in the areas far from the wells. The idea is to consider the seismic as pro-viding correct information of the large scale variability of the depth variable that we usually call the trend: this technique is known as the External Drift Kriging.

For applying this technique, it is necessary to know this external drift information both at the well locations and at the nodes of the final grid. Consequently, a preliminary consists in interpolating the seismic depth at the nodes of the final grid.

5.4.1 Structural Analysis of the Seismic Data

The Statistics / Exploratory Data Analysis feature is used to calculate the Variogram Map on the seismic depth variable. All the parameters are shown in the next graphic. Tolerances equal to 1 on Lags and Direction are used in order to have a better representation of possible anisotropies.

In the variogram map area you can activate a direction using the mouse buttons, left one to select a direction, right one to select Activate Direction in the menu. This variogram map clearly shows a short scale anisotropy, with a direction of maximum continuity about N15°E. The two principal axes have been activated and the experimental variograms confirm this feature. We can observe also a regional trend in the E-W direction.

You have the option to save these experimental variograms using the option Save in ParameterFile... in the Application menu. This set of directional variograms is saved in a new Parameter File Seismic.

Note - Pay attention to the fact that the angular tolerance on each directional variogram is equal to approximately 15° (180° divided in 36 angles, with a tolerance of 1 sector on each side of the direction of interest). Computing standard experimental variograms with a reference direction of N15°E and default angular tolerance (45° divided by the number of directions) could lead to slightly different results.

290

(snap. 5.4-1)

The Statistics / Variogram Fitting procedure is now used in order to fit an anisotropic model on these experimental variograms. In the Model Initialization frame, select Spherical. Then click Con-straint to allow the Anisotropy and lock the spherical sill to 1350. Click Fit to apply the automatic fitting.

By default, a Global Anisotropy is set to an angle consistent with the experimental calculation (equal to 75° in trigonometric convection in this case).

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(snap. 5.4-2)

292

(snap. 5.4-3)

The model is stored in the Standard Parameter File Seismic pressing the Run (Save) button.

(fig. 5.4-1)

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5.4.2 Estimation of the Seismic Depth

The seismic depth can now be estimated at the nodes of the output grid file Grid using the standard (Co-)Kriging panel. A new variable Depth from Seismic (Background) is created in the Output File; there is no need to calculate the standard deviation of the estimation. The variogram model Seismic is used.

The only specificity of this estimation is in the choice of the Neighborhood parameters. Due to the large number of data, a Moving Neighborhood is strongly advised; it is called Seismic. As the data is not regularly spread (high density along lines), a large number of angular sectors (12) is recom-mended to ensure that the neighboring information will surround the target node as regularly as possible. To reduce statistical instabilities, an optimum number of 2 samples per angular sector (hence 24 points) is selected within a neighborhood circle of 7000m radius, and the minimum num-ber of samples is set to 3 samples. The radius could be extended above the longest correlation dis-tance (11km) but, due to the high sample density, this would not improve the estimation. It would simply allow us to go further in the extrapolated areas, which is not our goal.

In order to avoid having clustering data, an advanced parameter setting the "Minimum Distance Between two Selected Samples" to 500m is also used.

294

(snap. 5.4-4)

Non Stationary & Volumetrics 295

(snap. 5.4-5)

296

(fig. 5.4-2)

From the resulting map, it can be observed that:

Non Stationary & Volumetrics 297

l The map using the previous color scale almost covers the whole area.

l The top of the structure (where the wells are located) has a seismic depth around 2150m, while the well information produces a value around 2200m.

Before using Depth from seismic (Background) as an external drift function, it is recommended to verify the correlation between this variable and the depth information at wells. To achieve that, a kriging estimation of the seismic depth variable is performed into the Wells point file, using the same variogram and neighborhood configuration as previously; a new variable Depth from seismic (Background) is created.

The scatter diagram between the two variables is displayed hereafter. The regression line (bold line) and the first bisector (thin line) are indicated. Both variables are highly correlated, and this correlation is linear. Furthermore, the global shift of approximately 50m between seismic depth and depth at wells is obvious.

(fig. 5.4-3)

5.4.3 Estimation with External Drift

The previous map, Depth from seismic (Background), can now be considered as an external drift function for the construction of the final depth map, estimated from the well information. As krig-ing honors the information used as data, this method will ensure the following two goals:

2150 2200 2250 2300 2350

Depth from seismic (Background)

2150

2200

2250

2300

2350

depth at wells

rho=0.978

298

l To provide a surface which closely fits to the depth values given at the wells, avoiding misties.

l To produce a depth map which resembles the seismic map (at least far from the control wells).

This technique may be used with several background variables (external drifts). However, the bun-dled version Interpolate / Estimation / External Drift (bundled) described here allows only one background variable. The method requires the background variable to be known at the well loca-tions: this is automatically provided by a quick bilinear interpolation run on the background vari-able. The Unique Neighborhood is used. The final question concerns the Model which must be inferred knowing that the seismic information is used as External Drift. The procedure offers the possibility of calculating it internally using the polynomial basic structures for the determination of the optimal generalized covariance. In presence of outliers, the procedure often finds a nugget effect as the optimal generalized covariance; it is therefore useful to ask the procedure to exclude the nugget effect component from the trial set of generalized covariances.

(snap. 5.4-6)

The resulting non stationary model is printed during the process, before the kriging with External Drift actually takes place.

====================================================================== | Structure Identification | ====================================================================== Data File Information: Directory = Non stationary File = Wells Variable(s) = depth at wells Target File Information: Directory = Non stationary File = Wells Variable(s) = depth at wells Seed File Information:

Non Stationary & Volumetrics 299

Directory = Non stationary File = Wells Variable(s) = depth at wells Variable(s) = KRIG_DATA Type = POINT (87 points) Neighborhood Name = Unique - UNIQUE .../... Drift Identification ==================== The drift trials are sorted by increasing Mean Rank The one with the smallest Mean Rank is preferred Please also pay attention to the Mean Squared Error criterion T1 : 1 f1 T2 : 1 x y f1 Trial Error Error Rank T2 2.348e-02 7.485e+01 1.460 T1 3.488e-02 7.165e+01 1.540 Results are based on 22 measures

Covariance Identification ========================= The models are sorted according to the scores (closest to 1. first) When the Score is not calculated (N/A), the model is not valid as the coefficient (sill) of one basic structure, at least, is negative

S1 : Order-1 G.C. - Scale = 2033.624m S2 : Order-3 G.C. - Scale = 2033.624m Score S1 S2 1.019 1.100e+02 0.000e+00 1.915 0.000e+00 2.834e+03 N/A 1.131e+02 -3.394e+00

The following graphic representation is performed using the same items as previously:

300

(fig. 5.4-4)

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5.4.4 Conclusions

Although both maps have been derived with the same set of constraining data (the 87 wells):

l the results are similar in the area close to the conditioning wells: in both maps, the top of the structure is reached at 2200m,

l the external drift map is more realistic in the extrapolated area as it resembles the seismic back-ground variable,

l the reliability of the map is estimated to be better on the external drift map: the area where the standard deviation is smaller than 15m is larger.

The next graphic shows the horizontal position of a section line and its respective cross section. Depth is measured in meters, and the horizontal axis measures the distance along the trace AA'.

(fig. 5.4-5)

This graphic is obtained using a Section in 2D Grid representation of the Display facility, applied to the 4 variables simultaneously. The parameters of the display are shown below.

A

A'

320 325 330 335 340 345

X (km)

5

10

15

20

25

Y (km)

Depth from seismic (Background)

Depth from wells (quick stat)

Depth from wells (Non Stat)

Depth from wells (Ext.Drift)

0 5 10 15

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2400

2350

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302

(snap. 5.4-7)

Then, to define the trace you plan to display, you can either:

l enter its coordinates, using the Trace... button in the Contents tab illustrated above,

l digitize the trace in a second display corresponding to a geographic view of the area (basemap or grid); once this graphic is displayed, select Digitize Trace with the right button of your mouse, then select the vertices of the trace with the left button. Once you have finished, click on the right button to terminate and then ask to Update Trace on Graphics with the right button.

Coming back to the Contents window of the trace display, you can modify in the Display Box tab the definition mode for the graphic bounds as well as the scaling factors.

Finally, using Application / Store Page, save this template (call it trace for instance) in order to eas-ily reproduce this kind of cross-section later.

Non Stationary & Volumetrics 303

5.5 Assessing the Variability of the Reservoir Top

5.5.1 External drift bundled simulations of the top reservoir

Using the same model as for the estimation, we consider the Depth from Seismic (Background) as a trend for the actual top reservoir depth. By using this background variable as an external drift variable in the simulation process we will get images of the depth with similar behavior described by the seismic data and honoring the depth at wells.

l Use the Interpolate / Conditional Simulations / External Drift (bundled) menu to perform 100 simulations of the top reservoir depth. Ask to calculate the model without nugget effect, use a Unique neighborhood and set the number of turning bands to 500. The process will create a macro-variable called Simu Top with seismic for this case.

(snap. 5.5-1)

304

l Display a few simulations using the Display menu, with the previous color scale (simu #001).

5.5.2 Analyzing the local variability using Macro Variable Statistics

It is common practice to describe the local variability of a reservoir top using the Tools / Simulation Post-Processing facility. Indeed, the latter allows to compute standard deviation, probability and quantile maps. An example of this application is presented in the Property Mapping case study.

Non Stationary & Volumetrics 305

The purpose of this paragraph is to illustrate another way to analyze the local variability of a reser-voir top. Indeed, the Statistics /Statistics/ Macro Variable Statistics facility enables you to analyze the local distribution of simulated values and to compare them with values of interest such as OWC or neighbouring measured values.

Once the panel is open, the first thing to do is to click on Application / Load Files. Here, you can enter the macro variable to be analyzed, a grid reference variable and also an auxiliary Points file, containing for instance the well intercepts with the top of the reservoir.

(snap. 5.5-2)

Press OK. The following Basemap is displayed:

306

(snap. 5.5-3)

You can change the Basemap graphic preferences, for instance the color scale, by clicking on Appli-cation / Graphic Parameters for... / Basemap.

The local distribution of simulated values can now be obtained simply by clicking on a particular grid node. The selected node is automatically outlined (by default with a bold red line) and the his-togram containing all the simulated depth values for this particular node is displayed.

Usual histogram calculation parameters may be modified from the Application / Calculation Parameters window. The hereafter histogram is obtained with 21 classes ranging from 2280 to 2301m. Several particular values are then superimposed to the histogram, such as:

Non Stationary & Volumetrics 307

l the mean of all simulated values,

l the index of the simulation outcome currently displayed on the basemap (CIV),

l a particular value to be entered by the user below the histogram (URV),

l quantiles of interest.

The values to be displayed can be modified in the Application / Graphic Parameters.

(snap. 5.5-4)

Coming back to the Basemap, you will notice that a right-click produces the following menu.

(snap. 5.5-5)

308

You can then clear the current selection or append additional nodes. Note that instead of selecting an individual node, you can select blocks of nodes by modifying the Selection Neighborhoodparameter below the Basemap.

Finally, this menu allows you to select an auxiliary point, for instance a well close to your current selection. Once you have clicked on an auxiliary point, the corresponding symbol changes from the default «+» to a circle. The depth value, read from the auxiliary Points file, is then automatically superimposed on the histogram, the valued being displayed in the legend (PRV).

(snap. 5.5-6)

Non Stationary & Volumetrics 309

5.6 Volumetric Calculations

Now, geostatistical simulations are going to be used to estimate the distribution of oil/gas volumes above a water contact. These volumetric calculations will be derived from successive simulations of the reservoir top and of the reservoir thickness. The results will be compared to the volumes cal-culated above the spill point.

The depth of the intercepts with the top of the structure is contained in the variable depth at wells, the reservoir thickness is stored in the variable thickness at wells.

5.6.1 Simulation of the reservoir thickness

In Exploratory Data Analysis, displaying an histogram of the thickness at wells shows the exist-ence of an aberrant negative value, due to an undesired negative sign that has to be removed to avoid unrealistic results. This operation can be performed with the File / Variable Editor facility. Inside the panel, select the variable where the value has to be modified; then, select the sample of interest, modify its value to zero and Save (see next page). The next graphic shows the histogram with the modified negative value. The default omnidirectional experimental variogram shows a sta-tionary behavior.

(fig. 5.6-1)

Furthermore, computing a scatter diagram between the depth and thickness variables would show that these variables are not correlated.

22

256

271230

252 295 232

188

169157

0 1 2 3

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0

10

20

30

Variogram : thickness at wells

0 10 20 30

thickness at wells

0.00

0.05

0.10

0.15

Frequencies

Nb Samples: 87Minimum: 0.00Maximum: 29.15Mean: 15.49Std. Dev.: 5.06

310

(snap. 5.6-1)

At this stage 100 stochastic simulations of the thickness have to performed, without an external drift variable and in a stationarity assumption. One possibility is to adjust a variogram model of thickness at wells and perform directly a Conditional Turning Bands procedure, but there is a risk to obtain negative values of thickness. To tackle this point a gaussian anamorphosis modeling of thethickness at wells variable is performed; this approach allows to constrain the lower and upper thickness values.

Open the Statistics / Gaussian Anamorphosis Modeling window and enter the input raw variable thickness at wells in the Data area by pressing the Input... button. Then switch on the toggle Gaus-sian Transform and enter the new output variable name thickness at wells (gaussian). Then click on the interactive fitting... button. The window Fitting Parameters pops up.In the Windows area, clicking on the first icon called Anamorphosis pops up the experimental anamorphosis and the default point model. Click on Application / Graphic Bounds in the menu bar and enter the next val-ues

Horizontal Axis Min: -3.5

Horizontal Axis Max: 3.5

Vertical Axis Min : -5

Vertical Axis Max : 37

Non Stationary & Volumetrics 311

These values only adjust the display of the anamorphosis window. Now click on Interactive Fitting Parameters... in the Anamorphosis Fitting area and enter the following values:

312

(snap. 5.6-2)

The last window authorizes to obtain raw values between 0 and 35.

Non Stationary & Volumetrics 313

Click on the toggle Fitting Stats, the following statistics are displayed :=== Fitting Statistics for thickness at wells ===

Experimental mean = 15.49Theoretical mean (Discr) = 15.55Experimental variance = 25.58Theoretical variance (Discr) = 27.36

Interval of Definition: On gaussian variable: [-2.53 ,2.53] On raw variable: [0.00 ,29.15]

(snap. 5.6-3)

314

(snap. 5.6-4)

Finally give a name to the new anamorphosis function, Thickness at wells and press Run.

As the thickness simulations will be performed on this gaussian transform (before back-transforma-tion in raw scale using the anamorphosis function), it is now requested to evaluate the spatial struc-ture of the thickness at wells (gaussian) variable.

An experimental variogram of this variable is first calculated with 12 lags of 300m and saved in a Parameter File (using the Application menu of the variogram graphic window). called Thickness at wells (gaussian). Using the Statistics / Variogram Fitting window, a new variogram model is cre-ated, with the same name than the experimental variogram. This new model is edited (using manualedit/edit) and modified in order to improve the quality of the fit. A spherical basic structure with a range of 950m and a sill equal to 1.04 is chosen. This model is saved by pressing the Run (Save) button.

Non Stationary & Volumetrics 315

(fig. 5.6-2)

Using the Interpolate / Conditional Simulations / Turning Bands facility, perform 100 simulations of the thickness at wells (gaussian) in Unique Neighborhood using the previous model and activat-ing the Gaussian Back Transformation... option.

(snap. 5.6-5)

The new output macro variable is called Simu Thickness.

15

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0.00

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Variogram : thickness at wells (gaussia

316

(snap. 5.6-6)

You can use the Exploratory Data Analysis to check the outputs and verify that the simulated thick-ness are greater or equal to zero.

The next graphic shows two realizations of the Simu Thickness output.

Non Stationary & Volumetrics 317

(fig. 5.6-3)

(fig. 5.6-4)

To visualize the simulated reservoirs, you can create a new macrovariable corresponding to the base of the reservoir and represent the top and the base in a cross-section. To achieve that:

318

l with Tools / Create Special Variable create a macro variable of length type to store 100 simula-tions of the depth of the reservoir base, called Simu Base reservoir,

l using the File / Calculator, calculate the sum of the simulated top Simu Top with seismic and the last simulated thickness Simu Thickness.

(snap. 5.6-7)

Non Stationary & Volumetrics 319

l To display a cross-section, you can use the previous template Trace and replace the Cross-sec-tion in 2D contents by a realization of the top and the base of the reservoir (hereafter the top and the base of simulation number 42).

(fig. 5.6-5)

5.6.2 Calculation of Gross Rock Volumes

In order to restrict the calculations to the main structure (where most of the wells have been drilled) it is recommended to use the polygon contained in the ASCII file polygon.hd. This polygon is imported in Isatis using the File / Polygons Editor panel:

l Create a new polygon file, called Polygon for Volumetrics, in the Application / New Polygon File menu.

l You may display the Wells data, with Application / Auxiliary Data.

l Ask for an ASCII import of the file polygon.hd.

l The polygon, called P1, is displayed on the top of your wells data. Finally, Save and Run your polygon file.

Using the Tools / Volumetrics panel, the GRV will be calculated for the reservoir limited by the sim-ulated surfaces of the Top and Bottom and the Gas water contact (GWC) at the constant depth of 2288m. Enter the macro variables names for the reservoir top and bottom.

Pay attention to match the indices between these macrovariables. To achieve that, switch ON the toggle Link Macro Index to: Top Surface when you choose the second macro variable.

Top (Simu #42)

Base (Simu #42)

A B

0 1 2 3 4 5

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A

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320

Note - To be able to use in the Volumetrics panel the macro-variables previously created, you need to ensure that these macro-variables are of length type. If it is not the case, an error is produced; then, you have to go in the Data File Manager, click on the macro-variable and ask to modify the Format, with the right-button of the mouse. You have the possibility to specify that you want your macro-variable to be of length type and the unit, in meter in the present case.

(snap. 5.6-8)

l In Risk Curves / Edit specify that you are interested in the distribution of the volumes and choose an appropriate format. Also switch on the Print Statistics toggle. Click on Close and then on Run. You obtain the following statistics and quantiles for the P1 polygon:Statistics on Volume Risk Curves================================Polygon: P1Smallest = 367.44Mm3Largest = 500.49Mm3Mean = 422.32Mm3St. dev. = 26.50Mm3

Quantiles on Volume Risk curves===============================Global Field Integration P90.00 = 476.23Mm3 P50.00 = 705.81Mm3 P10.00 = 1003.03Mm3

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Quantiles on Volume Risk curves===============================Polygon: P1 P90.00 = 390.70Mm3 P50.00 = 422.87Mm3 P10.00 = 456.19Mm3

(snap. 5.6-9)

(fig. 5.6-6)

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l You can also derive specific thickness maps from this procedure, such as iso-frequency maps (for instance P10 and P90 maps), iso-cutoff maps (to derive for instance the probability for the thickness to exceed 10 or 20m) or statistical maps (thickness mean or standard deviation).

5.6.3 Determination of the spill point

The Tools / Spill Point panel enables the delineation of a potential reservoir. Considering a topo-graphic map (given on a regular grid) where the depth is counted downwards positively (the top of a structure corresponds to the lowest depth value), the aim of this procedure is to find the elevation (Spill Elevation) of the deepest horizontal plane which subdivides the field into areas inside the res-ervoir and areas located outside the reservoir.

Firstly, enter the macrovariable containing the top reservoir simulations, and create new output variables for the spill point, the mean height above spill and the probability to have a reservoir (above the spill). Isatis pops up a Display Grid Raster of the Macro for Depth/Elevation. It is advised to enter in Application / Map Graphic Parameters... and select your depth Color Scale.

(snap. 5.6-10)

The inside / outside reservoir constraints have to be digitized on the depth map before the Run.

Non Stationary & Volumetrics 323

(snap. 5.6-11)

324

To do this, you have to click with the mouse right-hand button on the graphic, and ask to:

m Digitize as: Inside for the first constraint inside the reservoir structure (green circle)

m Digitize as: Outside for the second constraint, located outside the reservoir (blue circle)

In the Application menu of the depth map graphic, you may ask to Print Information on Con-straints. They should be approximately located as follows:

Constraints characteristics (2 points) Rank X Y 1 334834.08m 15526.15m Inside 2 339925.24m 18263.27m Outside

Switch on the Map of the Mean Height above spill, Map of the Reservoir Probability, the Distribu-tion of Spill Elevations and the Distribution of Reservoir Volumes buttons. Set up the units to Mm3 by clicking on the Print Parameters... button.

Click on Run. Isatis will pop up the requested results; for what concerns the grid displays, it is advisable to enter in the Application / Map Graphic Parameters... menu and customize the Color Scale.... A grey color scale is used to represent the Reservoir Probability Map and a rainbow color scale to represent the mean height above the spill point.

(snap. 5.6-12)

(snap. 5.6-13)

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(snap. 5.6-14)

326

(snap. 5.6-15)

Spill Point calculation results =============================== Num : the relative rank of the outcome Macro : the absolute rank of the outcome in the MACRO variable IX0,IY0 : the coordinates of the Spill point (in grid nodes) ACC : the acceptation criterion YES if the outcome is valid or the rank of the (first) violated constraint Spill : Elevation of the Spill point Thick : Maximum thickness of the Reservoir Res. Vol.: Volume of the Reservoir (Unknown is not included) Unk. Vol.: Unknown volume Num Macro IX0 IY0 ACC Spill Thick Res. Vol. Unk. Vol. 10 10 17 15 Yes 2265.379m 69.424m 547.13m3 0.35m3 82 82 15 17 Yes 2266.296m 70.477m 597.62m3 1.22m3 78 78 38 7 Yes 2266.994m 72.872m 584.76m3 6.22m3

Non Stationary & Volumetrics 327

97 97 14 16 Yes 2267.007m 71.591m 561.78m3 23.84m3 4 4 16 15 Yes 2267.394m 72.691m 558.00m3 4.45m3 63 63 14 16 Yes 2267.524m 72.423m 606.63m3 32.50m3 93 93 13 16 Yes 2272.503m 75.501m 746.57m3 44.72m3 7 7 2 11 Yes 2272.744m 77.715m 794.38m3 29.30m3 69 69 16 13 Yes 2274.536m 79.723m 732.06m3 0.17m3 88 88 14 18 Yes 2275.137m 81.492m 756.87m3 2.34m3 58 58 15 15 Yes 2275.549m 77.512m 779.35m3 1.57m3 37 37 12 16 Yes 2275.675m 82.212m 818.17m3 2.37m3 .../... Statistics on Reservoir Volumes==============================Total count of outcomes = 100Count of selected outcomes = 100Count of Valid selected outcomes = 100

Spill Thick Res. Vol. Unk. Vol.Mean (All) 2285.568m 89.958m 1082.29m3 21.54m3Mean (Valid) 2285.568m 89.958m 1082.29m3 21.54m3St. dev (All) 8.896m 9.146m 277.33m3 25.93m3St. dev (Valid) 8.896m 9.146m 277.33m3 25.93m3Minimum (All) 2265.379m 69.424m 547.13m3 0.17m3Minimum (Valid) 2265.379m 69.424m 547.13m3 0.17m3Maximum (All) 2304.631m 109.831m 1826.83m3 125.52m3Maximum (Valid) 2304.631m 109.831m 1826.83m3 125.52m3

In this case the output print is sorted by spill elevations. Three spill elevations are below 2270m, the minimum acceptable spill elevation value for this reservoir. To identify the rank of these simula-tions:

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l Ask in Print Parameters to sort by Spill Elevations by increasing order (as it was done before) and click on Print Results: you can identify easily the corresponding simulations that do not sat-isfy our criteria. The corresponding indices: 10, 82, 78, 97, 4 and 63 have to be masked in the Macro variable.

l In the Data File Manager, click on the Macro variable Simu Top with seismic and, with the right button, ask to mask the indices 10, 82, 78, 97, 4 and 63 in the menu Variable / Edit Macro Indices. You can check that these indices do not belong anymore to the list of valid indices by asking Information on the Macro variable.

(snap. 5.6-16)

Rerunning the spill point application gives the following distribution of volumes and spill point depths:

Non Stationary & Volumetrics 329

(fig. 5.6-7)

The spill point depth lies between 2270.09 and 2311.33m.

In this Spill Point application, the gross rock volumes are calculated between the top reservoir and the contact at the spill point depth for each simulation. In order to take into account the bottom sur-face of the reservoir, it is compulsory to come back to the Volumetrics application: replace the con-tact definition by the new Spill point macro variable and leave the rest of the window unchanged.

(snap. 5.6-17)

Statistics on Volume Risk Curves================================Polygon: P1Smallest = 298.10Mm3Largest = 587.46Mm3Mean = 411.29Mm3St. dev. = 66.28Mm3

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Quantiles on Volume Risk curves===============================Polygon: P1 P90.00 = 328.60Mm3 P50.00 = 406.66Mm3 P10.00 = 500.06Mm33

The distribution of volumes calculated from 98 simulations (the simulations #66 and 80 being still masked off) can be compared with the distribution obtained in first place with the constant contact, which was close to the average of the spill point depth.

(fig. 5.6-8)

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6 Plurigaussian

This case study shows how to apply the plurigaussian approach to sim-ulate geological facies within two oil reservoir units. The aim of this study is to introduce the geologist with the different techniques and concepts in order to better control the lateral and vertical variability of facies distribution when dealing with complex geology. The reservoir is composed of a carbonatic/siliciclastic depositional environment characterized by a high variability in facies changes. The study explains how to integrate geological assumptions in the modeling, through the use of proportional curves (geographical trends of facies), lithotype rules (facies transitions) and gaussian functions (average geological bodies dimensions). Last update: Isatis version 2012

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6.1 Presentation of the Dataset

The information is composed of two separate ASCII files:

l The file wells.hd contains the facies information. This file is organized in a line type format; it means that it is composed of a header (name and coordinates of each collar) and the core sam-ples (coordinates of the core ends, and an integer value which corresponds to the lithofacies code).

l The file surface.hd contains three boundary surfaces called surf1, surf2 and surf3. They are defined on a rotated grid (Azimuth 70 degrees equivalent to 20 in mathematician convention).

6.1.1 Loading the wells

The wells.hd ASCII file is imported using the File / Import / ASCII panel, within a new directory data; the collar information is stored in the file Well Heads whereas the core information is stored in the file Wells.

(snap. 6.1-1)

The data represents a total of 10 wells and 413 samples. A basic statistics run in the File / Data File Manager utility on the Wells file shows that the dataset lies within the following geographical area:

XMIN= 95.05m XMAX= 2905.49m YMIN= -63.58m YMAX= 3779.84m ZMIN= -35.20m ZMAX= 20.50m

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Quantitative information about the variable lithofacies is provided by the Statistics / Quick Statis-tics application. The statistics tell us that this integer variable lies between 0 and 28. The average of 7.61 is not very informative, instead it is relevant to consider the distribution of this discrete data; the variable lies between 0 and 13 or takes the values 22 or 28. For each integer value, the utility provides the number of samples and the corresponding percentage:

Integer Statistics Calculation: lithofacies Integer Value Count of samples Percentage 0 8 2.06% 1 5 1.29% 2 26 6.68% 3 14 3.60% 4 35 9.00% 5 24 6.17% 6 31 7.97% 7 40 10.28% 8 39 10.03% 9 50 12.85% 10 19 4.88% 11 33 8.48% 12 47 12.08% 13 16 4.11% 22 1 0.26% 28 1 0.26%

6.1.2 Loading the surfaces

The same Import ASCII facility is used to store the surfaces in a new grid file Surfaces of the direc-tory data. The basic statistics (using the File / Data File Manager utility) give the following grid characteristics:

NX= 90 X0= 25.00m DX= 50.00m NY= 90 Y0= -775.00m DY= 50.00m Rotation: Angle=20.00(Mathematician) XMIN= -1496.99m XMAX= 4206.63m YMIN= -775.00m YMAX= 4928.62m

This grid clearly covers a larger area than the wells. Finally we calculate statistics on the three sur-faces (using Statistics / Quick Statistics application) and check that these surfaces are not defined on the whole grid (of 8100 cells), as shown in the following results:

Statistics: ------------------------------------------------------------------------------ | VARIABLE | Count | Minimum | Maximum | Mean | Std. Dev | Variance | ------------------------------------------------------------------------------ | surf1 | 7248| -10.20| -6.40| -8.68| 1.34| 1.80| | surf2 | 7248| -18.60| -15.40| -16.86| 0.70| 0.49| | surf3 | 7248| -26.00| -18.80| -22.25| 1.92| 3.68|

Note - The z variable corresponds to an elevation type (increasing upwards).

6.1.3 Standard Visualization

Once the datasets are loaded, it is advised to visualize them in 3D. The graphical environment of Isatis do not allow superimpose displays of data coming from 2D Grids and 3D Lines. To tackle this problem, we will transform the 2D grid surfaces into 3D point surfaces. Open Tools / Copy Vari-ables / Extract Samples ...

334

(snap. 6.1-2)

This process transforms a 2D grid variable surf1 from the Surfaces file into a new file called Surf1 3Dpoints with an output variable called surf1 2D points. Despite its name, this Surf1 3Dpointsfile is still in 2D; in order to transform it in a 3D file the variable surf1 2D points has to be changed into a z coordinate. To achieve that, this variable surf1 2Dpoint has to be informed in the whole extension of the grid, which is not the case for all the surfaces. In the calculator, enter a constant value to the undefined values of the surf1 2D points variable and call the output variable surf1 z.

Plurigaussian 335

(snap. 6.1-3)

The last calculator command can be read as follows: If v1 is not defined (~ffff) then store a value of -50 into the new variable v2, else store the value of v1 into v2.

This new variable v2 (surf1 z) has been created with a float type; before transforming it into coor-dinate, we have to change this type to a float length extension:

m Enter into the Data File Manager editor, select the variable of interest and ask for the For-mat option.

m Click on the Unit button and switch on the Length Variable option; finally select the Length Unit, meters in the present case.

m Now the 2D file may be changed into a 3D file by selecting the surf1 z variable as the new z coordinate value, using the option Modify 2D-3D.

Repeat the same operation for the two other surfaces.

Now we can merge displays coming from lines (wells) and points files (surfaces). Create a new Display by clicking on Display / New Page:

336

l Choose Perspective for the Representation Type, then in the Display Box tab switch off the Automatic Scales toggle and set the z scaling factor to 25. Change also the definition mode of the display box, in order to be able to specify the min and max values along the three axes; the Calculate button may help you to initialize the values.

(snap. 6.1-4)

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l Double-click on Lines in the Available Representations list. Select the variable lithofacies to be displayed with a default Rainbow color scale. Select the Well Names variable as the variable to be displayed from the linked file. In the Lithology tab, change the Representation Size to 0.1cm.

(snap. 6.1-5)

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l For each 3D point surface, ask for a Basemap representation and select the corresponding 3D point file. Each surface is customized with a different pattern color. Click on Display.

l The order of the representations in the display may be modified using the Move Back and Move Front buttons in the Contents window.

(fig. 6.1-1)

6.1.4 Visualization With the 3D Viewer

The Isatis 3D Viewer allows to easily create advanced 3D displays, overlaying efficiently different types of information and with more flexibility than with the standard displays.

For instance, the view below is obtained by a simple drag and drop of different objects from the Study Contents part of the Viewer (left) to the Representations part (middle). Each drag and drop operation leads to an immediate the display in the main part of the viewer, with default parameters that may be modified afterwards. In the present case:

l the 3D lines representation of the Wells file is performed with a radius proportional to the litho-facies and the same Rainbow color scale than with the standard display.

l the three surfaces from the Surfaces file are successively copied into the Surfaces representa-tion type. Resulting iso-surfaces are displayed with appropriate color scales that can be parame-trized by the user.

l The Zscale is set to 50 and the compass is shown.

Note - Detailed information about the 3D Viewer parameters may be found in the On-Line documentation.

Plurigaussian 339

(snap. 6.1-6)

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6.2 Methodology

The wells information containing the lithofacies has been imported into Isatis, as well as the three boundary surfaces. All the identified lithofacies do not necessarily need to be treated. Isatis handles the possibility of grouping the lithofacies variable into lithotypes; in this case study the variable lithotypes corresponds to groups of lithofacies related to the same depositional environment.

The next step consists in creating the 3D Structural Grid that will cover the whole field. Boundary surfaces can be used to split the structural grid into different units. All the nodes of each unit will form a new grid called Working Grid. These working grids will be created using the Tools / Discret-ization & Flattening facility. They may be treated differently, for example their vertical discretiza-tion may be different than the mesh of the 3D structural grid (0.2m) or they can be flattened according to a reference surface. In the next graphic we have represented a working grid using a reference surface, automatically it will be flattened and it will correspond to the vertical origin, and Isatis will assign to this new grid the X, Y and Z mesh values of the Structural Grid. Facies simula-tion will be performed into these working grids. At the end the different working grid simulations will be merged and back transformed into the 3D structural grid. This is illustrated in the next graphic, which shows a Y- Z 2D cross-section.

(fig. 6.2-1)

The well discretization of facies will then be achieved using a constant vertical lag. Even if it is not compulsory for the vertical lag to be equal to the Z mesh of the working grid, it is advised to use the same value in order to have one conditional facies value for node.

The plurigaussian simulation needs the proportions of the lithotypes to be defined for each cell of the working grid, using the Statistics / Proportion Curves panel. Transitions between lithotypes will then be specified within the Statistics / Plurigaussian Variogram panel, that will ultimately be used to perform the conditional Plurigaussian simulation itself.

Plurigaussian 341

6.3 Creating the Structural Grid

This step consists in creating an empty 3D grid which will be used to split the whole reservoir into units using the boundary surfaces. This 3D grid must have the same horizontal characteristics than the 2D grid containing the surfaces data. For the third dimension, we adjust the 3D grid extension to the minimum of the first surface (Surf3= -26) and the maximum of the last surface (Surf1) and set the vertical mesh to 0.2m to cover all the field. Finally the 3D grid must be rotated (around the Z-axis) by 20 degrees in mathematical convention or Azimuth=70 (geological convention) to be con-sistent with the 2D surfaces grid.

(snap. 6.3-1)

The 3D structural grid is created in a file simu of a new directory reservoir. This 3D structural grid will be used to split the field into two adjacent units, each of them being defined by the nodes between two boundary surfaces called `top' and `bottom'. These grid nodes will be stored into two new grids, the working grids for each unit. The units are called upper and lower.

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6.4 Creating the Working Grid for the Upper Unit

Now that wells, surfaces and structural grid data have been either imported or created, we need to create a `working grid' and `discretize' the wells for a specific unit (upper) in order to perform the plurigaussian simulation process. This process will then be repeated for the Lower unit.

This application corresponds to the menu Tools / Discretization & Flattening. A new parameter file called UnitTop is created to capture the names of all the variables of interest together, with most of the working parameters. Click on (NEW) Proportions... to give the name. The parameters, sepa-rated in several tabs, are discussed in the next paragraphs.

6.4.1 Input Parameters

In this panel, we specify if the simulation is conditional (matching some control input data) or not.

(snap. 6.4-1)

If data are available, we must identify the information used as control data, i.e. the variable lithofa-cies contained in the file data / Wells. Note that the facies variable can be converted into a new integer variable called a lithotype (group of facies) and this new variable could also be stored in the

Plurigaussian 343

input data file. To help identifying the wells in the future displays, the variable Well Name (con-tained in the linked header file WellHeads) that contains the name of the wells is finally defined.

If no data is available, the plurigaussian simulation is non conditional.

6.4.2 Grids & Geometry Parameters

In this panel, we define the way the simulated unit (working grid) will be distorted from its struc-tural position into a working position, prior to simulation. We also define all the necessary informa-tion to allow the final back-transformation within the initial structural grid.

The 3D structural grid file reservoir / simu will contain the new pointer variable ptr_UnitTopwhich will serve for the back-transformation of the 3D working grid into the 3D structural grid.

Note - When processing several units of the same 3D structural grid, we must pay attention to use different pointer variables for the different units.

(snap. 6.4-2)

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The unit is characterized by its top and bottom surfaces: for this unit, we use the surface surf1 for the top surface and surf2 for the bottom surface, both surfaces contained in the file Surfaces of the directory data.

We must also define the way the 3D structural grid is transformed in the working system: we refer to the horizontalisation step (flattening). This transformation is meant to enhance the horizontal correlation and consists in transforming the information back to the sedimentation stage. The parameters of this transformation are usually defined by the geologist who will choose between the two following scenarios:

l The horizontalisation parallel to a chosen reference surface. In this scenario (which is the one selected for this unit) we must define the surface which serves as the reference: here surf2.

l A vertical stretch and squeeze of the volume between the top and bottom surfaces: this is called the proportional horizontalisation. In this scenario, there is no need to define any reference sur-face.

We could also store the top and bottom surfaces after horizontalisation in the 2D surface grid in order to check our horizontalisation choice. This operation is only meaningful in the case of parallel horizontalisation.

Finally we must define the new 3D working grid where the plurigaussian simulation will take place (new file WorkingGrid in the new directory UnitTop). The characteristics of this grid will be derived automatically from the geometry of the 3D structural grid and the horizontalisation param-eters. In the case of proportional horizontalisation, we must provide the vertical characteristics of the grid which cannot be derived from the input grid. In the case of parallel horizontalisation, the vertical mesh is equal to the one of the structural grid and the number of meshes is calculated so as to adjust the simulated unit.

Some cells of the 3D working grid may be located outside the simulated unit: they will be masked off in order to save time during the simulation process (new selection variable UnitSelection).

This grid will finally contain the new macro variable defining the proportion of each lithotype for each cell, which will be used during the plurigaussian simulation (macro variable Proportions). At this stage, these proportions are initialized using constant values for all the cells; they are calculated as the global proportions of discretized lithotypes. This macro variable will be updated during the proportion edition step.

6.4.3 Lithotypes Definition

In this panel, we define the way the lithofacies input variable is transformed into the lithotype vari-able to be simulated. This operation is meant to reduce the number of lithotypes to be simulated (compared to the large amount of different lithofacies) and with the possibility to regroup them according to the geologist knowledge. This transformation may simply be skipped if the lithofacies already refers to the lithotype data.

In our case, we define 4 lithotypes whose definition and attributes will be provided in the two next panels:

Plurigaussian 345

l Lithotype Conversion panel

For each lithotype, we define the range of values of the lithofacies variable as the union of one or several intervals. Here each lithotype corresponds to a single interval of values of the lithofa-cies variable, specified by its lower and upper inclusive bounds. A lithofacies value which does not belong to any interval is simply discarded.

(snap. 6.4-3)

l Lithotype Attributes panel

A color and a name are attributed to each lithotype. These colors can be selected using the stan-dard color selector widget or they can be downloaded from an already existing palette. More-over this procedure enables us to establish a palette and a color scale which can be used afterwards to represent the simulated results using the Display Grid Raster facility: for simplic-ity, both objects have the same name LithotypesUnitTop.

346

(snap. 6.4-4)

6.4.4 Discretization Parameters & Output

In this panel, we define all the parameters related to the discretization of the well data (lithofacies). The discretized wells will be used as input data for the plurigaussian simulation and to calculate proportions. This discretization is meant to convert the information into data measured on support of equivalent size. The discretization is performed independently along each well: a well is sliced into consecutive cores of equal dimension (here Lag = 0.2m).

This value is set by default to the vertical mesh of the working grid, so as to keep, on average, one sample per grid cell.

In case of deviated (or horizontal) wells, it is crucial to compensate for the large distortion ratio between horizontal and vertical cell extensions. The operation consists in dividing the horizontal distances by the distortion ratio before slicing using the constant lag in this "corrected" space. The value of this distortion ratio should be consistent with the ratio of the horizontal to the vertical cell extensions (50m/0.2m=250).

In our case (lag=0.2m), a vertical and a horizontal well will produce one discretized sample per cell. The ending core is kept only if its dimension is larger than a minimum length threshold (here 10% of the lag, i.e. 0.02m).

A slicing core may contain pieces of several initial samples, each sample being assigned to a given lithotype. Therefore as a result of the slicing, we build in the Output tab a new file Discretized-Wells in the directory UnitTop where the macro variable Proportions will contain the proportions of the different lithotypes for each sample in a line type format.

Note - Pay attention to the fact that the discretization process creates two Proportions macro variables; the first one, defined in the DiscretizedWells, will be used to calculate and edit the second one, defined in the WorkingGrid and that will serve as the input model for the plurigaussian simulation.

Plurigaussian 347

Several subsequent procedures cannot handle proportions and require a single lithotype value to be assigned to each sample instead: this is why we also compute the "representative" lithotype for each sample (variable Lithotype). Several algorithms are available for selecting this representative litho-type:

l Central: the representative lithotype is the one of the sample located in the middle of the sliced core.

l Most representative: the representative lithotype corresponds to the one which has the larger proportion over the sliced core.

l Random: the representative lithotype is taken at random according to the proportions of the dif-ferent lithotypes present within the sliced core.

We can also store the actual length of the sliced cores. A new linked header file is also created which contains the name of the wells (Variable WellName). This variable is compulsory. If no vari-able has been provided in the Input panel, a default value is automatically generated.

(snap. 6.4-5)

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6.4.5 Run

Click on Run. All the information previously defined is stored in the New Proportions Parameter File... and used to perform the operation. As complementary information, this facility provides the following printouts concerning:

Line #1 : 9 initial samples intersected by the unit First Intersection Point : x = 2719.30m y = 2146.71m z = - 9.80m Last Intersection Point : x = 2647.38m y = 2181.37m z = -16.82m .../... Line #10 : 11 initial samples intersected by the unit First Intersection Point : x = 2010.75m y = 347.77m z = -7.00m Last Intersection Point : x = 2010.75m y = 347.77m z = -16.40m Description of the New Working Grid =================================== File Name : UnitTop/WorkingGrid Mask Name : UnitSelection NX= 90 X0= 25.00m DX= 50.00m NY= 90 Y0= -775.00m DY= 50.00m NZ= 51 Z0= 0.10m DZ= 0.20m Number of valid nodes = 308368/413100 Type of system : Parallel to a Reference Surface Surfaces File : data/Surfaces Top Surface : surf1 Bottom Surface : surf2 Reference Surface : surf2 Description of the 3D Structural Grid ===================================== File Name : reservoir/simu Pointer to Working Grid : ptr_UnitTop NX= 90 X0= 25.00m DX= 50.00m NY= 90 Y0= -775.00m DY= 50.00m NZ= 99 Z0= -26.00m DZ= 0.20m Number of valid nodes = 301033/801900 Statistics for the Discretization ================================= Input Data: ----------- File Name : data/Wells Variable Name : lithofacies Total Number of Lines = 10 Total Number of Samples = 413 Analyzed Length = 246.65m Initial Lithotypes Proportions: Conglomerate = 0.040 Sandstone = 0.348 Shale = 0.309 Limestone = 0.304 LithoFacies to Lithotype Conversion: ------------------------------------ Conglomerate = [7,7] Sandstone = [9,9] Shale = [10,10]

Plurigaussian 349

Limestone = [12,12] Discretization Options ---------------------- Discretization Length = 0.20m Minimum Length = 0.02m Distortion Ratio (Hor/Vert) = 250 Lithotype Selection Method = Central Discretization Results: ----------------------- File Name : UnitTop/DiscretizedWells Lithotype Name : Lithotype Proportions Name : Proportions[xxxxx] Total Number of Lines = 10 Total Number of Samples = 426 Number of Informed Samples = 343 Discretized Lithotype Proportions: Conglomerate = 0.045 Sandstone = 0.336 Shale = 0.303 Limestone = 0.315 Assigned Lithotype Proportions: Conglomerate = 0.047 Sandstone = 0.338 Shale = 0.297 Limestone = 0.318

To visualize the selection unit where the simulation will take place and superimpose the discretized wells:

l In a new Display page, switch the Representation type to the Perspective mode;

l Select, with a Symbols representation, the grid file UnitTop / WorkingGrid. In the Grid Con-tents area, switch to the Excavated Box mode and center IX and IY to the index number 45. In the Data Related Parameters area customize two Flags:

m the first flag with a lower bound of 0 and upper bound of 0.5, a red point pattern with a size of 0.1 has been used,

m for the second flag we have used bounds from 1 to 1.5 in order to catch the selection values (1), and gray circles of 0.1 size to represent the upper unit selection have been used.

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l Select a new Lines representation and select the line file UnitTop / DiscretizedWells. Select Lithotype for Lithology #1. In the Lithology tab, it is advised to use the LithotypesUnitTopcolor scale and customize the representation size to 0.1cm.

l Click on Display.

(fig. 6.4-1)

Note - You can also display the Lithotype variable in a literal way by using another item, for example Graphic Left #1: Lithotype.

Plurigaussian 351

6.5 Computing the Proportions

The principal task in this stage is to estimate lithotypes proportions at each cell of a working grid by an edition of the proportions at wells after the discretization and flattening process previously per-formed.

This application corresponds to the menu Statistics / Proportion Curves. Proportions must be ana-lyzed and estimated, usually in a different way for each unit, since they are related to a specific dep-ositional environment. These proportions are an essential ingredient of the plurigaussian model. The application is displayed as a main graphic window which represents the field base map in a horizontal projection.

6.5.1 Loading Data

Obviously, in its first use, the graphic window is left blank. The first operation is to use the Load Data option in the Application Menu to select the relevant information.

(snap. 6.5-1)

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When entering the name of the Proportion Parameter File, all the other parameters are defined auto-matically and classically do not have to be modified. We can now review the parameters of interest for this application:

l the discretized wells (UnitTop / DiscretizedWells) where the macro variable Proportions is specified,

l the linked header file WellHeads containing the well names (Variable WellName),

l the 3D working grid (UnitTop / WorkingGrid) where the macro variable Proportions will be modified, within the selected area (Selection variable UnitSelection)

Once these parameters have been defined, the main graphic window represents the 10 wells in the rotated coordinate system of the working grid. If we look carefully, we can see that some wells are deviated (W1, W5 and W9): their traces are projected on the horizontal plane.

(snap. 6.5-2)

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In the lower right corner of the graphic window, a vertical proportion curve (or VPC for short) rep-resents the variation of the global proportions along the vertical axis. It is displayed with a cross symbol, which represents the VPC's anchor useful for edition purposes.

This application offers two modes, indicated at the bottom, depending whether we operate using the polygons or the VPC. The Graphic Menu of this window depends on the selected option. In the case of Polygon Edition, the following menu options are available:

- Select All Polygons

- Create Regular Polygons...

- Create Polygon(s)

In the case of Vertical Proportion Curves Edition, the following menu options are available:

- Deselect

- Select All VPCs

- Select Incomplete VPC(s)

- Display & Edit...

- Editing

- Apply 2D constraint..

- Completion...

- Smoothing...

- Reset From Raw Data

- Delete VPC(s)

- Print VPC(s)

We can define the graphic window characteristics in the panel Graphic Options of the ApplicationMenu. These parameters will be illustrated in the subsequent paragraphs:

m The representation of the wells and the VPC on horizontal projections.

m The parameters concerning the VPC specific windows.

m The miscellaneous graphic parameters such as the polygon display parameters, the graphic bounds (for projections) and the order of the lithotypes.

Here, this panel is used in order to define the options for the VPC display windows: we switch ON the flag for Displaying Raw VPC and OFF the one asking for normalization.

354

(snap. 6.5-3)

6.5.2 Display global statistics

In this step, we visualize some global statistics on the proportions. First, we select the global VPC in the lower right corner by picking its anchor (cross symbol) and use the option Display & Edit in the Graphic Menu in order to display this VPC in a separate specific graphic window. This graphic shows the VPC projected from the wells to the working grid in a normalized mode; this VPC can be edited. Note that in our case we have another VPC to the right; this VPC corresponds to the raw mode VPC Global Raw Proportions that was specified previously in the Graphic Options panel (without normalization).

Plurigaussian 355

m The raw mode: for each (vertical) level, the (horizontal) bar is proportional to the number of samples used to calculate the statistics. You can display the numbers by switch on the Dis-play Numbers option in the Graphic Option panel. Each bar is subdivided according to the proportions of each lithotype, represented using its own color. The order of the lithotypes is defined in the Graphic Options panel.

m The normalized mode: the proportions are normalized to sum up to 1 in each level (except the levels where no sample is available). Note that the first and last levels of this global pro-portion curve are left blank as they do not contain any sample.

(fig. 6.5-1)

A second feature is obtained using the option Display Pie Proportions in the Application Menu. It creates a separate window where each well is represented by a pie located at the well header loca-tion. The pie is subdivided into parts whose size represents the proportion of each lithotype calcu-lated over the whole well (Calculated From Lines option). We can normalize the proportions by discarding any information which does not correspond to any lithotype. Here instead, we have cho-sen to take them into account: they are represented as a white fictitious complementary lithotype.

356

(fig. 6.5-2)

For particular usage, some lithotypes can be regrouped: this new set is then displayed as a fraction of the pie.

The 10 wells are displayed using the pie proportional chart where each lithotype is represented with a proportional size, calculated over the whole well. This application is used to check that the first lithotype (called Conglomerate) represented in red is present only in the 6 wells located in the northern part of the field (W1, W2, W6, W7, W8 and W9) and absent in the south, hence the non stationarity.

l Creating polygons

In this step, we turn the option of the main graphic window into the Polygon Edition mode. Using the Create Polygon(s) option of the Graphic Menu, we digitize two polygons. When a polygon is created, a vertical proportion curve is automatically calculated and displayed (in its normalized mode) at the polygon anchor position (located by default in the center of gravity of the polygon).

We can now select one VPC (or a group of them) and modify it (or them) using the features demon-strated hereafter.

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(snap. 6.5-4)

l Edition of a VPC

In our case, we select the VPC corresponding to the northern polygon and use the Display & Editoption of the Graphic Menu in order to represent it on a separate window. As this has already been discussed when visualizing the global VPC, we have chosen (in the Graphic Options panel) to rep-resent the VPC in the raw version on the right and in the normalized version on the left.

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(fig. 6.5-3)

We can easily check that, here again, the top and bottom levels of this VPC are not informed. Before using this VPC in a calculation step, we need to complete the empty levels.

To complete empty level, we use the Application / Completion option. By default this algorithm first locates the first and last informed layer Number of Levels = 1. If an empty layer is found between the first and last informed layers, the proportions are linearly interpolated. In extrapola-tion, the proportions of the last informed layer are duplicated. An option offers to replace the pro-portions of the last informed layer by the ones calculated over a set of informed layers; their number is defined in the interface. The result is immediately visible in the normalized version of the VPC in the left part of the specific graphic window and in the main graphic window. Note that this comple-tion operation could have been carried out on a set of VPC (without displaying them on separate graphic windows).

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(fig. 6.5-4)

Smoothing the vertical transitions is advised, and may be achieved with the option Application / Smoothing. It enables the application to run a low-pass filtering algorithm on the normalized ver-sion. This procedure requires the VPC to be completed beforehand. This procedure can be applied several times on each selected VPC: here 3 passes are performed. Once more, the results are visible in the normalized version of the VPC displayed in the left part of the specific graphic window. The corresponding VPC is also updated in the main graphic window.

(fig. 6.5-5)

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The same procedure applied on the southern VPC leads to the following graphic representation.

(fig. 6.5-6)

Note that the main difference between the two VPC, even after the completion and smoothing steps, is the absence of the first lithotype in the VPC corresponding to the southern polygon. As these VPC will serve as conditioning data for the subsequent interpolation phase: their contents as well as their location are essential.

l Computing proportions on the 3D grid

We recall that the proportions of the different lithotypes over the cells of the working grid have been initially set to a constant value corresponding to the global proportions calculated using all the discretized samples. The aim of this application is to calculate these proportions more accurately, enhancing the absence of the first lithotype in the south, for example.

For that purpose, we use the Compute 3D Proportions of the Application Menu. This procedure requires all the VPC used for calculations to be completed beforehand.

This application offers three possibilities for the calculation:

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l Copying the global VPC (displayed in the lower right corner of the main graphic window): the proportions are set to those calculated globally over each layer. This crude operation is slightly cleverer than the initial global proportions as the calculations are performed layer by layer: therefore the vertical non stationarity is taken care of.

l Inverse squared distance interpolation. This well-known technique is applied using the VPC as constraining data. For each level and each lithotype, the resulting proportion in a given cell is obtained as the linear combination of the proportions in all the VPC, for the same lithotype and the same layer. The weights of this combination are proportional to the inverse squared distance between the VPC and the target cell.

l Kriging. This technique is used independently for each layer and each lithotype, using the VPC as constraining information. A single 2D model (ModelProportions) is created and used for all the lithotypes: we assume therefore the intrinsic hypothesis of the multivariate linear model of coregionalization.

The model can be defined interactively using the standard model definition panel. Here it has been set to an isotropic spherical variogram with a range of 5000m. There is no limitation in the number and types of basic structures that can be combined to define the model used for estimat-ing the proportions. The sill is meaningless unless several basic structures are combined.

(snap. 6.5-5)

l Display the proportions from the 3D working grid

The proportions have been calculated over all the cells of the 3D working grid; it is now time to visualize them using the Display 3-D Proportions option of the Application Menu.

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This feature is specific to the display of proportions. The figure consists in an horizontal projection: each cell of the horizontal plane is displayed as a VPC obtained considering the proportions of all the lithotypes for all the levels of the grid column.

The following operation can be performed:

m Sampling. This option is relevant when the number of grid cells is large. We must simply specify the characteristics of a coarser grid (horizontally). When the step of the coarser grid is set to 1, no sampling is performed and the entire grid is visualized.

m Averaging. Before visualization, the VPC are averaged layer by layer in moving windows. The extension of the moving window is specified by the user.

m Finally we can choose to select a vertical window specifying the top and bottom levels to be visualized.

For this first display, the 3D working grid is sampled by step of 10, with the origin set at rank 5: only 9 cells are presented out of the 90 cells of the working grid:

(snap. 6.5-6)

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(fig. 6.5-7)

Note - Pay attention to the fact that the represented grid is rotated (by 20 degrees).

The resulting graphic shows that two conditioning VPC are obviously reproduced at their location. However, the first lithotype still shows up in the southern part of the display of the estimated pro-portions, because of the weak conditioning of the kriging step based on two VPC only.

Enhancing the conditioning set of information is therefore advised:

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l This can be achieved by increasing the number of VPC which serve as constraining data for the kriging step. A first solution is to digitize more polygons as one VPC is attached to each poly-gon, but this may lead to poorly defined VPC.

l The other solution considered here is simply to duplicate each VPC several times in its calcula-tion polygon: in VPC Edition mode, right click on the basemap and select the Duplicate One VPC option. You then have to pick one VPC and move its duplicate at the desired location.

l Each VPC is duplicated twice in its polygon, hence 6 VPC in total.

(fig. 6.5-8)

These VPC are used through the same computing process (Compute 3D Proportions in the Applica-tion Menu), using the kriging option with the same model as before; the printout gives the propor-tions:

Computing the proportions on the 3D Grid ======================================== Number of levels = 51 Number of lithotypes = 4 Experimental Proportions - Global VPC

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Number of active samples = 48 Proportion of lithotype #1 = 0.047 Proportion of lithotype #2 = 0.367 Proportion of lithotype #3 = 0.236 Proportion of lithotype #4 = 0.350 - Regionalized VPC(s) Number of VPC used = 6 Number of active samples = 306 Proportion of lithotype #1 = 0.091 Proportion of lithotype #2 = 0.320 Proportion of lithotype #3 = 0.219 Proportion of lithotype #4 = 0.369 Proportions calculated on the simulation grid Number of cell along X = 90 Number of cell along Y = 90 Number of cell along Z = 51 Number of calculated cells = 308370 Proportion of lithotype #1 = 0.027 Proportion of lithotype #2 = 0.264 Proportion of lithotype #3 = 0.259 Proportion of lithotype #4 = 0.450

The results are displayed using the Display 3D Proportions option of the Application Menu. As expected, the first lithotype does not show up in the southern area anymore.

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Some of the resulting VPC seem to be incomplete. This is due to the fact that, for these cells, the whole vertical column of the grid does not lie within the unit: it is truncated by the unit limiting sur-faces and therefore some cells are masked by the unit selection.

The final step consists in using the Save & Run option of the Application Menu which updates the Proportions Parameter File. Remember that the last edited proportion model will serve as input for the plurigaussian simulation.

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6.6 Lithotype Rule and Gaussian Functions

This phase is specific to the plurigaussian simulations as it is used to define the models of the two underlying gaussian random functions, as well as the lithotype rule which is used to convert the results from the bi-gaussian domain into lithotypes. This application corresponds to the menu Sta-tistics / Plurigaussian Variograms.

The principal idea under the plurigaussian approach is to split an initial global rectangle called `lithotype rule' into sub-rectangles an assign one lithotype to each sub-rectangle. You can split the rectangles and assign the lithotypes in different ways. By this way you are able to better control lithotype transitions.

The lithotype rule is usually represented by a diagram where the horizontal axis stands for the first gaussian random function G1 and the vertical axis for the second gaussian random function G2. By adding successively horizontal and vertical limits, we split the diagram into rectangles and assign one lithotype to each rectangle, for instance:

(fig. 6.6-1)

Apart from the conditioning data and the 3D grid proportion curves, the plurigaussian simulation will honor the lithotype rule and the variographic properties of the two gaussian functions.

In geological terms, this means that we can force lithotypes to follow transitional or erratic varia-tions, intrusions, erosions of lithotypes into the whole unit or into a group of lithotypes. Further-more we have the possibility to control anisotropies, horizontal and vertical extensions (ranges) and behaviors (type of variogram) for the two axes of the lithotype rule (for two groups of lithotypes).

We must first define the name of the Proportion Parameter File (UnitTop) which contains all the rel-evant information. In particular, it contains the information on:

m The well data information (UnitTop / DiscretizedWells): in this application, we use the assigned lithotype value (Variable Lithotype) at each sample rather than the proportions.

m The 3D Working grid (UnitTop / WorkingGrid) which contains the macro variable of the last edited proportions of the different lithotypes in each cell (Proportions).

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(snap. 6.6-1)

l Lithotype rule

Click on the Define button. The initial lithotype rule has to be split in sub-rectangles, each litho-type corresponding to a rectangle. The choice of the lithotype rule should be based on all geo-logical information about the unit. The geological model of the unit is important to assign the lithotype transitions. The application also produces a set of "histograms" (on the right part of the window) showing the vertical frequency of transitions between lithotypes along the wells. Click on Cancel.

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(snap. 6.6-2)

Click on Print Transition Statistics...Transition matrices =================== L1 = Conglomerate L2 = Sandstone L3 = Shale L4 = Limestone Downward probability matrix --------------------------- Number L1 L2 L3 L4 L1 16 0.625 0.375 0.000 0.000 L2 116 0.009 0.853 0.129 0.009 L3 102 0.000 0.059 0.853 0.088 L4 99 0.000 0.000 0.000 1.000 Upward probability matrix ------------------------- Number L1 L2 L3 L4 L1 11 0.909 0.091 0.000 0.000 L2 111 0.054 0.892 0.054 0.000 L3 102 0.000 0.147 0.853 0.000 L4 109 0.000 0.009 0.083 0.908

For example, from the Downward probability matrix print out (From top to bottom) we see that L1 (in red) only has vertical contact with L2 (in orange) with a 37.5% of frequency. The same calculation from bottom to top (Upward probability matrix) shows us that this L1 still has only contact with L2 but now with only 9.1% of frequency.

The facies transition can be also read from the histograms in the Lithotype Rule Definitiongraphic. The left column shows the whole set of lithotypes and the right column plots the litho-types that have a different zero frequency transition value between the respective lithotype of the left column and the rest of them.

In our case the lithotype from the left column corresponds to L1 (red one) and as it was said before we know that it has only one contact transition with L2 (no matter the direction), then only one histogram bar will be plotted with a frequency equal to one.

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Note - A lithotype bar is plotted in the histogram if it has a different zero frequency transition no matter the type of calculation (Upward or Downward). The frequency transitions are not respected in the histogram, instead they are averaged and normalized.

From the set of histograms we conclude that:

- L1 only has contact with L2,

- L2 has contact with L1, L3 and L4,

- L3 has contact with L2 and L4,

- L4 has contact with L3 and L2.

From the VPC analysis we have found that L1 only occurs at the top and L4 at the bottom of this unit. We have also observed that L1 is only present in the northern area of the field. In the present case, by now we will not use external information to confirm our assumptions or to cus-tomize the lithotype rule, and we will work without any geological model delineation.

In the Lithotype Rule Definition panel click switch on Add Vertically and split the lithotype rule set as default (L1). Repeat this action to split vertically the L2 area. Now switch on Add Hori-zontally and split in two the L3 area. The next graphic shows the obtained lithotype rule:

(fig. 6.6-2)

This lithotype rule is consistent with all the properties mentioned above, but remember that these diagrams are only informative as they are calculated on few samples and only along the wells. The lithotype rule as defined previously is echoed on the main panel.

l Definition of the models for the underlying gaussian random functions

Analog information, well logging, well correlation, sequential stratigraphy, etc. can give an idea of global dimensions of lithotypes (extensions) and can be integrated in the variogram model.

The next step is used to define the models for the underlying gaussian random functions G1 and G2. Note that, if all the limits of the lithotype rule are vertical, the second gaussian random function does not play any role and therefore there is no need in defining the corresponding model.

Since we have an horizontal limit in the lithotype rule, (L3-L4), we must define two new param-eter files containing the models for the two underlying gaussian random functions:

m g1Top (Horizontal Axis) will rules L1, L2 and L3-L4

m g2Top. (Vertical Axis) will rules only L3 and L4

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Each model can be edited using the standard model definition panel. For the time being, enter the following structures for the variograms of the two gaussian functions:

l g1Top: cubic variogram with a range of 2000m along X and Y, and 2.5m along Z,

l g2Top: exponential variogram with a (practical) range of 2000m along X and Y, and 3m along Z.

Note that, by default, the basic structures are anisotropic with a rotation equal to the rotation angle of the grid (20 degrees). In our case, the basic structures are isotropic (in the XOY plane) and this rotation angle is ignored. The quality of the fitting will be evaluated below.

l Control displays

This application offers the possibility of displaying the thresholds calculated for both gaussian random functions. They are represented in a form similar to the lithotype rule but this time, each axis is scaled in terms of cumulated gaussian density. In our case and as the two underlying gaussian random functions are not correlated, the surface directly represents the proportion of a lithotype. Here the working grid is sampled considering only one cell out of 10, which brings the number of cells down to 9*9. The level 25 (out of 51) is visualized.

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(fig. 6.6-3)

It is interesting to see that lithotype 1 (red) does not show up and that lithotype 4 (blue) progres-sively disappears towards the north east. External information will be used later to inform that lithotype 4 (blue) belongs to the Deep Platform environment and that lithotype 1 belongs to a coastal environment (upper part of the field). This puts in evidence the N-S lithotype prograda-tion.

We can visualize a non conditional simulation performed in the planes of the 3D working grid. For that sake, we must enter the seed used for the random number generator.

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(snap. 6.6-3)

For better legibility, it is possible to enlarge the extension of the grid along the vertical axis by specifying a Z scaling factor, here the distortion factor is set to 150. The next figure represents a YOZ section (X Index=60). We recall that this simulation is performed in the 3-D working grid. We can also visualize the horizontal section at Z index=25.

(fig. 6.6-4)

l Fitting variograms

The last tool corresponds to the traditional graphic variogram fitting facility. However, in the case of plurigaussian model fitting, it is rather difficult to use as:

m The variograms are calculated experimentally on the lithotype indicators. When choosing the models for both underlying gaussian random functions, we must fit simultaneously all the simple and cross-variograms: for 4 lithotypes, there are 4 simple variograms and 6 cross-variograms.

m The equation relating the variogram model to the variogram of the lithotype indicator uses the lithotype proportion: the impact of a strong non stationarity is difficult to evaluate on the model rendition.

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In addition in our case, we can calculate the variograms along the wells (almost vertically) but with lot of difficulty horizontally because of the small number of wells. The application allows the calculation on the fly of experimental simple and cross variograms in a set of directions (up to 2 horizontal and 1 vertical). We must first define the characteristics of these computations:

m Computation Parameters: the lag values and the number of lags in each calculation direction. By default the lags are set equal to the grid mesh in each direction.

(snap. 6.6-4)

m Horizontal tab. The horizontal experimental variogram is calculated as the average of vario-grams calculated layer by layer along a reference direction within angular tolerance. Here the reference direction is set to 70 degrees and the angular calculation tolerance to 45 degrees. There is no restriction on the vertical layers to be scanned.

(snap. 6.6-5)

m Vertical tab. The vertical experimental variogram refers to calculations performed along the vertical axis within a vertical tolerance. In addition, we can restrict the calculation to con-sider pairs only within the same well.

(snap. 6.6-6)

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Now, we can define a set of graphic pages to be displayed. Each page is identified by its name and by its contents composed of a set of lithotype indicator simple or cross-variograms. For each variogram, we can specify the line style and color. In the next figure, we define the page Horizontal which contains the four simple lithotype indicator variograms for two directions (Horizontal 1 and Horizontal 2). The Horizontal 1 direction corresponds to the E-W axe of the working grid. We first define a New Page and enter its name in the popup window, then select Horizontal 1 and 2 from the Directions list, then select the L1, L2 L3 and L4 lithotype from the Lithotypes list and press the arrow button: the list of variograms to be calculated is displayed in the Curves List (Hor1: simple[1], ...). We do the same for a new Vertical page.

The next figure shows the indicator simple variograms of the four lithotypes for the horizontal and vertical plane (with lithotype color scale): experimental quantities are displayed in dashes for Hor1 direction and points-dashes for Hor2 whereas the model expressions are displayed in solid lines.

(fig. 6.6-5)

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(fig. 6.6-6)

So far we have all the parameters needed to perform a plurigaussian simulation apart from the neighborhood definition. However before arriving to the final step we present how to integrate external information or geological assumptions to the lithotype rule and the two underlying gaussian functions in order to simulate lithofacies within conceptual or geological model fea-tures.

As it was said before lithotype L1 (conglomerate) is associated to a coastal environment. Geo-logical information put in evidence that lithotype L1 has a regional trend measured in Azi-muth=80 degrees. Lithotype L1 would be related to coastal facies (Northern part of the deposit).

L2 and L3 (sandstones and shales) belong to a shallow marine environment and show an oblique sigmoidal stratification. The dipping angle of this lithotype varies between 0.4 - 0.5 degrees. They have a progradation from the shore line ( ~ North to South), by consequence the layers are oriented to the same Azimuth=80 degrees.

L4 is related to the deep platform environment and it has a good horizontal correlation (regional presence).

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(fig. 6.6-7)

Note - The previous graphic is represented in working grid coordinates. The structural grid has a rotation, then you must pay attention when dealing with spatial correlations in the working grid.

In order to take into account this new information we will change the lithotype rule to fit the next graphic.

(snap. 6.6-7)

Note that this lithotype rule allows L1 to have a contact with L3, this feature is not consistent with the vertical transition of lithotypes from the wells, but as it was said before the transition histograms are only informative. In the other hand, it is now possible to control the regional trend of the coastal associated lithotype L1 with one of the gaussian random function, G1 (hori-zontal), for this case. We will call this function G1 Coastal Top. The characteristics of this model are:

- Global horizontal rotation = Azimuth=80, (equivalent to Az=10)

- Type = Cubic

- Ranges (Along U rotated= 2000, Along V rotated=700, Along Z=0.5m)

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Note - As this gaussian will principally affect lithotype L1 we have used a Z range value equivalent to its average thickness from the wells. In order to simulate the horizontal trend of the coastal environment we have used a range value U axis greater than V.

For the second gaussian that will only rule L2 and L3 we take into account the dipping angle of progradation and an anisotropy direction equivalent to the G1 gaussian function but with a range value along the V axis greater that the U axis due to the fact that progradation occurs orthogonal to the coastal trend. We will call this function G2 (L2-L3) Top. The characteristics of this model are:

- Local rotation for anisotropy = Azimuth=80, (equivalent to Az=10). Vertical rot= 0.4

- Type = Cubic

- Ranges (Along U rotated=1000, Along V rotated=1500, Along Z=1.5m)

(snap. 6.6-8)

Since L1 has a geological correlation with L3 (facies transition from coastal to ramp) we have used a correlation value of 0.6 between the two gaussian functions. You can compare the indica-tor variograms and the display of non conditional simulations to the previous model, as well as the impact of the correlation factor on the non conditional simulations.

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(snap. 6.6-9)

The next graphic shows non-conditional simulations using different correlation values. (From left to right: 0, 0.3, 0.9)

(fig. 6.6-8)

Clicking on Run finally saves the plurigaussian variograms.

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6.7 Conditional Plurigaussian Simulation

(snap. 6.7-1)

This application corresponds to the Interpolate / Conditional Simulations / Plurigaussian menu and performs plurigaussian simulations (only one in this case study). First, define the name of the Pro-portion Standard Parameter File (UnitTop) which defines all the environment parameters such as:

l The input discretized line structure (UnitTop / DiscretizedWells) and the assigned lithotype variable Lithotype.

l The 3D working grid (UnitTop / WorkingGrid) with the input macro variable Proportions and the output macro variable containing the simulated lithotypes Simupluri litho (2nd version).

In particular, the parameter file indicates if the plurigaussian simulation should be conditioned to some data or not. In addition, we must define the specific parameters, such as:

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l The neighborhood (Moving) using the standard neighborhood definition panel: the neighbor-hood search ellipsoid extensions are 10km by 10km in the horizontal plane and 20m along the vertical; it is divided in 8 angular sectors with an optimum of 4 points per sector.

l The parameters for reconstructing the underlying gaussian random functions at the constraining data points.

l The parameters for simulating the underlying gaussian random functions on the grid.

Finally click on Run.

A second plurigaussian simulation is performed, with updated parameters of the utility Statistics / Plurigaussian Variogram for the gaussian functions g1Top and g2Top and their respective lithotype rule. These models have different anisotropy angles and the correlation between the two gaussian functions is setup to zero. The same neighborhood is used and the new output is called Simupluri litho (1st version).

6.7.1 Displaying results of the conditional simulation

As the simulation has been performed in the 3D working grid, it is recommended to visualize the results in this flattening mode. This is done using a standard Raster representation in a new display page. At this stage it is clever to use the color scale LithotypesUnitTop specially designed in order to represent the lithotypes of interest.

The next figure represents the layer 37 projected on the horizontal plane for the two output models Simupluri litho (1st version) and Simupluri litho (2nd version). We can clearly see the mask cor-responding to the limits of the unit.

On this plane, we digitize a section (the diagonal of the 3D working grid) represented as a dashed line. Using the graphic option (right mouse button) Update Trace on Graphics, this section is passed to a second Display page which represents Cross-sections in 3D.

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(fig. 6.7-1)

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6.8 Simulating the Lithofacies in the Lower Unit

The same workflow is applied for the lower unit, located just below the previous unit in structural position. The upper and lower units are processed independently, even if some lithotypes may be common.

We review the different steps and only highlight the main differences with the previous unit.

6.8.1 Creating the unit and discretizing the wells

We setup the environment for the plurigaussian simulation using the Tools / Discretizing & Flat-tening application. The Input panel is the same for both units.

(snap. 6.8-1)

The new Proportion Parameter File (UnitBottom) is used to store all the information required by the plurigaussian simulation for the current unit, such as:

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l The 3D structural grid (reservoir / simu) where the final results will be stored, is the same than for the upper unit. However, we must pay attention to use a different name for the back-transfor-mation pointer than the one used for the upper unit (Variable ptr_UnitBottom).

l For this unit, the horizontalisation is performed using a distortion proportional between the Top surface (surf2) and the Bottom surface (surf3), contained in the 2D surface grid file (data, File Surfaces). In this case, there is no need to specify any reference surface.

l The new 3D working grid (UnitBottom / WorkingGrid) is used to store the macro variable containing the proportions (Variable Proportions). Note that, in this proportional flattening case, the grid mesh along vertical axis (0.2m) is defined arbitrarily. The number of meshes (27) is defaulted according to the mesh extension of the structural grid and the unit thickness.

l A new file to store the discretized wells (UnitBottom / DiscretizedWells) with the macro vari-able for the proportions (Variable proportions) and the assigned lithotype (Variable lithotype). The linked header file (UnitBottom / WellHeads) contains the names of the wells (Variable WellName).

Four lithotypes are defined, as illustrated in the next window. We also define the corresponding name and color for each lithotype and create a new Color Scale and Palette that will be used to rep-resent the lithotype simulated grid using the traditional Display/Grid/Raster facility (Lithotype-sUnitBottom).

(snap. 6.8-2)

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(snap. 6.8-3)

(snap. 6.8-4)

Once pressed the Run button, the printout shows the following results:

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Description of the New Working Grid =================================== File Name : UnitBottom/WorkingGrid Mask Name : None NX= 90 X0= 25.00m DX= 50.00m NY= 90 Y0= -775.00m DY= 50.00m NZ= 27 Z0= 0.10m DZ= 0.20m Type of system : Proportional between Top and Bottom Surfaces Surface File : data/Surfaces Top Surface : surf2 Bottom Surface : surf3 Description of the 3D Structural Grid ===================================== File Name : reservoir/simu Pointer into Working Grid : ptr_UnitBottom NX= 90 X0= 25.00m DX= 50.00m NY= 90 Y0= -775.00m DY= 50.00m NZ= 99 Z0= -26.00m DZ= 0.20m Number of valid nodes = 198700/801900 Statistics for the Discretization ================================= Input Data: ----------- File Name : data/Wells Variable Name : lithofacies Total Number of Lines = 10 Total Number of Samples = 413 Analyzed Length = 153.56m Initial Lithotype Proportions: Continental sandstone = 0.327 Continental shales = 0.198 Very shallow packstones = 0.439 Shallow wackstones = 0.036 Lithofacies to Lithotype Conversion: ----------------------------------- Continental sandstone = [2,2] Continental shales = [3,3] Very shallow packstones = [8,8] Shallow wackstones = [11,11] Discretization Options ---------------------- Discretization Length = 0.20m Minimum Length = 0.02m Distortion Ratio (Hor/Vert) = 250 Lithotype Selection Method = Central Discretization Results: ----------------------- File Name : UnitBottom/DiscretizedWells Lithotype Name : lithotype Proportions Name : proportions[xxxxx] Total Number of Lines = 10 Total Number of Samples = 270 Number of Informed Samples = 241 Discretized Lithotype Proportions:

Plurigaussian 387

Continental sandstone = 0.351 Continental shales = 0.166 Very shallow packstones = 0.448 Shallow wackstones = 0.035 Assigned Lithotype Proportions: Continental sandstone = 0.365 Continental shales = 0.166 Very shallow packstones = 0.436 Shallow wackstones = 0.033

At the end of this discretization procedure, the proportions in the 3D working grid are defined as constant over all the cells, equal to the discretized lithotype proportions (as defined above).

6.8.2 Computing the proportions

The second step consists in modifying these constant lithotype proportions over the field using the Statistics / Proportion Curves application. For this purpose, we first load the UnitBottom Propor-tions parameter file and represent the global proportions calculated over the 10 wells using the Dis-play Pie Proportions feature.

(fig. 6.8-1)

W1

W2

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W8

W9

W10

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2D Point Proportion

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388

This figure does not show any particular feature linked to the geographical position of these propor-tions.

Lithofacies L3 (Very shallow packstone - in orange) is associated to a shallow platform environ-ment. Lithofacies L1 and L2 are associated to a continental environment but we do not have more external information. For this reason we consider the proportions of the lithotypes as stationary in the horizontal extension of the field.

We now focus on the global VPC (calculated using the samples from all the wells) and displayed in the bottom right corner of the main graphic window. The next figure (obtained using the Display & Edit option of the Graphic Menu) shows the VPC in the raw version (on the right) and in the modi-fied version (on the left).

(fig. 6.8-2)

Note that, in our case where the horizontalisation has been performed in a proportional manner, there is no need for completing the VPC. The initial global VPC has been smoothed (3 iterations) as in the upper unit. The Compute 3D Proportions option is used to simply duplicate the global VPC in each column of cells in the 3D working grid.

Plurigaussian 389

(snap. 6.8-5)

This utility produces the following results:

390

Computing the proportions on the 3D Grid ======================================== Number of levels = 27 Number of lithotypes = 4 Experimental Proportions - Global VPC Number of active samples = 27 Proportion of lithotype #1 = 0.359 Proportion of lithotype #2 = 0.171 Proportion of lithotype #3 = 0.438 Proportion of lithotype #4 = 0.031 Proportions calculated on the simulation grid Number of cell along X = 90 Number of cell along Y = 90 Number of cell along Z = 27 Number of calculated cells = 218700 Proportion of lithotype #1 = 0.359 Proportion of lithotype #2 = 0.171 Proportion of lithotype #3 = 0.438 Proportion of lithotype #4 = 0.031

Note that, as expected, the proportions are the same on the global VPC than over the whole grid. They are slightly different from the one calculated on the discretized samples in the previous appli-cation, due to the smoothing step. Do not forget to Save and Run this parameter file.

The proportions can be visualized using the Display 3D Proportions utility, which produces a figure where all the VPC are exactly similar.

Plurigaussian 391

(fig. 6.8-3)

6.8.3 Lithotype rule and gaussian functions

The lithotype rule is defined using the Statistics / Plurigaussian Variogram application.

(snap. 6.8-6)

392

The models of the two underlying gaussian random functions are:

m g1Bot: anisotropic exponential variogram with a (practical) range of 1500m along X and Y and 2m along Z.

m g2Bot: anisotropic cubic variogram with a (practical) range of 1500m along X and Y, and 2m along Z.

The next graphic shows a non-conditional simulation using the previous parameters.

(snap. 6.8-7)

We clearly see the influence of the variogram types and lithotype rule which produce the following transitions:

- spotted between lithotype #1 (yellow) and lithotype #3 (orange),

- spotted between lithotype #2 (green) and lithotype #3 (orange),

- spotted between lithotype #3 (orange) and lithotype #4 (blue),

- smooth between lithotype #1 (yellow) and lithotype #2 (green).

6.8.4 Conditional plurigaussian simulation

We now run the conditional plurigaussian simulations using the application Interpolate / Condi-tional Simulations / Plurigaussian. Only one simulation is performed using the same neighborhood and specific parameters than for the Upper unit.

Plurigaussian 393

(snap. 6.8-8)

The next figure represents the layer 26 projected on the horizontal plane for the output model and a cross-section.

394

(fig. 6.8-4)

Plurigaussian 395

6.9 Merging the Upper and Lower Units

The different units are now simulated in separate 3D working grid files. It is now time to merge all these simulation outcomes and to back transform them in the 3D structural grid, using the Tools / Merge Stratigraphic Units facility.

(snap. 6.9-1)

We first define the different 3D units to be merged: they are characterized by the corresponding Proportions Standard Parameter Files (UnitTop and UnitBottom) which contain all the relevant information such as the name of the pointer variable (not shown in the interface) or the name of the macro variable containing the lithotype simulations.

396

We also define the 3D structural grid (reservoir / simu) where the merged results will be stored in a new macro variable (lithotype).

The number of simulated outcomes in the different 3D working grids may be different. The princi-ple is to match the outcomes by their indices and to create a merged outcome in the structural file using the same index.

In the bottom part of the window, the procedure concatenates the list of all the lithotypes present in all the units. We must now define a set of new lithotype numbers which enable us to regroup litho-types across units: this is the case here for the lithotype Shale present in both the upper and the lower units, which is assigned to the same new lithotype number (3).

We can then use the Colors option in order to define the name and color attributes for the new litho-types and define the corresponding new palette and Color Scale (LithotypesReservoir).

(snap. 6.9-2)

You can then Run the Merge procedure.

The next graphic shows the merged simulation. The upper part correspond to a plan view represen-tation of the resulting simu grid in raster mode (Z node or level = 72). A section is discretized in this view and updated in all graphics in order to be taken into account within the application Display / Grid / 3D Fences / Unfolded.

Plurigaussian 397

(fig. 6.9-1)

We can see the two units merged and back transformed in the structural position with the limiting surface between them. The final statistics consist in running the Statistics/Quick Statistics applica-tion on the lithotype variable of the resulting 3D structural grid (reservoir / simu) in order to get the global statistics on the different lithotypes:

Integer Statistics: Variable lithotype[00001] Integer value Count of samples Percentage 1 6545 1.31% 2 81184 16.25% 3 106415 21.29% 4 136680 27.35% 5 73203 14.65% 7 89028 17.82% 8 6678 1.34%

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Structural Grid – Plan view, IZ=72 Lithofacies simulation

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Structural Grid – Cross section Lithofacies simulation

LithotypesConglomerate

Sandstone

Shale

Limestone (upper)

Continental sandstone (lower)

Very shallow packstones (lower)

Shallow wackstones (lower)

398

Oil Shale 399

7 Oil Shale

This case study illustrates the use of faults on a 2D data set containing two variables: the elevation of the bottom of a layer and its thickness. Last update: Isatis version 12.0

400

7.1 Presentation of the Dataset

The data set consists in two ASCII files:

l the first one (called oil_shale.hd) contains the sample information, i.e:

m the name of the borehole,

m its coordinates,

m the depth of the bottom of the layer (counted positively downwards) called elevation,

m the thickness of the layer.

l the second one (called oil_fault.hd) contains the coordinates of 4 segments which constitute the main fault system as digitized by the geologist.

7.1.1 Loading the Data

The data is loaded using the Files / Import / ASCII from the file oil_shale.hd into a new Directory (Oil Shale) and a new File (Data).

(snap. 7.1-1)

Oil Shale 401

We can check the contents of the file by asking for some basic statistics on the variables of interest (all expressed in meters):

7.1.2 Loading Faults

The faults are defined as portions of the space which can interrupt the continuity of a variable.

In other words, when a fault is present:

l at the stage of the calculation of experimental variograms, the pair made of two points will not be considered as soon as the segment joining them intersects a fault.

l at the estimation phase, a sample will not be used as neighboring data if the segment joining it to the target intersects a fault.

In 3D, the faults are defined as a set of triangular planes. In 2D, the faults represent the projection on the XoY plane of possible 3D faults. Therefore, we can distinguish two categories of faults:

l the set of broken lines which corresponds to the trace of vertical 3D faults,

l the closed polygon which is the projection of a set of non vertical 3D faults: special options are dedicated to this case.

In this case study, the geologist has digitized one major fault which corresponds to a single broken line composed of four vertices. It is given in the ASCII file called oil_fault.hd.

# FAULTS SAVING: Directory: Oil Shale File: Data # # # max_priority=127 # # field=1 , type=name # field=2 , type=x1 , unit=m # field=3 , type=y1 , unit=m # field=4 , type=x2 , unit=m # field=5 , type=y2 , unit=m # field=6 , type=polygon # field=7 , type=priority # # # #+++++++----------++++++++++----------++++++++++----++++ 1 5000.00 69000.00 13000.00 63000.00 0 1 1 13000.00 63000.00 13000.00 54000.00 0 1 1 13000.00 54000.00 19700.00 45700.00 0 1 1 19700.00 45700.00 37000.00 69000.00 0 1

Variable name Number of

valid samples

Minimum Maximum

X 191 637.04 55018.00

Y 191 27.84 68039.04

elevation 190 1299.36 2510.03

thickness 168 27.40 119.48

402

The faults are loaded using File / Faults Editor utility. This procedure is composed of a graphic main window. In its Application menu, we use, in order, the Load Attached File, to define the file where we want to load the faults (Directory Oil Shale, File Data already created in the previous paragraph) and the ASCII Import option to load the faults.

Pressing the Import button starts reading the fault which is now represented on the graphic window together with the data information. The single fault is given the "name" 1.

(fig. 7.1-1)

When working in the 2D space, this application offers various possibilities such as:

- digitizing new faults,

- modifying already existing faults,

- updating the attributes attached to the faults (such as their names).

An interesting attribute is the priority which corresponds to a value attached to each fault segment: this number indicates whether the corresponding segment should be taken into account (active) or not, with respect to a threshold priority defined in this application.

In order to check the priority attached to each segment of the fault, we select Edit Fault in the graphic menu, select the fault (which is now blinking) and ask for the Information option.

Polyline Fault: 1 Priority:[1,1] Nber of Segments: 4

1

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Oil Shale 403

This statement tells us that the designated fault, called 1, is composed of four segments whose pri-ority are all equal to 1. This means that, if the threshold is left to 127 (value read from the ASCII file containing the fault information), all the segments are active.

If we decrease the threshold down to 0 in the main graphic window, the fault is now represented by dashed lines signifying that no segment is active and the data set would then be treated as if no fault had been defined. By giving different priorities to different segments, we can then differentiate the severity of each segment and set it for the next set of actions.

As we want to use the faults in this case study, we modify the threshold to 1 (in fact any positive value) and use SAVE and RUN in the Application menu to store the fault and the threshold value together with the data information.

7.1.3 Creating the Output Grid

The initial task consists in defining the grid where the results will be stored, using File / Create Grid File. We will minimize the extrapolated area, avoiding extending the field of investigation down to the furthest sample points in the South. The following parameters are used:

l X origin: 0m, Y origin: 15000m,

l X and Y mesh: 1000m,

l X nodes number: 58, Y nodes number: 55.

This procedure allows a graphic control where the final grid is overlaid on the initial data set.

(fig. 7.1-2)

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404

7.2 Exploratory Data Analysis

The structural analysis will be performed in terms of variograms calculated within the Statistics / Exploratory Data Analysis module, using the target variables elevation and thickness from the data file.

Check that, when displaying a Base Map on any of the target variables, the fault is active.

(fig. 7.2-1)

The next task consists in checking the relationship between the two target variables thickness and elevation: this is done using a scatterplot where the regression line is represented.

The two variables are negatively correlated with a correlation coefficient of -0.72. An interesting feature consists in highlighting the samples located on the upper side of the fault from the base map: they are represented by asterisks and correspond to the smallest values of the elevation and also almost always to the largest values of the thickness.

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thickness

Oil Shale 405

(fig. 7.2-2)

Because of this rather complex correlation between the two variables (which depends on the loca-tion of the samples with regard to the faulting compartment), we decide to analyze the structures of the two target variables independently.

Due to the high sampling density a preliminary quick interpolation may help to understand the main features of the phenomenon. Inside Interpolate / Interpolation / Quick Interpolation a Linear Model Kriging is chosen to estimate each variable using a Unique neighborhood.

(snap. 7.2-1)

1250 1500 1750 2000 2250

elevation

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thickness

rho=-0.724

406

(snap. 7.2-2)

(fig. 7.2-3)

Note - These displays are obtained with a superimposition of a grid in raster representation, in isolines and the fault. Details are given at the last section of this case study.

From the last graphic it is clear that the thickness is anisotropic with the elongated direction of the anisotropy ellipse close to the NW-SE direction.

Oil Shale 407

Note - A variogram map calculation applied to thickness and elevation datasets would lead to similar conclusions about the main directions of anisotropy.

Two directional variograms are then calculated with 15 lags of 2km each and an azimuth rotation of 45 degrees (N45).

(fig. 7.2-4)

We save this set of two directional variograms in the Parameter File called Oil Shale Thickness.

Note - By displaying the variogram cloud and highlighting several variogram pairs, the user may note that none of these pairs crosses the faults.

We reproduce similar calculations on the elevation variable, for 15 lags of 2km each, but this time the rotation is slightly different: Azimuth 30 degrees (N30).

N45

N135

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200

300 Variogram : thickness

408

(fig. 7.2-5)

We note that variograms of the two variables have similar behaviour even if the directions are slightly different. The set of directional variograms is saved in the Parameter File called Oil Shale Elevation.

N30

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Variogram : elevation

Oil Shale 409

7.3 Fitting a Variogram Model

The procedure Statistics / Variogram Fitting will be used twice in order to fit a model to each set of experimental directional variograms previously calculated. Each model is stored in a new Parame-ter File bearing the same name as the one containing the experimental quantities: they will still be distinguished by the system as their type is different. The thickness and Elevation variograms are fitted using a single basic structure: a power variogram.

Click Experimental Variogram and select oil Shale Elevation. Skip the model initialization frame and fom the Automatic Fitting tab click Strucutres to add a Power Model.

410

(snap. 7.3-1)

Oil Shale 411

(snap. 7.3-2)

Click Constraint to allow the anisotropy and then press Fit.

412

(snap. 7.3-3)

In order to check the automatic fitting on the two directions simultaneously, we use the Global Win-dow. The model produced is satisfactory. Press Run(Save) to save the parameter file.

Repeat the process for Thickness.

Oil Shale 413

(fig. 7.3-1)

(fig. 7.3-2)

N45

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Variogram : thickness

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7.4 Estimation

7.4.1 Estimation of Thickness

The estimation will be performed with the procedure: Interpolate / Estimation / (Co-)Kriging.

Note - Being a property of the Input File, the fault system will be automatically taken into account in the estimation process.

(snap. 7.4-1)

We will store the results in the variables:

Oil Shale 415

l Thickness (Estimate) for the estimation,

l Thickness (St. dev.) for the corresponding standard deviation.

After having selected the variogram model, we must define the neighborhood which will be used for the estimation of both variables. It will be saved in the Parameter File called Oil Shale. In order to decluster the information, we will use a large amount of data per neighborhood (3x8) taken within a large neighborhood circle (30km).

Finally, in order to avoid too much extrapolation, no target node will be estimated unless there is at least 4 neighbors within a circle of radius 8km.

(snap. 7.4-2)

416

(snap. 7.4-3)

The first task is to check the consistency of these neighborhood parameters graphically using the Test button in the main window: a secondary graphic window appears representing the data, the fault and the neighborhood parameters. Pressing the Left Button of the mouse once displays the tar-get grid. Pick a grid node with the mouse again to start the estimation: each active datum selected in the neighborhood is then highlighted and displayed with the corresponding weight (as a percent-age). Using the Domain to be estimated item in the Application menu cross-hatches all the grid nodes where no estimation will be performed (next picture).

Oil Shale 417

(fig. 7.4-1)

Note - Although the sample locations are the same, the graphics obtained for the two variables will not necessarily be similar as the number of active data is not the same (190 values for elevation compared with only 168 for the thickness): the samples for which the current variable is undefined are represented as small dots instead of crosses.

Before visualizing the results, we run the same process with the elevation variable, modifying the name of the Parameter File containing the model to Oil Shale Elevation; we store the results in the variables:

l Elevation (Estimate) for the estimation,

l Elevation (St. dev.) for the corresponding standard deviation.

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7.5 Displaying Results

The thickness kriging result is visualized using several combinations of the display capabilities.

You are going to create a new Display template, that consists in an overlay of a grid raster, grid iso-lines and thickness data locations. All the Display facilities are explained in detail in the "Display-ing & Editing Graphics" chapter of the Beginner's Guide.

Click on Display / New Page in the Isatis main window. A blank graphic page is popped up, together with a Contents window. You have to specify in this window the contents of your graphic. To achieve that:

l Firstly, give a name to the template you are creating: Thickness. This will allow you to easily display again this template later.

l In the Contents list, double click on the Raster item. A new window appears, in order to let you specify which variable you want to display and with which color scale:

m In the Data area, in the Grid file select the variable Thickness (Estimate),

m Specify the title that will be given to the Raster part of the legend, for instance Thickness,

m In the Graphic Parameters area, specify the Color Scale you want to use for the raster dis-play. You may use an automatic default color scale, or create a new one specifically dedi-cated to the thickness variable. To create a new color scale: click on the Color Scale button, double-click on New Color Scale and enter a name: Thickness, and press OK. Click on the Edit button. In the Color Scale Definition window:

- In the Bounds Definition, choose User Defined Classes.

- Click on the Bounds button and choose 18 classes between 30 and 120, then click on OK.

- In the Colors area, click on Color Sampling to choose regularly the 18 colors in the 32 colors palette. This will improve the contrast in the resulting display.

- Switch on the Invert Color Order toggle in order to affect the red colors to the large Thickness values.

- Click on the Undefined Values button and select Transparent or Blank.

- In the Legend area, switch off the Automatic Spacing between Tick Marks button, enter 10 as the reference tickmark and 10 as the step between the tickmarks. Then, specify that you do not want your final color scale to exceed 6 cm. Switch off the Automatic Format button and set the number of digits to 0.

- Click on OK.

Oil Shale 419

(snap. 7.5-1)

m In the Item contents for: Raster window, click on Display current item to display the result.

m Click on OK.

l Back in the Contents list, double-click on the Isolines item to represent the thickness estimate in isolines:

m In the Data area, in the Grid file select the variable Thickness (Estimate),

m In the Data Related Parameters area, choose two classes of isolines:

- from 30 to 120 by steps of 10 with a solid line and labels,

- from 50 to 100 by steps of 50 with a double thickness line and labels,

m In the Graphic Parameters area, switch off the Visibility toggle

m Click on Display current item to display the result and then OK.

420

l Double-click on the Basemap item to represent the thickness values. In the Data area, select Data / thickness as the proportional variable. In the Graphic Parameters area, choose a size of 0.1 and 0.2 for the lower and the upper bounds. The samples where the thickness variable is not defined will be represented with blue circles. Click on Display Current Item to check your parameters, then on Display to see all the previously defined components of your graphic. Click on OK to close the Item contents panel.

l Double-click on the Faults. In the Data area, select file Data, the fault being a property of this file. Change the Faults Style to a double thickness red line. Click on Display to see all the defined components of your graphic. Click on OK to close the Item contents panel.

l In the Item list, you can select any item and decide whether or not you want to display its leg-end. Use the Move Back and Move Front button to modify the order of the items in the final Display.

l The Display Box tab allows you to decide whether you want display all the contents or just the area containing specific items. Select the mode: Containing a set of items, then click on the Ras-ter item and then on Display.

l Close the Contents window. Your final graphic window should be similar to the one displayed hereafter.

(fig. 7.5-1)

l Before closing the graphic window, click on Application / Store Page to save its contents, allowing you to reproduce easily this graphic later.

Multi-layer Depth Conversion With Isatoil 407

8 Multi-layer Depth Con-version With Isatoil

This case study illustrates the workflow of the Isatoil module, on a data set that belongs to a real field in the North Sea. For confidentiality reasons, the coordinates and the characteristics of the information have been modified. For similar reasons, the case study is only focusing on a subset of the potential reservoir layers. Last update: Isatoil version 5.0

408

8.1 Introduction

The main goal of Isatoil is to build a complete geological model in a layer-cake framework. This is done when the surfaces corresponding to the tops of the different units are established. The layer-cake hypothesis assumes that each unit extends between two consecutive surfaces. The order of the units remains unchanged over the whole field under study. One or several units may disappear over areas of the field: this corresponds to a pinch-out.

A secondary process produces the values of the petrophysical variables as a two-dimensional grid within each unit. Some units may be considered as outside the set of reservoirs and therefore they do not carry any valuable petrophysical information.

Finally the program may be carried over, with the estimation tool (Kriging) being replaced by Sim-ulations in order to reproduce the variability of each one of the parameters involved. This procedure allows a non-biased quantification of the volumes located above contacts and within polygons which delineate the integration areas.

Before reading this case study, the user should carefully read the technical reference dedicated to Isatoil, which describes the general terminology. This technical reference is available in the On-Line Help.

Note - The aim of Isatoil is to derive a consistent layer-cake 2D block model: in other words, it will be used in order to determine the elevations of a set of surfaces. Therefore elevation will be regarded as the variable rather than a coordinate, while all the information will be considered as 2D. Isatis will also be used during this case study, whenever Exploratory Data Analysis and particular types of graphics are required. The user should already be acquainted with the various Isatis applications, therefore we shall only mention in this documentation the names of the Isatis panels that will be used - e.g. Isatis / Statistics / Exploratory Data Analysis.

8.2 Field Description

The field of interest is reduced to a single set of layers contained in a rectangular area of about 14

km2 located in the North Sea.

It contains several surfaces which delineate the reservoir units. Each surface is given a set of desig-nation codes (area, layer and zone codes) which indicates its order in the layer-cake.

The geological model contains 4 units that are vertically contiguous: Upper Brent, Lower Brent, Dunlin and Statfjord.

The top and bottom surfaces of these units correspond to seismic markers that are clearly identified and measured in time. The time maps will be converted into depth in the layering phase: the corre-sponding depth surfaces are referred to as Layers.

Multi-layer Depth Conversion With Isatoil 409

Because of its high quality, the seismic marker corresponding to the top of the Upper Brent forma-tion has been selected as the Top Layering surface, from which the whole layer-cake sequence will be derived.

The second unit (Lower Brent) has been subdivided into 4 production Zones. These zones do not correspond to visible seismic markers and thus have no time map associated.

Note - Other units were originally subdivided into zones but this option has not been retained in this Case Study for the sake of simplicity.

The entire layer-cake sequence is truncated at the top by the Base Cretaceous Unconformity (BCU) as well as two other erosional surfaces (named ERODE1 and ERODE2). These surfaces correspond to Upper Limits. This field does not have any Lower Limit.

The geological sequence is summarized in the following table where the name, the type and the designation codes of the different surfaces are listed; the area is skipped as it is always equal to 1.

8.2.1 Introducing the data

The data consists of:

l the well geometry file which contains the intercepts of the wells with the different surfaces

l the well petrophysics file which contains the values of the petrophysical variables sampled within the zones

l the grid file where all the results will be stored

8.2.1.1 The well geometry file

This file contains data organized as lines with two sets of information.

Surface Name Surface Type Layer Zone

BCU Upper Limit 0 0

ERODE 1 Upper Limit 0 1

ERODE 2 Upper Limit 0 2

Upper Brent - B1 Top Layering 1 0

Lower Brent - B4 Layer 2 0

Lower Brent - B5A Zone 2 1

Lower Brent - B5B Zone 2 2

Lower Brent - B6 Zone 2 3

Dunlin - D1 Layer 3 0

Statfjord - S1 Layer 4 0

Base Statfjord - BS Layer 5 0

410

The header file contains the following variables:

l the name of the well

l the coordinates (in meters) of the well collar

The base file is two-dimensional. Each sample contains the following variables:

l the 2D coordinates (in meters) of the intercept of each well with the surfaces constituting the geological model

l the depth (in meters) of the intercept of each well with the surfaces constituting the geological model

l the indices for the area, the layer and the zone designation codes

8.2.1.2 The well petrophysics file

This file contains the following variables:

l the 2D coordinates (in meters) of the point where the petrophysical parameters are sampled. There may be more than one sample along the same well within a given unit, for instance if the well is highly deviated or even horizontal.

l the depth (in meters) of the petrophysical measurements

l the values of the petrophysical measurements

l the indices for the Area, the Layer and the Zone designation

8.2.1.3 The grid file

This file - organized as a regular 2D grid - will contain all the results of the study. To be more accu-rate Isatoil will use its definition to create another grid file (if necessary) in order to store auxiliary results during volumetrics calculation.

This file also contains several input variables:

l the Top Layering surface - Upper Brent B1 - from which the entire layer-cake sequence will be derived

l the Upper Limit surfaces - ERODE1 & ERODE2 -

l the time variables for the different seismic markers

l the trend surfaces used as external drift for the petrophysical parameters

The input variables read from the ASCII grid file must follow the terminology as defined in the Reference Guide.

Multi-layer Depth Conversion With Isatoil 411

8.3 Loading the Data

8.3.1 Initializing a new study

The following steps should be carried out first to make sure that you will be able to compare the results that you will obtain in Isatoil with the statistics and illustrations reported in this Case-Study.

First of all make sure to create a fresh new Study in the Data File Manager

Then go to Preferences / Study Environment and setup the following parameters:

l Default Distance Unit: meters

l Units for the X & Y axes on the graphics: kilometers

8.3.2 Loading the data

All the data used in this Case Study is provided in ASCII files compatible with the formats required by the software. These files can be immediately loaded by Isatoil, they are located at the following place on your disk:

m $GTX_HOME/Datasets/Isatoil (UNIX)

m C:\Program Files\Geovariances\Isatis\Datasets\Isatoil (Windows)

In this case study the three data files will be loaded using the standard ASCII import facility (File / Import / ASCII). For convenience all the files will be stored in the same Directory named Reser-voir.

You user should refer to the Isatis manual for a detailed description of the standard import applica-tions that are shared by Isatis & Isatoil.

8.3.2.1 Loading the well geometry file

The contents of the ASCII file (wells.hd) will be stored in two linked files because of the Line orga-nization of the samples. The reading mode to be used when loading this datafile is Points+Lines. Use Wells header as the name of the output Points file, and Wells Lines (Geometry) for the output Lines file.

You can check with the Data File Manager that the directory named Reservoir should contain two files:

l a header file (Wells header) with 50 samples and the following variables:

m SN+ Sample Number (READONLY) contains the rank of the well

m X-Top and Y-Top are the coordinates of the well tops

m well name contains the name which distinguishes one well from another - this is a 3-charac-ter alphanumerical variable -

412

l the base file (Wells Lines (Geometry)) with 972 samples and the following variables:

m SN+ Sample Number (READONLY) contains the rank of the sample in the file - from 1 to 972 -

m LN+ Line Number (READONLY) which contains the rank of the line to which the sample belongs - from 1 to 50 -

m RN+ Relative Number (READONLY) which contains the rank of the sample in the line to which it belongs - from 1 to 50, 50 being the count of samples in the longest well -

m X-Coor and Y-Coor are the coordinates of the intercepts

m Z-Coor is the depth of the intercepts

m Area Code, Layer Code and Zone Code are the designation codes for the geological sequence

Some basic statistics show that the data set is constituted of 50 wells (or 972 intercepts) and that the depth of the intercept (variable Z-Coor) varies between 2337m and 3131m.

Note that the field extension (in the XOY plane) is different for the two files:

This indicates that the wells are not strictly vertical, which one can check out on the following XOY projection, performed with Display/Lines in Isatis.

Variable Header File Base File

Minimum along X (m) 37. -317.

Maximum along X (m) 3675. 4141.

Minimum along Y (m) 48. -777.

Maximum along Y (m) 4919. 4919.

Multi-layer Depth Conversion With Isatoil 413

(fig. 8.3-1)

Horizontal projection of the geological well information (with the well names)

In the same application, the wells may also be represented in a perspective volume which gives a better understanding of the well trajectories in 3D.

Note - This operation is not straightforward since the well information has been loaded as 2D data. The well file must be temporarily modified into 3D lines: the elevation variable is transformed into a coordinate - in the Data File Manager - for the sake of the 3D representation.

One can check that most of the wells are highly deviated - squares indicate the tops of the wells and triangles the bottoms.

414

(fig. 8.3-2)

Perspective view of the wells

8.3.2.2 Loading the well petrophysics file

The data is available in the ASCII file named wells_petro.hd

The reading mode to be used when loading this datafile is Points.

Use Wells (Petrophysics) as the name of the output Points file.

The file named Wells (Petrophysics) in the directory named Reservoir will contain the following variables:

l SN+ Sample Number (READONLY) contains the rank of the sample in the file - from 1 to 408 -

l X-Coor and Y-Coor are the coordinates of the samples

l Area Code, Layer Code and Zone Code are the designation codes for the geological sequence

l Porosity and Net to Gross Ratio are the measurements of the petrophysical variables

Note - Note that there is no variable corresponding to the third coordinate. As a matter of fact the petrophysical parameters are assumed to be vertically homogeneous. Therefore it suffices to know the unit to which the measurements belong (as well as the X & Y coordinates) in order to perform the corresponding 2D estimation or simulations.

The data set consists of 408 samples. The following basic statistics are reported for the two petro-physical variables - using Statistics / Quick Statistics in Isatis - Please note that both variables are not necessarily defined for the same samples - Count indicates the number of samples at which the variables are defined.

Multi-layer Depth Conversion With Isatoil 415

The following picture shows the distribution of Porosity (crosses) and Net to Gross ratio (circles) on an horizontal projection - using Display/Points/Proportional in Isatis - The grey spots correspond to samples where one of the variables has not been measured.

(fig. 8.3-3)

Distribution of petrophysical variables

At this stage it is interesting to notice the lack of dependency between the two petrophysical vari-ables. Let's recall that these variables will be processed independently in Isatoil.

The validation is performed on the whole data set - regardless of the geological unit - and illustrated on the following Scatter Plot - using Statistics / Exploratory Data Analysis in Isatis.

The 287 samples at which both variables are sampled lead to a correlation coefficient of 0.59 with a dispersed cloud which enforces the validity of the hypothesis. The regression line is also repre-sented.

Variable Count Minimum Maximum Mean St. Dev.

Porosity 290 0.0010 0.3400 0.2404 0.0451

Net to Gross ratio 307 0. 1.000 0.5777 0.3436

416

(fig. 8.3-4)

Scatter plot between Porosity and Net to Gross Ratio

8.3.2.3 Loading the grid file

The data is available in the ASCII file named grid_data.hd.

The reading mode to be used when loading this datafile is Grid.

Use Wells (Grid) as the name of the output 2D Grid file.

The parameters of the grid are defined in the header of the ASCII file:

The following table gives the list of input variables defined on the grid. Note that the variable names comply with the Isatoil naming convention.

Along X Along Y

Origin (m) 0. 0.

Mesh (m) 50. 50.

Count 73 79

Maximum (m) 3600. 3900.

Multi-layer Depth Conversion With Isatoil 417

Variable Name Variable Type Surface Name

depth_1_0_0 Depth of Upper limit BCU

depth_1_0_1 Depth of Upper limit ERODE 1

depth_1_0_2 Depth of Upper limit ERODE 2

depth_1_1_0 Depth of Top Layering Upper Brent - B1

time_1_1_0 Time for Top Layering Upper Brent - B1

time_1_2_0 Time for layer Lower Brent - B4

time_1_3_0 Time for layer Dunlin - D1

time_1_4_0 Time for layer Statfjord - S1

time_1_5_0 Time for layer Base Statfjord - BS

trendporgauss_1_2_2 Trend for Porosity (normal transf.) Lower Brent - B5B

trendporosity_1_2_2 Trend for Porosity Lower Brent - B5B

418

8.4 Master File Definition

The Master File Definition panel drives all the remaining applications. The access to the other Isa-toil panels is frozen as long as the Master File has not been properly defined.

(snap. 8.4-1)

8.4.1 Definition of the data files

You must first define the input data that will be used by Isatoil, and that has already been loaded in the database:

l The well geometry file

Multi-layer Depth Conversion With Isatoil 419

In the Header File select the following variable:

m well name as the Zonation Well Name.

In the Line File select the following variables:

m Z-Coor as the Zonation Intercept Depth.

m Area Code as the Zonation Area Code.

m Layer Code as the Zonation Layer Code.

m Zone Code as the Zonation Zone Code.

l The well petrophysics file

In the Point File select the following variables:

m Porosity as the Porosity Variable.

m Net to Gross Ratio as the Net/Gross Variable.

m Area Code as the Petrophysical Area Code.

m Layer Code as the Petrophysical Layer Code.

m Zone Code as the Petrophysical Zone Code.

l the grid file:

This file defines the geometry of the grid where the results will be stored, and which also con-tains the time maps, Top Layer surface, Limit surfaces, etc.

You must simply select the file at that stage.

8.4.2 Verbose output

This additional parameter offers to printout all the relevant information for each calculation step. This option is used when detailed analysis of the results is required - e.g. auditing.

For convenience, in the rest of this documentation, we will assume that the Verbose option flag is switched off.

When this option is activated it produces two types of printout:

l the message each time a data point is discarded, for one of the following reasons:

m Error when calculating the thickness by comparison to the reference surface: the reference surface is not defined

m Error when calculating the thickness by comparison with an extrapolated (time) surface: the surface is not defined

m Error when calculating velocities by scaling by a time thickness: the thickness is null or not defined. This may happen essentially in the vicinity of a pinchout.

420

m Finding duplicates. If two intercepts (with the same layer) are located too close (less than one tenth of the grid mesh away), the points are considered as duplicates: their coordinates are printed and only the first point is kept, the second one is discarded.

When a point is discarded, the following message is produced with the references of the dis-carded information, followed by the final count of active data:

Discarding point in the following step : Calculating Layer Proportions (Degenerated Well) Well 113 (Zone id.=140) Coordinate along X = 1599.87m Coordinate along Y = 4134.25m Depth = 2772.030 - Initial count of data = 59 - Final count of active data = 52 the dump of the active information available in the following format: List of active information used for Correlations of Depth using the following variable(s): - Variable 1: Lower Brent - B4 - Variable 2: Dunlin - D1 - Variable 3: Statfjord - S1 - Variable 4: Base Statfjord - BS Rank - Name - X - Y - Initial - Data - Pr1 - Pr2 - Pr3 - Pr4 - Trend1 - Trend2 - Trend3 - Trend4 1 3 1965.27m 649.64m 2435.310 1.057 1.00 0.00 0.00 0.00 2313. 0 0 0 2 3 1965.27m 649.64m 2544.110 1.178 0.44 0.56 0.00 0.00 2313. 2398 0 0 3 3 1965.27m 649.64m 2813.410 1.373 0.20 0.26 0.53 0.00 2313. 2398 2573 0 4 4 2408.25m 3422.11m 2927.000 1.362 0.13 0.17 0.33 0.37 2279. 2353 2491 2649 5 113 1668.08m 3070.14m 2772.050 1.341 0.17 0.27 0.56 0.00 2304. 2389 2566 0 6 119 827.59m 3060.44m 2498.780 1.412 1.00 0.00 0.00 0.00 2373. 0 0 0 7 120 1162.99m 3212.98m 2456.170 1.183 1.00 0.00 0.00 0.00 2352. 0 0 0 8 120 1827.41m 2868.45m 2483.780 1.033 0.40 0.60 0.00 0.00 2299. 2379 0 0 9 120 1915.18m 2825.42m 2478.170 1.015 0.42 0.58 0.00 0.00 2297. 2374 0 0

For this calculation phase (a layering phase which processes 4 variables simultaneously) the dif-ferent columns represent:

m Rank: the rank of the active data.

m Name: the name of the well to which it belongs.

m X and Y: the 2-D coordinates of the information.

m Initial: the initial value, as found in the data base. In this case of layering, the data consist of the depth of the intercept.

m Data: the data after it has been pre-processed for usage in the next calculation step. In this case of layering, data are converted into velocities.

m Pr1,...,4: percentage spent in each layer. The percentage is set to 0 if the layer is not reached. In this case of layer (in velocity), the value represents the time percentage spent in each layer located above the intercept.

m Trend1,...,4: trend used as an external drift for each layer. In this case of layering, the time of each layer is used as its external drift. The trend value is not printed for a layer which is not reached.

Multi-layer Depth Conversion With Isatoil 421

8.4.3 Setup of the geological sequence

Note - In this Case Study most of the surfaces constituting the original North Sea environment have been kept with their original names in order to provide realistic results, however the layer-cake sequence has been reduced - both in terms of covered area and in the number of units - and the coordinate reference system has been transformed for confidentiality.

To achieve the Master File definition you must build up the list of surfaces that will be used and you will also define the sets of parameters - geometry, petrophysics, faulting, etc. - possibly attached to these surfaces.

Please refer to the Reference Guide for detailed explanations on the meaning of the various param-eters that can be accessed while editing a surface.

We recommend to start defining the geological sequence from the top (BCU) to the bottom (Base Statfjord). This is the order in which they will be presented in the list. For a start just define the dif-ferent surface names and make sure to give them the proper types and codes.

The following table recalls the correspondence between the different surfaces and the designation codes employed in Isatoil. There is only one Area which covers the entire area of interest - hence the Area code = 1.

When the list is completely initialized you will need to Edit the different surfaces separately in order to give them their parameters and constraints for computation.

8.4.3.1 Geometry definition

The following table summarizes the geometrical parameters that must be defined for the surfaces in this Case-Study:

Area Layer Zone Surface Name Surface Type

1 0 0 BCU Upper Limit

1 0 1 ERODE 1 Upper Limit

1 0 2 ERODE 2 Upper Limit

1 1 0 Upper Brent - B1 Top Layering

1 2 0 Lower Brent - B4 Layer

1 2 1 Lower Brent - B5A Zone

1 2 2 Lower Brent - B5B Zone

1 2 3 Lower Brent - B6 Zone

1 3 0 Dunlin - D1 Layer

1 4 0 Statfjord - S1 Layer

1 5 0 Base Statfjord - BS Layer

422

The first 4 surfaces - Top Layering and Limit surfaces - cannot be calculated by Isatoil, they are already stored on the Grid file and will be used as data. All the other surfaces will be calculated.

m T2D indicates whether the Time to Depth conversion will be performed using intermediate Velocity or directly in terms of Thickness.

m EDL indicates the type of external drift information possibly used during the Layering stage

m EDZ tells the type of external drift information possibly used during the Zonation stage.

8.4.3.2 Faulting definition

Fault Polygon files are located at the following place on your disk:

m $GTX_HOME/Datasets/Isatoil (UNIX)

m C:\Program Files\Geovariances\Isatis\Datasets\Isatoil (Windows)

The following table summarizes the faulting parameters that must be defined for the surfaces in this Case-Study. The Unit must be set to Meter whenever polygons are used.

Surface Name Calculated T2D EDL EDZ

BCU No

ERODE 1 No

ERODE 2 No

Upper Brent - B1 No

Lower Brent - B4 Yes Velocity Time No

Lower Brent - B5A Yes No

Lower Brent - B5B Yes No

Lower Brent - B6 Yes No

Dunlin - D1 Yes Velocity Time No

Statfjord - S1 Yes Velocity Time No

Base Statfjord - BS Yes Velocity Time No

Multi-layer Depth Conversion With Isatoil 423

l Count designates the number of fault polygons in the file. Some polygons which do not lie within the rectangular area of interest will be automatically discarded.

8.4.3.3 Contact definition

The following table summarizes the geometrical parameters that must be defined for the surfaces in this Case-Study. The OWC and GOC values are assumed to be constant and are defined for the first index only.

Note - Note the particular case of Lower Brent - B4 where no GOC is provided. The only fluids that can be encountered in this zone are Oil and Water.

Surface Name Surface Type Faulting Fault Polygon file Count

BCU Upper Limit No

ERODE 1 Upper Limit No

ERODE 2 Upper Limit No

Upper Brent - B1 Top Layering Yes b1.pol 22

Lower Brent - B4 Layer Yes b4.pol 22

Lower Brent - B5A Zone No

Lower Brent - B5B Zone No

Lower Brent - B6 Zone No

Dunlin - D1 Layer Yes d1.pol 22

Statfjord - S1 Layer Yes s1.pol 23

Base Statfjord - BS Layer Yes bst.pol 22

Surface Name Surface Type Calculated GOC (m) OWC (m)

BCU Upper Limit No No No

ERODE 1 Upper Limit No No No

ERODE 2 Upper Limit No No No

Upper Brent - B1 Top Layering No 2570 2600

Lower Brent - B4 Layer Yes No 2600

Lower Brent - B5A Zone Yes No No

Lower Brent - B5B Zone Yes 2570 2600

Lower Brent - B6 Zone Yes No No

Dunlin - D1 Layer Yes No No

Statfjord - S1 Layer Yes No No

Base Statfjord - BS Layer Yes No No

424

8.4.3.4 Petrophysics definition

The following table summarizes the petrophysical parameters - Porosity and Net to Gross Ratio - that must be defined for the surfaces in this Case-Study:

l Calc indicates whether the petrophysical variable must be calculated or not,

l Norm indicates if the variable must be Normal Score transformed before the simulation pro-cess,

l ED indicates if the estimation (or simulation) should take an external drift into account.

8.4.3.5 Saturation definition

The following table summarizes the saturation parameters that must be defined for the various sur-faces in this Case-Study:

Porosity Net to Gross Ratio

Surface Name Surface Type Calc Norm ED Calc Norm ED

BCU Upper Limit No No No No No No

ERODE 1 Upper Limit No No No No No No

ERODE 2 Upper Limit No No No No No No

Upper Brent - B1 Top Layering Yes Yes No Yes Yes No

Lower Brent - B4 Layer Yes No No Yes No No

Lower Brent - B5A Zone No No No No No No

Lower Brent - B5B Zone Yes Yes Yes Yes Yes No

Lower Brent - B6 Zone No No No No No No

Dunlin - D1 Layer No No No No No No

Statfjord - S1 Layer No No No No No No

Base Statfjord - BS Layer No No No No No No

Gas Oil

Surface Name Surface Type A B C A B C

BCU Upper Limit 0 0 0 0 0 0

ERODE 1 Upper Limit 0 0 0 0 0 0

ERODE 2 Upper Limit 0 0 0 0 0 0

Upper Brent - B1 Top Layering 0.329 1.145 0.949 0.663 0.918 0.106

Lower Brent - B4 Layer 0.604 1.135 0.357 0.107 1.661 0.106

Lower Brent - B5A Zone 0 0 0 0 0 0

Lower Brent - B5B Zone 0.332 1.763 0.587 0.714 0.756 0.856

Multi-layer Depth Conversion With Isatoil 425

8.4.3.6 Constant definition

The following table summarizes the constant parameters that must be defined for the various sur-faces in this Case-Study:

8.4.3.7 Surface Statistics and verification

The Stats-1 and Env-1 buttons can be used in order to report individual statistics about the selected surface in the list.

l Stats-1 will produce the basic statistics of all the information regarding the selected surface.

l The following example will be obtained for the Upper Brent B1 surface - obviously after the calculations have been performed -

General Statistics ================== Layer : Upper Brent - B1 (Identification : Area = 1 - Layer = 1 - Zone = 0 - Top Layer - Faulted) Grid - Time value : Nb = 5767 - Min = 2155.364 - Max = 2399.886 Grid - Depth value : Nb = 5767 - Min = 2304.683 - Max = 2561.189 Wells - Depth value : Nb = 13 - Min = 2340.490 - Max = 2505.970 Wells - Porosity : Nb = 8 - Min = 0.247 - Max = 0.302 Wells - Net/Gross : Nb = 8 - Min = 0.769 - Max = 0.905

Lower Brent - B6 Zone 0 0 0 0 0 0

Dunlin - D1 Layer 0 0 0 0 0 0

Statfjord - S1 Layer 0 0 0 0 0 0

Base Statfjord - BS Layer 0 0 0 0 0 0

Volume correction factors Color

Surface Name Surface Type Gas Oil

BCU Upper Limit 0 0 Black

ERODE 1 Upper Limit 0 0 Black

ERODE 2 Upper Limit 0 0 Black

Upper Brent - B1 Top Layering 110 1.31 Yellow

Lower Brent - B4 Layer 105 1.44 Red

Lower Brent - B5A Zone 0 0 Pink

Lower Brent - B5B Zone 110 1.42 Purple

Lower Brent - B6 Zone 0 0 Orange

Dunlin - D1 Layer 0 0 Green

Statfjord - S1 Layer 0 0 Blue

Base Statfjord - BS Layer 0 0 White

426

l Env-1 will list the parameters of the selected surface.

The following example is obtained for the Upper Brent B1 surface:

General Environment =================== Layer : Upper Brent - B1 (Identification : Area = 1 - Layer = 1 - Zone = 0 - Top Layer - Faulted) Geometry : Must not be calculated (always read from the file) Porosity : Calculation : No special option Will be normal score transformed (before simulations) Net to Gross : Calculation : No special option Will be normal score transformed (before simulations) Saturation : Not defined Contacts : Segment Variable : None Vol. fact. : Gas correction factor : 0.000000 Oil correction factor : 1.528000 Displaying the Data

At this stage of the Case-Study no surface has been calculated yet. However the reference depth surface - Top Layering - as well as the different time surfaces have been loaded, therefore we can already perform various types of graphical representations of this data. Obviously these representa-tions will also apply to the results that will be obtained later in the project.

8.4.4 Map representations

The Display / Map application lets you visualize one of the following types of variables:

m Time

m Depth

m Isochrone - time interval -

m Isopack - depth interval -

m Velocity

m Porosity

m Net to Gross ratio

Let us first use Display / Map to visualize maps of some of the surfaces that are already available on the final grid.

8.4.4.8 Representing a time surface

We will first display the time map of the layer Upper Brent- B1. By clicking on the Check button we can see that time varies from 2155 ms to 2400 ms on this layer, and that time is defined on the entire grid - 5767 nodes -

Multi-layer Depth Conversion With Isatoil 427

(snap. 8.4-2)

We choose to represent:

l The time variable as a colored image - using the automatic Color Scale named Rainbow -

l The corresponding legend

l The time variable as a series of contour lines - click on the Edit button to access the contour lines definition window -

(snap. 8.4-3)

In this example, the variable is smoothed prior to isoline representation - using 3 passes in the filtering algorithm - and two sets of contour lines are represented:

m the multiples of 10 ms using a red solid line

m the multiples of 50 ms using a black solid line and representing the label on a pink back-ground

428

l The well information: the corresponding Edit button is then used to define the characteristics of the point display.

(snap. 8.4-4)

In this example the intercepts with the target surface - Upper Brent - B1 - are represented with a "+" sign and the well name is displayed in a rectangular white box.

l The fault polygons

Multi-layer Depth Conversion With Isatoil 429

Click on RUN to obtain the following map:

(fig. 8.4-1)

8.4.4.9 Representing an isochrone map

An Isochrone surface - i.e. the interval between two time maps - is not stored in the database. It is calculated at run time whenever is necessary, for instance for display. Let us display the isochrone surface between the Upper Brent - B1 - the First Variable - and the Base Statfjord - BS - the Second Variable - time maps. We can Check that this isochrone varies between 343 ms and 507 ms.

The isochrone is represented as a colored image - using the same Rainbow Color Scale as before - but without contour lines.

The Fault polygons flag enables to display the polygons corresponding to the upper surface - in black - while the Auxiliary Fault polygons flag activates those of the lower surface - in red - The well display only represents the intercepts with the upper surface.

430

(fig. 8.4-2)

This display clearly shows the shift of the non-vertical fault planes through their intersections with two time surfaces located around 250 ms apart. It also shows the impact of the faulting on the isoch-rone map.

Note that in the upper-right corner the three faults intersect the Base Statfjord - BS level although they are not visible on the Upper Brent - B1 level - at least within the grid extension -

8.4.4.10 Representing a depth map

The Top Layering reference surface - Upper Brent - B1 - is a depth variable that we can also dis-play at this stage. We can take into account the Upper Limit surfaces - BCU, ERODE1 & ERODE2 - which truncate the reference surface, hence the following map where the truncated area appears in black:

Multi-layer Depth Conversion With Isatoil 431

(fig. 8.4-3)

In terms of statistics we can check that 2135 grid nodes are defined out of 5767.

8.4.5 Time Section representation

Any Time Section through the geological sequence can displayed, given 2 points that define the extremities of the section's segment. All the time surfaces are interpolated along this section while the layers are being represented with the colors that have been defined in the Master File.

Several representation options are still locked at this stage, however we can:

l represent the legend

l draw the section which corresponds to the first bisector of the field - click on the Automatic button to initialize the segment's coordinates -

l switch OFF the Automatic Vertical Scale and instead use the following parameters:

m default Minimum and Maximum elevations - click on Automatic -

m a Vertical Scale Factor of 300 to exaggerate the thickness for better legibility.

The figure below clearly shows the impact of (at least) one non-vertical fault.

432

(fig. 8.4-4)

We can add the traces of the fault polygons corresponding to each layer on top of the previous sec-tion. The intersection between the vertical section and the fault polygons attached to a given layer is represented as vertical lines - with the same color coding as the layer - This helps checking that fault polygons are indeed matching with the time maps.

(fig. 8.4-5)

There is an interactive link between the map and section representations, so that you can:

l Display the location of the current section on time maps, depth maps, etc.

l Any map can be used in order to digitize a new segment while the sections are being refreshed simultaneously.

Multi-layer Depth Conversion With Isatoil 433

8.5 Building the Reservoir Geometry

Building the reservoir geometry consists in estimating the surfaces which divide the layer-cake into layers and zones which are vertically homogeneous against petrophysical parameters.

This operation makes use of the information regarding the intercept locations (from the well geom-etry file), the fault polygons and possible variables (read from the grid file) which are used as exter-nal drifts.

Building the geometrical frame of the sequence is essentially performed in two nested steps:

l first a Layering which estimates the seismic units - layers -

l then a Zonation which subdivides each seismic unit into several zones

8.5.1 Layering

8.5.1.1 Correlation for Layering

The Geometry / Seismic Layering / Correlations application allows us to check the hypothesis con-cerning the correlation between layer thickness and the trend surfaces used as external drift - if applicable -

Note - In this Case Study we have specified that Layering should be performed through velocity - rather than directly in depth - using the time maps as external drift surfaces.

The application represents - in separate graphics - the behavior of the interval velocity against time, for each of the four layers constituting the sequence.

(snap. 8.5-1)

434

The system first derives the interval velocities from the apparent velocity information at wells (deduced from the Top Layering reference surface). For layer #N the interval velocity is obtained by:

l subtracting the thickness of all the layers located above layer #N - the thicknesses are simply estimated by their trend -

l dividing the residual thickness of layer #N by the time interval

Obviously the deeper the surface the less accurate - and often the less numerous - the represented data.

Select the following parameters for representing the well names:

l switch ON the flag which indicates that names will be posted on the graphics

l select the symbol (style and size) to be posted - e.g. a 0.2 cm star -

l select the background color for the label's box - e.g. white -

l choose some angle for writing the well name - e.g. 0. -

(snap. 8.5-2)

The following graphics show a good organization of the well data around the regression line for B4 and D1, and a more dispersed cloud for Statfjord S1 and Base Statfjord, as expected.

Multi-layer Depth Conversion With Isatoil 435

(fig. 8.5-1)

From the 52 active data remaining, the equation of the trend for each layer is produced in the mes-sage area:

Compression stage: - Initial count of data = 59 - Final count of active data = 52 The following trends are defined from the data Variable 1: Lower Brent - B4 = -15.50940 + Trend * ( 0.00708) Variable 2: Dunlin - D1 = -0.88130 + Trend * ( 0.00090) Variable 3: Statfjord - S1 = -1.23549 + Trend * ( 0.00109) Variable 4: Base Statfjord = -1.34698 + Trend * ( 0.00110)

436

8.5.1.2 Model Fitting for Layering

We must now build with Geometry / Seismic Layering / Model the geostatistical model which will be used to estimate - or simulate - all the layers simultaneously.

(snap. 8.5-3)

The experimental simple and cross-covariances are calculated in an isotropic manner and for a given number of lags - e.g. 10 lags of 1000 m -

Let us click on Edit and define the following set of Basic Structures:

l a first Spherical structure with a range of 1000 m

l a second Exponential structure with a range of 3000 m

Let us switch ON the flag Automatic Sill Fitting so that Isatoil will compute the set of optimal sills - for all simple and cross-structures - by minimizing the distance between the experimental covari-ances and the model.

Note - The matrix of the sills must fulfill conditions for definite positiveness.

By switching ON the flag Printout we will obtain the following report in the Message Window:

l for each pair of variables, the array of experimental covariances and the corresponding values in the model:

Multi-layer Depth Conversion With Isatoil 437

Printout of the experimental and theoretical variograms Covariance for variable Lower Brent - B4 Rank Pairs Distance Experimental Theoretical 1 42 325.67m 0.015105 0.019366 2 172 937.45m -0.001934 0.002372 3 136 1958.54m -0.004176 0.000099 4 64 2919.49m -0.007718 0.000038 5 6 3793.50m -0.004702 0.000016 Cross-covariance for variables Dunlin - D1 and Lower Brent - B4 Rank Pairs Distance Experimental Theoretical 1 86 265.58m 0.004259 0.006290 2 290 975.03m 0.000550 0.000586 3 196 1985.13m -0.002676 0.000038 4 92 2897.20m 0.000475 0.000015 5 8 3820.12m -0.005808 0.000006 .../...

l the parameters defining the model - i.e. for each basic structure, the coregionalization matrix, the coefficients of the linear model and the eigen vectors and values - :Number of basic structures = 2 S1 : Spherical - Range = 1000.00m Variance-Covariance matrix : Variable 1 Variable 2 Variable 3 Variable 4 Variable 1 0.0351 -0.0112 -0.0168 0.0032 Variable 2 -0.0112 0.0112 0.0055 0.0026 Variable 3 -0.0168 0.0055 0.0080 -0.0014 Variable 4 0.0032 0.0026 -0.0014 0.0020 Decomposition into factors (normalized eigen vectors) : Variable 1 Variable 2 Variable 3 Variable 4 Factor 1 0.1859 -0.0700 -0.0892 0.0119 Factor 2 -0.0228 -0.0792 0.0090 -0.0436 Factor 3 -0.0006 0.0002 -0.0014 -0.0003 Factor 4 0.0000 0.0000 0.0000 0.0000 Decomposition into eigen vectors (whose variance is eigen values) : Variable 1 Variable 2 Variable 3 Variable 4 Eigen Val. Var. Perc. Factor 1 0.8524 -0.3211 -0.4091 0.0544 0.0476 84.42 Factor 2 -0.2430 -0.8459 0.0957 -0.4651 0.0088 15.57 Factor 3 -0.3821 0.1043 -0.9013 -0.1756 0.0000 0.00 Factor 4 -0.2616 -0.4130 -0.1056 0.8660 0.0000 0.00 S2 : Exponential - Scale = 3000.00m Variance-Covariance matrix : Variable 1 Variable 2 Variable 3 Variable 4 Variable 1 0.0007 -0.0001 0.0004 0.0007 Variable 2 -0.0001 0.0000 -0.0000 -0.0001 Variable 3 0.0004 -0.0000 0.0003 0.0004 Variable 4 0.0007 -0.0001 0.0004 0.0007 Decomposition into factors (normalized eigen vectors) : Variable 1 Variable 2 Variable 3 Variable 4 Factor 1 0.0259 -0.0030 0.0163 0.0262 Factor 2 0.0000 0.0000 0.0000 0.0000 Factor 3 0.0000 0.0000 0.0000 0.0000 Factor 4 0.0000 0.0000 0.0000 0.0000 Decomposition into eigen vectors (whose variance is eigen values) : Variable 1 Variable 2 Variable 3 Variable 4 Eigen Val. Var. Perc. Factor 1 0.6421 -0.0736 0.4033 0.6477 0.0016 100.00 Factor 2 0.7666 0.0617 -0.3379 -0.5426 0.0000 0.00 Factor 3 0.0000 -0.1628 -0.8458 0.5082 0.0000 0.00 Factor 4 0.0000 -0.9820 0.0887 -0.1669 0.0000 0.00

438

By switching ON the flag Printout we will obtain the following graphics:

(fig. 8.5-2)

Each view corresponds to one pair of variables - e.g. D1 vs B4 - Only wells that intercept both lay-ers are retained, and the experimental quantity is then averaged at distances which are multiple of the lag - up to the maximum number of lags - The experimental curves are represented in black while the model appears in red. The values posted on the experimental curves correspond to the numbers of pairs averaged at the given distance.

Note - For better legibility only 6 of the actual 10 views are represented here.

The geostatistical model is stored in a Standard Parameter File named Model_area which will be automatically recognized when running the Base Case or the simulations later on.

Multi-layer Depth Conversion With Isatoil 439

8.5.1.3 Base Case for Layering

Let us now perform the estimation - a.k.a Base Case - of all the layers simultaneously.

The Geometry / Seismic Layering / Base Case application automatically loads the geostatistical model - which was stored in the Standard Parameter File named Model_area - and uses all the rel-evant information to perform the one-step cokriging of all the seismic layers.

The calculated surfaces are stored in the result Grid using the Isatoil naming convention - depth_area_layer_zone.

(snap. 8.5-4)

Basic statistics on the estimated surfaces are reported at the end of calculation:

Statistics on the base case results =================================== Layer : Lower Brent - B4 (Id : Area = 1 - Layer = 2 - Zone = 0 - Faulted) Grid - Depth value : Nb = 5767 - Min = 2362.685 - Max = 2692.186 Layer : Dunlin - D1 (Id : Area = 1 - Layer = 3 - Zone = 0 - Faulted) Grid - Depth value : Nb = 5767 - Min = 2380.481 - Max = 2799.331 Layer : Statfjord - S1 (Id : Area = 1 - Layer = 4 - Zone = 0 - Faulted) Grid - Depth value : Nb = 5767 - Min = 2540.824 - Max = 3096.612 Layer : Base Statfjord (Id : Area = 1 - Layer = 5 - Zone = 0 - Faulted) Grid - Depth value : Nb = 5767 - Min = 2724.232 - Max = 3363.551

By switching ON the flag named Replace estimation with one simulation Isatoil will perform a geo-statistical Simulation instead of a Kriging, using the Turning Bands method. The results are stored in the same variables as for the Base Case and they can be visualized to get a feeling for the amount of variability of simulation outcomes.

The following parameters are required by the simulation process:

l the seed used for the generation of the random numbers

l the number of turning bands used in the non-conditional simulation algorithm - Turning Bands method.

Note - Since the simulated results are stored in the same variables as the Base Case, always make sure to run the Base Case one more time before moving to the Zonation phase.

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8.5.1.4 Representing the results of Layering

Let us now look at the results of the Layering phase by displaying the surfaces maps with Display / Map. The following maps of the Base Case results - as well as the Top Layering surface Upper Brent - B1 - share the same Color Scale:

(fig. 8.5-3)

Multi-layer Depth Conversion With Isatoil 441

We can also visualize the estimated surfaces along a vertical section, by using the Display / Cross Section application. Similar to the Display / Time Section representation described before, this type of section is here performed in depth - with a wide range of available options -

(snap. 8.5-5)

Let us draw a section along the segment defined by the two points (X=605,Y=1119) and (X=3060,Y=2932). We shall activate the truncation of the estimated surfaces by the Limit Surfaces and also ask to represent the well information - names and intercepts - on top of the section. By set-ting the Maximum distance to the fence equal to 40 m, this section only shows three wells - 145, 152 & 191 -

The following figure shows the cross-section as well as two maps corresponding to the surfaces Statfjord - S1 and Dunlin - D1

442

(fig. 8.5-4)

The influence of the major fault - which is clear on this section - is inherited from the time maps that have been used as external drifts.

8.5.2 Zonation

For the sake of simplicity in this Case Study, the zonation has been restricted to the Lower Brentunit only. Moreover external drift will not be used during the zonation.

8.5.2.5 Model Fitting for Zonation

The modeling of the Lower Brent unit involves the following surfaces:

m Lower Brent - B4 which is the top surface of the layer

m Lower Brent - B5A

m Lower Brent - B5B

Multi-layer Depth Conversion With Isatoil 443

m Lower Brent - B6

m Dunlin - D1 which is the bottom surface - i.e. the top of the next layer -

The top and bottom surfaces that were estimated during the layering stage are now considered as known input data. By adding the bottom surface as an extra constraint - through an original Collo-cated Cokriging method - the Zonation ensures that the sum of the thickness of the four zones will match the total thickness of the unit.

The Geometry / Geological Zonation / Model application will be used to compute experimental simple and cross-covariances for a given number of lags - e.g. 10 lags of 500 m -

Let us click on Edit and define the following set of Basic Structures for the model:

m a first Spherical structure with a range of 2000 m

m a second Nugget Effect structure

Let us switch ON the flag Automatic Sill Fitting so that Isatoil will compute the set of optimal sills - for all simple and cross-covariances - by minimizing the distance between the experimental cova-riances and the model.

The following statistics - truncated here - are reported when the model is established:

l for the spherical component .../... Coregionalization matrix (covariance coefficients) : Variable 1 Variable 2 Variable 3 Variable 4 Variable 1 27.3009 1.6692 -2.2599 5.3611 Variable 2 1.6692 10.1627 -0.7517 -5.4312 Variable 3 -2.2599 -0.7517 2.0555 2.8640 Variable 4 5.3611 -5.4312 2.8640 9.1236

l for the nugget effect: .../... Coregionalization matrix (covariance coefficients) : Variable 1 Variable 2 Variable 3 Variable 4 Variable 1 14.0727 -10.6434 -2.2063 13.4752 Variable 2 -10.6434 75.1475 -75.7311 3.6356 Variable 3 -2.2063 -75.7311 89.6306 -18.2533 Variable 4 13.4752 3.6356 -18.2533 45.1714

The model is automatically saved in a Standard Parameter File named Model_1_2 which will be automatically recognized when running the Base Case or the simulations later on.

8.5.2.6 Base Case for Zonation

Let us now perform the estimation - a.k.a Base Case - of all the zones in the Lower Brent layer.

The Geometry / Geological Zonation / Base Case application automatically loads the geostatistical model - which was stored in the Standard Parameter File named Model_1_2 - and uses all the rele-vant information to perform the one-step cokriging of all the zones.

The calculated surfaces are stored in the result Grid using the Isatoil naming convention - depth_area_layer_zone -

444

The following basic statistics are reported for the three estimated zones - based on 64 active data out of 77 intercepts -

8.5.2.7 Representing the results of Zonation

The same vertical section that was visualized after the Layering may now be represented with the extra surfaces corresponding to the zones of the Lower Brent unit:

(fig. 8.5-5)

This section does not represent the fault surfaces (as interpolated within the package for chopping the zones) due to the small extension of the polygon fault at the vicinity of the cross-section seg-ment.

8.5.3 Running a simulation (instead of an estimation)

The program offers a possibility to check that the estimation process (using linear cokriging method) produces the smoothed picture of the surfaces. Therefore, to get accurate volume estimates above contacts (which is by definition a non-linear operation), we must use the simulation tech-nique instead, which reproduces the variability of the surfaces this time.

For volumetrics, the usual procedure consists of drawing a large number of simulations, to calculate the volume for each one of them and to present a risk curve.

As a preliminary task, we will simply run the base case procedures for the layering and the zona-tion, but replacing the estimation procedure by the simulation one: this facility produces a single simulation outcome. It requires the definition of:

Name Minimum Maximum

Lower Brent - B5A 2365 2727

Lower Brent - B5B 2380 2779

Lower Brent - B6 2380 2794

Multi-layer Depth Conversion With Isatoil 445

l the seed used for the random number generator: 324321

l the number of turning bands which is the essential parameter of the simulation technique used: 100.

The following figure shows the map of the thickness between tops surfaces of Lower Brent - B5Aand Lower Brent - B6 (isopack), either for the simulated version (on the left) or the estimated ver-sion (on the right). Although the spread of values is different (up to 84.3m for the simulation and 71.5m for the estimation - using the check button in the display window), the same color scale is used (lying between 50m and 85m). Any thickness smaller than 50m is left blank: this is the case for the fault traces for example.

(fig. 8.5-6)

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8.6 Filling the Units With Petrophysics

This step is dedicated to filling each unit of interest with petrophysical information (porosity or net to gross ratio). Petrophysical variables have been defined in the Master File for the following units:

l Upper Brent - B1

l Lower Brent - B4

l Lower Brent - B5B

Since the two petrophysical variables are assumed to be independent one from the other - and also from one unit to another - we must study 6 different variables separately.

8.6.1 Normal Score Transform

Petrophysics / Normal Score Transformation

In some cases the distribution of a petrophysical variable is far from the normal distribution. There-fore in order to be compatible with the simulation technique (Turning Bands) which requires a mul-tinormal framework, the data must be normal scored beforehand. The geostatistical models are then derived and the simulation process is performed on the transformed data. The results are finally back-transformed into the raw scale.

The need for a normal score transform is defined for each petrophysical variable and for each unit in the Master File. This is the case for:

l Porosity and Net to Gross ratio for Upper Brent - B1

l Porosity and Net to Gross ratio for Lower Brent - B5B

Therefore, the same process must be performed 4 times although it is only described once here - for the Net to Gross ratio of Lower Brent - B5B.

Multi-layer Depth Conversion With Isatoil 447

This procedure offers several possibilities such as:

l defining the authorized interval for the variable: the Net to Gross variable will be defined between 0 and 1 while the porosity between 0 and 0.4: this definition is essential in order to avoid the back transformed results to reach unexpected values.

l defining additional lower and upper control points which modify the experimental cumulative density function for extreme values: this option is necessary when working with a reduced num-ber of active data, however it will not be used in this case study.

l choosing the count of Hermite polynomials for the fitted anamorphosis function (set to 30)

l display the experimental and theoretical probability density function and/or bar histogram (the count of classes is set to 20)

(snap. 8.6-1)

The next paragraph informs us of the quality of the normal score transform as it produces:

l statistics on the gaussian transformed data (optimally, the mean should be 0 and the variance 1)

448

Absolute Interval of Definition: On gaussian variable: [ -10.0000 , 10.0000] On raw variable: [ 0.0000 , 1.0000] Practical Interval of Definition: On gaussian variable: [ -1.7962 , 1.8000] On raw variable: [ 0.0000 , 0.9364] Statistics on Gaussian Variable (Frequency Inversion): Minimum = -1.567957 Maximum = 1.567957 Mean = -0.000000 Variance = 0.816242 Std. Dev. = 0.903461

l and statistics on the difference between the initial data values and their back and forth trans-formed values

Statistics on Z-Zth: Mean = -0.005603 Variance = 0.001493 Std. Dev. = 0.038639

Finally the next figure shows the comparisons between experimental (in blue) and theoretical (in black) probability density function (on the left) and bar histogram (on the right).

(fig. 8.6-1)

The anamorphosis model (for each petrophysical variable and for each unit) is automatically saved in a Standard Parameter File whose name follows the naming convention (Psi_Poro_1_2_2 or Psi_Net_1_2_2 for example).

If the printout option is switched on, the (normalized) coefficients of the different Hermite polyno-mials are printed out:

Multi-layer Depth Conversion With Isatoil 449

Normalized coefficients for the Hermite polynomials 0 1 2 3 4 0+ 0.723538 -0.195402 -0.090866 -0.006407 0.031800 5+ 0.029486 -0.002152 -0.026471 -0.012341 0.017568 10+ 0.018136 -0.008480 -0.018879 0.001060 0.016816 15+ 0.004225 -0.013377 -0.007465 0.009474 0.008972 20+ -0.005671 -0.009125 0.002297 0.008298 0.000479 25+ -0.006826 -0.002595 0.004991 0.004064 -0.003021

8.6.2 Correlations for the petrophysical variables

The Petrophysics / Correlations application enables to check the compatibility of a petrophysical variable with the external drift which will be used during the estimation (or simulation) stage. In our case, only the porosity variable of the Lower Brent - B5B makes use of an external drift.

In this case study, the results which rely only on 13 active data are rather poor. The linear regression line is (trends are defined from the data):

Variable 1: Lower Brent - B5B = 0.29590 + Trend * ( -0.00003)

(fig. 8.6-2)

8.6.3 Model fitting for the petrophysical variables

Each petrophysical variable of each unit must be modelled separately, using the same application Petrophysics / Model.

450

Whenever a variable has been normal score transformed, two individual models must be fitted:

l one model on the raw variable is used for the estimation

l one model on the gaussian variable is used for the simulation

The only difference with the other geostatistical modelling panels is that:

l the current process is monovariate

l there is no restriction to strict stationarity. Variograms are used instead of covariances and non-bounded theoretical models - e.g. a linear - are authorized.

The following table summarizes the structures that have been fitted automatically, based on experi-mental quantities calculated with 10 lags of 300m each usually:

8.6.4 Base case for the petrophysical variables

The estimation of each petrophysical variable defined in the Master File is achieved with the Petrophysics / Base Case application.

During the estimation process, a post-processing test is performed in order to truncate the resulting values between 0 and 1. The same operation is also performed in the simulations.

Unit Variable Type Sill Range

B1 N/G Raw Nugget 0.0022 No

B1 N/G Gaussian Nugget 0.8048 No

B1 Porosity Raw Exponential 0.0004 2000 m

B1 Porosity Gaussian Nugget

Exponential

0.6754

0.2827

No

2000 m

B4 N/G Raw Nugget

Spherical

0.0008

0.0003

No

1000 m

B4 Porosity Raw Nugget

Linear

0.0002

0.0006

No

10000 m

B5B N/G Raw Spherical 0.0764 2000 m

B5B N/G Gaussian Nugget

Spherical

0.1426

1.1238

No

2000 m

B5B Porosity Raw Spherical 0.0006 2000 m

B5B Porosity Gaussian Spherical 1.1927 2000 m

Multi-layer Depth Conversion With Isatoil 451

The following statistics are obtained on 5767 samples:

8.6.5 Map representation for the petrophysical variables

We can use Display / Map to display the petrophysical base-case results.

In comparison with the maps already drawn before, note that the well information now comes from the Well Petrophysics file, therefore the samples do not carry the well name anymore.

The next figure shows the estimation of the Porosity on the Lower Brent - B5B unit:

(fig. 8.6-3)

Unit Variable Minimum Maximum

B1 Porosity 0.251 0.302

B4 Porosity 0.290 0.298

B5B Porosity 0.193 0.266

B1 N/G 0.854 0.854

B4 N/G 0.952 0.983

B5B N/G 0.037 0.931

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8.7 Volumetrics

This section introduces the calculation of accurate volumes based on the results of geostatistical estimation and/or simulations. There are several levels of details in the reported volumes, since the volumetrics algorithm takes into account the following parameters:

l volumes are computed separately for each Unit of the sequence,

l volumes are calculated separately for Oil and Gas, above the relevant contacts,

l volumes are computed either as Gross Rock or Oil in Place - if petrophysics is used -,

l volumes are reported separately per areal Polygons of interest.

8.7.1 Polygon definition

In this Case-Study we have considered three polygons, the vertices of which are stored in the ASCII file named polzone.dat. Each polygon starts with a star character (*) typed in the first column fol-lowed by a number of lines which contain the coordinates the polygon vertices:

* 126.43 1803.66 490.45 2167.29 916.61 1936.70 1227.35 1803.66 1404.91 996.58 339.51 996.58 144.96 1733.48 * 259.61 2859.07 490.45 2167.29 1227.35 1803.66 1582.48 2663.96 943.24 3621.81 259.61 3222.70 253.23 2923.09 * 339.51 996.58 969.88 349.14 1236.23 358.01 1502.58 233.85 1955.37 393.49 1689.02 1014.32 1404.91 996.58

The polygon coordinates are expressed in meters in this example. A polygon does not need to be closed - since Isatoil will automatically close it if necessary -

The following illustration has been obtained with Isatis. The polygons have been loaded from the ASCII file named polzone.hd - which contains the proper header organization - and have been dis-played with Display / Polygons on top of a time map of Dunlin - D1.

Multi-layer Depth Conversion With Isatoil 453

(fig. 8.7-1)

Note - The formats of polygon files for Isatis and Isatoil are different. It is not necessary to load the polygons inside the Isatis database unless you wish to perform a graphic representation such as above.

454

8.7.2 The volumetrics algorithm

(snap. 8.7-1)

Each volume results from the integration within a unit:

l between a top and a bottom

l between a lower and an upper contact:

m for gas contents: the lower contact is the OWC and there is no upper contact

m for oil contents: the upper contact is the GOC and the lower contact is the OWC

l of the thickness for gross volume

l of the product of the thickness by the petrophysical parameters for the storage in place volume

All these operations correspond to non-linear operations (as soon as contacts are involved). A skilled geostatistician knows that applying a non-linear operation on the result of a linear process (such as kriging) leads to biased estimations. It is recommended to run simulations instead.

Multi-layer Depth Conversion With Isatoil 455

Each simulation produces a realistic outcome and therefore a plausible volume result. Then, draw-ing several simulations will lead to the distribution of possible volumes from which any types of statistics can be derived:

l on the volumes: mean, P5, P50 (median) or P95

l on a pixel basis in order to produce probability and quantile maps

The general principle consists of calculating one or several block models and to derive the different volumes (per polygon, per layer). A block model is a set of layers and petrophysical variables, all these surfaces (either geometrical or petrophysical) being calculated consistently. Each block model is the result of the following six nested elementary operations:

l defining the Limit Surfaces (always provided as external variables)

l defining the Top Layering Surface (always provided as external variable)

l performing the layering phase

l performing the zonation phase

l painting each unit with porosity

l painting each unit with net to gross ratio

Each operation has two possible status, according to the flag Already calculated:

l ON: it must not be performed and the resulting surface(s) should already exist in the grid file with a name which follows the naming convention.

l OFF: it must be performed during the Volumetrics procedure. The resulting surface(s) are usu-ally not stored in the grid file (see the Simulation Parameters panel for exception).

The surface(s) (either calculated or read from the grid file) can be the result of one of the two fol-lowing procedures:

l the base case which involves the kriging technique

l a conditional simulation

Note - In particular, this allows the user to derive volume from kriged estimates, regardless of the bias of the result.

8.7.3 Volumetrics using the Base Case

Note - Although the following operation is not recommended because of the bias on the calculated volumes, we will first evaluate the volumes of oil and gas for each layer and for each polygon, based on the base case results.

The base case has already been performed for:

456

l the layering phase

l the zonation phase

l the petrophysical phase for both Porosity and the Net to Gross ratio.

Therefore we can switch ON the Already calculated flags for all the phases (including the Petro-physical steps) together with the Base Case option.

This estimation will serve as a reference, therefore the values of the GOC and OWC for each unit are set to the following constant values in the Master File:

l the GOC is fixed to 2570m

l the OWC is fixed to 2600m

Isatoil returns the following figures - which are expressed in 106 m3 - per polygon and per zone:

l GRV is the gross rock volume which only depends on the reservoir geometry

l IP is the volume in place obtained as the product of the geometry, the petrophysical variables and the volume correction factor

It also produces the same results:

l per polygon and per layer: regrouping all the zones of a layer

l per area: regrouping all the zones and layers of the same area

l per polygon: regrouping all the zones, layers an areas

l globally: regrouping all zones, layers, areas and polygons

Note - Note that when the results of several are regrouped, the program simply adds the results of each individual polygon without checking that the polygons do not overlap.

Layer Polygon Gas Oil

GRV IP GRV IP

B1 1 110.78 1987.26 7.65 2.28

2 155.31 2716.52 0.12 0.04

3 67.32 1159.69 0.40 0.12

B4 1 0. 0. 27.05 6.52

2 0. 0. 50.04 12.09

3 0. 0. 29.27 7.03

B5B 1 0.16 1.92 1.78 0.35

2 7.08 90.73 6.41 1.59

3 3.43 47.93 2.60 0.53

Multi-layer Depth Conversion With Isatoil 457

The last result will serve as a global reference result:

8.7.4 Randomization of the contacts

The next operation is to return to the Master File menu in order to modify the initial contact values:

Where T(2600,-5,+2) means a triangular law with a minimum of 2595, a maximum of 2602 and a mode of 2600.

For each volume calculation, the value of the contacts is drawn at random according to the law as defined in the Master File panel (for each layer, each fluid and each index). These random numbers use a random number generator which depends on the seed number that can be defined in the Sim-ulation Parameters panel (the other parameters of the panel will be discussed later): changing the seed number will alter the following Volumetrics results, even when based on the base case process.

(snap. 8.7-2)

When selecting the Verbose Output option in the Master File panel, the volumetrics procedure produces the values of the contacts:

Gas BRV 344.07

Gas IP 6004.04

Oil BRV 125.32

Oil IP 30.54

Surface Type GOC(m) OWC(m)

Upper Brent B1 Top Layering 2570 T(2600,-5,+2)

Lower Brent B4 Layer No T(2600,-3,+2)

Lower Brent B5B Zone 2570 U(2598,2602)

458

Random generation of contacts for layer Upper Brent - B1 GOC : Index-1 = 2570.000000 Index-2 = 0.000000 Index-3 = 0.000000 OWC : Index-1 = 2599.918702 Index-2 = 0.000000 Index-3 = 0.000000 Random generation of contacts for layer Lower Brent - B4 OWC : Index-1 = 2601.204858 Index-2 = 0.000000 Index-3 = 0.000000 Random generation of contacts for layer Lower Brent - B5B GOC : Index-1 = 2570.000000 Index-2 = 0.000000 Index-3 = 0.000000 OWC : Index-1 = 2600.848190 Index-2 = 0.000000 Index-3 = 0.000000

The global results of the base case are compared with the reference values obtained with the con-stant contacts of the previous paragraph:

8.7.5 Volumetrics using simulations

The next task consists in replacing the base case process by geostatistical simulations so that the Volumetrics results do not suffer from the bias that we already discussed. Choosing between base-case and simulations can be done for each single step involved in Volumetrics calculation:

l Limit surfaces

l Top Layering reference surface

l Layering

l Zonation

l Porosity

l Net to Gross ratio

When the flag Already calculated is switched ON, Isatoil reads the results from the grid file using the relevant naming convention. For example, the depth corresponding to the zone (3) of the layer (2) inside area (1) must be stored under:

l depth_1_2_3 for the Base Case

l depth_1_2_3[xxxxx] for the Simulations

When the flag Already calculated is switched OFF, the base-case or the simulation outcomes are computed at RUN time.

When simulations have been selected for a given step, the user can specify the number of outcomes that will be calculated or read from the grid file.

Constant contacts Randomized contacts

Gas GRV 344.07 344.07

Gas IP 6004.04 6004.04

Oil GRV 125.32 126.05

Oil IP 30.54 30.71

Multi-layer Depth Conversion With Isatoil 459

The Simulation parameters panel is used to indicate:

l the number of turning bands that must be used in order to generate an outcome which repro-duces correctly the variability as defined in the geostatistical model. On one hand, this number should be large for a good quality, on the other hand it should not be too large as the time con-sumption of each simulation is directly proportional to the number of bands. In this case study, this value is set to 500.

l should we match or combine the simulations? When two nested phases have to be simulated with 3 outcomes for each one, this flag tells the system if the final count of scenarios should be 3 (match option) or 9 (combine option). When match is required, the number of outcomes obtained is the smallest number of outcomes defined for the various simulation steps. When combine is selected, the final number of outcomes is obtained as the product of the individual numbers of outcomes.

8.7.5.1 Matching simulations

In this first run, we use Already calculated surfaces (base case) for the Top Layering and the Lim-its. All the other steps are simulated using 10 outcomes each.

The simulations are matched so that the final count of scenarios is 10.

The following statistics can be compared with those obtained using the base case and the random-ized contacts.

8.7.5.2 Combining simulations

In this second run, we choose to reduce to 5 the count of simulations - for each one of the 4 steps - but to combine the simulations in the end. Therefore the final count of outcomes is 625.

Gas Oil

GRV IP GRV IP

Base Case 344.07 6004.04 126.05 30.71

Mean 341.11 6260.19 124.85 30.63

St. dev. 7.79 247.35 5.94 2.22

P90 332.30 5981.47 117.71 28.72

P50 342.30 6418.19 126.61 30.94

P10 356.20 6584.76 133.03 34.26

460

The following results are obtained:

These results bring two comments:

l the volume obtained using the base case is not necessarily close to the one (say the median or P50) obtained with simulations. This is due to the bias that we have mentioned before. In the case of the Gas IP in particular, the difference between the P50 (6345.) and the base case (6004) is almost twice as large as the standard deviation (158).

l the gain in accuracy has one severe drawback: CPU time consumption. As a matter of fact, the volumes obtained on 625 simulation outcomes cost much more than one single volume obtained using the base case

In order to avoid running several times the simulations for a given configuration of parameters, the results of the RUN can be stored in some Histogram file (e.g. histo). The contents of this file can be used in the Volumetrics / Histogram application.

Note - Although this file is in ASCII format, it can only be interpreted properly by Isatoil itself. It is useless and not recommended to try reading these figures with another software...

8.7.5.3 Volume distributions

The detailed volumetrics results - which have been stored in the Histogram file - can be used in the Volumetrics / Histograms application in order to display volumes in the form of distribution curves.

Once the Histogram file has been read Isatoil shows the list of all available items, each of them being named upon two things:

l the polygon number - 1,2 or 3 since 3 areal polygons have been used -

l the unit number - e.g. Upper Brent - B1 -

Gas Oil

GRV IP GRV IP

Base Case 344.07 6004.04 126.05 30.71

Mean 340.47 6352.31 126.35 31.78

St. dev. 6.33 158.55 7.40 1.86

P90 334.33 6152.96 116.72 29.38

P50 338.16 6345.31 127.17 31.94

P10 351.66 6561.83 135.32 34.09

Multi-layer Depth Conversion With Isatoil 461

We can select the type of the volume to be displayed among the following options:

l Gas Pore Volume

l Gas in Place

l Oil Pore Volume

l Oil in Place

Finally, in our case, we get 625 possible consistent block systems: for each block system, the pro-gram has calculated the volumes of 21 different items, for 4 different materials.

The Histogram utility enables the user to select one or several item(s) of interest and to extract the values of the 625 realizations. When several items have been selected (say Polygon 1 for Upper Brent - B1 and Polygon 2 for Lower Brent B5B), the value for each realization is the sum of the two individual volumes.

(snap. 8.7-3)

This first illustration shows the volumes obtained on Polygon 1 in the unit Upper Brent - B1.

462

In the next figure, we represent:

- the Gas GRV in the upper left corner

- the Oil GRV in the upper right corner

- the Gas IP in the lower left corner

- the Oil IP in the lower right corner

(fig. 8.7-2)

The previous figure requires the following comments:

l The Gas GRV figure clearly shows a step function with 5 values:. This emphasizes that the out-comes result from the combination of:

m 5 outcomes of the Layering stage

m 1 (this layer does not include any zonation)

m 1 (GRV does not involve any petrophysical variable)

hence the 5 different volume values.

Multi-layer Depth Conversion With Isatoil 463

l Similarly, the Oil GRV figure shows several step functions with edges not as sharp as in the Gas GRV. This is due to the fact that the OWC contact of this layer is randomized.

l In the Gas IP figure, the outcomes result from the combination of:

m •5 outcomes of the Layering stage

m •1 (this layer does not include any zonation)

m •5 outcomes for the Porosity variable

m •5 outcomes for the Net to Gross variables

...hence the 125 different volume values

l the same type of results for the Oil IP figure

For the sake of the demonstration, we also show the Gas GRV figure for the Polygon 1 in the layer Lower Brent - B5B. The figure clearly shows 25 different volumes this time, obtained from the combination of 5 outcomes from the Layering stage and 5 outcomes from the Zonation stage.

(fig. 8.7-3)

The last illustration consists of cumulating all the volumes over all the units and all the polygons, so as to provide one value for each type of material. This compares to the statistics given in the previ-ous paragraph.

464

(fig. 8.7-4)

(fig. 8.7-5)

This is particularly interesting to show the bias of the volume established on the base case: in the case of Gas in Place (lower left), this volume (6004) is far from the mean simulated volume.

8.7.5.4 Production of Maps during the Volumetrics process

When running the Volumetrics process (essentially when performing the simulations) it may be interesting to check the spread of these outcomes on a basis of each grid node, in order to produce maps.

The principle is to specify a new grid file name where the procedure will write these maps, switch-ing on the option "Saving Results for Map Production". If the file already exists, its contents is emptied first before the new variables are created: no warning is issued.

When the name of the new grid file has been entered, you must use the Definition button in order to specify the set of maps to be stored (snap. 8.7-1).

Multi-layer Depth Conversion With Isatoil 465

(snap. 8.7-4)

This procedure offers the possibility of defining several calculations that will systematically be per-formed on all the units of the block system, regardless of their contents in Gas and Oil.

The first set of maps concerns mean and dispersion standard deviation maps, calculated for:

l the Depth of the Top Reservoir: the Reservoir is only defined where either gas or oil is present

l the Gas Reservoir Thickness: for each grid cell, this represents the height of the column within the Gas Reservoir

l the Gas Pore Volume: for each cell, this represents the product of the height within the Gas Res-ervoir scaled by the petrophysical variables

l the Oil Reservoir Thickness

l the Oil Pore Volume

The user can also ask for Probability Maps of the Reservoir Thickness. Here again, the Reservoir is only defined where either Gas or Oil is present. When the flag is switched on, you must use the Definition button to specify the characteristics of the probability maps.

The probability map gives the probability that the reservoir thickness be larger than a given threshold for each grid cell. For example the threshold 0m gives the probability that the reservoir exists. You may define up to 5 thresholds.

466

(snap. 8.7-5)

The user can also ask for Quantile Maps of the Depth of the Top Reservoir. Here again, the Reser-voir is only defined where either Gas or Oil is present. When the flag is switched on, you must use the Definition button to specify the characteristics of the probability maps.

For a grid cell located within the reservoir, the quantile map gives the depth of the top which cor-responds to a given quantile threshold (defined in percent). For example the threshold 0% gives the smallest depth for the top reservoir. You may define up to 5 thresholds.

(snap. 8.7-6)

Note - None of these maps can be considered as a simulation outcome - they do not honor the geostatistical structure of the variable - therefore any volume calculation based on them would be biased.

These special maps obey the following naming convention. Their generic name is of the form:

Code-code_number : variable_type

Multi-layer Depth Conversion With Isatoil 467

where:

l code_number stands for the designation code of a unit, as defined in the Master File - e.g. 122 for the Lower Brent - B5B -

l variable_type indicates the type of calculation that has been performed, chosen among the fol-lowing list:

m Mean Depth - average of the Depth of the Top reservoir -

m Mean Gas Thickness

m Mean Gas Pore Volume - product of thickness * petrophysical variables * volume correction factor -

m Mean Oil Thickness

m Mean Oil Pore Volume

m St. dev. Depth - standard deviation of the Depth of the Top reservoir -

m St. dev. Gas Thickness

m St. dev. Gas Pore Volume

m St. dev. Oil Thickness

m St. dev. Oil Pore Volume

m Proba of thickness larger than threshold: probability that the thickness of the Reservoir (Gas + Oil) is larger than the given threshold value

m Depth quantile quantile: value of the depth corresponding to the quantile

468

8.7.5.5 Visualizing the special maps

The following figures show some results obtained for the Layer Lower Brent - B5B using Display / Auxiliary Grid.

(fig. 8.7-6)

Code-122: mean of the depth of Lower Brent - B5B

Multi-layer Depth Conversion With Isatoil 469

(fig. 8.7-7)

Code-122 : Standard deviation of the depth of Lower Brent - B5B

470

The next figures compare the quantiles maps (for quantiles 10%, 50% and 90%) and the mean map. The calculations are slightly different for quantile and mean maps calculation. If we consider N outcomes and concentrate on a given grid node:

l quantile. The N values of the depth are considered (when there is no reservoir, the value is set to a non-value). Then these values are sorted and the p-quantile corresponds to the value ranked: p*N/100. If the result corresponds to a non-value, then the reservoir does not exist in the quan-tile map. Therefore, when the quantile increases, the depth of the reservoir top increases and, as the contact remains unchanged, the reservoir extension shrinks down.

(fig. 8.7-8)

Code-122: Depth Quantile 10.000000%

Multi-layer Depth Conversion With Isatoil 471

(fig. 8.7-9)

Code-122: Depth Quantile 50.000000%

(fig. 8.7-10)

Code-122: Depth Quantile 90.000000%

472

l mean. Among the N values, only those where the reservoir exists are stored and averaged.

(fig. 8.7-11)

Code-122: Mean Depth

The next figures compare the probability maps for the Depth of the Top Reservoir, in the layer Lower Brent - B5B, to be above 10m, 20m and 30m.

Multi-layer Depth Conversion With Isatoil 473

(fig. 8.7-12)

Code-122: Proba of Thickness Larger than 0.00m

474

(fig. 8.7-13)

Code-122: Proba of Thickness Larger than 5.00m

Multi-layer Depth Conversion With Isatoil 475

(fig. 8.7-14)

Code-122: Proba of Thickness Larger than 10.00m

8.7.5.6 Saving the simulations

The Volumetrics procedure offers the additional possibility of storing the simulation outcomes, for each one of the variables processed, in the main Grid File.

This option can be used in an "expert way" in order to run a second time the volumetrics process with the options Already Calculated switched ON...

For another use, this option is not recommended for the following reason. In order to save time, the simulations (and the kriging) are only performed in the places which can serve during the whole process of nested phases, i.e.:

l at each grid node as long as it belongs to at least one polygon

l at the closest node to each one of the intercepts with layers and zones

Therefore each simulation outcome is calculated on a limited number of cells.

Moreover, the calculated surface is not intersected by the Limiting Surfaces before storage.

476

This is the reason why the result simulation outcome of the depth of the top of Lower Brent - B5Bunit is difficult to interpret:

(fig. 8.7-15)

Multi-layer Depth Conversion With Isatoil 477

8.8 Tools

Isatoil offers several procedures for checking the results and understanding the calculations. A quick review of these tools will be given in this section.

Most of these tools require the definition of a particular point that will serve as a target: this point can be picked from a graphic representation. We will arbitrarily select the following target point:

X=780m Y=2349m

8.8.1 Inform the Target Point

This procedure, in the "Tools" menu, allows you to check the contents of the variables stored in the main Grid File. The target point is first located within the grid and the value of each variable is cal-culated by bilinear interpolation.

(snap. 8.8-1)

In the case of our target point, we obtain the following results:

Unit Time Depth Porosity N/G Trend(s)

BCU 2384.084

ERODE 1 2307.114

ERODE 2 1187.447

Upper Brent B1 2291.331 2399.005 0.295 0.854

Lower Brent B4 2370.886 2510.750 0.293 0.964

Lower Brent B5A 2545.005

Lower Brent B5B 2599.685 0.250 0.703 2370.886

Lower Brent B6 2614.688

Dunlin 2460.001 2619.640

Statfjord 2629.971 2899.720

Base Statfjord 2769.980 3137.597

478

Note the following remarks:

l the time values are only defined for layers (not for zones)

l the depth variable is defined everywhere (in m). They do not take into account the order rela-tionships between the layers: this is only performed at the output stage.

l the porosity and Net to gross ratio are only calculated for the units where at least one contact is defined

l the trends (for porosity and normal transform of porosity) are defined for the unit where the porosity variable requires:

m a normal scale transform before simulation

m an external drift (provided by the user) for processing

An additional flag allows you to display the Simulated Results. When using this option after the Volumetrics last procedure (running simulations and storing the outcomes in macro variables), the 5 simulated outcomes are listed for the calculated variables (Depth (layers and zones), Porosity, Net/Gross).

8.8.2 Estimate the Target Point

The Tools / Estimate the Target Point application enables to perform the estimation at the Target Point of one the following items:

- Layering

- Zonation

- Petrophysics

This procedure presents even more interest when the Verbose option flag is switched ON in the Master File: then the data information taken into account at each step is listed exhaustively.

At this point, it is important to distinguish the results obtained with this procedure from the ones obtained in the previous section. Here the estimation is actually performed at the target point loca-tion whereas, in the previous paragraph, the value was derived from the values estimated at the four surrounding grid nodes (by a bilinear interpolation).

Multi-layer Depth Conversion With Isatoil 479

(snap. 8.8-2)

8.8.2.1 Estimating a Layer value

The layering estimate gives the following results:

Compression stage: - Initial count of data = 59 - Final count of active data = 52 Estimation at the Target Point ============================== X-coordinate of the target point = 780.00m Y-coordinate of the target point = 2349.00m Depth for Top = 2399.005 Estimate #1 = 1.404 (Lower Brent - B4) Estimate #2 = 1.222 (Dunlin)Estimate #3 = 1.648 (Statfjord - S1) Estimate #4 = 1.699 (Base Statfjord)

As it was requested in the Master File, the calculation for the layering stage are performed in terms of interval velocity, hence the values of the estimations for the four intervals of the layering

8.8.2.2 Estimating a Zonation value

The Zonation of the Lower Brent - B4 layer (this designates all the zones located below that top layer) gives the following results:

Compression stage: - Initial count of data = 77 - Final count of active data = 64 Estimation at the Target Point ============================== X-coordinate of the target point = 780.00m Y-coordinate of the target point = 2349.00m Depth for Top = 2510.750 Depth for Bottom = 2619.640 Value for Pre-Faulted Thickness = 108.890 Estimate #1 = 32.899 (Lower Brent - B5A) Estimate #2 = 55.049 (Lower Brent - B5B)

480

Estimate #3 = 15.252 (Lower Brent - B6) Estimate #4 = 1.781 (Dunlin) Sum of estimates = 104.980 Results after the Collocation correction Estimate #1 = 34.252 (Lower Brent - B5A) Estimate #2 = 54.681 (Lower Brent - B5B) Estimate #3 = 15.003 (Lower Brent - B6) Estimate #4 = 4.954 (Dunlin) Sum of estimates = 108.890

Here the results correspond to thicknesses of the zones. The calculations are performed in two steps:

l direct estimation of the thicknesses

l correction in order to account for the total thickness of the layer (collocation correction)

8.8.2.3 Estimating petrophysical variables

The porosity estimation of the Lower Brent - B5B unit gives the following results:

Estimation at the Target Point ============================== X-coordinate of the target point = 780.00m Y-coordinate of the target point = 2349.00m Estimate #1 = 0.250 (Lower Brent - B5B)

The Net to Gross ratio estimation on the same unit gives:

Estimation at the Target Point ============================== X-coordinate of the target point = 780.00m Y-coordinate of the target point = 2349.00m Estimate #1 = 0.703 (Lower Brent - B5B)

8.8.2.4 Verbose output

The Layering calculation is performed again, but switching ON the Verbose Output flag in the Master File. This case has been selected as it covers all the interesting features of the output:

.../... List of active information used for Estimation of Depth using the following variable(s): - Variable 1: Lower Brent - B4 - Variable 2: Dunlin - D1 - Variable 3: Statfjord - S1 - Variable 4: Base Statfjord Rank - Name - X - Y - Initial - Data - Pr1 - Pr2 - Pr3 - Pr4 - Trend1 - Trend2 - Trend3 - Trend4 1 3 1965.27m 649.64m 2435.310 1.057 1.00 0.00 0.00 0.00 2313 0 0 0 2 3 1965.27m 649.64m 2544.110 1.178 0.44 0.56 0.00 0.00 2313 2398 0 0 3 3 1965.27m 649.64m 2813.410 1.373 0.20 0.26 0.53 0.00 2313 2398 2573 0 4 4 2408.25m 422.11m 2927.000 1.362 0.13 0.17 0.33 0.37 2279 2353 2491 2649 5 113 1668.08m 070.14m 2772.050 1.341 0.17 0.27 0.56 0.00 2304 2389 2566 0 6 119 827.59m 060.44m 2498.780 1.412 1.00 0.00 0.00 0.00 2373 0 0 0 7 120 1162.99m 212.98m 2456.170 1.183 1.00 0.00 0.00 0.00 2352 0 0 0 8 120 1827.41m 868.45m 2483.780 1.033 0.40 0.60 0.00 0.00 2299 2379 0 0

Multi-layer Depth Conversion With Isatoil 481

9 120 1915.18m 825.42m 2478.170 1.015 0.42 0.58 0.00 0.00 2297 2374 0 0 10 121 1163.91m 301.18m 2452.170 1.086 1.00 0.00 0.00 0.00 2353 0 0 0 11 129 1085.61m 416.12m 2467.400 1.213 1.00 0.00 0.00 0.00 2358 0 0 0 12 132 1251.11m 919.74m 2458.860 1.078 1.00 0.00 0.00 0.00 2338 0 0 0 13 132 1285.38m 811.34m 2563.580 1.128 0.44 0.56 0.00 0.00 2339 2429 0 0 14 145 1110.81m 556.81m 2514.440 1.344 1.00 0.00 0.00 0.00 2379 0 0 0 15 145 1064.02m 590.65m 2628.170 1.381 0.51 0.49 0.00 0.00 2380 2459 0 0 16 147 1226.54m 393.18m 2456.330 1.378 1.00 0.00 0.00 0.00 2340 0 0 0 17 147 1198.95m 484.77m 2561.870 1.226 0.45 0.55 0.00 0.00 2342 2435 0 0 18 148 1827.40m 702.87m 2388.930 0.853 1.00 0.00 0.00 0.00 2302 0 0 0 19 148 2057.15m 336.04m 2493.740 1.169 0.43 0.57 0.00 0.00 2290 2361 0 0 20 149 2957.57m 356.44m 2840.000 1.291 0.23 0.08 0.29 0.39 2283 2316 2433 2589 21 152 1195.30m 949.32m 2500.520 1.139 1.00 0.00 0.00 0.00 2376 0 0 0 22 152 1249.02m 077.51m 2605.040 1.219 0.53 0.47 0.00 0.00 2370 2450 0 0 23 152 1509.20m 787.16m 2789.920 1.403 0.20 0.25 0.55 0.00 2322 2400 2572 0 24 155 3536.50m 315.45m 2813.000 1.152 0.27 0.05 0.36 0.33 2273 2292 2444 2582 25 156 2186.83m 501.30m 2698.350 1.319 0.23 0.18 0.59 0.00 2295 2345 2506 0 26 157 3032.15m 054.42m 2841.570 1.289 0.17 0.15 0.29 0.39 2264 2323 2436 2588 27 163 1888.61m 078.96m 2420.540 0.717 1.00 0.00 0.00 0.00 2318 0 0 0 28 163 1861.60m 117.71m 2536.310 1.053 0.44 0.56 0.00 0.00 2319 2405 0 0 29 165 1155.13m 673.93m 2515.010 1.466 1.00 0.00 0.00 0.00 2371 0 0 0 30 165 1101.41m 650.76m 2624.150 1.336 0.48 0.52 0.00 0.00 2374 2461 0 0 31 166 1508.61m 169.48m 2456.530 0.810 1.00 0.00 0.00 0.00 2350 0 0 0 32 170 2527.97m 546.14m 2974.090 1.351 0.18 0.12 0.38 0.32 2299 2356 2532 2683 33 171 556.79m -47.33m 2601.910 1.486 1.00 0.00 0.00 0.00 2430 0 0 0 34 173 2128.88m 426.30m 2779.210 1.304 0.23 0.23 0.55 0.00 2311 2384 2561 0 35 174 1908.26m 122.07m 2417.160 0.716 1.00 0.00 0.00 0.00 2315 0 0 0 36 174 1935.48m 037.06m 2523.020 1.008 0.44 0.56 0.00 0.00 2316 2400 0 0 37 178 3037.82m 440.93m 2830.480 1.253 0.15 0.17 0.35 0.33 2257 2322 2459 2588 38 180 1500.90m 868.04m 2424.580 1.292 1.00 0.00 0.00 0.00 2317 0 0 0 39 180 1467.48m 978.30m 2533.400 1.180 0.41 0.59 0.00 0.00 2323 2413 0 0 40 182 1796.31m 109.01m 2452.170 1.017 1.00 0.00 0.00 0.00 2334 0 0 0 41 182 1783.65m 41.02m 2565.320 1.156 0.46 0.54 0.00 0.00 2337 2425 0 0 42 183 1840.94m 638.18m 2426.250 0.815 1.00 0.00 0.00 0.00 2324 0 0 0 43 183 1839.25m 632.94m 2537.920 1.087 0.45 0.55 0.00 0.00 2324 2409 0 0 44 191 2094.11m 836.57m 2455.290 0.895 0.49 0.51 0.00 0.00 2299 2366 0 0 45 191 2129.59m 211.76m 2722.490 1.283 0.22 0.21 0.57 0.00 2300 2364 2535 0 46 193 3141.96m 354.71m 2822.910 1.268 0.22 0.11 0.28 0.39 2271 2314 2422 2575 47 201 945.72m 639.46m 2504.690 1.488 1.00 0.00 0.00 0.00 2366 0 0 0 48 201 881.27m 761.02m 2614.100 1.322 0.46 0.54 0.00 0.00 2368 2460 0 0 49 202 1920.85m 808.40m 2365.780 0.558 1.00 0.00 0.00 0.00 2303 0 0 0 50 203 1930.07m 850.41m 2367.500 0.570 1.00 0.00 0.00 0.00 2303 0 0 0 51 204 2357.63m 533.98m 2944.710 1.384 0.17 0.12 0.34 0.36 2299 2352 2503 2664 52 205 2292.26m 475.18m 2952.830 1.411 0.12 0.19 0.32 0.37 2282 2363 2498 2656 Estimation at the Target Point ============================== X-coordinate of the target point = 780.00m Y-coordinate of the target point = 2349.00m Depth for Top = 2399.005 Estimate #1 = 1.404 (Lower Brent - B4) Estimate #2 = 1.222 (Dunlin) Estimate #3 = 1.648 (Statfjord - S1) Estimate #4 = 1.699 (Base Statfjord)

482

We recall that Layering is performed by a cokriging procedure using 4 variables (layers) simulta-neously. The list of the information contains the following information:

l Rank designates the rank of the sample in the list

l Name is the name of the well which provided this intercept information. This information is not available in the case of Petrophysical variables.

l X - Y gives the coordinates of the intercept

l Initial is the depth value of the intercept, as read from the Well File

l Data is the value which is actually entered in the cokriging system: in the case of the Layering, this corresponds to an apparent velocity value calculated from the Top Layering surface down to the surface which contains the intercept.

l Pr* give the weighting coefficients which denote the percentage of time spent in each layer. Note that a layer located below the intercept surface corresponds to a zero weight.

l Trend* are the values that are used as external drift for each variable

The Pr* weight indicates if a layer (or a zone) lies between the intercept and the surface that serves as a reference, or set to 0 otherwise If the procedure works in depth, this weight is simply an indica-tor (0 or 1); if it works in velocity, the weight corresponds to the percentage (in time) that the layer thickness represents in the total distance from the intercept to the reference surface: the weights add up to 1. This weight is not available in the case of petrophysical variables.

The Trend* values are only displayed if the variable(s) to be processed require external drift(s).

8.8.3 Inform Depth at Well Locations

The Tools / Inform Depths at well locations application enables to compare the depth of the inter-cepts contained in the Well File, with the value that can be back-interpolated from the base case results (or simulation outcomes) stored in the Grid File. The back-interpolation uses a bilinear inter-polation from the four grid nodes surrounding the intercept location.

You may choose either to concentrate on the intercepts with one layer (or zone) in particular, or to review all the intercepts contained in the Well File.

Multi-layer Depth Conversion With Isatoil 483

(snap. 8.8-3)

The following printout is obtained when checking the base case results on the Lower Brent - B5Blayer:

Information on Well Location(s) (Bilinear interpolation) ======================================================== Layer : Lower brent B5B (Identification : Area = 1 - Layer = 2 - Zone = 2) Point: X= 1965.27m; Y= 649.64m; Data= 2525.110; Depth value = 2525.179 Point: X= 1778.15m; Y= 2892.79m; Data= 2481.390; Depth value = 2469.974 Point: X= 1435.80m; Y= 3427.97m; Data= 2503.980; Depth value = 2515.955 Point: X= 1617.06m; Y= 3512.49m; Data= 2472.870; Depth value = 2496.956 Point: X= 1277.78m; Y= 3831.64m; Data= 2543.590; Depth value = 2546.401 Point: X= 1073.01m; Y= 1584.21m; Data= 2606.220; Depth value = 2608.123 Point: X= 1203.30m; Y= 2470.50m; Data= 2544.980; Depth value = 2542.129 Point: X= 1238.08m; Y= 1051.87m; Data= 2588.180; Depth value = 2587.588 Point: X= 1865.64m; Y= 1111.99m; Data= 2520.070; Depth value = 2516.301 Point: X= 1109.77m; Y= 654.31m; Data= 2607.140; Depth value = 2604.150 Point: X= 464.31m; Y= -117.23m; Data= 2704.740; Depth value = N/A Point: X= 1930.98m; Y= 1050.76m; Data= 2507.830; Depth value = 2508.816 Point: X= 1930.17m; Y= 1053.23m; Data= 2505.030; Depth value = 2508.670 Point: X= 1471.79m; Y= 2963.77m; Data= 2518.720; Depth value = 2515.207 Point: X= 1785.84m; Y= 52.65m; Data= 2546.240; Depth value = 2546.405 Point: X= 1839.51m; Y= 633.56m; Data= 2521.030; Depth value = 2520.635 Point: X= 891.47m; Y= 2741.75m; Data= 2597.020; Depth value = 2594.410 Point: X= 1918.16m; Y= 1854.49m; Data= 2463.280; Depth value = 2451.716 Point: X= 1898.10m; Y= 1994.28m; Data= 2456.820; Depth value = 2458.152 Point: X= -275.56m; Y= 1591.47m; Data= 2747.290; Depth value = N/A

where:

l X - Y designate the intercept location coordinates

l Data refers to the depth information read from the Well File

l Depth value is the back-interpolated value

The back-interpolated value is not defined (N/A) when at least one of the grid nodes surrounding the intercept location is not defined: this is the case for the intercept located at (X=464.31m; Y=-117.23m) which lies outside the grid.

484

8.8.4 Cross-Validate the Well Information

This tool enables you to perform the cross-validation (blind test) of the data. The procedure consists in masking a data value, re-estimating it from the remaining information and comparing the initial information to the estimation. The procedure is performed, in turn, on all the information samples.

You can choose to perform the cross-validation test of:

- Layering

- Zonation

- Petrophysical

You can perform the cross-validation either on all the possible surfaces or restrict the test to a given subset of surfaces. In the petrophysical case, you must finally specify the target variable.

(snap. 8.8-4)

The following printout is obtained when cross-validating the porosity information of the Lower Brent - B5B layer:

.../... Estimation at the Data Points ============================= X - Y - True Value - Estimation - Layer Name 1069.52m 1586.71m - 0.231 - 0.237 - Lower brent B5B 1201.76m 2475.57m - 0.267 - 0.236 - Lower brent B5B 1242.07m 1061.08m - 0.220 - 0.207 - Lower brent B5B 1864.16m 1114.09m - 0.212 - 0.237 - Lower brent B5B 1106.90m 653.09m - 0.193 - 0.214 - Lower brent B5B 1932.38m 1046.48m - 0.237 - 0.213 - Lower brent B5B 1470.03m 2969.66m - 0.233 - 0.243 - Lower brent B5B 1785.02m 48.33m - 0.216 - 0.214 - Lower brent B5B 1839.40m 633.30m - 0.213 - 0.211 - Lower brent B5B 887.55m 2749.13m - 0.245 - 0.258 - Lower brent B5B 1768.50m 547.24m - 0.205 - 0.211 - Lower brent B5B 1787.09m 2888.33m - 0.228 - 0.239 - Lower brent B5B 1280.84m 3823.25m - 0.260 - 0.223 - Lower brent B5B

Multi-layer Depth Conversion With Isatoil 485

where:

l X-Y designate the coordinates of the data point

l True Value is value read from the data file (here the Petrophysical Well File), possibly con-verted into velocity in the Layering case

l Estimation is the result of the estimation

l Layer Name gives the identification of the information: this is mainly relevant in the multivari-ate case (Layering or Zonation)

8.8.5 Cleaning procedure

This tool (located in the File menu) allows you to delete some results produced by Isatoil. In partic-ular, it allows you to get rid of the following items:

l variables corresponding to the base case results - stored in the Grid File -

l simulation outcomes that might have been stored in the Grid File by the Volumetrics procedure

l Standard Parameter Files containing the models for the covariance and the distributions (anamorphosis).

The use of this procedure ensures that only variables resulting from calculations are deleted. In par-ticular, it does not delete depth variables corresponding to the Top Layer or the Limit surfaces, or any surface which is not calculated by Isatoil, as specified in the Master File.

The procedure offers the possibility either to clean all the results (from one of the above items men-tioned above) or to restrict the deletion to the ones relative to a given unit.

Use the Check button to check the number of files which will be deleted before actually cleaning them!

486

(snap. 8.8-5)

The following printout is obtained when cleaning all the files relative to the Lower Brent - B5B sur-face (all three items selected):

List of the Base Case result files to be deleted - depth_1_2_2 - porosity_1_2_2 - netgross_1_2_2 List of the Simulation result files to be deleted - depth_1_2_2[xxxxx] - porosity_1_2_2[xxxxx] - netgross_1_2_2[xxxxx] List of the Model Standard Parameter Files to be deleted - Model_Poro_Raw_1_2_2 - Model_Poro_Gauss_1_2_2 - Model_Net_Raw_1_2_2 - Model_Net_Gauss_1_2_2 List of the Anamorphosis Standard Parameter Files to be deleted - Psi_Poro_1_2_2 - Psi_Net_1_2_2

Geostatistical Simulations for Reservoir Characterization 421

9 Geostatistical Simula-tions for Reservoir Char-acterization

This case study is based on a public data set used by Amoco during the 80’s. The dataset has been kindly provided by Richard Chambers and Jeffrey Yarus.

It demonstrates the capabilities of Isatis in Reservoir Characterization using lithofacies and porosity simulations. Volumetrics calculations are performed on 3D models.

Last update: 11.0.0

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Geostatistical Simulations for Reservoir Characterization 423

9.1 Introduction

3D earth modeling is a key issue for reservoir characterization. Moreover, the uncertainties on the reservoir structure, the contacts and the rock properties may be assessed through the simulations that preserve the geological features of the reservoir. In this case study, one purpose is the optimal use of the available data: the wells with information on key horizons markers, lithofacies, porosity and the facies proportions.

The reservoir is located in North Cowden area (Texas). There are three main facies (siltstone, anhy-drite and dolomite). The carbonates were deposited during high sea-level stands and the siltstone during low stands when the carbonate platform was exposed to sub-aerial conditions. The silt is actually eolian sediment whose source is from the northwest and it was reworked into «sheet-like» deposits during subsequent sea-level rise.

In this case study, several geostatistical methods from Universal Kriging to facies simulations (Plurigaussian Simulation) and continuous simulations as Turning Bands are performed.

The main steps of the workflow are:

l Simulations of the surfaces delimiting the top and bottom of the reservoir, using the information from wells.

l Facies simulations (TPGS for Truncated Plurigaussian Simulation). It requires the building of a stratigraphic grid (flattening), within which variogram calculations and simulations are per-formed. The 3D vertical proportions matrix (VPC) is computed. A 2D Proportion map com-puted from a seismic attribute is used to constrain the 3D matrix proportions.

l Simulations of the average porosity in the reservoir.

l 3D simulations of porosity are achieved independently for each facies, then a cookie cutting procedure constrained by facies simulations provides the final porosity simulations.

Several types of simulations are used (surfaces simulations, SIS, TPGS, 3D porosity simulations). Therefore, different models are available. To evaluate these models, volumetric calculations based on the simulations of the different parameters provide stochastic distributions of volumes.

In conclusion, this case study explores some of the possibilities that Isatis offers to improve the res-ervoir characterization.

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9.2 General Workflow

1. Structural Modeling: Creation and simulations of the top and bottom surfaces of the reservoir from the well data;

The top and bottom of the reservoir are stored for each wells. The purpose is to interpolate or simulate the top and bottom surfaces of the reservoir from wells data. Eventually, the distribu-tion of the GRV is performed using these surfaces and a constant contact.

2. Discretization and Flattening: Transformation from real space to the stratigraphic space This step is crucial as it determines the lateral continuity of facies as expected from a sedimen-tary deposition of the facies. A flat working grid is created with a resolution of 50mx50mx1m.

3. Computing Proportion Curves: Computing curves from the well data over the working grid.

The vertical proportion curves are calculated from the wells discretized in the stratigraphic space. Then a 3D matrix of proportion is created for further use in SIS and Plurigaussian Simu-lations. Finally, the computation of the proportions is performed using a 2D proportions con-straint: kriging of mean proportion (siltstone). This proportion constraint was estimated by external-drift kriging. The drift was a map of acoustic impedance (AI) extracted from the seis-mic filtered cube. This proportion constraint will be used for the PGS.

4. Lithotypes Simulations: Simulations with PGS of the Lithotypes.

This step aims at deriving the variogram models of two Gaussian random functions that are sim-ulated and truncated to get the simulated lithotypes. The thresholds applied on the different lev-els follow the so-called lithotype rules. Then Plurigaussian simulations are performed and transferred to the structural grid

5. 3D Porosity Simulation: Simulation of porosity with Turning Bands and Cookie Cutting

The porosity is simulated using Turning Bands for each lithotypes, then the porosity is condi-tioned from the lithotypes simulations (Cookie Cutting). The cookie cutting method is the com-bination of the facies and porosity simulations. The porosity is simulated at each node of the grid located between the top and the bottom of the reservoir layer as if these nodes were in the facies of intent. In the final model only the porosity of the facies actually simulated at each node will be kept. Finally the HPCV is computed using Volumetrics.

6. 3D Volumetrics: Volumetrics of the 3D simulations

The HPCV is computed using Volumetrics. The results from the previous steps (top, bottom and porosity simulations) are all used in conjunction in order to compute the volumetrics. In addi-tion we assume that OWC depth is known.

Geostatistical Simulations for Reservoir Characterization 425

9.3 Data Import

Firstly, a new study has to be created using the File / Data File Manager facility; then, it is advised to verify the consistency of the units defined in the Preferences / Study Environment / Units win-dow. In particular, it is suggested to use:

l Input Output Length Options:

Default Unit... = Length (m) Default Format...= Decimal (10,2)

l Graphical Axis Units: X Coordinate = Length (km) Y Coordinate = Length (km) Z Coordinate = Length (m)

The data are stored in three ASCII files:

l The first named wells.hd contains the data available at the wells: depth, porosity, selection reser-voir (Sel Unit S2).

l The second named surfaces.hd contains the surfaces delimiting the reservoir on a grid with res-olution of 50mx50m.

l The third named 3D grid.hd contains a seismic impedance acoustic cube in a grid with a resolu-tion of 50mx50mx1m.

Import these files into Isatis using the ASCII file import (File/Import/ASCII). These files are avail-able in the Isatis installation directory under the Datasets/Reservoir sub-directory:

Each ASCII file already contains a header.

Enter a directory name and a file name for each imported file:

426

l For the wells, Directory: 3D wells; File: 3D wells; Header: 3D wells header (snap. 9.3-1).

l For the surfaces, Directory: 2D Surfaces; File: Surfaces (snap. 9.3-2).

l For the structural grid, Directory: 3D Grid; File: Structural Grid (snap. 9.3-3).

(snap. 9.3-1)

Geostatistical Simulations for Reservoir Characterization 427

(snap. 9.3-2)

(snap. 9.3-3)

428

9.4 Structural Modeling

The reservoir is limited by the horizons named S2 (top) and S3 (bottom) (2D Surfaces/Surfaces/sel Unit S2).

Considering the surfaces are known through the intercepts with the wells, simulations of these sur-faces are achieved. A kriging, called «base case» is performed to compare the estimations with the surfaces already imported.

Copy the top elevation (Maximum Z) and the bottom elevation (Minimum Z) of the wells in the file Wells/3D Wells Header using Tools/Copy Statistics/line->Header Point. Apply the selection sel Unit S2.

(snap. 9.4-1)

9.4.1 Kriging of the Surfaces

l Exploratory Data Analysis

This step describes the structural analysis performed on the top and bottom reservoir marker.

Using Statistics/Exploratory Data Analysis, display the cross-plots Minimum Z/X-UTM andMaximum Z/X-UTM at the wells. To do so, select the variables Minimum Z and X-UTM as input, highlight the variables and then click on the cross-plot representation (second icon from left). Then do the same for Maximum Z and X-UTM.

Geostatistical Simulations for Reservoir Characterization 429

(snap. 9.4-2)

(fig. 9.4-1)

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rho=-0.907

430

(snap. 9.4-3)

(snap. 9.4-4)

The crossplot of the top and bottom surfaces at wells according to X shows the existence of a trend depending on X (East-West).

1500 2000 2500 3000

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rho=-0.880

Geostatistical Simulations for Reservoir Characterization 431

Compute the omnidirectional variogram of Minimum Z and then the variogram of Maximum Z. The experimental variograms are both computed with 12 lags of 125 m.

(snap. 9.4-5)

(fig. 9.4-2)

The variograms of Minimum Z and Maximum Z also show a strong non-stationarity (fig. 9.4-2). Therefore a non-stationary model seems the most appropriate. In that purpose, a Universal Kriging approach (UK for short) will be applied. It amounts to decompose explicitly the variable of interest

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into its trend and a stationary residual. A variogram will be fitted to the residuals. The kriging amounts to krige the residuals and add the estimates to the trend model.

l Non Stationary Modeling

To build the non-stationary model, the first step consists in modelling the trend by means of the least square polynomial method fit based on a model a+ b*X.

Before modeling the trend you have to create a 2D copy of the 3D Wells Header using Data File Manager/File/Copy. Call the new file 2D Wells Header and modify it in 2D (Data File Manager/Modify 2D-3D).

For each variable, store the global trend in a new variogram model using Statistics/Modeling/Global Trend Modeling. A variable corresponding to the residuals is also created.

Geostatistical Simulations for Reservoir Characterization 433

(snap. 9.4-6)

Store the global trend in a variogram model and then fit the variogram of the residuals. By adding the variogram model of residuals to the model initialized at the trend modeling stage, the required non-stationary model is obtained (for example: Maximum Z no-stationary and Minimum Z no stationary). In that purpose, run Statistics/Variogram Fitting on the model of the residuals.

Below is the example for the residuals of maximum Z. The variogram model is the same for Maxi-mum Z and Minimum Z.

You can automatically initialize your model (using Model Initialization) or edit your-self the model with the following parameters:

- a Cubic structure with Range =1800m, sill= 50.

Save the model under Maximum Z no Stationary.

434

(snap. 9.4-7)

Geostatistical Simulations for Reservoir Characterization 435

(fig. 9.4-3)

Run Interpolate/Estimation/(Co)-Kriging using the non-stationary model and a unique neighbor-hood.

0 100 200 300 400 500 600 700 800 900 1000

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IsatisWELLS/2D Wells Headers- Variable #1 : Residuals maximum ZExperimental Variogram : in 1 direction(s)D1 : Angular tolerance = 90.00 Lag = 110.000m, Count = 10 lags, Tolerance = 50.00%Model : 1 basic structure(s)S1 - Cubic - Range = 1800.000m, Sill = 50

Jun 03 2008NCU-test

436

(snap. 9.4-8)

The results of the kriging are called respectively Maximum Z kriging and Minimum Z kriging. These base cases are very closed to the surfaces already stored in the 2D grid file (see hereafter the correlation cross plot between SURF 3: S2 and Maximum Z kriging)

Geostatistical Simulations for Reservoir Characterization 437

(fig. 9.4-4)

Note - An alternative approach would be to model the top surface and the thickness of the unit, avoiding the risk of getting surfaces crossing each other.

9.4.2 Simulations of the Surfaces

Three reasons lead to simulate directly the surfaces with the same non-stationary model, without any normal score transformation:

438

l The non-stationarity that is somehow contradictory with the existence of an unique histogram,

l The even density of wells in the gridded area, that controls the distribution by means of condi-tioning of simulations,

l The distribution is closed to be symmetrical.

(fig. 9.4-5)

Using Interpolate/Conditional Simulations/Turning Bands, perform the simulations for Minimum Z (Simu Minimum Z[xxxxx]) and simulations for Maximum Z (Simu Maximum Z[xxxxx]).

Geostatistical Simulations for Reservoir Characterization 439

(snap. 9.4-9)

Using Tools/Simulation Post-processing, calculate the average of 100 simulations in order to com-pare it to the kriged values. The match is almost perfect (fig. 9.4-7), which was expected as the mean of numerous simulations (over 100) tends towards the kriging.

In order to define the geometrical envelope of the S2 Unit where facies and porosity simulations are achieved, store the maximum of the simulated top (Maximum Z Top) and the minimum of the simulated bottom (Minimum Z Bottom). The use of the envelope ensures that all grid nodes will be filled with a porosity value.

(fig. 9.4-6)

440

(fig. 9.4-7)

(snap. 9.4-10)

Using Tools/Create Special Variable create a new macro variable with 100 indices, name it Thick-ness. Using File/Calculator, compute the thickness from the simulations of Maximum Z and the simulations of Minimum Z. Check with Statistics/Quick Statistics that the surfaces do not cross each other (there are not negative values).

Geostatistical Simulations for Reservoir Characterization 441

442

Geostatistical Simulations for Reservoir Characterization 443

9.5 Modeling 3D Porosity

In this part, the goal is to compute the simulations of the 3D porosity constrained by the facies sim-ulations. Porosity simulations are computed for each facies.

The steps are the following ones:

l 3D facies modeling (Plurigaussian Simulations)

l 3D porosity simulations conditioned by facies simulations. («cookie cutting simulations»)

9.5.1 Simulations of lithofacies

9.5.1.1 Flattening

First, create the working grid (stratigraphic grid) where the simulations are performed and a new parameter file essential for the simulations. This parameter file contains:

l The information relative to the lithotype (regrouping of the original lithofacies);

l The names of files and variables used in the construction process;

l The parameters used in the calculation of the vertical proportion curves.

Before going to Discretization & Flattening window you need to convert the 3D wells into cores lines. Use the panel Tools/Convert Gravity Lines to Core Lines:

The «old» gravity files are saved, the new core lines are named 3D Wells and 3D Wells Header.

In the Data File Manager, set the variable Well Name as Line Name in the 3D Wells Header: right click on the Well Name variable and choose Modify into Line Name.

In the File Manager, change the format of the variable Maximum Z Kriging, Maximum Z Top and Minimum Z Bottom.

444

(snap. 9.5-1)

l Go to Discretization and Flattening.Create a new proportion file S2 Unit and fill the 5 tabs.

(a) Input Parameters

To populate the facies simulation with porosity, define an auxiliary data file enquiring the porosity variable.

Geostatistical Simulations for Reservoir Characterization 445

(snap. 9.5-2)

(b) Grid and Geometry

(snap. 9.5-3)

446

Take the Maximum top and Minimum bottom (the envelope) as Top Unit Variable and Bottom Unit Variable. The reference variable is Maximum Z kriging (the Base Case corresponding to Surf 3: S2). The kriging of the top surface is used as the reference variable because it is consistent geologically speaking. This is not the case for the envelope.

Note - The S2 top surface has been chosen as the reference surface because the base of the S2 unit shows downlapping layers as the platform built eastward into the Midland basin.

(c) Lithotype Definition

In the S2 Unit, consider the lithofacies 1 (siltstone), 2 (anhydrite) and 3 (dolomite) and assign them to lithotype 1 to 3. In this case, the data already contain the lithotype formations.

(snap. 9.5-4)

For further display, create a dedicated colour scale by using Lithotype Attributes.

(snap. 9.5-5)

(d) Discretization Parameters

Geostatistical Simulations for Reservoir Characterization 447

The wells are discretized with a vertical lag of 1 m, which corresponds to the vertical mesh of the stratigraphic grid. There is a distortion ratio of 50 (50/1: ratio of mesh (x,y) and mesh (z)).

(snap. 9.5-6)

(e) Output

In the output tab, enter the discretized wells file and the header file. Define the output variables.

(snap. 9.5-7)

After running the bulletin, read carefully the informations printed in the message window, for checking the options and the discretization results.

It is possible to visualize the discretized wells in the new stratigraphic framework with the display menu using Lines representation.

448

Note - The envelope (Maximum Z Top and Minimum Z bottom) is used to be sure that inside the reservoir unit (S2 Unit) all the grid nodes will be filled by a porosity value when performing the porosity simulations.

9.5.1.2 Computing Proportions Curves

The task is to estimate the proportions of each lithotype at each cell of the working grid, from the experimental proportions curves at the wells in the stratigraphic reference system.

The different operations are achieved by several applications of the menu Statistics/Statistics/Pro-portions Curves.

(a) Loading the Data

Click Load Data in the Application menu of the graphic window, as shown below:

(snap. 9.5-8)

2D Proportion Constraints are specified: the proportion variable is kriging mean proportion silt-stone calculated before.

The graphic window displays the wells projected on the horizontal plane and the global proportion curve in the lower right corner.

Change the Graphic Options by using the corresponding Application menu.

Geostatistical Simulations for Reservoir Characterization 449

(snap. 9.5-9)

450

(snap. 9.5-10)

(b) Display the global statistics

Among different possibilities, visualize the global proportion curves by picking its anchor and use the Display & Edit menu.

Geostatistical Simulations for Reservoir Characterization 451

(snap. 9.5-11)

Using the Application menu and the option Display Pie Proportions, each well is represented by a pie subdivided into parts with a size proportional to the lithotype’s proportion.

(snap. 9.5-12)

(c) Create Polygons and calculate corresponding VPC

Digitalize four polygons after having activated the Polygon Edition mode, in order to split the field into four parts.

452

(snap. 9.5-13)

Coming back to the Vertical Proportion Curves Edition mode, perform the following actions: Dis-play&Edit, completion by 3 levels and Smoothing with 3 passes. An other method using the Editing tool is described in the section (e) Edition Mode.

Note - You can see that the Raw VPC present gaps at the top and the bottom. These gaps are explained by the fact that, in the display, the top corresponds to the maximum of the Top unit variable (here maximum Z Top) and the bottom corresponds to the minimum of the bottom unit variable (here minimum Z bottom). The wells information does not fill the total length between the defined top and bottom. These gaps may be an issue as an extrapolation is performed to fill them (especially at the top). An other method would be to use the simulations of both surfaces two by two to create the VPC. It would require to create as many VPC as there are couple of simulations. This would be rather inconvenient.

Geostatistical Simulations for Reservoir Characterization 453

(d) Compute the proportions on the 3D grid

The task is to assign to each cell proportions of lithotype that take into account the gradual change from south to north. In the menu Application/Compute 3D Proportions, choose the kriging option and an arbitrary variogram like a spherical scheme with an isotropic range of 2 km and a sill of 0.207. Do not forget to select the option: Use the 2D Proportions constraints.

454

(snap. 9.5-14)

To visualize the interpolated proportions, use Application/Display 3D proportions with the sam-pling mode (step 5 for instance along X and Y).

(fig. 9.5-1)

In order to update the parameter file use the menu Application/SAVE and RUN.

9.5.1.3 Determination of the Gaussian Random Functions and their variograms for plurigaussian simulations

This phase is specific to the simulation using the plurigaussian technique and is achieved by means of the menu Statistics/Modeling/Plurigaussian Variograms.

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Geostatistical Simulations for Reservoir Characterization 455

The aim is to assign the lithotypes to sets of values of a pair of Gaussian Random Function (GRF), i.e. by means of thresholds applied to the GRF. The transform from GRF to the categorical litho-types is called the lithotype rule. It is necessary to define it first in order to represent the possible transitions between the facies as they can express in geological terms the deposition process.

L1 = Siltstone, L2 = Anhydrite, L3 = Dolomite.

(snap. 9.5-15)

The first GRF horizontal (G1) will rule L2, L1 and L3. It is represented by a spherical scheme with ranges of 300 m for U, 300 m for V and 5 m for Z and a sill of 0.5.

The second GRF (G2) will rule L1, L2 and L3. It is represented by a spherical scheme with ranges of 1200 m for U, 2700 m for V and 5 m for Z and a sill of 0.5.

(snap. 9.5-16)

456

Run non-conditional simulations along the 3 main sections of the stratigraphic space by using Dis-play Simulations. By changing the coefficient of correlation, visualize the effect on the spatial orga-nization of the facies.

Visualize the thresholds applied on the 2 GRFs by using Display Threshold.

By using the variogram fitting button, calculate variograms on the lithotypes indicators in two hori-zontal directions and along the vertical.

(snap. 9.5-17)

The figure below shows the variograms for the horizontal directions and the vertical one. The dot-ted lines correspond to the experimental variograms and the plain lines to the model.

Geostatistical Simulations for Reservoir Characterization 457

(snap. 9.5-18)

(snap. 9.5-19)

9.5.1.4 Conditional Plurigaussian Simulation

Run 100 simulations using Interpolate/Conditional Simulations/Plurigaussian.

458

For the conditioning of the simulation to data, use a standard moving neighborhood (moving Facies). It is defined by a search ellipsoid with radii of 1.2 km x 3 km x 20 m and 8 sectors with an optimum of 4 points by sector. Display the simulation in the flat space using Display New Page with a raster representation or a section in a 3D grid representation.

(snap. 9.5-20)

Geostatistical Simulations for Reservoir Characterization 459

(snap. 9.5-21)

Finally, transfer the plurigaussian simulations from the working grid to the structural grid by using Tools/merge stratigraphic Units (Facies S2 Unit PGS).

The 3D viewer may be used to visualize the simulations.

9.5.2 Porosity 3D Simulations

9.5.2.5 Porosity Simulations for Each Facies

(a) Macro selection lithotype

Go to File/Selection/Macro and create a macro selection of three alpha indices (Siltstones, Anhy-drite and Dolomite) conditioned by the lithotype in the Auxiliary/variable/S2 UNIT. The purpose is to select the porosity for each facies to perform simulations. First create the three macro indices by using ’NEW’. Then define the rule for each facies by selecting the variable Lithotype(phi). For Siltstone the condition is equals

460

(snap. 9.5-22)

(b) Transformation in the Gaussian domain

In order to perform simulations, the transform from real space to gaussian space is necessary. That is done using the Gaussian Anamorphosis Modeling (Statistics/Gaussian Anamorphosis Modeling).

An anamorphosis (with 30 Hermite polynomials) is stored for porosity conditioned for each litho-type (ex Phi Siltstone (snap. 9.5-23)).

Geostatistical Simulations for Reservoir Characterization 461

(snap. 9.5-23)

(snap. 9.5-24)

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(c) Variogram fitting

Go to EDA/Variogram and compute an experimental variogram of the Gaussian Porosity for each lithotype (Gaussian Phi Siltstone, Gaussian Phi Anhydrite, Gaussian Phi Dolomite).

The experimental variograms are computed with 10 lags of 110 m horizontally and 20 lags of 1 m vertically. Check for possible anisotropies on N0/N90.

(snap. 9.5-25)

(snap. 9.5-26)

Hereafter are the characteristics of the variogram models:

m Gaussian Phi Siltstone: 2 basic structures: An anisotropic spherical model, with ranges of 680 m along U; 800 m along V; 7 m along W (sill: 0.94). An anisotropic cubic model with ranges of 700 m along U, 1600 m along V and 2 m along W (sill: 0.06).

Geostatistical Simulations for Reservoir Characterization 463

m Gaussian Phi Anhydrite: An anisotropic spherical model with an horizontal range of 450 m along U, 850 along V and a vertical range of 4,4 m. (sill: 1)

m Gaussian Phi Dolomite: An anisotropic spherical model with ranges of 476 m along U; 1072 m along V and 4.7 m along W (sill 0.5). An anisotropic cubic model with ranges of 323 m along U, 497 m along V and 5.4 m along W (sill 0.5).

An example is given below for Gaussian Phi Siltstone:

(snap. 9.5-27)

(snap. 9.5-28)

(d) Simulations

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l For each lithotype, run the Turning Bands simulations (Interpolate/Conditional Simulations/Turning Bands).

(snap. 9.5-29)

Do not forget to use the gaussian back transform option in the simulation parameters.

Geostatistical Simulations for Reservoir Characterization 465

(snap. 9.5-30)

Below is the neighborhood used for Turning Bands. The same standard neighborhood is used for the porosity of the different lithotypes (Phi Siltstone, Phi Anhydrite, Phi Dolomite).

(snap. 9.5-31)

9.5.2.6 Conditioning Porosity Simulations to Facies Simulations

(a) Create special variable

Create the macro variable Porosity [xxxxx] in order to store the results of the independent porosity simulations per facies conditioned by facies simulations. (Tools/Create Special Variable)

466

(snap. 9.5-32)

(b) Calculator

The transformation created in the calculator allows to inform the macrovariable Porosity [xxxxx]from porosity simulations conditioned by lithotype (Phi Siltstone [xxxxx], Phi Anhydrite [xxxxx], Phi Dolomite [xxxxx]) with facies simulations (PGS [xxxxx]).

(snap. 9.5-33)

Geostatistical Simulations for Reservoir Characterization 467

Then the macro variable Porosity[xxxxx] is transferred from the Working flat Grid (3D working Grid) to the 3D real space (3D, Structural grid) using Tools/Merge Stratigraphic Units.

(snap. 9.5-34)

Post-processing with simulations can be performed.

9.5.3 Compute Volumetrics using Porosity 3D

Use Tools/volumetrics to perform an estimation of HCPV.

The input parameters for HCPV are identical to the previous ones. The only different variable is the 3D Porosity (Porosity[xxxxx]).

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(snap. 9.5-35)

Geostatistical Simulations for Reservoir Characterization 469

(fig. 9.5-2)

The volumetrics computed using the 3D Porosity simulations are generaly higher than those com-puted using the 2D mean porosity simulations.

P90

P50

P10

58 59 60 61 62 63 64

Volumes (Mm3)

0

10

20

30

40

50

60

70

80

90

100

Frequencies

470

9.6 Conclusion

This case study handles different technics available in Isatis (surface simulations, facies simulations and volumetrics). The volumetrics outcomes are interesting to study as they are relevant concerning the use of the porosity (3D porosity simulations).

The use of the envelope in the discretization and flattening has an influence in the computation of 3D proportion matrix. It is necessary to extrapolate the proportions at the top and bottom of the VPC. This extrapolation has of course an influence on the facies simulations, therefore on the porosity simulations and finally on the resulting volumes.

The volumes calculated with the 3D porosity simulations are generaly higher than those calculated using a 2D mean porosity simulations. This can be explained by the extrapolation of siltstone at the top during the computation of the 3D proportions curves.

To conclude, this case study presents a possible workflow for Reservoir Characterization. In this purpose, several methods are applied (Turning Bands simulations, Plurigaussian Simulation, Uni-versal Kriging). It deals with structural, facies and property modeling. An interesting topic is the use of a 2D proportion constraint. It shows how to account for the uncertainty on surfaces and prop-erties (e.g. Porosity) together.