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A combined process-based and geostatistical methodology for simulation of

realistic heterogeneity with data conditioning

H.A. Michael, H. Li, A. Boucher, T. Sun, S. Gorelick, and J. Caers

Abstract

The goal of simulation of aquifer heterogeneity is to produce a model of the subsurface that

represents a system such that it can be used to understand or predict flow and transport

processes. Modeling requires incorporation of data and geologic knowledge, as well as

representation of uncertainty. Geostatistical techniques allow for data conditioning and

uncertainty assessment, but models often lack geologic realism. Simulation of physical geologic

processes of sedimentary deposition and erosion (process-based modeling) produces detailed,

geologically realistic models, but because the formation is built forward in time, data

conditioning is limited. The use of process-based models as training images for multiple-point

geostatistical simulation could produce geologically-realistic, conditioned models that

incorporate uncertainty; but the non-stationarity, non-repetitiveness, and structural complexity of

process-based models are challenges to this direct integration. An approach to reservoir

modeling that combines geologic process models, and object-based, multiple-point, and two-

point geostatistics to produce geologically-realistic realizations that incorporate geostatistical

uncertainty and can be conditioned to data is presented. The method is described as follows.

First, the geologic features of process-based model output are analyzed statistically. The statistics

are used to generate multiple realizations of reduced-dimensional features using an object-based

technique. These realizations are used as multiple alternative training images in multiple-point

geostatistical simulation, a step that can incorporate local data. Lastly, a two-point geostatistical

technique is used to produce conditioned maps of depositional and erosional thickness.

Successive realizations of individual geologic layers are generated in depositional order, each

dependent on previously-simulated geometry, and stacked to produce a three-dimensional facies

model that mimics the architecture of the process-based model. This method can be expanded to

other geologic systems for simulation of geologically-realistic, fully conditioned aquifer models.

1 Introduction

The spatial distribution of subsurface properties exerts an important control on groundwater flow

and solute transport (e.g., Gomez-Hernandez and Wen, 1998; Sharp et al., 2003; Zinn and

Harvey, 2003; Wang and Bright, 2004; Edington and Poeter, 2006; Feyen and Caers, 2006;

Fleckenstein et al., 2006; Jankovic et al., 2006; Ronayne and Gorelick, 2006; Swanson et al.,

2006). The development of methods to characterize and model this heterogeneity has been a

focus of hydrogeologists and petroleum engineers over the past several decades, leading to

establishment of the modern field of geostatistics as well as advancements in understanding and

modeling of the physical processes that produce geologic formations. Many and varied methods

to create models of geologic properties based on qualitative and quantitative knowledge of

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spatial property distributions, local characteristics, and depositional processes have been

developed (see e.g., Kolterman and Gorelick, 1996; Anderson 1997; Paola, 2000; de Marsily et

al., 2005; Eaton, 2006), each with unique advantages and disadvantages.

Geostatistical methods for modeling heterogeneity use information on the spatial structure,

represented as statistics, of the medium to produce models, or realizations, with the same

statistical structure. Many equally probable realizations can be produced, allowing some

understanding of the uncertainty in a given realization, assuming the underlying model is correct.

Traditional methods that incorporate only the correlation between two points in space are

computationally efficient and easily conditioned to data, but the high-entropy fields that are

produced rarely capture realistic, large-scale geologic continuity. Recently developed multiple-

point geostatistical (MPS) techniques use patterns obtained from training images (TIs), which are

a representation of the believed underlying structure, geometry, and statistics of the simulated

formation in two or three dimensions. These methods reproduce higher-order statistics than two-

point techniques, resulting in a better representation of spatial patterns and continuity. Multiple-

point techniques that employ sequential, pixel-based simulation, such as SNESIM (Strebelle,

2002), FILTERSIM (Zhang et al., 2006) or SIMPAT (Arpat and Caers, 2007), like two-point

methods, are easily conditioned to data, though large simulations can be computationally

intensive.

Object-based methods also produce continuous patterns of properties that are geologically

realistic (e.g., Haldorsen and Lake, 1984; Matheron et al., 1987; Deutsch and Wang, 1996;

Syversveen and Omre, 1997; Holden et al., 1998, Lantuejoul, 2002). These methods, also called

marked-point or Boolean techniques, consist of placing pre-defined three-dimensional

‘geobodies’ probabilistically within a model domain. Simulation is fast, but data conditioning is

difficult because entire geobodies are placed stochastically within a simulated field, so

conditioning often must proceed as trial-and-error (e.g., Lantuejoul, 2002; Allard et al., 2005),

significantly increasing CPU times.

Process-based techniques simulate aquifer heterogeneity in a forward manner, in contrast to

geostatistical techniques, which require prior knowledge of aquifer structure in the form of

variograms, training images, or sets of object geometries and spatial statistics, for example.

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Process-based models are based on knowledge of the processes that created the geology, rather

than its existing structure. The simulation models are formulated mathematically and the genesis

of geologic formations is simulated over time. Process-based models can produce very realistic

geometries that are not limited by assumptions of spatial structure. However, obstacles to

widespread application of geologic process models are the required computational intensity and

difficulty in conditioning to local data such as core, well-log, and geophysical data (e.g., seismic

and ground-penetrating radar).

Approaches to reservoir modeling that combine techniques have been developed prior to this

work. Geostatistical methods are often combined to simulate heterogeneity in a multiple-step

process, first using a method that produces large-scale heterogeneities, and then a method to

reproduce finer-scale variations in geologic properties (e.g., Lu et al., 2002; Damsleth et al.,

1992; Deutsch and Wang, 1996; Jones and Foreman, 2003; Falivene et al., 2006; Zappa et al.,

2006; Al-Khalifa et al., 2007). Object-based methods have also been combined with two-point

techniques for easier data conditioning (e.g., Holden et al., 1998; Shmaryan and Deutsch, 1999;

Viseur, 1999; Oliver, 2002; Vargas-Guzman and Al-Qassab, 2006). Process-based models based

on rules, not governing differential equations, have been combined with object-based and two-

point techniques (e.g., Xie et al., 2001; Pyrcz and Deutsch, 2004; Teles et al., 2004; Pyrcz and

Strebelle, 2006; Reza et al., 2006), though with limited success in data conditioning.

The objective of this work is to develop a modeling approach that generates a geologically

realistic aquifer model that represents well the primary heterogeneity controlling groundwater

flow and solute transport and can be conditioned to data. Output from a process-based model

(PBM) based on governing physical differential equations is used as a database from which to

draw statistics and deterministic rules. Object-based, multiple-point, and two-point geostatistical

techniques are then combined to produce a three-dimensional model of heterogeneity that closely

approximates that of the PBM and true geology, and can be conditioned to local data. This

methodology attempts to borrow the best of each technique: a process-based method for

geological realism, an object-based technique for simplifying shapes into mathematically

treatable objects, and pixel-based multi-point and two-point methods for data conditioning.

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2 Combination of Methods: Approach to Simulation

The simplest combination of simulation methods that would produce realistic, conditioned

heterogeneity would be the use of the results of a process-based model, described in Section 2.1,

directly as a training image for multiple-point geostatistical simulation in three dimensions.

However, this direct combination is difficult, as described in Section 2.2. Sections 2.3-2.6 detail

a methodology which incorporates multiple simulation techniques, developed to overcome

limitations of each method in order to produce geologically-realistic 3D reservoir models that are

conditioned to local data.

2.1 The Process-Based Model

A model that uses equations that govern the physical processes of deposition and erosion in the

development of a deepwater turbidite system was developed at ExxonMobil (Tao Sun, personal

communication). The output of this proprietary code, a realization of a simulation, is used in this

work. Deposition and erosion of sediments are simulated through time, dependent on fluid flow

velocity in the overlying water. Turbulent flow velocity is determined by numerical solution of

differential equations similar to those developed by Parker et al. (1986). The fluid flow is

coupled to empirical equations describing erosional and depositional processes on the bed.

Erosion is a function of flow velocity and grain size distributions (see e.g., Garcia and Parker,

1991), and deposition is the product of settling velocity (e.g., Dietrich, 1982) and sediment

concentration. The model simulates subsidence, but not compaction (though this can be included

in post-processing).

Initial conditions are topography, bed grain size distribution, and suspended sediment

composition and concentration. Boundary conditions for flow are specified, and temporally and

spatially variable forcing is input as one or more sediment sources with specified flow velocity

and sediment concentration. The grid is regular horizontally (X-Y space) and irregular vertically

(Z space). Simulation is forward through time, beginning from the initial conditions, and cells

are built up and removed as deposition and erosion occur. Information such as grain size

distribution and time of deposition is retained in each cell as it is deposited.

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Figure 1. Process-based model results. Colors represent grain sizes. Scale in panel B (blue = 0.1,

red = 1.5) represents colors in panels A and B; scale in panel C represents colors in panels C-O

(blue = 1, red = 1.5). Full range of grain sizes is 0-2. A) View of entire model, sediment source

location designated by arrow. B) Zoom view of model. C) View as in B, only grain sizes greater

than 1 are shown. D-O) Geobodies 1-12, respectively, view and colors as in panel C.

One realization of the process-based model (PBM) was used to develop the methodology

described in this work. A single sediment source with open lateral and distal boundaries was

specified in the PBM simulation. The realization consists of a system of channels and lobes

originating from the sediment source, with sediments generally fining distally. The geometry of

the deposits is illustrated in Figure 1. For visualization purposes, only the greatest 50% of grain

sizes are shown for each depositional period. Twelve individual channel-lobe geobodies were

identified in the realization, with separation based on formation through time: lobes generally

prograded and then stepped back; the next progradation, in the same or a new direction, was

designated the start of a new lobe.

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Geobodies were deposited immediately after one another with three exceptions. During each of

these intervals, a thin layer of fine-grained material was deposited over the model domain, with

some areas of sediment bypass and subsequent erosion. These layers, shown in Figure 2, are

referred to here as intermediate fine-grained units.

Figure 2. Intermediate fine-grained unit of the process-based model realization. Colors represent

grain size (blue=0.1, red=1.5). Left panel is entire field, right panel is zoom view. Missing

sections are areas of erosion.

2.2 Direct use of Process-Based Model Output as Training Images: Requirements and

Limitations of Current MPS Methods

Three limitations of currently-available multiple-point geostatistical algorithms make it difficult

to use PBM results directly as training images in MPS. These are the requirement of stationary

model assumptions for training images, the necessity of sufficient pattern repetition, and the

difficulty in reproduction of complex pattern geometry in three dimensions (Ortiz, 2008).

Stationarity

Geostatistical models are often decomposed into a stationary residual and trend. This is true for

two-point as well as multi-point techniques (Caers and Zhang, 2002). In MPS, statistics are

extracted from the training image, hence an assumption of stationarity over the entire TI domain

is required to borrow such statistics consistently. An example of an MPS simulation obtained

using a non-stationary TI is shown in Figure 3. The trends in channel orientation and thickness in

TI are not replicated in the realization, only stationary features are retained.

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Figure 3. Non-stationary training image and realization produced with the multiple-point

geostatistical algorithm, SNESIM (Strebelle, 2002). The realization does not display the trend in

patterns seen in the training image.

Heterogeneity simulated with geologic process models, as in actual aquifers, is non-stationary,

which limits the use of process-based model output as training images.

Pattern Repetition in Training Images

Training images must contain enough statistics in the pattern variability to be useful in

simulation. Similarly, that set of statistics must be large enough that it is unbiased and

sufficiently represents the range of variability. Thus, TI domains must be large, certainly larger

than the geologic features of the image, and often larger than the field to be simulated. Thus, if

simulation of geologic processes produces models with large features that are few in number, a

single PBM realization the size of the MPS simulation grid would likely not contain enough

pattern replicates for use as a TI. This limitation would be amplified were non-stationary pieces

of the PBM output extracted and used directly as TIs.

Three-dimensional simulation with MPS

A third limitation of MPS that makes direct use of process-based realizations as training images

difficult is the quality of reproduction of patterns and continuity in 3D (Ortiz, 2008). The MPS

algorithms use a template to scan the training image. The size of the template affects both pattern

reproduction (generally larger templates reproduce patterns better, though potentially with less

variability) and computational efficiency (larger templates require more memory and processor

time). In 2D, a template size can usually be found that produces satisfactory realizations in a

reasonable time. However, in 3D, memory requirements are much larger for the same template

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dimension, so simulation times often limit template size, resulting in realizations that do not

accurately reflect the characteristics of the training images. An example is given in Figure 4. The

long, single channels of the training image are not replicated in the realizations, which exhibit

disconnected and branching channels.

Figure 4. Three-dimensional multiple-point geostatistical simulation. SNESIM-generated

realizations do not display the connectivity or single-channel geometry of the training image.

2.3 Object-Based Simulations as Training Images

If process-based simulations are not used directly as training images, they can be used indirectly

as sources of information on structural geometry and spatial statistics. This type of information is

incorporated easily into object-based simulation techniques. Further, a reduction in the

dimension of the training image used in MPS can improve the reproduction of continuous

features, and the use of multiple training images can increase pattern repetition. A methodology

to create multiple realizations of layer thickness generated by object-based simulations using

geometries and statistics extracted from the PBM realization is presented here. These object-

based realizations form 2D training images that are then used in traditional MPS simulation to

achieve data conditioning. This approach incorporates information from geologic process models

while overcoming the limitations to their use as training images in MPS. The various steps

required to implement this approach are detailed in the next sections.

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Geobody geometry and spatial statistics

Object-based modeling requires a mathematical representation of geobody geometry, resulting in

simplification of the complex shapes present in the PBM realization. However, the goal of this

work is not to reproduce every detail simulated by a process-based model, but to replicate

geologic structures that most control groundwater flow. Thus, as a first approximation, lobe and

channel shapes have been simply defined, as illustrated in Figure 5.

Figure 5. Simplification of process-based model geobodies into parametric objects. A)

Separation of Geobody 1 into channel and lobe portions, the anchor point designates the

intersection of the two. B) Determination of parameters from geobody: channel and lobe length

and width, migration distance (X-distance from sediment source to anchor point), and

progradation distance (Y-distance from sediment source to anchor point). C) Simplified shapes

and position that approximate that of Geobody 1. D) Geobody fully defined by length and width.

The anchor point, defined as the joint between the channel and lobe, can geologically be

considered as the point at which channel deposition becomes unconfined and empties into a

lobate structure. The exact location of this point is critical for extracting statistics from the PBM

realization, though in practice its identification is subjective. The location of the anchor point

determines the orientation of the lobe: the major axis is taken to follow the angle between the

sediment source and the anchor point. The channel is defined only by its width, connecting the

sediment source to the anchor point. The thickness of the geobody is assigned based on a

deterministic rule, decreasing linearly with distance from the maximum thickness at the anchor

point.

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The statistics drawn from the PBM realization are listed in Table 1. In addition to statistics,

qualitative information is taken from the PBM realization, incorporated into the model as

deterministic rules. Examples are geobody and fine-grained deposit thickness trends and the

dependence of anchor point location on prior topography.

Table 1. Parameters used in simulation, statistics used to define them, and the source of the

statistics.

Parameter Statistics Source*

Width CDF of widths Geobody analysis

Length CDF of lengths Geobody analysis

Maximum thickness CDF of thicknesses Geobody analysis Lobe

Erosion thickness Assumed, deterministic Estimate

Width CDF of lengths/widths Geobody analysis

Length Defined by anchor point location --

Maximum thickness Equal to lobe maximum thickness -- Channel

Erosion thickness Assumed, deterministic Estimate

X CDF of migration distances Geobody analysis Anchor Point

Y CDF of progradation distances Geobody analysis

Frequency Proportion of intermediate fine-grained layers

Fine-grained layer analysis Fine-grained

intervals Thickness between lobe deposition

CDF of interval time, constant deposition rate

Fine-grained layer analysis

Erosion Erosion frequency Proportion of intermediate fine-grained interval missing near the sediment source

Fine-grained layer analysis

* All analyses are of the process-based model output.

Simulation of single channel-lobe geobodies

The geobody geometry established above requires only simulation of a ‘marked-point’. This

term refers to a point that is placed randomly in space along with markers that describe the

geobody geometry. Here, the marked-point is the anchor point, which is located stochastically

according to a Poisson process with a spatially variable intensity function (see Lantuejoul, 2002).

The geometric markers are channel width, lobe width, lobe length, and maximum lobe thickness,

all drawn at random from each CDF. This information together comprises a realization of the

object-based model.

Placement of the anchor point depends on directly-obtained spatial statistics and rule-based

statistics obtained from the PBM as well as geologic knowledge. The migration and progradation

CDFs are each converted into probability density function (PDF) maps: each cell over the XY-

domain is associated with a probability based on its distance from the sediment source. The rule-

based component is incorporated by constructing a probability map for anchor point placement

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based on the underlying topography. The rule used in the example simulations is that geobodies

are more likely to be deposited in locations of low topography. For conditional simulations, a

facies probability map is also generated and combined, as described in Section 3.2. The

individual probability maps (shown in Figure 7) are combined, using the Tau model (Journel,

2002) into a single probability map that is equivalent to the intensity function of the Poisson

process. Simulation of the anchor point location proceeds according to an acceptance-rejection

algorithm of Lantuejoul (2002, Alg. 7.7.2).

Figure 7. Probability maps for migration distance, progradation distance, topography, and facies

type (geobody or non-geobody) that are combined to produce the Poisson process intensity

function. Data points are marked on the facies probability map: yellow diamonds indicate coarse-

grained data, and black stars indicate fine-grained data at locations corresponding to the wells in

Figure 13.

Once the anchor point is simulated, the channel-lobe geobody object is generated from the

associated geometric markers, which are drawn from prescribed pdfs. This process can take

place many times for the same initial topography, with different results each time. Examples of

channel-lobe objects are illustrated in Figure 8.

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Figure 8. Two object-based realizations. Top panels: colors represent layer thickness. Bottom

panels: corresponding training images used in SNESIM, colors represent thickness category.

2.4 Multiple-Point Geostatistical Simulation

Multiple-point simulation is incorporated into the modeling methodology because it is a fast way

to integrate local constraining data with the statistics and rules obtained from the object-based

model while maintaining the structure and continuity of the geobodies. The images simulated

with object-based modeling hold the information of the process-based model, translated into a

form simple enough to be used by MPS.

The multiple-point simulation algorithm used in this work is SNESIM (Strebelle, 2002), which

has specific advantages and limitations relative to other MPS algorithms. One particular

restriction is that SNESIM simulates only categorical variables. Thus the training images of

Figure 8 that exhibit continuously variable thickness are used and simulated as thickness

categories only.

The requirements and limitations of MPS that prevent direct use of the PBM realization as

training images (variability, non-stationarity, and adequate reproduction of training image

features) present similar problems in using object-based TIs. However, the simplicity of the 2D

object-based TIs, combined with some algorithm modifications and simulation constraints,

detailed below, enable the use of MPS despite its limitations.

Variability: use of Multiple Training Images

Each simulated layer contains only one channel-lobe geobody, so each training image displays

only one such geobody as well. Also, the size of the training image is exactly the size of the

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simulation grid, so the general rule that TIs should be large and display multiple similar patters is

not followed. If only one training image with one channel-lobe pattern is used, the resulting MPS

simulation will generally be an exact replica of the TI. This undermines the advantage of MPS,

which is that it introduces variability so that data can be integrated while maintaining the

integrity of the prior statistics and conceptual model. This difficulty was overcome by

implementing the use of multiple alternative but similar training images into the SNESIM

algorithm. This enables the use of numerous and variable geobody patterns without violating the

conceptual model (as would be the case if more than one lobe were simulated on a single training

image) and produces simulations with more variability than in the case where only one training

image would be used.

Non-Stationarity: use of Self-Consistent TIs

A second consideration is the non-stationary model assumptions required for training images of

this kind. If multiple training images are used, for example the set in Figure 8, patterns from one

may be inconsistent with patterns from another, even though each individually is consistent with

the conceptual model and statistics. Thus, an MPS simulation, which requires stationarity in the

training images, may produce an inconsistent realization that incorporates patterns from very

different images. This is overcome by using a set of training images that are self-consistent in the

sense that they have a similar orientation, but different geometrical characteristics (channel and

lobe dimensions, for example). This is achieved by generating an unconstrained set of (N) TIs,

drawing one at random, and then selecting a subset of (n) TIs with the most similar orientation as

the set of TIs for use in MPS. This limits the set of training images to those that are similar

enough that a consistent MPS realization can be produced.

Reproduction of Features: Constraining the Simulation Area

If simulation is constrained well enough, MPS can produce non-stationary model realizations

that adequately reproduce the features of the training images. Two constraints that do not depend

on local data are incorporated into the methodology. The first is assignment of channel thickness

category to the cell at the sediment source location. The second is restriction of MPS simulation

to an area smaller than the entire grid. This simulation area is determined by producing many (m)

object-based realizations that fall into an angle range narrower than the entire 180-degree range.

Examples of unconstrained SNESIM realizations, a simulation area, and a constrained realization

are shown in Figure 10.

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Figure 10. Unconstrained SNESIM realizations, constraints, and constrained realization with

four thickness categories. A and B) Examples of unconstrained realizations. Marginal

probabilities are inconsistent with those of the training images: lobe facies proportions are too

high. C) Simulation area (gray) and fine-grained category (blue). D) Realization constrained by

area of C and sediment source location data point.

2.5 Erosion and Intermediate Fine-grained Units

Erosion is simulated within the process-based model, and is generally associated with geobody

deposition. Though the depth and location of erosion is not retained in the PBM realization,

evidence of it is observed as breaks in the fine-grained units between lobes (Figure 2) and

discontinuous surfaces along the tops of channel-lobe geobodies (where another channel-lobe

eroded and then was deposited). Because erosion is not retained, it is difficult to extract erosion-

specific statistics from the process-based model. Instead, a rule is applied, specifying that erosion

is limited to only the area of the depositional geobody simulated by MPS, with erosion depth

greatest in locations of greatest depositional thickness and highest underlying topography. The

topographic dependence attempts to mimic the physical process of erosion along a turbidity

current, which does not cut evenly into undulating topography, but rather erodes more from

higher elevations along the path than from lower elevations due to the momentum of the

turbidity current, which tends to maintain its trajectory.

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As indicated in Section 2.1, the process-based model results include some time intervals during

which only fine-grained material was deposited. Fine-grained units deposited between times of

geobody deposition are simulated according to the CDF of interval times between depositional

periods obtained from the PBM realization. Thickness is assigned by rule: decreasing linearly

with distance from the sediment source.

2.6 Simulation of Continuous Depositional and Erosional Thicknesses

Multiple-point simulation with SNESIM produces thickness categories; simulation of realistic

geometry requires smoothing of both depositional and erosional thicknesses. Direct sequential

simulation (DSSIM) using simple kriging with a locally-varying mean (Journel, 1994) is chosen

for simulation of continuous thicknesses. The pixel-based geostatistical algorithm is very fast

and generates continuous-valued non-Gaussian random fields requiring only knowledge of the

univariate distribution and co-variance model of the variable being simulated.

For lobe deposition, the locally-varying mean (LVM) map is the result of MPS simulation. For

erosion and intermediate fine-grained units, the LVM map is obtained as in Section 2.5.

Examples of LVM maps are shown in Figure 12.

Figure 12. Locally-varying mean maps for selected layers of realization U1. A) Geobody

deposition thickness. B) Erosional thickness corresponding to geobody in A. C) Intermediate

fine-grained unit thickness.

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2.7 Sequential Layer Simulation, Erosion, and Stacking

The processes described in Sections 2.1-2.5 result in simulation of a single thickness or erosional

map (as illustrated in Figure 11). The geologic formation to be modeled consists of multiple

stacked geobodies, a result of deposition and erosion through time. A simulated formation is

similarly produced with this methodology by repeated simulation of maps to create a full 3D

multi-layer stacked model. Simulation begins from the bottom of the model, in the sequence of

deposition. The value of performing progressive simulation is that, as in natural sedimentary

systems, deposition and erosion associated with a particular geobody are dependent on the

underlying conditions.

The number of depositional layers, or stacked geobodies, is specified for a particular simulation.

Beginning with an initial topography, a set of training images is generated and a simulation area

designated as described in Sections 2.3 and 2.4. The migration and progradation probability

maps used to produce the Poisson process intensity function are constant over all simulations,

but the topographic probability map changes for each new geobody deposition simulation, as

does the facies probability map (see Figure 7), as described in Section 3.

Once the set of TIs is simulated based on the combined probability map, a categorical geobody

thickness map is generated with SNESIM. The thickness map is smoothed with a moving

average, and then input into DSSIM as a locally-varying mean for simulation of continuous

thickness. From this map and the underlying topography, an erosion map is generated with

DSSIM. The simulated erosion and deposition are stacked onto the initial topography: erosion

cuts into the previous layer and deposition is stacked on top.

After each layer is simulated, a fine-grained interval deposition time is drawn from the CDF.

When this time is greater than a threshold, an intermediate unit is simulated as described in

section 2.5. That unit is then stacked on top of the previously-simulated material. Examples of

training images and depositional and erosional simulated thicknesses are shown in Figure 11.

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Figure 11. Simulation examples. A, B) Object-based training images. C) SNESIM-simulated

categorical geobody thickness. D) DSSIM-simulated depositional thickness. E) DSSIM-

simulated erosion depth. F) DSSIM-simulated intermediate fine-grained unit thickness.

The stacking process is repeated for each depositional layer. In each step, a new topography is

used to generate new training images, and then geobody deposition, erosion, and intermediate

units are simulated. This cycle continues until the desired number of geobody layers are

produced, and the result is a layered system, built in the sequence of geologic deposition, that is

self-consistent (each layer is dependent on prior deposition and erosion) and consistent with

information drawn from the process-based model.

3 Data Conditioning

Only a methodology for unconditional simulation is presented in Section 2. The primary

advantage of this methodology, aside from simulation time, is the ability to condition the model

to data. Data is integrated in every step of the simulation, though exact matching is accomplished

using the pixel-based multiple-point and two-point geostatistical techniques. The model

presented here is conditioned to well facies data only: sequences of coarse-grained and fine-

grained material. However, the methodology can be adapted to condition to any information,

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including seismic and other geophysical data, that can be incorporated with pixel-based

geostatistical techniques (see e.g., Gilbert et al., 2004).

3.1 Data Processing

For initial testing of the methodology, and to ensure consistency (inconsistency between data and

model is not the subject of this paper, as this is consequential to any technique), well data were

drawn from the PBM realization. Twelve locations near the sediment source were selected

arbitrarily as well data locations, these are shown in Figure 13A. Each “well log” (Figure 13B)

was converted from a distribution of grain sizes to facies by choosing a grain size of 50% of the

maximum as the threshold between non-geobody (fine-grained) and geobody (coarse-grained)

facies (the same as chosen for distinguishing channel-lobe geobodies in Figure 1 from non-

geobody facies and for obtaining associated statistics). The well data can be analyzed assuming

different levels of interpretation, or knowledge of the system. In this case, it was assumed that

individual depositional layers can be identified in well data based on fining patterns or dating

techniques, so well log intervals were separated according to each of the 15 (12 geobody and 3

intermediate) depositional periods identified. It was assumed that this interpretation is exact, no

error was made (in reality one would have to account for interpretation error). Each well log has

at most 15 data intervals, but most have many fewer because not all intervals are present in the

depositional record in every location. While the interpretation of intervals requires more effort

than needed in a traditional geostatistical modeling study, it does allow the inclusion of

additional well-log interpretation that can generally not be included in traditional multi-point or

two-point simulation methods.

The data input into the model are first forward modeled for erosion. An additional thickness, a

random proportion of the maximum erosion thickness, is added to coarse-grained intervals at the

spatial frequency of erosion. This frequency, the proportion of any surface that is subsequently

eroded, was estimated as the missing proportion of the near-sediment source intermediate fine-

grained units found in the PBM. The entire thickness of the forward-modeled interval is

simulated by deposition, and the added erosional thickness is subsequently eroded during the

modeling process. This is equivalent to allowing the model to simulate erosion at random, but

ensures that data can be exactly matched.

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Figure 13. Well locations and well logs obtained from process-based model output. A) Well

locations. Model colors indicate topographic elevation. B) Facies category along each well log

used as conditioning data in simulation. Vertical exaggeration is 10:1.

Prior to simulation of each layer, the facies category and thickness of the bottommost interval of

each well log is taken as the set of data points for each simulation. After each simulation of

deposition and erosion, the data set is re-processed and a new set of data is obtained from the

new bottommost interval (with the data matched previously removed). During this process,

individual data points may be designated as “must match”, “probable match” or “bypass”. The

probability of matching each point in the next data set is calculated as the number of intervals

left in the well log divided by the number of lobe intervals remaining to simulate. If that

probability is equal to 1, the interval must be matched in the subsequent layer simulation.

3.2 Consistency between Training Images, Simulation Area and Well Data

In conditional simulations, well data is first taken into account in the object-based simulation.

The angle of the training images constrains the simulation area for each MPS simulation, so the

presence and absence of geobody facies at particular locations must be taken into account prior

to the MPS data integration step, when selecting the training image angles. A probability map for

anchor point location is created by performing indicator kriging on the binary (1 for coarse-

grained, 0 for fine-grained) facies present in the lowest non-simulated interval in the well data

(the “next data set”) to produce a probability map. This map is combined with the migration,

20

progradation, and topographic probability maps (Figure 7) using the Tau model to produce the

intensity function for the inhomogenous Poisson process simulation, as described in Section 2.3.

Thus, training image geobodies are more likely to be placed in areas where geobody facies are

identified in the well data than where fine-grained facies are identified.

Well data also constrain the location of the simulation area for MPS designated by the object-

based simulations. If the matching probability is 1 for any coarse-grained data point (meaning

that it must be matched in the next simulation) and if that point is located outside of the

simulation area determined from the object-based modeling, the simulation area is rejected (not

considered for simulation). The training images and simulation area are then re-simulated until

the simulation area includes the data point.

3.3 Facies, Erosion, and Thickness Matching with MPS and DSSIM

Multiple-point geostatistical simulation is the means by which facies are matched exactly in this

methodology. Data points falling within the simulation area are assigned an MPS category based

on facies (coarse- or fine-grained) and interval thickness for coarse-grained data points, both

easily matched in MPS (Figure 14A). The exact thickness of each interval is then matched in the

DSSIM step by assigning hard data values to the data points (Figure 14B). Because DSSIM uses

a model of spatial correlation, the data points affect the simulated thicknesses of nearby points

according to the input variogram. Thus, data is integrated into the position and shape of the

geobody (through MPS) as well as the shape of the geobody thickness (through DSSIM). Data is

also incorporated into simulation of the thickness of intermediate fine-grained units (Figure

14D). Forward-modeled erosional depth is also exactly matched during erosion DSSIM (Figure

14C).

21

Figure 14. Categorical and continuous depositional and erosional thickness generated with

SNESIM and DSSIM, conditioned to well data. A) SNESIM-generated thickness category map.

Black and yellow diamonds are sand interval data located inside and outside the simulation area,

respectively. White and yellow circles are fine-grained interval data located inside and outside

the simulation area, respectively. B) Thickness map generated with thickness values assigned to

categories in A as a locally-varying mean map. Data interval markers as in A. C) Erosion depth

map generated with DSSIM. Black diamonds are data locations inside the erosional (lobe) area.

D) Fine-grained unit thickness generated with DSSIM. White and yellow circles are fine-grained

data intervals to match and to bypass, respectively.

4 Validition: Visual and Statistical comparison between ‘Stacked’ Model and Process-

Based Simulation

Two unconditional and two conditional stacked model realizations were generated for

comparison to the process-based model realization. A list of the parameters and values used to

generate the realizations is given in Table 2.

In the case of this deepwater turbidite stacked channel-lobe system, the features that are known

to control flow and potential fluid yield most (e.g., Larue and Hovadik, 2006) are the volume and

thickness of the sand bodies and the extent to which they are hydraulically connected. This

means that the proportion of coarse-grained material, geobody shape and stacking pattern, and

22

fine-grained unit existence and continuity are essential features to compare between the process-

based and stacked model realizations for evaluation of the methodology.

Table 2. List of stacked model user-controlled parameters, descriptions, and values used to

produce realizations U1, U2, C1, and C2.

Parameter Description Value used in Realizations

General

Grid dimensions Length in X-direction [m], length in Y-direction [m], number of cells in X-direction, number of cells in Y-direction

600, 500, 2, 2

Sediment source location I,J (cell) location of channel starting point 150, 250

Number of lobes Total number of depositional lobe layers to be simulated 12

Number or categories Number of categories to divide object-based simulated thickness maps into

4

τmg, τpg, τele

Tau parameters for combining probability maps into Poisson intensity function maps: anchor point migration (mg) and progradation (pg) and surface elevation (ele)

1, 1, 2

Object-based simulation

N Number of anchor points simulated from which TIs are drawn

12

n Number of TIs simulated 4

M Number of extra object-based model realizations used to create a simulation area

6

SNESIM

Servosystem Factor that controls the constraint on replication of the input histogram

0.2

Number of multi-grids Number of telescoping grids used in SNESIM simulation 5

Deposition

Bypass threshold Depositional thickness value below which deposition simulated with DSSIM is considered bypass (thickness=0 m)

0.01

Interval deposition rate Average rate of deposition assumed for intermediate fine-grained units [m/year]

1.4 x 10-5

Erosion

Erosion threshold Erosion depth value (<0) above which erosion simulated with DSSIM is considered zero [m]

-0.001

Erosion depth Maximum erosion depth in the model [m] 1.0

Conditioning

Erosion frequency Frequency used to forward model erosion depths into the well data, obtained from analysis of the PBM

0.19

τdata

Tau parameter for combining probability maps into Poisson intensity function maps: data-derived lobe location probability

1

Figure 15 compares cross-sections from an unconditional simulation to the corresponding cross-

sections in the process-based model realization. The colors represent facies: blue is the fine-

grained (non-geobody) facies, and yellow is the coarse-grained (geobody) facies. The bottom

23

geometry of the models is the same. Inspection of the cross sections indicates that there are

similarities in the stacking patterns, the thickness of the sand intervals, and the degree of

connectivity between them.

Figure 15. Comparison of cross sections of the process-based model realization and a

conditional 12-lobe stacked realization (C1). Colors represent thickness categories (blue is the

fine-grained facies, yellow is the coarse-grained facies). A, B) Cross sections perpendicular to

flow for the PBM and realization C1, respectively. Vertical exaggeration is 10:1. C, E) Cross

sections parallel to flow direction for the PBM. D, F) Cross sections parallel to flow for

realization C1.

The stacking pattern and geobody shapes can be compared by visualizing only the coarse-

grained material. Figure 16 allows comparison of the PBM realization with the two

unconditional (U1 and U2) and two conditional (C1 and C2) realizations of the stacked model.

The sequence of colors is consistent among the five panels. Though generated similarly, the

distribution of geobodies in the realizations of the stacked model are different. The lobe locations

are variable between the realizations, and in some cases not as uniformly distributed as the PBM

realization. Also, the lobe shapes are not parameterized in such a way that they accurately reflect

the shape of the lobes in the PBM realization: the lobe geometry in the stacked model

realizations appears to be somewhat wider than that of the PBM realization. Such inconsistencies

can be corrected by re-defining and parameterizing the channel-lobe objects if this is deemed

relevant for the flow application at hand.

24

Figure 16. Channel-lobe geobodies from the process-based model realization and four

realizations of the 12-lobe stacked model (U1, U2, C1, and C2).

The connectivity between coarse-grained geobodies is important for flow behavior. This is

controlled in part by the existence and erosion of fine-grained units that are deposited between

intervals of geobody deposition. An example of an eroded fine-grained unit from a stacked

model realization is visually compared to that of the process-based model in Figure 17 (note that

the simulation domain for the stacked model is smaller than the domain of the entire PBM,

which is shown in Figure 1A and Figure 2). The PBM realization shows more erosion, but with

similar patterns. The extent of erosion can be controlled in the stacked model by adjusting the

maximum erosional depth, which was arbitrarily assigned as 1m in this case. More information

on erosional statistics obtained from the process-based model would also improve simulation of

erosion.

25

Figure 17. Intermediate fine-grained units simulated with stacked and process-based models.

White areas indicate erosion. Color is uniform, representing fine-grained facies. A) Fine-grained

unit from the process-based model realization. B) Fine-grained unit from a stacked model

realization.

Many features of the stacked models, both unconditional and conditional, compare well visually

to the PBM realization. However, a statistical comparison provides a more quantitative measure

of consistency. The statistics drawn from the PBM realization that were used as input into the

stacked model can be compared to those of the stacked model results in quantile-quantile plots

(Figure 18). The data from the two unconditional realizations and from the two conditional

realizations are pooled together and plotted against the data from the PBM realization. Because

there are only 12 geobodies in each realization, a relatively small sample size, some deviation

from the 1-1 line (which would indicate an exact statistical match to the PBM) is expected. The

migration and progradation distances and channel lengths match the PBM relatively well, with

values that oscillate above and below the 1-1 line. Channel width and lobe length statistics

deviate for high values, and high values of stacked model lobe widths are considerably lower

than those of the PBM.

26

Figure 18. Quantile-quantile plots for statistics of the process-based and four stacked model

realizations (U1, U2, C1, and C2).

Statistics which reflect the bulk features of the model realizations are listed in Table 3. The

proportions of sand in the stacked models are slightly higher than that of the PBM realization,

while the total volumes are lower, particularly for the unconditional simulations. This means that

the stacked models contain less fine-grained material in particular, a volume which can be

controlled by parameters of the model that are specified by the user but not derived from the

PBM. In practice, the volume represented in analog models such as process-based models need

not be the same as the actual reservoir. This methodology allows adjustment of the volume of the

generated realization to that desired for a particular aquifer system.

The statistics of the coarse-grained and fine-grained deposits can also be compared among the

model realizations (Table 3). The number and thickness of continuous vertical intervals of

coarse- and fine-grained material (which may include more than one depositional layer) at each

27

grid cell are indicators of the nature and connectivity of the deposits. An interval is considered

individual if it is bounded vertically by a different facies or the top or bottom of the model. The

total number, mean number, and maximum, mean, and standard deviation of interval thicknesses

for both coarse- and fine-grained material compare very well among the model realizations.

These statistics provide information similar to that of vertical variograms, which do not give

useful information on any of the five realizations due the small size and non-stationarity of the

models.

Table 3. Statistical comparison of the process-based model realization and four realizations of

the stacked model.

Unconditional Conditional

PBM U1 U2 C1 C2

Total Sand Proportion (Net to Gross) 0.26 0.35 0.31 0.28 0.30

Total Volume [105 m

3] 79 42 48 67 68

Coarse-Grained Deposits

Total Number of Intervals* 9300 9908 8674 8899 9344

Maximum Thickness [m] 6.2 6.2 4.9 4.9 4.9

Mean Thickness [m] 1.4 0.9 1.1 1.3 1.4

Standard Deviation Thickness [m] 1.1 0.8 0.7 0.8 0.8

Mean Number of Vertical Intervals 0.5 0.5 0.5 0.5 0.5

Fine-Grained Deposits

Total Number of Intervals* 27034 26235 25371 25391 25086

Maximum Thickness [m] 5.8 5.7 5.9 5.6 5.9

Mean Thickness [m] 1.3 0.6 0.8 1.2 1.2

Standard Deviation Thickness [m] 0.9 0.6 0.7 1.0 0.9

Mean Number of Vertical Intervals 1.4 1.4 1.4 1.4 1.3 * Total number for the stacked model, which has 2m x 2m cells, is divided by four for comparison to the process-based model, which has 4m x 4m cells.

The statistics, reported as overall numbers in Table 3, vary spatially over the non-stationary

model realizations. The trend in average coarse-and fine-grained interval thickness with linear

distance (in any direction) from the location of the sediment source for each model realization

are shown in Figure 19. The stacked model trends compare well to those of the PBM realization,

but the peak values vary in some cases. The unconditional simulations do not produce the thick

intervals of coarse-grained material near the sediment source. This could be due to

parameterization of the geobodies, or to the erosion rules. The conditional simulations produce

better results, likely because they are constrained by wells near the source. The peak average

thicknesses of fine-grained intervals of the stacked model realizations correspond well to the

28

PBM realization peak in location, but the values are somewhat higher. However, the trends with

increasing distance correspond well.

Figure 19. Average thickness of individual vertical intervals of coarse-grained and fine-grained

material vs. distance from the sediment source for the process-based model realization and four

realizations of the stacked model.

A similar spatial comparison of sand proportion is shown in Figure 20. The trends and peaks in

the average proportions of all four stacked model realizations compare very well to that of the

PBM realization. Plots of point-wise sand proportion with distance also compare well. The

principal differences are comparatively few data between 0 and 0.1 in the stacked model

realizations, and a greater number of higher values at distances greater than 200m from the

source. The smoother appearance of the patterns in the graphs for the stacked model realizations

29

may be due to the larger number of values: there are four times as many horizontal grid cells in

the stacked realizations within the same model area.

Figure 20. Point-wise and average sand proportion vs. distance from the sediment source for the

process-based model realization and four realizations of the stacked model.

5 Assumptions, Simplifications, and Practical Limitations

The methodology developed in this work combines techniques to take advantage of the positive

aspects of each approach (object-based, multiple-point, two-point, and process-based methods).

However, the integration of techniques in a multiple-step process introduces assumptions and

30

simplifications as well as input data requirements that together could be restrictive for practical

application.

The systems to which this simulation method can be applied are limited by the type of system,

the available local information, and the geometry of heterogeneity. The depositional environment

must be known and well-simulated with a process-based model. Estimates of relevant

parameters, such as sediment grain sizes, and knowledge of initial and boundary conditions must

be available for input into the process-based simulation. Because an object-based technique is

used, the depositional process must result in distinct geometry of heterogeneity, allowing

identification and parameterization of geobody objects.

Assumptions that may significantly affect results are made throughout the simulation process.

These assumptions may change for different simulated systems, and sensitivity to them can be

tested, but they are necessary for simulation to proceed. For example, by taking a set of statistics

from the PBM realization, it is assumed that the particular parameter is stationary over the

model. If a CDF of channel widths is made from all identified channels, it must be assumed that

the widths do not change in any predictable way during the depositional process (i.e., channel

widths should not decrease as topographic gradient increases). Deterministic rules or trends

developed from analysis of the process-based model results are another type of assumption. If

only one set of model output is analyzed, it may be difficult to discern such rules, making their

adoption subjective and potentially incorrect. The important characteristics of the process-based

model output to be reproduced during simulation are another significant assumption. If the

primary controlling characteristics are left out (perhaps small-scale flow barriers control flow but

are not explicitly represented in the object-based modeling or MPS), then simulation can never

produce a realization that would exhibit the flow properties of the actual system.

While the aim of this paper is to present a practical methodology for combining several

techniques for simulating aquifer heterogeneity and integrating various types of geological data,

many aspects of the simulation methodology can be improved with further effort. First, working

with a set of process-based model realizations (instead of only one) simulated with identical

inputs as well as a range of inputs that represents their uncertainty would make the statistical

analysis more robust and would allow better recognition of trends and identification of rules.

31

Retaining information on erosion within the process model would also improve statistics and

representation of erosion in this methodology. Parameterization of the channel-lobe geobodies

that results in more geologically-realistic object-based realizations would result in more realistic

simulations.

Despite the limitations, and perhaps with some improvements, the method can be applied with

success to systems which are well-constrained by knowledge of depositional processes, without

requiring extensive information on the structure of heterogeneities. For particular types of

systems, the methodology developed has the potential to simulate geologically-realistic aquifer

heterogeneity that efficiently reproduces the essential features present in a computationally-

expensive process-based model with the added capacity of conditioning to local data.

6 Conclusions

A methodology is presented that incorporates multiple methods for modeling geologic

heterogeneity in order to take advantage of particular aspects of each. The results of process-

based simulation of a deepwater turbidite system are analyzed, and geobody shapes, relevant

statistics, and rules and trends extracted. That information is used in object-based modeling to

reduce the dimension of the simulation by creating many equally-probable two-dimensional

maps of geobody thicknesses. These object-based realizations are then used as sets of training

images in multiple-point geostatistical simulation, which is used to produce maps of geobody

geometry and thickness. Two-point geostatistical simulation is then used to simulate the

thickness of each depositional layer and erosion associated with geobody deposition. Simulation

of individual geobodies and erosion occurs in stratigraphic succession, beginning at the bottom

of the formation, with each layer dependent on the topography of the previously simulated

layers. The methodology allows for data conditioning, exactly matching thickness and facies data

obtained from wells.

Though assumptions are made and limitations exist in simulation, the resulting stacked models

display many of the features of the process-based model that were targeted for reproduction.

Geobodies overlap in similar ways, and erosion cuts through fine-grained units, increasing

32

connectivity between coarse-grained geobodies. It is expected that differences between the two

can be reduced with simple adjustments to model parameters.

The proposed methodology is a promising technique for simulating geologically-realistic aquifer

heterogeneity. The procedure retains the speed and conditioning capability that make

geostatistical techniques practically applicable, but also includes the non-stationarity and

geologic realism of models that are based on forward simulation of geologic processes.

Acknowledgements

The authors thank James K. Miller, Craig S. Calvert, and ExxonMobil Corporation for providing

simulated data and assistance with the project. This work was supported by the Stanford Center

for Reservoir Forecasting and by NSF grant EAR-0207177. Any opinions, findings, or

recommendations expressed in this material are those of the authors and do not necessarily

reflect the views of the National Science Foundation.

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