building geostatistical models constrained by dynamic data...

SPE 35478FE!!D

Society of Petroleum Engineers

Building Geostatistical Models Constrained by Dynamic Data - A Posteriori Constraints

G. Blanc, SPE, D. Guerillot, SPE, D. Rahon, F. Roggero, IFP and the HELIOS Reservoir group

Copyright 1W6, Smefy of Peboleum Engineam, Inc.

7hIs paper was prepared for presentatim at the SPBNPF EuroPan Conference held m

Stavanger, Norway, 16-17 April 1995.

Th,s papw was selected for presentatmn by an SPE Commmee Iolbwmg rewew 01 mfcfmaton

contwned m an abstract submitted by the aulhor[ s). Cdents of Ihe papef, as presented, havenot been reviewed by the %xtety of Petroleum Engmears and are subject to correction by the

author(s) The matertal, as presenled does not necassary reflact any pesdmn of the Scmety of

Petroleum Eng(neers, IIS oHlcels or membars. Papefs presanted at SPE mealings are subpct

10 publlcat$on review by Ed!tor!al Committals of the Soxty of Petroleum Engmears Parmm-

mon to copy IS restricted 10 an abslract of not more than 300 words. Illustrabon may not be

copied The abstract should contain conspicuous acknwladgmenl o! where and by whom the

papar was praaented Write L!branan, SPE, P O 8333836, Richardson, TX 75W33-2836 USA,lax 01-214-952.9435

AbstractThis paper describes one solution to the problem of

constraining geostatistical models by well-test results whichhas been obtained within the frame of the HELIOS researchproject conduced by Elf and IFP in the field of reservoir engi-neering.

To do so, a numerical simulation program devoted tothe simulation of well-tests has been coupled with a nonlinearconstrained optimization program to make an inversion loop.The resulting software package provides the reservoir engineerwith a tool to compute the set of optimal facies or rock-typeproperties and the well skin which give the best fit between thesimulated and the measured well pressure during the test dura-tion. Any petrophysical facies or rock-type property can beassigned either a conslant value everywhere or a Gaussiandistribution defined by its mean and its standard deviation.

On one hand the numerical program is able to com-pute not only the pressure and its derivative but also the gradi-ents of this pressure with respect to the petrophysical facies orrock-type properties and with respect to the well skin. Thegradien[s used by the optimization program to control thesearch algorithm can also be used as a stand alone diagnosistool to analyze the simulation results.

Several examples are given which show the efficiencyof the various algorithms. These examples also demonstratethe wide range of applicability of the software package toanalyze and to interpret well-tests as well as to integrate dy-namic data in geostatistical modeling.

As a conclusion, the paper sets forth several new re-search axes to extend this work toward the inversion of theshape of the geostatistical images themselves.

IntroductionGeostatistical modeling of reservoir heterogeneity is nowwidely used by geoscience engineers to fill in reservoir simu-lation grids. These geostatistical models are made of faciespixels or geologic objects and are built by using simulationalgorithms that reflect the main statistics of the geology ofdeposits, Integration of dynamic data together with the geologyenhances the quality of the geostatistical modeling and pro-vides the reservoir engineer with a better basis for reservoirsimulation and management. The uncertainty of simulatedproduction scenarios is then reduced, allowing more realisticeconomic evaluations,

In this paper the dynamic data considered are re-stricted to well-test results. However, the tools and the meth-odology presented here apply to other dynamic data includingfield-production data.

The problem of constraining geostatistical simulationby well-test results can be considered in two different ways:

● The geostatistical simulation algorithm can be modi-fied in order to integrate the dynamic data into thenumerical processing. The purpose is to generate afacies model that features a given average permeabil-ity within the investigation area of each well, Severalattempts have been presented, which use simulatedannealing ‘-3or sequential simulation 4. This is an “aPriori “ process.

● Once a geostatistical grid has been generated, thepetrophysical parameters of the grid can be computedso that the corresponding numerical well-test modelmatches the well-test behavior. This is an “a Posteri-or” process that applies to pixel or objet based geo-statistical modeling as well as to deterministic model-ing. More generally, the inverse problem of findingthe field properties such as permeabilities and porosi-ties by matching the field behavior is known as His-tory Matching and has received a great deal of atten-tion for long time 5“’6.This concept has even been

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2 BUILDING GEOSTATISTICAL MODELS CONSTRAINED BY DYNAMIC DATA - A PRIOR CONSTRAINTS

extended to the matching of future extreme reservoirdevelopment scenarios ‘7.These two process are currently being investigated

within the framework of the HELIOS project “ in the field ofreservoir engineering. The present paper addresses the a Pos-terior process and gives an overview of the current status ofthe project.

Inversion loopThe basic tool that has been developed to reach the objectiveof the project is the inversion loop, which is diagramed inFig. 1.

The aim of the inversion process is to start from aninitial guess for the parameters and to improve it by integratingdynamic data inan automated loop. Thequality of the fit be-tween measured and computed data is evaluated by using anobjective function. The proposed approach is based on Baye-sian formalism “ “~’ which includes a Priori information com-ing from the geological model in the objective function. Thisapproach is well suited for integrating static data from geologytogether with dynamic data from well-tests.

The heart of the loop is made of the numerical well-test simulation program. This program computes the well pres-sure behavior for a given set of the parameters that the user islooking for as well as the gradients of the well pressure withrespect to these parameters. Well pressure and gradients are[hen used by the optimization program to compute a betterestimate of the parameter set that will lead to a better fit. Thesecomputations are run again and again in the loop until theconvergence criterion is reached.

The three different components of the inversion loopare: the numerical simulation program, gradient computationand the optimization program. They are detailed here under.The main features of the user interface that was developed tomake it easier to use the inversion loop and to visualize theresults are also presented.

On top and outside the loop are the geological model-ing program (geostatistical simulation program or deterministicsimulation program) and the analytical well-test interpretationprogram (Fig, 1). The first one provides the grid system offacies. The second one gives a pre-interpretation of the well-test, which helps the user to check the data consistency and toderive a first guess of the petrophysical properties within thearea investigated by the test.

Numerical well-test simulationThe pressure behavior of a single-phase flow in a 3D reservoiris governed by the following equation:

a # f3(P+pgz)

ax, p ax,)=OC, ;+q ........................... (1)

* The HELIOS reservoir project is a research project operatedjointly by Elf Aquitaine Production and Institut Franfais duPetrole in the field of reservoir engineering,

where :P is the pressurei is the axis index.r, coordinates (xl, x2, X3,stands for x, y, z)k, is the directional permeabilityK is the fluid viscosityp is the fluid densityg is the gravity

$ is the porosityc, is the total rock and fluid compressibilityq is a sink term

When the geometry of the reservoir becomes compli-cated or when the reservoir cannot be considered as homoge-neous, analytical solution methods no longer provide the well-test pressure behavior. In such situations, numerical simulationprogram must be used instead. Reservoir models based ongeostatistical simulation belong to this class of problems.

SIMTESTW is a numerical simulation program “ thatuses the 7-point finite-difference scheme of Equation (1). Thegrid-block system is Cartesian irregular and dead cells are usedto simulate complex shapes.

Discretization of Equationgrid-block i and at time step n, is:

;;[(f’;-?’)+P+, -%)]=

(1) in space and time, for a

I*\............. . ... . . .,

@c,ViFy’ - ~“”’

+~,pQpIn –tn-’

where:j is the index of all neighbors of grid-block iTJ is the transmissivity between grid-blocks i and j

(harmonic average of hxk products)V, is the volume of grid-block i

Qp is the rate of the well p8,, is equal to 1 when grid-block i contains the well p and

Oelsewhere

P“, is then the solution vector of the following matrix equation:

A“ Xp”= & .......... ................................... ..................(3)

where:A’ is the 7-band matrix of the system of discretized

equations at time nP“ is the vector of the pressure map at time nR is the vector column of the right hand side of the sys-

tem of discretized equations at time n

This program can handle vertical, horizontal or com-plex wells operated at an imposed rate or bottomhole pressure.The management of the well completion is flexible enough to

20

BUILDING GEOSTATISTICAL MODELS CONSTRAINED BY DYNAMIC DATA - A PRIORI CONSTRAINTS 3

allow [he user to open or close any grid-block to the flow dur-ing the simulation time.

The rate allocation Q,,, for any grid-block b, can be

explicitly specified by the user or automatically controlledthrough the use of a numerical productivity index (NPJ,),which relates the grid-block pressure Ph to the bottomholetlowing pressure P,,(according to:

Q/)/’”, =F’~— —NPIh

................................................(4)

NP/,, can be either specified by the user or computed using thePeaceman ‘q”~”method. For example, the Peaceman method fora vertical well located in a regular rectangular grid system isbased on the following equations :

2nh#rk,NPlh = .,.,,,.,.,.. . ....... .................. (5)

~ln(~)(1

w,,,, = rw x e “$ .................................................(6)

[[tr’~’+($r””rr,, = 0.28 ........... (7)

(;)’’4+[;)”4

where :h is the grid-block thicknessk, and k, are the directional permeabilitiesi-t, is the well radiusS is the well skin& and Ay are the grid-block sizes

Each well can be assigned a wellbore storage Crelates the well rate Q to the sand face rate Q,, appliedreservoir as follows:

whichto the

Q,, =Q+cx> .............................. .... ...... . (8)

where :AP,,, is the pressure drop during At

The simulation program can handle any distributionof imposed pressure or rate boundary conditions.

The input data set of the program is made of severalASCII files that handle the definition of static and dynamicdata as well as the grid files generated by the most populargeostatistical simulation programs. A Windows based userfront end is available on a PC to help the user to input data andto generate the proper ASCII files.

Upon user request several output ASCII or binaryfiles can be generated. Well status reports, pressure maps, aswell as bottomhole flowing pressures and derivatives and ratesare such a kind of output. A Windows based post processor isalso available on a PC to display pressure maps and well data.

Another very powerful capability provided by thenumerical simulation program is the ability to compute andthen to interpret the moments of the pressure variation duringthe simulation history. The first attempt to derive the area ofinvestigation from the first three moments presented by B.Noetinger 2’for heterogeneous reservoirs has been extended toanisotropic ones by G. Blanc e~ al ‘2. Besides the fact that thismatter is beyond the scope of this paper, it must be noted thatthis kind of tool adds to the interpretation of simulation resultsas will be stated in the conclusion for future research.

The program has been extensively tested using a widerange of test cases, which include both analytical problems(limited, multilayer and composite reservoirs), and some morecomplex problems based on geostatistical simulated faciesgrids.

Pressure gradientsThe optimization algorithm needs the pressure gradients withrespect to the parameters to be optimized. There are severalway to determine these gradients.

The first one is the numeric difference method. Thegradient of the bottomhole pressure Pw with respect to theparameter d is the difference between two simulated pressuresPK, and Pw,corresponding to two close values, d, and d,, of theparameter d, with:

ap PWI(4)-pw2(~2) ,,,,, (9)grad(f(d, t)) = G(~,,) = # =

d, - d2

This method is a brute-force method that is difficult tomanage because the quality of the computed gradients dependson the selection of the differentiation interval d,-dz.

Antherion ‘‘ et al. proposed a so called analyticalmethod to compute the gradients which is based on the differ-entiation of Equation (2) and Bissel ‘zapplied it to the problemof reservoir history matching. This method has been extendedby Rahon et al. ‘3 to allow the computation of the gradients ofPwf with respect to the petrophysical properties of the facies(or rock-type) [hat make up the reservoir as well as gradientsof Pwfwith respect to the well skin,

If we consider the facies permeabilities Kp, for exam-ple, the gradient comes from the derivation of Equation (2).The details of the method, as well as several examples of thevalidation, have been proposed for presentation at the 1996SPE Fall meeting. However, the main point lies in the fact thatthe gradient vector is the solution to the matrix system:

An xG; = C; .......................................................(lO)

21


where:A“ is the same 7-band matrix as the one used for pressure

system (3) at time nG;; is the vector of the gradient map with respect to the

parameter d at time nC:; is the vector column of the right hand side of the sys-

tem of discretized equations of the gradient for pa-rameter d at time n. This vector is built using the gra-dient vector at time n-/ and the pressure vector at cur-rent time n

aB” aAn ~nc:;=— —

aK,k - aK,k.................................(11)

Depending on the numerical method selected, part ofthe computer time needed to solve Equation ( 10) can be saveddue to the fact that the A“ matrix is the same as the one used forpressure calculation.

The gradient algorithm has been implemented in thenumerical simulation program, which can then deliver therequired gradient unknowns on the output file together withthe well pressure results.

The list of the parameters for which the bottomholepressure gradients are currently computed is:

● Horizontal K,, and vertical K, permeabilities for anygiven number of facies. The horizontal permeabilitiescan be considered as a constant within each facies orcan be assigned a Gaussian distribution. In the lattercase, the gradients of the mean and of the standarddeviation of each facies distribution are computed.When needed, the software package is able to managea constant K,/K,,, ratio whilst computing the gradientsfor one of these parameters.

● Porosities for any given number of facies;● Well skin;● Layer thickness for any number of groups of grid-

block.Example 2 gives an overview of the usefulness of the

gradients computed for three different facies and for the wellskin to interpret numerical simulation results.

Inversion algorithmsLet us define a least-squares objective function as follows:

fi:2(cf)= + x Wk(p:,, - P;, )2 .......................(12)L k=l,m) ‘

where:L2 is a reference to the least-squares formulationno is the number of data points measuredW, is a weight (greater or equal to O) applied to every

data pointo is the index that refers to observed or reference datac is the index that refers to computed data

The quality of the matched model can be increased byaccounting for all prior geological information. This can beachieved by including a term related to the parameters in theprevious objective function, which now becomes:

1.2 _ : [.2F- Z(qm + F:2) .............................(I3)

with :

F/%f)=:,=~npw,(dfl -d)2 ..,, ............(14)

where:np is the number of parametersw, is a weight (>0) applied to every parameterp is the index that refers to the prior information

If the uncertainties for measured pressures as well asfor the prior model are assumed to be Gaussian, the Ieast-squares objective function has a statistical meaning X throughthe posterior probability density function (pdf), by applyingthe Bayesian formalism:

proh(d)= Cfexexp(–F”2) ...................(15)

The minimum of the least-squares objective functionis related directly to the maximum of the pdf.

Moreover, each parameter can be bounded to staywithin a domain having a physical meaning. To do so, a La-grange multiplier method is applied.

When the gradients of the pressure with respect to theparameters are available, optimization become more efficient.To minimize the objective function, several algorithms havebeen included in the inversion loop :

.

.

.

.

.

The steepest-descent algorithm, is robust far from thesolution but has a slow convergence rate near the so-lution;The Gauss-Newton algorithm, is very efficient forsimple problems and has a quadratic convergence ratenear the solution;The Levenberg-Marquardt algorithm, is an improve-ment of the previous algorithms and provides morerobustness. This is the default one.The Fletcher-Powell algorithm, combines steepest-descent robustness and Gauss-Newton speed near thesolution.The mapping method, allows the user to define a gridfor the parameter space and to compute the objectivefunction at every point on the grid. This method canbe used as a preinterpretation tool to identify the oc-currences of local optimums.

User interfaceTo make the inversion loop easy to use, a user interface hasbeen developed. This interface runs on a PC with Windows(Fig, 2).

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The main purpose of this interface is to help the userinput the data. Three different kinds of data are managed bythe user interface through the menu system, which is describedin Fig. 3:

● The MODEL menu, concerns the following data:+ Description of the grid-block system;+ Facies map (including map editing and import

facilities);+ Well characteristics (type, location and comple-

tion);+ the time discretization parameters.

● The TEST menu, concerns the following data:+ Pressure and derivative (including editing and

import facilities);+ Rate history;--) Static data.

A preinterpretation window (Fig. 8), allowsthe user to perform a unit-slope graphic analysis to getthe wellbore storage as well as a graphic analysis ofthe derivative stabilization to get an average perme-ability within the area of investigation of the test.

● The OPTIMIZATION (or inversion) menu, concernsthe following data and items:

-) Type of execution required, either a new execu-tion or a restart;

+ General data, i.e. the optimization algorithm se-lected and the associated control parameters ifany;

+ The parameters of the inversion together withtheir first guess, the a priori model if any and theconstraints to be applied to these parameters.

The OPTIMIZATION menu also controls the outputof all the ASCII files required by the numerical simu-lation program and by the optimization algorithm.Execution script can also be generated, to launch theinversion loop on a user specified computer (PC, SunHP and IBM workstations, Futjitsu 4 processor main-frame). The RUN item in this menu allows the user tolaunch the execution script and to control the inver-sion loop behavior on the PC screen.Another purpose of the user interface is to display the

results of the inversion loop for each iteration of the optimiza-tion process using the PLAYBACK menu. The reference pres-sure and the computed one are displayed on the same log-loggraph for comparison purpose together with a table of thecurrent values of the parameters. All these graphs can be dis-played one after the other back and forth, and they can also beanimated in an endless loop. All the gradients can also bedisplayed using the same process.

Application casesThree different examples are presented here to illustrate sev-eral applications of the software package:

● The first example is one of the test cases of the soft-ware package, which illustrates the optimization proc-

ess efficiency. Reservoir permeability and well skinare the parameters of the inversion.

● The second one is a buildup simulation run on a pixelbased geostatistical model made of three different fa-cies.

● The last one is a drawdown simulation for an objectmodel of a meandering system.

Example 1- Homogeneous reservoirCase description

Example 1 reservoir model is a monolayer homogeneous strip320 m wide by 3000 m long represented in Fig. 4. The reser-voir thickness is constant over all the model and equal to 18 m.The porosity is equal to 0.3.

The well, which has a radius of 7.85 cm, fully pene-trates the reservoir. It is located as indicated in Fig. 4.

The mathematical model is made of 1738 grid-blocks:● 22 grid-blocks along the X axis;● 79 grid-blocks along the Y axis.

The well-test is a DST made up of 6 flow periodsfollowed by a 48 hour buildup (Fig. 5). The pressure data wererecorded using several quartz gauges.

The reservoir permeability and the well skin are theparameters of the inversion process. The data recorded duringthe buildup period were selected as the reference data for theoptimization (actually only 286 data points were sampled fromthe original recorded data set).

Seven different sets of reservoir permeability and wellskin guesses were used to test the convergence of the optimi-zation process using the Levenberg-Marquardt algorithm.These sets are given in Table 1. The constraints applied to thepermeability and to the skin are also given in Table 1 and areshown on Fig. 6.

ResultsThe convergence of the optimization process is very fast asshown in Table 1. Very few iterations are necessary to reachthe optimal set of parameters corresponding to a permeabilityof 774.9 mD and to a skin of -0.902. The quality of the fit forRun 1, which is illustrated in Fig. 5, corresponds to an averageerror of 1.76x10< bar between the reference data and the com-puted buildup.

The initial guess has no influence on the optimizationresults within the validity domain as shown by Fig. 4. Thelower bound constraint on the skin, which is activated for runs3, 4 and 5, for which the initial permeability guesses are low, iseffectively satisfied by the algorithm and has little influence onthe convergence speed.

Example 2 - Three facies reservoir (geostatistical pixelmodel)

Case descriptionThe reservoir model used for Example 2 is a square singlelayer of 2500 m x 2500 m with a constant thickness of 15 m.

The geological model is made up of three differentfacies and was built using the Sisimpdf geostatistical simula-

23


tion program which is part of the GSLIB library “. This simu-lation program is based on the indicator method applied tocategorical variables. The geostatistical properties of thesefacies are summarized in Table 2 and the corresponding faciesmap is shown in Fig.7.

The porosities of the three facies are equal to 0.3.Their total compressibilities are also constant and equal to5.5x10” I/bar.

The well, which has a radius of 7.85 cm, fully pene-trates the reservoir and has a wellbore storage of 10“2m]/bars.It is located 975 m from the left boundary of the reservoir and1675 m from the bottom one as indicated in Fig. 7. The reser-voir in the well area is made of facies 2.

The mathematical model is made of 3364 grid-blocks:● 58 irregular grid-blocks along the X axis;● 58 irregular grid-blocks along the Y axis.

The well-test is a synthetic drawdown run with thenumerical simulation program. For this numerical simulationthe permeabilities of the three facies and the well skin arereported in Table 2. A constant rate of 100 m3/d was main-tained during 5X10s s ( 138.9 hours). Thus the reference pres-sure data set corresponds to the 90 time steps of the simulation.This reference pressure and its corresponding derivative areshown in Fig. 8.

The permeabilities of the three facies and the wellskin are the parameters of the inversion process.

The initial guess for the permeabilities of the threefacies has been set at 600 mD for all of them and constrainedto be > I and < 2000. This guess roughly corresponds to thepermeability derived from the minimum of the pressure de-rivative after the end of the wellbore storage effect as shownby Fig. 8. The initial guess for the skin effect was set at O andwas constrained to be >-5 and <5.

The Levenberg-Marquardt optimization algorithmwas selected.

Results - Gradients analysisThe results of the gradient computation corresponding to theinitial guess parameter set are represented in Fig. 9.

The plot of the well pressure gradients related to thepermeabilities clearly shows that the permeability of facies 2 isthe one which has the greatest influence on the well pressure,then comes the permeability of facies I and the one of facies 3.This result is not surprising if we remember that the well islocated within an area where facies 2 is predominant. Howeverthis is the kind of information that is very helpful for analyzingthe results of a numerical simulation. The gradients can beused as a diagnosis tool by the reservoir engineer even if aninversion process is not used.

The second part of Fig. 9 is a plot of the well pressuregradient with respect to the well skin. During the period domi-nated by the wellbore storage (i.e. the 500 first seconds of thedrawdown as indicated by the pressure derivative in Fig. 8),the sand face rate grows from O up to the well rate and so doesthe pressure difference due to the skin. This explain why thegradient grows from O up to a constant value.

Results - In verswn processThe convergence of the optimization process is very fast asshown in Table 3 and Fig. 10. Six iterations are necessary toreach the optimal set of parameters corresponding to the per-meabilities of the three facies and to the well skin. The qualityof the fit, which is illustrated in Fig. 10, corresponds to a verylow average error of 1.43xl 04 bar per data between the refer-ence data and the computed buildup. Such a degree of accu-racy can be only obtained when matching synthetic data.

Example 3- Two facies object based multilayer reservoirCase description

The reservoir is a turbiditic sandy channel (facies I ) withwhich a system of sandy levees (facies 2) is associated. Eachlayer is 5 meters thick. The channel geometry is shown in Fig.11 (the layer located at the top of this figure is the higher one,and so on).

The porosities of the two facies are equal to 0.3. Theirtotal compressibilities are also constant and equal to 1.92x104I/bar (Table 4).

The well is located at the center of the coordinatesystem and it has a radius of 7.85 cm, The well partially pene-trates the reservoir and is open only in the top layer. The well-bore storage is 101 m’hr,

The mathematical model is made of 21844 irregulargrid-blocks :

● 127 irregular grid-blocks along the X axis;● 43 irregular grid-blocks along the Y axis;● 4 layers.

The well-test is also a synthetic drawdown ran withthe numerical simulation program as for Example 2. For thisnumerical simulation the permeabilities of the two facies andthe well skin are the ones reported in Table 4. A constant rateof 100 m3/d was maintained during IOCs (277.8 hours). Thusthe reference pressure data set corresponds to the 159 timesteps of the simulation. This reference pressure and its corre-sponding derivative are shown in Fig. 12,

Two inversions were performed with this syntheticdrawdown.

For the first one (Example 3a), the horizontal reser-voir permeabilities of the two facies were the parameters of theinversion process. The anisotropy ratio between Kh and Kuwas constrained, as reported in Table 4.

For the second one (Example 3b), both horizontal andthe vertical reservoir permeabilities of the two facies were theparameters of the inversion process.

The Levenberg-Marquardt optimization algorithmwas selected.

Results - Example 3aThe inversion of the horizontal permeabilities of the two faciesis very accurate as illustrated in Fig. 12 and Table 5. Afterfour iterations the solution is very close to the optimal one.The anisotropy constraint that is satisfied as every iteration,does not alter the convergence of the process.

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Results - Example 3bWhen the anisotropy constraint is released and when the verti-cal permeabilities are included in the list of the parameters, theof the inversion process is still very fast for the horizontalpermeabitlities of the two facies as illustrated in Fig. 13 andTable 6. However the convergence of the vertical parameterinversion is a little bit slower and then the final objective func-tion is much higher than for Example3a.

ConclusionsThe solution to the problem of constraining geostatis-

tical models by well-test results, which has been presentedhere, is very promising as shown by the application examples.Therefore, the resulting software package combined with ananalytical interpretation package of well-tests provides thereservoir engineer with a powerful tool for analyzing complexgeologic representations

Moreover, this approach can be extended forpolyphasic history matching applications 2’.

The research project is now focusing on the inversionof geometric parameters such as the shape of the reservoirmodel as well as the shape of reservoir bodies. New mathe-matical simulation programs using flexible grids and allowingthe accurate simulation of complex wells “ are another re-search axis.

Nomenclature~=

Ax,Ay=

h=~=

NPI=

P=

q=

Q=

r,,=r=

s=t=

T=

v,=w.

(1.

~.

p=

Subscriptsi=‘,=

(]={)=

,Sf=b,,=

total compressibility, Lt’/m, I/barsgrid-block dimension, L,mreservoir thickness, L, mpermeability, L’, mD

numerical productivity index, L4t/mpressure, m/L t’, bar

sink term, L’/twell flow rate, L’/I, m’/d

well radius, L, cm

Pexernan equivalent radius, L. cmwell skin

time, t

transmissivity, L’/t

grid block volume, L’, m’weightporosity, fraction

fluid viscosity, m/Lt, cpfluid density, m/ L’, g/cm3

coordinate indexcomputed data pointparameterobserved data point

sand face

well

AcknowledgmentsThe authors thanks Elf Aquitaine Production and InstitutFrangais du P&role for their permission to publish this paper.All participants to the HELIOS reservoir project are thankedfor their valuable discussions on all aspects of the work. Spe-cial thanks are given to J. Poncet and to his colleagues fromElf Aquitaine Production who tested the software package andsuggested many improvements.

References1.

2.

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16.

Deutsch, C.V, and Journel, A.G.: “Annealing Techniques Ap-plied to the Integration of Geological and Engineering Data,”Report for the Stanford Center for Reservoir Forecasting, Stan-ford U. ( 1992).Ouenes, A.: “Application of Simulated Annealing to ReservoirCharacterization and Petrophysics Inverse Problem,” Ph.D. dis-sertation, The New Mexico Inst. of Mining and Technology,Socor’ro,NM ( 1992).Sagar, R.K. et al.: “Reservoir Description by Integrating WellTests Data and Spatial Statistics ,“ SPERE (December 1995)267.Haas, A. and Ncttinger, B.: “3D Permeability Averaging forStochastic Reservoir Modelling Constrained by Well-Tests,”presented at the Herriot-Watt and Stanford University ReservoirDescription Forum held in Puebbles Hydro (U.K.), Sept. 1995.Jacquard, P.: “Th40rie de I’interpretation des mesures de pres-sions”, Revue de l’/FP, 1961, pp 297-334 (in French).

Jacquard P., Jain C.: “Permeability distribution from field pres-sure data,” SPEJ, Dec. 1965, pp. 281-294.

Jahns, HO,: “A rapid method for obtaining a two dimensionalreservoir description from well response data, ” SPE.J, Dec1966, pp. 315-327,Chavent G., Dupuy M., and Lemonnier P.: “History matchingby use of optimal control theory,” SP,JU, pp. 74-86, Feb. 1975.Randy Hwan, R.: “Well Test Analysis by Reservoir SimulationCoupled with a History Matching Program,” SPE paper 30217presented at the Petroleum Computer Conference held inHouston, Tx, U.S.A., 11-14 June 1995.Anterion, F.: “History matching: a one day long competition:classical approaches versus gradient method, ” First internationalforum on reservoir simulation, Alpbach, Austria, September 12-[6, 1988,Anterion, F. Eymard, R. and Karcher, B.: “Use of parametergradients for reservoir history matching,” SPE 18433, Houston,TX, February 6-8, 1989.Bissel, R.: “Calculating Optimal Parameters For HistoryMatching,” 4th European Conference on the Mathematics of OilRecovery, Roros, Norway, 7-10 June 1994,Feitosa, G. er af.: “Determination of Permeability Distributionfrom Well Test Pressure Data,” JPT (Jul y ,1994) 607.Floris F.J,T., Bos C.F. M.: “Flow Constrained Reservoir Charac-terization using Bayesian Inversion, ” 4th ECMOR, Norway, 7-10 June 1994.Eide A. L., Holden L., Reiso E., Aanonsen S. l,, ‘“AutomaticHistory Matching by use of Response Surfaces and Experimen-tal Design,” 4th ECMOR, Roros, Norway, 7-10 june 1994,Hegstad B. K., Omre H., “The NTH-Approach to HistoryMatching,” preliminary note, Department of Mathematical Sci-

25


ences, Norwegian Institute of Technology, Trondheim, Norway,January 1994

17. Gu4rillot, D. and Roggero, F.: “Matching the future for theevaluation of Extreme Reservoir Development scenarios ,“ Pa- 23.per presented at the 8th European Symposium on Improved OilRecovery, Vienna (Austria), May 15-17, 1995.

18. Blanc, G. et al. : “ Numerical Well-tests Simulation in Hetero-geneous Reservoirs “. Poster presented at AAPG, Nice 95. 24.

19. Peaceman, D.W.:’’Interpretation of Wellblock Pressures inNumerical Reservoir Simulation,” SPEJ (June 1978) 183; 25Trans. AIME, 253.

20. Peaceman, D.W.: “Interpretation of Wellblock Pressures in 26.

Numerical Reservoir Simulation with Nonsquare Gridblockand Anisotropic Permeability,” SP.!U (June 1983) 531

21. Nmtinger B.: “A Pressure Moment Approach for Helping Pres-sure Transient Analysis in Complex Heterogeneous reservoirs, ” 27.

SPE Paper 26466 presented at the SPE Technical Annual Con-ference & Exhibition held in Houston, U.S.A., October 1995,521-529.

22. Blanc, G., Nmtinger B., Piacentino, L. and H41ios ReservoirGroup : “Contribution of the Pressure Moments to the interpre-

tation of Numerical Well-tests,” paper to be published atECMOR V Conference to be held in Leoben (Austria), Sep-tember 1996.Rahon, D., Blanc, G., Gu&illot, D., and H41ios ReservoirGroup: “Gradient Method Constrained by Geological BodiesFor History Matching,” paper 10 be published at ECMOR VConference to be held in Leoben (Austria), September 1996.Tarantola, A.: “Inverse Problem Theory - Methods for DataFitting and Model Parameter Estimation,” Elsevier Ed., 1987.Deutsch, C.V. and Joumel, A.G.: “Geostatistical SoftwareLibrary and User’s Guide”. Oxford University Press, 1992.Roggero, F, and Gu.%illot, D.: “Gradient Method and BayesianFormalism Application to Petrophysical Parameter Characteri-zation,” paper to be published at ECMOR V Conference to beheld in Leoben (Austria), September 1996.Ding, Y.: “A Generalized 3D Well Model for Reservoir Simu-lation,” SPE Paper 30724 presented at the SPE Technical An-nual Conference & Exhibition held in Dallas, U.S.A., 22-25October, 1995.

26


Table 1- Example 1- Results of the inversionprocess for the 7 runs

Run K Skin K Opt, Skin Her. Obj.# guess guess (mD) opt. # function

(mD) (ba~)

1 5MJ o 774.9 -0.902 4 0.1562 500 1 774.9 -0.902 4 0.1563 200 0 774.9 -0.902 7 0.1564 200 -2 774.9 -0.902 6 0.1565 300 3 774.9 -0903 6 0.1566 900 4 774.9 -0,902 6 0.1567 1100 -1.5 774.9 -0,902 4 0.156

Remarks:1- The validity domain for permeability K is: 10< K <2006

The validity domain for the skin is :-2<S<5P- For the computed optimum, average error between refer-

mce and calculated data sets is 0.0176 bar (mean of abso-ute value of differences)

Table 2- Example 2- Properties of the faciesmodel (for synthetic drawdown)

Facies # Proportion Seed Angle Permeability(fraction) (m) (0) (mD)

1 .750 100 0 302 ,125 100 -45 7003 .125 100 45 300

Remarka:1- Porosity of the three facies is 0.32- Total compressibility of the three facies is 10’ lhar3- Well skin is -1.5

Table 3- Example 2- Results of the inversion

Her. Permeability Skin Objective# (mD) Function

Facies 1 Facies 2 Facies 3 (ba#)

o 600.00 600.00 600.00 0.000 4,05E+OI1 1.06 703.09 1.16 -1.450 2,86E+032 86.88 611.61 92,58 -0.300 3.99E+O03 44.67 671.84 73.36 -1.317 6.43E-014 29.26 728.27 216,02 -1.331 1.19E-015 29.97 701,53 293,14 -1.494 2.92E-036 30.00 699.74 300,00 -1.502 1,82E-06

Remarka :For the computed optimum, the average error between refer-ence and calculated data sets is 1.43 10“ bar (mean of absc-Iute value of differences)

Table 4- Example 3- Properties of the faciesmodel (for synthetic drawdown)

Facies # Type Permeability (mD)Horizontal Vertical Anisotropy

Kv/Kh

1 Channel 500.00 50 1/102 Levee 100.00 1 1/100

Remarks:I - Porosity of the two facies is 0.3? - Total compressibility of the two facies is 1.9210’ 1/bar? - Well skin is -1.5

Table 5- Example 3a - Results of the inversion

Iter. # Permeability Objective(mD) Function

Facies 1 Facias 2 (bat)

o 10.00 900.00 1.07E+011 27.58 506.67 5.70E-012 177.69 514.89 5.65E-013 95.26 491.14 3.64E-024 100.11 500.64 1.33E-045 99.99 499.98 1.05E-076 100.00 500.00 1.71 E-09

?emarks:‘or the computed optimum, the average error between refer-mce and calculated data sets is 2.25 10“ bar (mean of abso-ute value of differences)

Table 6- Example 3b - Results of the inversion

Iter. # Permeability(mD)

Facies 1 Facies 2

Kh

o12345678

40.0088.2095.6789.1088.7289.5991.4196.61

103.56

Kv

0.4002.6141.0142.5722.6122.6022.5012.1901.741

Kh

400.00465.47501.84466.99466,17467.41470.68481.28497.09

Kv

FunctionObjective

(ba#)

40.00657,29

1,0091.93

364,31179.4543.5710.656.50

8.37E+O08.09E-022.81 E-018.23E-027.48E-026.60E-025.99E-023.16E-026.13E-05

?emarks:‘or the computed optimum, the average error between refer-mce and calculated data sets is &36 10’ bar (mean of abso-ute value of differences)

27


/ AnelytlcalWELL \

-%’’”’”F

INVER910N

OU*8S

“d

T!r

~ SIMTESTSlmulatlon

I.WI,:%WMI,& gradients

L+!+(E)Figure I - General structure of the eofhvare package - Inveralon loop

IlloOk..ll-o mlPRESSURE

I

Figure 2- User interface

28


File menuNew project

Open ProjectSave ProjectSave asClose ProjectPreferences

klodel menuGeneral Data.Numerical Data.Map Data

General Data.Map Primary AssignmentDx-Dy Data...Facies Map.Thickness Map.Kb...Kv.Porosity. .

Well Data...Time Data.Refined Grid.

rest menuPressure DataRate Data.Display well historyStatic data.Interpretation

-130

+150010

well

1500

E001~m

001

Fig. 4- Examplel Reservoir

[nvarsion menuChange modeGeneral Data. .Inversion Data

Kb...Kv...Porosity.Skin. .

Restart file...Restart Period. .Restart Data...Launch

Preferences. ..SIMTESTW

Save SIMTESTW Data. ..Save OPTIM Data. .Create Shell

SUN Unix Shell. ..IBM Unix Shell. ..HP Unix Shell ...FUJI Unix Shell. ..PC Dos Shell. .

ATHOSSave ATHOS Data...Save GRADIENT Data. ..Save OPTIM Data.Create Shell

SUN Unix Shell. ..IBM Unix Shell. ..HP Unix Shell ...FUJI Unix Shell. ..PC Dos Shell. ..

Run...

Figure 3-Userlnlerface -Menu system

1x Recorded pressu[e

—Computed pressure

x

x4m -

X

/ ‘ .

~lm-0

0

I I 0 23 40 m m lmTime (h)

o ml 001 01Buildup Time [h)’

10 lm

Figure 5- Exsmple l- Computed versus recorded dsts (Runl)

29


5(

3C

.s%

10

-10

-30

0

....... ... ....... . ..... . . . . . . .

—Run 1(500,0)

—Rum 2 (500,1)

—Run 3 (200,0}

_Run4 [200-21

—Run 5 (300,3)

—Run 6 (900,4)

—Run 7 (1100,-1 5)

Constmned area

500Pmrmblltty (mD]

IO(II 15m

Fig. 6- Example 1- Convergence of the inversion process

w. Erch”*** $uh.*

Pr*ssursmemanwlax)

,//

,,//’

/’//’

///’

/’/’

/’~.

I F======l

Figure 7- Exsmple 2- Fscies model Figure 8- Exsmple 2 - Oulck interpretation of the pressure/derlvstlve reference dsts


Initial gradientsof faclee parrssaabllltba

o OE+OO - .-.> ----- ----------

-1 OE-03 .

:

P% — Facies 1

.2 OE.03 — — Facies 2 —

--- Facies 3

.3.oE-03

0,25

0.20

015

0.05

0.00

Initiel gradientof akin

1 10 100 1coo loOco 103000 1OQo+m 1 10 100 1030 1Cooo

Tlnw [s) - {.1

Figure 9- Example 2- Gredlente at Initial guess

Inltld gums Itwalien n6r*tlen M

(mm}I I ‘mm~

m-n m n- [.)

Irdrstlul # 3 FM Irerallm

‘“”~ ‘“m~

~“m EEl.~‘“

i: ,, ,<,

I

100000 1Ooooco

,t~ ,.,!,, ,m ,m ,m ,m !- !0 !m ,m !m ,- ,.

n-m w= (9}

Inversion Loop - Evolulion of the Bottomhole Pressure and Bottomhole Pressure Derivative during the Iteration process

tE“!, , 4!,

,“... .

w.,

as

,,

,,Km

Inversion Loop - Evolution of the Paramatars and of the Norm during the ttaration procass

Figure 10- Example 2- Results of the optimization

31


I~ . .. .. . .. ..... . ... . . . . ... . . .. ... .. . . . ... ......... ...._- . .....-..-.. -.-—_-__.-—.....-.-. . . . . . . . . . . . . ....

1

I

Figure 11- Example 3- Facies Map of the 4 layers

Initial guess iterationloom

a

?2

; ~j~.

g

1 10 lIM Icm Im llx.xa 1&

Tk-m (s)

mRrinw#A PEmEAmJrEs (dJ)

‘m~‘ml,/’”’-’”-’---

‘:U0 1 2 3 4 5 6

Mmum #

Final Iteration

1 10 IIm lox !Om lm ICfm’m

Tim (s)

NORM(ha?)1 DE+r3?

IG-E+CO

1 OE.03

1 OE-M

10ECJ90 ! 2

W&n # 45 6

Figure 12- Exemple 3a - Results of the optimization

32


Initial guess iteration

=====”

~e low.

3,=— -Computed derIval Ibe

f IC13.

a

ta

8010.

2a m~

1 10 Ial 1C03 lixm lmxol Iccmm

Tim. (s)

R3M3MJw (I@)

IQ0

10. /’/

0! 234 5670

lk8mc#19

final iteration1C93COT

1000

II+ R8ierenca derium.e

—Computed pressure

0014 4

1 10 $03 lCOI lam) Imxa 11XCD33

1 *41

! WG

1 C&c+

Tinn (s)

NORM(ha?)

01234 567

nerdm s

Figure 13- Example 3b - Resulte of the optimization

33

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