geostatistical history matching under training-image based...

SPE 77429

Geostatistical History Matching Under Training-Image Based Geological ModelConstraintsJef Caers, SPE, Stanford University

Copyright 2002, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the SPE Annual Technical Conference andExhibition, held in San Antonio, Texas, 29 September - 2 October

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AbstractHistory matching forms an integral part of the reservoir model-ing work-flow process. Despite the existence of many historymatching tools, the integration of production data with seismicand geological continuity data remains a challenge. Geosta-tistical tools exists for integrating large scale seismic and finescale well/core data. A general framework for integrating pro-duction data with diverse types of geological/structural data islargely lacking. In this paper we develop a new method for his-tory matching that can account for production data constraintby prior geological data, such as the presence of channels, frac-tures or shale lenses. With multiple-point (mp) geostatisticsprior information about geological patterns is carried by train-ing images from which geological structures are borrowed thenanchored to the subsurface data. A simple Markov chain iter-atively modifies the mp geostatistical realizations until historymatch. The method is simple and general in the sense that theprocedure can be applied to any type of geological environmentwithout requiring a modification of the algorithm.

IntroductionProduction data brings an important, yet indirect constraint tothe spatial distribution of reservoir variables. Pressure data pro-vides information on the average pore volume and permeabilityconnectivity near wells, while fractional flow data informs theextent of permeability connectivity between wells. Productiondata rarely suffice however to characterize heterogeneous reser-voirs, a large amount of uncertainty still remains after historymatching of geostatistical models [1].

History matching is an ill-posed inverse problem attempt-ing to invert reservoir properties from measured flow and pres-sure data. Solutions to such inverse problems are rarely uniquewhich allows imparting other sources of data such as providedby seismic surveys and geological interpretation. The non-uniqueness of the history matching problem is well-known andvarious techniques have been developed that allow integratingproduction data with geological continuity information in finescale geostatistical models2;3;4;5;6;7. Most of these prior geo-logical models reproduce only the covariance as a measure ofgeological continuity. Covariance models are rarely sufficientto depict patterns of geological continuity consisting of stronglyconnected, curvi-linear geological objects such as channels orfractures, see for example [8] and [9]. Ideally one would liketo possess a single history matching algorithm that can handlediverse type of geological structures.

We propose a pixel-based history matching method that canaccount for a large variety of styles of geological continuity, notnecessarily limited to the two-point statistics of a variogrammodel, or to simplistic Boolean shapes. For that purpose, weborrow ideas from the area of multiple-point (mp) geostatis-tics. mp-Geostatistics relies on the concept of a training image.The training image quantifies, explicitly, patterns of geologicalheterogeneity relevant for the subsurface reservoir. A fast se-quential simulation algorithm, termed snesim (single normalequation simulation), has been developed that borrows thosepatterns from the training image and anchors them to local sub-surface data. Next, a simple one-parameter Markov chain pro-cess to changes the mp realizations until a history match. Thetransition matrix of this Markov chain is parameterized by a sin-gle parameter and modifies gradually and iteratively an initialgeological consistent geostatistical realization to match betterthe production data. The Markov chain is implemented suchthat the final model honors the imposed training-image basedgeological structure. We first review some important conceptsin mp geostatistics that allows defining a large variety of priorgeological models, then develop the proposed history matchingmethodology.

2 GEOSTATISTICAL HISTORY MATCHING USING TRAINING IMAGES SPE 77429

Multiple-point geostatistics

Borrowing structures from training images

The snesim algorithm Traditional to geostatistics, geolog-ical continuity is captured through a variogram. A variogrammeasures the degree of correlation/connectivity or converselyvariability between any two locations in space. Since the vari-ogram is only a two-point statistics, it cannot model curvi-linearstructures such as channels, nor can it model strong contigu-ous patterns of connectivities such as fractures. The represen-tation of such complex geological features requires multiple-point statistics, involving jointly more than two locations. Theidea behind multiple-point geostatistics is to infer spatial pat-terns using many spatial locations of a given geometric templatescanning a training image or reservoir analog8;9.

The corresponding algorithm, termed snesim, is proposedin [9,10]. It is essentially not different from existing more tra-ditional conditional simulation techniques [11,12], in that it se-quentially generates the numerical model, one grid cell afteranother. The difference comes from the probability distribu-tions from which these pixel values are drawn: in snesim theseprobabilities are actual proportions inferred from the trainingimage and made conditional to an mp data event. In traditionalsequential simulation these probabilities are derived by krigingusing a variogram model.

Sequential simulation then allows to generate a number ofequiprobable realizations which reproduce the training imagepattern of continuity and honor local well-log and seismic data.The snesim approach essentially replaces the variogram mod-eling by the construction of a training image. A training im-age would typically be constructed using an unconstrained 3DBoolean simulation, see Fig. 1.

At each node of the simulation grid, denote by P (AjB)the probability model from which the value at that grid cell isdrawn, where A could be the event ”channel present” at a givengrid cell location and B is the set of sample data and previouslysimulated grid cells used to constrainA. In the sequantial Gaus-sian simulation (sGs), P (AjB) is a Gaussian distribution withmean and variance determined by a set of (variogram-based)kriging equations. The snesim algorithm follows the sameprinciple of sequential simulation, but the probability modelP (AjB) is read from the training image rather than built bykriging from the variogram model. The snesim algorithm thenallows generating the patterns found on the training image (see[9] for details).

The various proportions P (AjB) are retrieved from thetraining image and stored in a dynamic search tree prior to start-ing the random path [9]. An example of the snesim methodol-ogy, using the training image of Fig. 1 is presented in Fig. 2.Note that the training image model need not have the same sizeas the actual zone being simulated.

Constraining to soft data The snesim algorithm allows forthe integration of secondary information. For example, seismicinversion procedures allow quantifying from amplitude datathe probability of presence of specific facies, see for example[13,18]. This probabilistic inversion result must then be inte-

grated with finer scale well data and geological prior models asdepicted by the training image. Using a notation similar to thatabove we denote the probability model derived from secondarydata as P (AjC), where A is the unknown property at each gridnode and C is the secondary data event observed in the neigh-borhood of that node, typically a window of seismic amplitudevalues [18].

In order to integrate that secondary information C into thesnesim algorithm we need to draw from the conditional distri-butionP (AjB;C) instead ofP (AjB), i.e. each simulated valueshould also depend on the secondary data. To combineP (AjB)and P (AjC) into P (AjB;C), we use the following expressionbased on an improved form of conditional independence 14

x

b=

c

a(1)

where

x =1� P (AjB;C)

P (AjB;C)

and

b =1� P (AjB)

P (AjB); c =

1� P (AjC)

P (AjC); a =

1� P (A)

P (A);

P (A) is the global proportion of A occurring, hence a can beinterpreted as a prior distance to the event A occurring, priorto knowing the information carried by the event B or C. In-deed if P (A) = 1 then the distance a = 0 and A is certain tooccur. Likewise, the values b and c state the uncertainty aboutoccurrence of A, given information B and C respectively. x isthe uncertainty when knowing both B and C. The combinedprobability P (AjB;C) is derived as follows

P (AjB;C) =1

1 + x=

a

a+ bc

Based on this expression, an algorithm termed cosnesim hasbeen developed (see [9,13,18]) which allows generating mod-els constrained to both the geological structure depicted by thetraining image (information B) and the secondary data C.

Solving the inverse problemMethodology We will consider only the case of a binary spa-tial variable described by an indicator random function model

I(u) =

�1 if a given facies occurs at u0 else

where u = (x; y; z) 2 Reservoir, is the spatial location of a gridcell. In a reservoir context i(u) = 1 could mean that channeloccurs at location u, while i(u) = 0 indicates non-channel oc-currence. In the mp geostatistics context, we denote by A theevent I(u) = 1 (”the event occurs”) and use D for the produc-tion data.

Next, define a non-stationary Markov chain on the entireset of random variables I(u);8u, starting from an initial modeli(o)(u), generating iterations i(l)(u) till convergence. Conver-gence is defined as matching the data D up to a given precision�. To define such Markov chain, consider the single random

SPE 77429 JEF CAERS 3

variable I(u) at a specific location u. Since I(u) is binary,define the four transition probabilities of a 2� 2 non-stationarytransition matrix, moving the chain from state (l) to state (l+1)at location u. The chain is parameterized by a single parameterrD , where rD 2 [0; 1] and depends on the data D as follows

PrfI(l+1)(u) = 1jD; i(l)(u) = 0g = rD PrfI(u) = 1g)(2)

PrfI(l+1)(u) = 0jD; i(l)(u) = 1g = rD PrfI(u) = 0g)(3)

PrfI(l+1)(u) = 1jD; i(l)(u) = 1g = 1� rD PrfI(u) = 0g

and for closure

PrfI(l+1)(u) = 0jD; i(l)(u) = 0g = 1� rD PrfI(u) = 1g

The first two transition probabilities (2) and (3) are the proba-bilities of changing states (facies) from step (l) to step (l + 1);rD is the relative probability of such change of state, relativeto the prior P (A). rD is a taken as a function of the condi-tioning data D. The degree of freedom rD allows moving themodel to matching closer the data D. At each iteration, a one-dimensional optimization is carried out to find the value r

opt

D

that matches best the data D. For any given rD 2 [0; 1] at anygiven current iteration (l) and for all grid cells u the conditionalprobability P (AjD) is obtained using the above transition ma-trix as

P (AjD) = rDP (A) if i(l)(u) = 0 (4)

1� P (AjD) = rD(1� P (A)) if i(l)(u) = 1 (5)

rD is the same for all grid cells u. The resulting probabil-ity P (AjD) is then combined with thee training image-derivedprobabilityP (AjB) using Eq. (1) whereP (AjC) is replaced by

P (AjD). Note that the resulting realization i(l+1)rD (u), honors

the prior geological continuity as depicted by the training im-age. Using a simulator of choice, the production data is forwardevaluated on i

(l+1)rD (u) and an optimal roptD can be selected us-

ing any simple one-dimensional optimization method (e.g. theDekker-Brent method, see [17]). This Markov chain is termed”non-stationary” since rD changes at each iteration (l).

Two limit cases exist for rD:

� rD = 0: then the probability for a change is according toEq. (2)

PfI(l+1)(u) = 1� ijD; i(l)(u) = ig = 0; 8i = 0; 1

that is I(l+1)(u) = I(l)(u). Iteration (l+1) does not pro-vide a better match to the data D, hence, one can eitherstop the iteration or change the random seed s. The latteramounts to starting the chain/iteration over with i(l)(u)as the initial guess.

� rD = 1: then

PfI(l+1)(u) = 1jD; i(l)(u) = 0g = P (A)

PfI(l+1)(u) = 0jD; i(l)(u) = 1g = 1� P (A)

hence the state is changed according to the prior proba-bility of the new state. The combined probability thenbecomes

P (AjB;D) = P (AjB)

Consequently the cosnesim algorithm will generate a newrealization i(l+1)(u), drawn independently of i(l)(u)

Fig. 3 shows an example of five models i(1)rD (u) generatedwith different values rD , starting from the initial model i(o)(u)shown in the top left corner. Note that the bottom right modeli(1)rD=1

(u) appears to be completely independent of i (o)(u).

Another interpretation of rD Eqs. (4) and (5) can be recom-bined into a single equation providing another interpretation ofrD

P (AjD) = (1� rD)i(u) + rDP (A) 2 [0; 1] (6)

The probability P (AjD) appears as a mixture of the current re-alization i(u) and the net-to-gross prior proportion P (A), thatmixture being controlled by the optimization parameter rD

� if rD = 1, P (AjD) = P (A) and P (AjB;D) =P (AjB), an independent new realization is generated

� if rD = 0, the current realization is retained as shownpreviously.

Algorithm summary The proposed algorithm to integrateproduction data (D) and mp geological information (B) pro-ceeds as follows

� Define a training image depicting the desired geologicalcontinuity (information B)

� Using the snesim algorithm: generate an initial modeli(o)(u);u 2 Res:, l = 0.

� Iterate, l = 1; : : : ; Lmax,

– Define a transition matrix Eqs. (2) and the proba-bility P (AjD).

– Perform a one-parameter optimization on rD thatprovides the best match to the data D, The latter isdone by finding the realization i(l)rD(u) that matched

best D. i(l)rD (u) is generated using the cosnesim al-gorithm based on Eq. (1).

– Make a final run of the cosnesim algorithm to gen-erate a model i(l+1)rD (u)


Application examples

Quarter 5-spot A set of applictation examples illustrate theproposed approach. Consider the 2D horizontal referencemodel in Fig. 4 consisting of diagonal elliptical bodies of highpermeability (750 mD) in a low permeability matrix (150 mD).We assume that the values of 150/750 mD are known, whilethe placement of the high permeability bodies is not known. Atraining image reflecting knowledge about the elliptical shapesis generated using a Boolean program ellipsim [15], see Fig. 1.

The production data D is generated by placing an injectorwell in the lower left corner which injects water in an initiallyoil saturated reservoir and a producer located in the upper rightcorner. We use a simple black oil model, unbalanced produc-tion with the injector at constant rate of 700 STB/day. No flowboundary conditions are assumed. Fluid properties and den-sity are assumed invariant with pressure. Capillary effects areignored. Connate water saturation is 0.15. Typical relative per-meability curves are used. Initial reservoir pressure is set at655 psi. Grid cells size is 10 ft. A finite difference simulator”Eclipse”16 is used. The target production data D is the frac-tional flow of water observed in the producing well as functionof time. It appears that water breaks through after about 15days as shown in Fig. (5). The task is to generate solutions thathonor the production data and the elliptical structures depictedby the training image of Fig. 1.

After 9 iterations a satisfactory match to the production datais found as shown in Fig. 5. While the initial model fails tocapture the connectivity of flow facies between injector and pro-ducer well, the final history match appears visually to have sim-ilar connectivity between injector and producer as the referencemodel.

To investigate the flexibility of the approach in terms ofprior models, we apply our approach to other types of geolog-ical heterogeneities. First consider the reference model in Fig.7 depicting a population of fractures. Production data D simi-lar to those obtained for the elliptical bodies case is generatedby forward simulation on the reference set, providing the frac-tional flow of water versus time measurements shown in Fig.8. Convergence to an acceptable history match is obtained af-ter only 3 iterations. The initial guess plus the 3 iterations areshown in Fig. 9 and the history match shown in Fig. 8. Thetraining image used to obtain these results is shown in Fig. 10.

A final example concerns a reference model containingchannels as shown in Fig. 11. The results are show in Figs.12 to 14.

Unknown facies permeability In previous cases, the perme-ability within facies is supposed to be known. We proposea simple hierarchical approach that alleviates this assumption.The following 2-step procedure is applied iteratively

step 1: Assume a certain permeability for each facies, and per-turb the facies geometry until no better match can beachieved

step 2: Freeze the facies geometry and perturb the permeabilitywithin each facies jointly.

The first step can be achieved with the method proposed in thispaper, the second step can be achieved with more traditionalhistory mactching technique, based on gradients. Figure 18shows the performance of this approach on the simple 2D case.Convergence is achieved after 2 iterations, each consisting ofstep 1 and 2.

5-spot case Finally, we present a larger case of a 100 � 100reference model shown in Fig. 15. Boundary conditions, gridand fluid specifications are the same as before, only now a full5-spot configuration is used. The permeability constrast is nowhigher with 1500mD for facies 1 and 50 mD for facies 0. TheInjector is located in the middle, producing well 1 in bottomleft, well 2 in bottom right, well 3 in top left, well 4 in top right.

Fig. 15 shows a good connection between the injector andproducing wells 1 and 4, a poor connection to producing wells2 and 3, exhibiting later breakthrough. To match the fractionalflow of all 4 wells jointly, our objective function measures thesquared difference between fractional flow at all 4 wells. Ahistory match is achieved after 8 iterations. The initial model,history matched model and some selected iterations are shownin Fig. 16. While the initial model appears to lack the strongconnectivity between injector and producing well 1, the finalhistory matched results shows a connectivity similar to the ref-erence model. Fig. 17 show that the history matched is largelysatisfactory for all wells.

Local perturbation The above algorithm achieves a globalperturbation of the facies model, by which it is understood thatthe model is changed, in probability, the same everywhere. Thismight not be desirable when matching a large area with a sig-nificant amount of wells. In such case, one would require theflexibility to locally perturb the model. The above algorithmallows this flexibility by making the perturbation parameter rDspatially varying. This can be achieved as follows

rD(u) = rD � �(u)

where �(u) is any arbitrary function 2 [0; 1], for example itcould be function of the distance to a well, or function of flowparameters, such as streamline densities or pressure gradients.The probability model now becomes

P (AjD) = i(u)(1� rD(u)) + rD(u)P (A)

In a demonstrative example, we imply take for �(u) a measureof proximity to the axis between injector and producer. This isillustrated for a simple two-well case in Fig. 19. An alterna-tive would be to determine the streamline-density by applyinga streamline simulator. A high streamlines density indicates aregion important to flow, hence the permeability in that regionhas a considerable impact in the flow response measured in theproducing well. Such an approach is elaborated in [19] and isbeyond the scope of this paper.

We history match the fractional flow observed in the pro-ducing well of Fig. 19 using a spatially varying rD(u) where�(u) is given by Fig. 19. As shown, a maximum perturbation isachieved on the axis between the injecting and producing well,gradually decreasing to zero away from this axis. No perturba-tion is done in the white region of Fig. 19. The reference model


is the same as in the case of the 5-spot, see Fig. 15. As shownin Fig. 19 a satisfactory match is achieved after 7 iteration. Fig.20 shows the initial model and some selected iterations. Fig.20 shows that geological continuity is well reproduced, withoutdiscontinuities, and that the initial model is only perturb signif-icantly between the two wells.

Nomenclatureu: (x; y; z)A: an event: for example channel occurs at uB: another event: for example channel occurs at u1 and shaleoccurs at u2P (AjB): probabibility that event A occurs given one knowsthat B occursD: production dataC: seismic data(l): iteration counter for outer loop

References[1] WEN, X-H., DEUTSCH, C. and CULLICK, A.S., Integrat-ing pressure and fractional flow data in reservoir modeling withfast streamline based inverse methods, paper SPE 48971 pre-pared for presentation at the 1998 SPE Annual Technical Con-ference and Exhibition, New Orleans, 27-30 September.[2] VASCO, D.W., YOON, S. and DATTA-GUPTA, A., In-tegrating dynamic data into high resolution reservoir modelsusing streamline-based analytic sensitivity coefficients, paperSPE 49002 prepared for presentation at the 1998 SPE An-nual Technical Conference and Exhibition, New Orleans, 27-30September.[3] HU, L.Y and BLANC, G., Constraining a reservoir faciesmodel to dynamic data using a gradual deformation model. pa-per B-01 prepared for presentation at the 6th European Confer-ence on Mathematics of Oil Recovery (ECMOR VI), Septem-ber 8–11, Peebles, Scotland, 1998.[4] WU, Z, REYNOLDS, A.C. and OLIVER, D.S, Condition-ing geostatistical models to two-phase production data. pa-per SPE 49003 prepared for presentation at the 1998 SPE An-nual Technical Conference and Exhibition, New Orleans, 27-30September.[5] TRAN, T., DEUTSCH, C.V. and XIE, Y., Direct Geosta-tistical Simulation With Multiscale Well, Seismic, and Produc-tion Data, paper SPE 71323 prepared for presentation at the2001 SPE Annual Technical Conference and Exhibition, NewOrleans, September 30 – 3, October.[6] HEGSTAD, B.K., and OMRE, H., An Inverse Problemin Petroleum Recovery: History Matching and Stochastic

Reservoir Characterisation, proceedings ECMI96 conference,Copenhagen, June 25-29, 1996.[7] CAERS, J., KRISHNAN, S., WANG, Y and KOVSCEK,A.R., A geostatistical approach to streamline-based historymatching. Paper submitted 2002.[8] CAERS, J. and JOURNEL, A.G., Stochastic reservoir mod-eling using neural networks trained on outcrop data. paper SPE49026 prepared for presentation at the 1998 SPE Annual Tech-nical Conference and Exhibition, New Orleans, 27-30 Septem-ber.[9] STREBELLE, S. Sequential simulation drawing structurefrom training images. Ph.D dissertation, Stanford University,Stanford, California, 2000.[10] CAERS, J. AND ZHANG, T. Multiple-point geostatistics:a quantitative vehicle for integrating geologic analogs into mul-tiple reservoir models. To be published in ”Integration of out-crop and modern analogs in reservoir modeling, AAPG mem-oir, 2002.[11] ISAAKS, E. The application of Monter Carlo methodsto teh analysis of spatially correlated data. PhD dissertation,STanford University, Stanford, California, 1990.[12] GOMEZ-HERNANDEZ, J., and SRIVASTAVA, S.ISIM3D: an ANSI-C three dimensional multiple indicator con-ditional simulation program. Computers and Geosscience 16,395–410, May 1990.[13] CAERS, J., AVSETH, P. and MUKERJI, T. Geostatisticalintegration of rock physics, seismic amplitudes and geologicalmodels in North-Sea turbidite systems The Leading Edge, 20,308-312 , March 2001.[14] JOURNEL, A.G. Combining knowledge from diverse in-formation sources: an alternative to Bayesian analysis. Mathe-matical Geology, 34, no 5, 2002.[15] DEUTSCH, C.V. and JOURNEL, A.G., GSLIB: Geosta-tistical Software Library and User’s Guide. Oxford Universitypress, 1998.[16] ECLIPSE 100 Reference Manual, Schlumberger-Geoquest, Houston, Texas, 1998.[17] PRESS, W.H., TEUKOLSKY, S.A, VETTERLING, W.Tand FLANNERY, B.P. Numerical recipes in C. Cambridge Uni-versity Press, 1992.[18] STREBELLE, S., PAYRAZYAN,K. and CAERS, J. Mod-eling of a deepwater turbidite reservoir conditional to seismicdata using multiple-point geostatistics. paper SPE 77425 pre-pared for presentation at the 2002 SPE Annual Technical Con-ference and Exhibition, New Orleans, Sept. 30 – Oct. 3.[19] CAERS, J. Methods for history matching under geologi-cal constraints In Proceeding to ECMOR VIII, European Con-ference on Mathematics of Oil Recovery, Freiberg, Germany,Sept. 3-6, 2002.


training image

East

Nor

th

0.0 150.0000.0

150.000

facies 0

facies 1

Figure 1: Example of a training image containing elliptical patterns

Single realization witrh snesim

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1.000

Well data

0.0 10.0 20.0 30.0 40.0 50.0

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Figure 2: Example of a single geostatistical realization constrained to the 8 well data on the right. Patterns are borrowed from thetraining image of Figure 1


initial model

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Figure 3: A single parameter rD defines the transition from an initial realization (top left) to another independent realization (rD = 1,bottom right)

Reference model

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Figure 4: Reference model


0 5 10 15 20 25 30 35 400

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timesteps (days)fw

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Figure 5: History match to the fraction flow data fw.

Initial model

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Figure 6: Some selected steps during the iterative inversion.


Reference model

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Figure 7: Reference fracture model.

0 5 10 15 20 25 30 35 400

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fw

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Figure 8: Data and history match results.


Initial model

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Figure 9: Initial models and iterations to reach a history match.

Training image

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Figure 10: Training image for the fracture model


Reference model

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Figure 11: Reference channel model.

Initial model

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th

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 6

East

Nor

th

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 7

East

Nor

th

0.0 50.0000.0

50.000

facies 0

facies 1

Figure 12: Initial model and selected iterations


0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

timesteps (days)

fw

referencematchinit

Figure 13: Data and history matched results.

Training image

East

Nor

th

0.0 250.0000.0

250.000

facies 0

facies 1

Figure 14: Training image for the channel model


Reference model

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Figure 15: Reference model for a five spot case: injector located in the middle, well 1 in bottom left, well 2 in bottom right, well 3 intop left, well 4 in top right

Initial model

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 2

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 4

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 6

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 7

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 8

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Figure 16: Initial model, history match and some selected iterations


0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

timesteps (days)

fw

Well 1

referencematchinit

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

timesteps (days)

fw

Well 2

referencematchinit

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

timesteps (days)

fw

Well 3

referencematchinit

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

timesteps (days)

fw

Well 4

referencematchinit

Figure 17: History matched results for all 4 wells


Figure 18: Hierarchical matching, in step 1, only facies are perturbed, while in step one facies geometry is frozen and only permeabilityvalues are perturbed. It takes two iteration till convergence.


0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

timesteps

fw

refmatchinit

Figure 19: (right) �(u) (left) Results

Initial

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 1

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 6

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Iteration 7, final

East

Nor

th

0.0 100.0000.0

100.000

facies 0

facies 1

Figure 20: Initial model and some selected iterations

geostatistical history matching under training-image based...

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