Copyright 2000, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 2000 SPE/DOE Improved Oil RecoverySymposium held in Tulsa, Oklahoma, 3–5 April 2000.

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AbstractThis study proposes and develops a streamline approach forinferring field-scale effective permeability distributions basedon dynamic production data including producer water-cutcurve, well pressures, and rates. The approach simplifies thehistory-matching process significantly.

The basic idea is to relate the fractional-flow curve ata producer to the water breakthrough of individualstreamlines. By adjusting the effective permeability alongstreamlines, the breakthrough time of each streamline is foundthat reproduces the reference producer fractional-flow curve.Then the permeability modification along each streamline ismapped onto cells of the simulation grid. Modifying effectivepermeability at the streamline level greatly reduces the size ofthe inverse problem compared to modifications at the grid-block level. The approach outlined here is relatively directand greatly reduces the computational work by eliminating therepeated inversion of a system of equations. It works well forreservoirs where heterogeneity determines flow patterns.Example cases illustrate computational efficiency, generality,and robustness of the proposed procedure. Advantages andlimitations of this work, and the scope of future study, are alsodiscussed.

IntroductionHistory-matching plays an important role in monitoring theprogress of displacement processes and predicting futurereservoir performance. Historical production data is routinelycollected and it carries much information, althoughconvoluted, that is useful for reservoir characterization anddescription of reservoir heterogeneity1,2. In this paper, theconcept of streamlines is applied to develop an automatic

method for inferring the permeability distribution of areservoir based on the history of pressure, flow rate, and watercut at producers.

The properties of streamlines are used in deriving theinverse method, and so a brief review of streamlinemethodology follows. Streamline and streamtube techniquesare approximate reservoir simulation methods proposed someyears ago3-6 that have undergone recent intense study7-9. Theyare most accurate when heterogeneity determines flow pathsand the recovery process is dominated by displacement(viscous forces) as opposed to gravity or capillarity10. Astreamline is tangent everywhere to the instantaneous fluidvelocity field and, for a symmetric permeability tensor,streamlines are perpendicular to isobars or iso-potential lines.Streamlines bound streamtubes that carry fixed volumetricflux, and in some cases, flow rate is assigned tostreamlines8,11. For this reason, we use the terms streamlineand streamtube interchangeably.

Displacement along any streamline follows a one-dimensional solution, and there is no communication amongstreamlines. Thus, the flow problem is decomposed into a setof one-dimensional flow simulations linked by commonboundary conditions. A streamline must start and end at asource to maintain continuity. In a streamline-based approach,pressure equations are solved independent of saturationequations. The decoupling of pressure equations fromsaturation equations speeds up significantly the simulation byreducing the number of times that the pressure field must beupdated and greatly reduces the number of equations to solve.

For unit mobility ratio and constant boundaryconditions, the streamline distribution remains unchangedthroughout the displacement process. Therefore, the pressurefield or streamline distribution only needs to be solved onceand saturation solutions can be mapped along streamlines. Fornon-unit mobility ratio, there are two common approaches totreat streamlines. One is to fix the streamline geometry andallow the flow rate to change during the displacementprocess12,13. The other is to update periodically the streamlinedistribution and distribute the flow rate equally among thestreamtubes14,15. In the second case, the pressure field andstreamline geometry must be updated periodically.

For complete descriptions of the various streamlineformulations and inclusive reviews of the history ofstreamlines for predicting reservoir flow, please refer to

SPE 59370

A Streamline Approach for History-Matching Production DataYuandong Wang, Anthony R. Kovscek, Stanford University

2 Y. WANG AND A. R. KOVSCEK SPE 59370

references7,10,12,16.In this study, the streamline simulator 3DSL by

Batycky et al11 is employed for forward simulation. In short,after solving the pressure field and the streamline distribution,3DSL assigns equal flow rate to each streamline. Then a one-dimensional saturation solution, either analytical or numerical,is solved along the streamlines. Periodically, the streamlinesaturation distribution is mapped onto the multidimensionalgrid, the pressure equation is resolved, and streamlinegeometry redetermined. There is no detectable deduction inaccuracy with this technique17.

Previous workMost approaches to history-matching field data manipulatepermeability at the grid-block level, and hence, demand agreat amount of computational work because there are manygrid-blocks in a typical simulation. Integration of productiondata with reservoir description remains an important issuebecause of the prevalence of production data and theinformation that it carries about the reservoir. In this briefreview, we focus on work that is most similar to our method tofollow. Other approaches to history matching are based uponsimulated annealing18, sensitivity coefficients19,20,21, andparameter estimation approaches22.

Sensitivity coefficient techniques compute thesensitivity of the objective function to the change ofpermeability of a cell or a set of cells and solve an inversesystem that can be very large and somewhat difficult toconstruct21,22. Sensitivity coefficient methods might also becomputationally expensive if the sensitivity coefficients areevaluated numerically by running multiple simulations. Chu etal21 developed a generalized pulse spectrum technique toestimate efficiently the sensitivity of wellbore pressure togridblock permeability and porosity. Other work employedsensitivity coefficients in the integration of well testinformation, production history, and time-lapse seismic data22.

Vasco et al20 combined streamlines and a sensitivitycoefficient approach while integrating dynamic productiondata. They employ streamlines to estimate sensitivitycoefficients analytically thereby greatly speeding up theprocedure. The streamline analysis allows them to "line up"the first arrival of injected fluid at production wells and thenmatch the production history. This technique remains a grid-block-level optimization approach as all of the cells from theflow simulation are used to describe reservoir heterogeneity.

Sensitivity coefficients have also been employed in ascheme to identify the geometry of geological features such asfaults and the dimensions of flow channels23. In essence, thetechnique is to minimize an objective function incorporatingsingle and/or two-phase production data. The parameters forminimization are the size and interfacial area of geologicalbodies rather than grid-block parameters.

Simulated annealing, as applied by Gupta et al18,perturbs permeability in a set of grid-blocks and evaluatesenergy objective functions or the degree of misfit betweensimulated and desired results. This process is stochastic and it

is not guaranteed that a perturbation will decrease the energylevel. The decision of whether or not to accept theperturbation is based on the change of energy caused by thisperturbation. Perturbations that increase the degree ofmismatch are accepted with a frequency that decreases withincrease in the error. Many iterations are usually required toobtain an acceptable solution. In general, the computationalcosts of incorporating production data using simulatedannealing become very large if a reservoir simulation must beconducted for each iteration.

Reservoir characterization approaches such asgeostatistics do not explicitly account for field productiondata. It is a major task to determine the geostatisticalrealizations that are consistent with injection and productiondata. Generally, this involves conducting flow simulations formany different geostatistical simulation realizations. Efficientconditioning of permeability fields to both a geostatisticalmodel and production data is discussed by Wu et al24. In otherwork, Wen et al19, 25 also present a geostatistical approach tothe inverse problem of integrating well production data. Theyadapt the sequential self-calibration (SSC) inverse techniqueto single-phase, multi-well, transient pressure and productionrate data. The SSC method is an iterative, geostatistically-based inverse method coupled with an optimization procedurethat generates a series of coarse grid two-dimensionalpermeability realizations. In later work, they combine SSCwith analytical computation of sensitivity coefficients usingstreamline distributions19. The output realizations correctlyreproduce the production data. In both instances, thisapproach is applied for single-phase flow.

An interesting question that arises with any match toproduction data is the accuracy of the match. Lepine et al26

combine error analysis with a gradient-based technique tocompute the uncertainty in estimates of future performancebased upon history-matched models. Their method helps toidentify and select the parameters of a given reservoir modelthat most sensitively affect the match.

The remainder of this paper presents the developmentof a streamline-based history-matching approach wherestreamline effective permeability is manipulated rather thangrid-block level data. An algorithm for mapping inversestreamline information onto the grid is also discussed. Next,several synthetic cases are used to explore the proposedapproach. Discussion and conclusions complete the paper.

MethodWe propose a two-step method to match dynamic productiondata and infer reservoir heterogeneity. The first step is tomodify the permeability distribution at the streamline levelbased on the difference between simulation results and fielddata for water cut, pressure drop, and flow rate. By matchingthe fractional-flow curve through manipulation of thepermeability field, we try to capture reservoir heterogeneity.The second step is to map the streamline permeabilitymodification onto the grid-blocks. Then flow simulation isperformed to check the match. The above process is iterated

SPE 59370 A STREAMLINE APPROACH FOR HISTORY-MATCHING PRODUCTION DATA 3

until convergence.In a multi-phase displacement process, each

streamline breakthrough contributes a small amount to thefractional-flow curve at a producer. Throughout, we use theterms water-cut and fractional flow interchangeably. Becauseof the equal flow rate property of streamlines in forward flowsimulation with 3DSL, every streamline breakthroughcontributes the same amount to the fractional flow. Therefore,by ordering the streamlines with respect to their breakthroughtime, we discretize the fractional-flow curve and relate varioussegments of the fractional-flow curve to the breakthrough ofindividual streamlines. In this sense, our proposed method issimilar to the work of Vasco et al20; however, we do notcompute sensitivity coefficients nor is our formulation of theinverse problem similar. When the fractional-flow curve of thesimulation result for a given permeability does not match thefield water-cut curve, we infer the streamlines responsible forthe difference between the fractional flow curves. Then basedon the relation between streamline breakthrough time andeffective permeability, a modification of effectivepermeability along streamlines can be computed to match theproduction data. The objective function, as defined below,indicates the error of the simulation result compared to thefield data, and therefore is minimized.

)( ,,1

, nqnp

N

nnt EEEE

p

++= ∑=

(1)

In Equation (1) the subscript n is the index of the producer, Np

is the total number of producers in the study, ntE , , npE , ,

and nqE , are errors in the dimensionless breakthrough time of

individual streamlines, pressure, and flow rate at producer i,respectively.

Because streamtubes do not communicate except atproducers, it is logical to organize subsystems for eachproducer. It is also computationally efficient if each produceris treated independently because solving several small subsystems is much cheaper numerically than solving a largesystem.

To match pressure and flow rate, the effectivepermeability of the entire region is modified. However, tocapture heterogeneity, we need to solve an inverse system tomatch the fractional flow curve. This is the most importantpart of this study and is therefore discussed in detail.

In the following development, the index of producern is omitted. The degree of mismatch between reference andhistory-matched results is computed as

∑=

=sl

BT

N

iit

slt E

NE

1

2,

1 (2)

where Nsl is the number of streamlines connected to theproducer. The error

it BTE ,

refers to breakthrough time of

streamline i as defined below R

iBTDC

iBTDit ttEBT ,,,,, −= (3)

where CiBTDt ,, and R

iBTDt ,,are the computed, C, and

reference, R, dimensionless breakthrough times (pore volumeinjected) for the ith streamline, respectively.

Inverse systemPermeability modification of the ith streamline alters thebreakthrough time of not only the ith streamline, but also theother streamlines. Therefore, all the streamlines must beconsidered and a system of equations has to be solved.

The system to solve for each producer is

−=

∆

∆∆∆

NBTt

BTt

BTt

BTt

NNNNNN

N

N

N

E

E

E

E

k

k

k

k

aaaa

aaaa

aaaa

aaaa

,,

3,,

2,,

1,,

3

2

1

321

3333231

2232221

1131211

LL

L

L

L

L

L

(4)

where iBTtE ,, is defined in Eq. (3), jk∆ is the modification

of effective permeability along streamline j required to get amatch, and aij is the sensitivity of breakthrough time(dimensionless) of the ith streamline to the effectivepermeability of the jth streamline. These derivatives aredefined as

j

iBTDij k

ta

∂∂

= ,, (5)

where iBTDt ,, is the dimensionless breakthrough time of

streamline i, and kj is the effective permeability alongstreamline j. Because streamlines are non-communicating, thederivatives can be approximated by applying Dykstra andParsons27 method for non-communicating layers of differentlength. The method relates the breakthrough time of differentlayers to the effective permeability of each layer. For unitmobility ratio and piston-like displacement, the approximationis exact.

The breakthrough time for streamline i is calculatedas

∑

∑

=

==sl

sl

N

kk

N

kkDk

iBTD

lA

xlAt

1

1,,

)(

,)(

φ

φ (6)

where l is the length of a streamline, kDx , is the

dimensionless position of the displacing phase front of the kth

streamline when the ith streamline breaks through. The term

kDx , can also be viewed as the fraction of pore volume of

streamline k swept when streamline i breaks through (i.e.,

1, =iDx ). For those streamlines that break through earlier

than streamline i, kDx ,

can be greater than 1. The

4 Y. WANG AND A. R. KOVSCEK SPE 59370

symbols kφ and kA represent the average porosity and

average cross-sectional area of streamline k respectively.They are defined as

∫= 1

0)( DD dxxAA (7a)

∫= 1

0)( DD dxxφφ (7b)

Now, define the ratio of pore volume of streamline kover the total pore volume as

T

kN

kk

kkD V

V

lA

lAV

sl==

∑=1

,

)(

)(

φ

φ (8)

where kV is the pore volume of streamline k, the subscript D

denotes dimensionless, and TV is the total pore volume.

Equation (6) can be rewritten as

∑=

=slN

kkDkDiBTD xVt

1,,,,

(9)

Then by applying the chain rule, Eq. (5) is evaluated:

j

kDN

kkD

j

kDN

k kD

iBTD

j

iBTD

k

xV

k

x

x

t

k

t slsl

∂∂

=∂

∂∂

∂=

∂∂

∑∑==

,

1,

,

1 ,

,,,, (10)

Dykstra and Parsons(1950)27 method provides

kDx , in terms of the effective permeability of all the

streamlines; hence, the summation above must be computedfor all k streamlines. The formula for calculating

kDx , for

unit mobility ratio is slightly different from that for non-unitmobility ratio.

For unit mobility ratio, the pressure field as well asthe streamline distribution remains unchanged throughout thedisplacement process for constant boundary conditions. Whenbreakthrough happens at streamline i, the front position atstreamline k is calculated by

i

kikkD k

kcx =,

(11)

where ikc is a constant related to the length of the streamline

i and k. Applying this definition and completing the partialderivative indicated in Eq. (7) yields

≠

=−=

∂∂

=∂∂ ∑

∑ ≠=

=

jiifkVc

jiifkVck

k

xV

k

t

ijDij

N

ikkkkDik

i

j

kDN

kkD

j

iD

sl

sl

,/

,1

,

,1,2

,

1,

, (12)

This procedure can be repeated for non-unit mobility ratios

given the standard Dykstra-Parsons result28, where kDx , is a

function of the permeability of streamlines i and k, and theend-point mobility ratio.

It is an approximation to treat streamlines as non-

evolving for the non-unit mobility ratio cases. Thisassumption works well if heterogeneity is a dominant factorduring displacement. Therefore, it is most applicable tounfavorable mobility ratios and heterogeneous permeabilityfields. Although streamlines evolve during a non-unit mobilityratio displacement, for a heterogeneity dominant reservoir, thestreamlines evolve little. An important fact is that the order ofbreakthrough of streamlines is preserved even though lengthand volume may change. That is, if a streamline passesthrough a high permeability channel at the start of thedisplacement process, that channel will remain a channelthroughout the entire process. The streamlines with thesmallest pore volume or time of flight always break throughearliest, no matter how they evolve.

Simplifying the inverse systemFor unit mobility ratio, the inverse system can be simplified bydefining relative or normalized parameters such as

−=

NBTt

BTt

BTt

BTt

NNNNNN

N

N

N

e

e

e

e

k

k

k

k

bbbb

bbbb

bbbb

bbbb

,,

3,,

2,,

1,,

3

2

1

321

3333231

2232221

1131211

LL

L

L

L

L

L

δ

δδδ

(13)

where iBTte ,, is normalized error in breakthrough time of

streamline i to be defined in Eq. (14), jkδ is the normalized

modification of effective permeability along streamline j, and

ijb is the derivative of itBT

e , with respect to jkδ . For the

purpose of discussing updates from one iteration to the next,

let 1+λjk be the new value of kj, and likewise, λ

jk be the

previous value. In equation form, the normalized variables areR

iBTDR

iBTDC

iBTDR

iBTDitit ttttEeBTBT ,,,,,,,,,, /)(/ −== (14)

λλ

λλ

δj

j

j

jjj

k

k

k

kkk

∆=

−=

+ 1 (15)

1,,

,,)(,,

+∂∂

=∂

∂= λ

λ

δ j

iBTDR

iBTD

j

j

t

ijk

t

t

k

k

eb iBTD (16)

Substitute Eq. (12) into (16),

≠≈=

=−≈

−

=

∑∑

∑∑

==

≠=

=

jiifN

kVc

kVc

k

Vc

kkVc

k

jiifkVckkkVc

k

b

slN

kkkDik

ijDij

i

jDij

N

kikkDik

i

N

ikkkkDik

iN

kikkDik

i

ij

slsl

sl

sl

,1

/

,11

/

1,

,,

1,

,1,2

1, (17)

With the normalization above, the elements of the

SPE 59370 A STREAMLINE APPROACH FOR HISTORY-MATCHING PRODUCTION DATA 5

matrix are now functions of only matrix size Nsl for a given setof streamlines. Therefore, the inverse of the matrix is alsosolely a function of Nsl. Because the matrix is symmetric,diagonal-element dominant, and positive definite, it is easy tocompute. Thus all elements in the inverse of the matrix areknown functions of Nsl. These known functions can bederived in general and do not vary from iteration to iteration.All of the elements in the inverse of the matrix are calculateddirectly. A good approximation of this inversion is a unitmatrix. This is verified by practice as discussed later. Here,the inverse process is simply a computation of the product ofthe inverted matrix and the error vector.

The above simplification works well for unit mobilityratio, as will be demonstrated. For non-unit mobility ratios, italso works to some extent, especially for cases where themobility ratio is close to unity or heterogeneity is thedominant factor.

In non-unit mobility ratio cases, the elements aij ofmatrix A are functions of mobility ratio M. If we repeat theabove process, we do not get a matrix as simple as B in Eq.(17). An alternative for non-unit mobility ratio cases is tosolve the system of equations in Eq. (4) by inverting thematrix A. However, the streamlines evolve during thedisplacement process. To obtain a good history match withsuch a procedure, we may need to select several differentstreamline distributions over the time period of interest. Eachof the distributions could be used to match a segment of thefractional flow curve. This is computationally intensive andso we have not implemented this alternative yet. The casestudies indicate that it is appropriate to apply the simplifiedsystem developed in Eq. (13) for unit mobility ratio to non-unit mobility ratio cases.

Steps of the inverse streamline approachThis approach can be implemented in a relatively simpleprocedure that involves no modification to the forwardstreamline simulator. This is especially helpful when sourcecode is not available. The steps include:1. Obtain an initial permeability field by guess or

geostatistical realization;2. Run a simulation on the initial permeability field. Check

whether the simulation results match the field data(reference data) including fractional-flow curve at theproducers, flow rate, and pressure. If not, modify thepermeability as in the following steps;

3. Work on the streamlines. Calculate the time of flight (orthe associated pore volume) for all streamlines. Sort thestreamlines in ascending order of the pore volume;

4. Compute the difference in fractional flow, flow rate, andpressure between the simulation result and the reference.Relate differences in the fractional-flow curve to thecorresponding streamline;

5. Solve the system described above (either Eq. (13) or Eq.(4)) to compute the modification of the effectivepermeability of each streamline required for a good matchto available data;

6. Map the modification at the streamline level onto grid-blocks honoring the effective permeability of eachstreamline;

7. Iterate steps 2 to 6 until a satisfactory match is achieved.

Example Applications

Several cases have been tested with this approach and theresults show that this method is robust and converges quickly.Five cases are presented below. First, a reference permeabilityfield is generated, and then 3DSL run with constant injectionrate, except for the last case, to obtain the reference productiondata. Water cut and pressure drop are the data from thereference case that we try to match. An initial permeabilityfield is guessed to start the procedure. The permeability fieldis either uniform or generated by a geostatistical sequentialGaussian simulation method29 where the field is conditionedto sparsely distributed data. All the cases discussed in thispaper deal with a quarter of a five-spot pattern, although ourmethod is not restricted to this pattern, nor is the methodrestricted to two-dimensional areal cases. For all of the cases,the injector is located at the lower left corner, and the producerat the upper right corner of the pattern. The pressure at theproducer is constant and the fluid is incompressible.

1. Large scale trends, Cases 1, 2 and 3The three cases discussed in this subsection show how a large-scale trend is retrieved. By large scale, we mean apermeability feature that nearly spans from injector toproducer. For all the three cases, the mobility ratio is unity, thewater injection rate is constant, and the initial guess for thepermeability field is uniform. The numerical solution of theBuckley-Leverett equation is mapped along the streamlines.

In Case1, the reference permeability field contains ahigh permeability channel connecting the injector andproducer as shown in Fig. 1 (a). In Fig. 1, light gray indicateslow permeability and dark gray high permeability. Startingfrom a uniform permeability field of 500 md, four iterationsare required to match water-cut and pressure data. Eachiteration includes running the flow simulation, computingerrors, modifying the permeability of the streamlines, andmapping the modification onto the grid-blocks. The inferredpermeability field is shown in Fig. 1 (b). Although somewhatmore diffuse, the feature linking injector and producer isreproduced. Figure 2 shows the evaluation of the match ofwater-cut, and Fig. 3 shows the normalized error for fractionalflow and pressure. After four iterations, the match is verygood. The inversion converges quickly and the relative erroris less than 1%. Most of this error is associated with thematch of the water-cut curve. The computational time is spentmainly in running the forward flow simulation. The inversionrequires only a few seconds because the solution of the inversesystem of equations is simplified.

The configuration of Case 2 is the same as that inCase 1, except a low permeability barrier replaces the highpermeability channel. The procedure is the same as above. A

6 Y. WANG AND A. R. KOVSCEK SPE 59370

very good match to production data is obtained, and therelative error of each parameter is less than 1% after oneiteration. Figure 4 shows the reference and inferredpermeability fields. Again the light gray indicates lowpermeability and dark high. The feature is reproduced, butsome detail is lost as in the previous case. Figure 5 plots thefractional flow for reference and simulation results, and Fig. 6plots errors in fractional flow and pressure. Note that thereproduction of water-cut behavior is excellent and achievedin two iterations.

In the first two cases, permeability features werealigned along flow paths between injector and producer. InCase 3, an off-flow-trend barrier exists as shown in thereference permeability field of Fig. 7a. With the sameprocedure as in Cases 1 and 2, a very good match to fractionalflow and pressure behavior is achieved with two iterations asillustrated in Figs. 8 and 9. However, the geometry and size ofthe barrier are not reproduced very well. Comparing thereference permeability and the inferred permeabilitydistributions, the inferred field has a diffuse low-permeabilityregion that nearly spans from the injector to the producer.This is the worst case that we can formulate to testreproduction of true permeability field. The reason for thisresult is explored in the discussion section.

Effect of mobility ratio, Case 4

In this case, the end point mobility ratio is an unfavorable 2.5such that the displacing phase is more mobile than thedisplaced phase. In forward flow simulations, the Buckley-Leverett solution is again mapped along the streamlines.

To generate more realistic reference and initialpermeability fields, scattered permeability data were chosenand the two permeability field realizations shown in Fig.10were generated by the routine SGSIM29. One realization, Fig.10a, is used as the reference, the other, Fig. 10b, is used as theinitial permeability field. The flow simulation results fromthese two permeability fields show obvious differences in theproducer fractional-flow curve as illustrated in Fig. 11 becausethe realizations are not constrained by production data.

Three iterations were performed to match thereference water-cut data, and the results are illustrated inFigures. 10 to 13. The match is excellent in terms ofreproducing fractional flow, Fig.11, and pressurecharacteristics. Figure 12 shows that most of the error remainsin matching the water-cut curve, but this error is less than 1%.The match of permeability distribution is acceptable, as shownby comparing Fig. 10 (a) and (c). Note that the inferredpermeability field is the best that we can achieve given theproducer information and initial guess.

The match obtained above, Fig. 10 (c), was testedwith other mobility ratios to demonstrate that the match doesnot depend on mobility ratio M. Flow simulations were runon the reference and inferred permeability fields at M=1,M=0.25, and M=5. Figure 13 shows the comparison. Itindicates that the permeability field inferred by matchingproduction data at one mobility ratio can be used to predict

displacement processes at other mobility ratios and that ourinverse approach is independent of the mobility of the injectedfluid. This is very useful when a tracer study is used tocharacterize reservoir heterogeneity and the inferredpermeability distribution is then used to predict the reservoir’sfuture performance.

3. Variable injection rate, Case 5In this case, the injection rate undergoes several step changesduring the course of the displacement process. The endpointmobility ratio is 2.5 and the fluids are again incompressible.The reference permeability field is generated, in adeterministic manner, with an off-trend low permeabilityregion as shown in Fig. 14a. Then six permeability valueswere collected, two at the wells, two at the other two corners,one inside the low permeability region and one outside the lowpermeability region. Sequential Gaussian simulation wasperformed on these data28. One realization is chosen as ourinitial guess for the permeability field (Fig. 14b).

With the procedure given above and fixing theinjection rate schedule to match the history of the referencecase, a very good match to the fractional flow curve and thepressure at the injector is obtained as shown in Figs. 15 and16, respectively. The error during iteration is given in Fig. 17.Only two iterations are needed to converge despite therelatively complicated injection conditions. The inferredpermeability field is shown in Fig. 14c. Visually the finalmatch appears to be closer to the reference permeability fieldof Fig. 14a than is the initial permeability in Fig. 14b.

DiscussionThe example applications above show that this approach isrobust and converges quickly. Each iteration significantlyreduces the errors in fractional flow, pressure drop, and flowrate. In all of the examples, an acceptable match to theproduction data is obtained within a small number ofiterations. One strength of this approach is that the input ofthe inverse problem is simply the usual output of thestreamline simulation. Namely, streamline coordinates,production data including fractional flow, flow rate, andpressure. No internal modifications to the streamlinesimulator are required for implementation. Therefore, thisapproach stands alone. Importantly, it appears to beexpandable to integrate geostatistical data or to honorgeological and seismic data. We are actively pursuing thistask.

Another strength is that repeated inversion of amatrix is not needed. The permeability modification for everyiteration is computed directly based on differences betweenthe streamline breakthrough time from field observation andthe simulation result. Therefore, this process takes very littlecomputational time. All of the examples illustrated here usedthe simplified inverse approach derived for unit mobility ratiocases. It appears that this idea works well for flow withunfavorable mobility ratios in heterogeneous reservoirs.

The computation of the error in water-cut for both

SPE 59370 A STREAMLINE APPROACH FOR HISTORY-MATCHING PRODUCTION DATA 7

unit and non-unit mobility ratio cases is performed usingfractional-flow curves from the simulation and the referenceresults. The pore volume associated with each streamline iscalculated for the purpose of sorting streamlines. Theaccuracy of computing streamline pore volume does not affectinversion results as long as the correct ordering of thebreakthrough time is obtained. Therefore, for non-unitmobility ratio cases, the method works well as long as theorder of streamline breakthrough time is preserved, eventhough the streamlines evolve.

Streamline permeability modifications are mappeddirectly to grid-blocks by propagating the indicated changethrough all of the grid-blocks that a particular streamlinepasses. For example, when the effective permeability of astreamline needs to be increased by 5 percent, then thepermeability of all the grid-blocks along that streamline areincreased by 5%. Mapping is not unique and differentmapping schemes can produce the same effective streamlinepermeability. Necessarily, however, the water-cut curve ismatched and the inferred field is independent of mobility ratio.In the mapping process, it is possible to enforce constraints tohonor observed geological information, if any.

Examples shown here are for two-dimensional flowproblems. We continue to work on the generalization to three-dimensional problems with gravity. Additionally, thisapproach was originally designed for incompressible two-phase flow or tracer flow studies. That is, the fractional flowcurve or tracer breakthrough curve at the producers is requiredto infer the heterogeneity of the reservoir. More study isneeded to extend this procedure to compressible single-phaseflow scenarios such as well test problems.

In Cases 1 and 2, the permeability feature is on trend,and we find that it is easy to retrieve the features of thedistribution. However, if the heterogeneity is off trend, thenthe permeability feature may not be retrieved as easily by thisapproach. Case 3, as shown in Fig. 7, indicates that off-trendbarrier is reproduced as diffuse, scattered regions of relativelylow permeability if no constraints of mapping are enforced.This result occurs because during the update of permeabilityalong a streamline, all of the grid-blocks along that streamlineare multiplied by the same factor. This problem can beovercome by applying geological information while mappingor generating a better initial permeability field. We arecurrently working on this extension. Nevertheless, theeffective permeability of each streamline in the inferredpermeability field is very close to that of the correspondingstreamline in the reference permeability field. As a result, thefractional flow curve and pressure drop obtained for theinferred permeability field are quite close to the referencevalues.

ConclusionsThis approach relates producer water-cut curves to thebreakthrough time of individual streamlines. The effectivepermeability along streamlines is modified directly to history-match the fractional flow curve, pressure drop, and flow rate

information. No matrix inversion is involved in the inverseprocess and therefore it is quite fast. The forward flowsimulation with 3DSL is also fast and so, for the examplesexamined, the entire process appears to be computationallyefficient.

The current work examined 2-D areal porous media,where the effect of gravity is not important, as well asincompressible two-phase flow. This approach works well forreservoirs where heterogeneity is a dominant factor. It alsoworks well for unfavorable mobility ratios because the effectsof heterogeneity are exacerbated by the unstable displacement.Although streamlines evolve for non-unit mobility ratio casesduring the displacement process, we find it feasible to chooseone streamline distribution and apply the simplified inversesystem derived for unit-mobility ratio cases.

Importantly, the history matching results areindependent of mobility ratio. A permeability field inferredby matching production data for one mobility ratio can beused to predict reservoir performance for displacementprocesses at other mobility ratios.

AcknowledgementThis work was supported by the Assistant Secretary for FossilEnergy, Office of Oil, Gas and Shale Technologies of the U.SDepartment of Energy, under contract No. DE-FG22-96BC14994 to Stanford University. Additionally, the supportof Stanford University Petroleum Research Institute (SUPRI-A) Industrial Affiliates is gratefully acknowledged.

NomenclatureA cross-sectional area of a streamtube, L2

c constantE absolute errore relative errorfw fractional flow of waterk permeability, L2

l streamline lengthM mobility ratiop pressure, M/(LT2)q flow rate, L3/TtD dimensionless timeV pore volume, L3

VD dimensionless volume (fraction of pore volume)xD dimensionless lengthφ porosity

Subscripts:n producer indexi, j, k streamline indexsl streamlineBT breakthroughD dimensionlessSuperscripts:C ComputedR Referenceλ iteration index

8 Y. WANG AND A. R. KOVSCEK SPE 59370

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