a newton-raphson iterative scheme for integrating multiphase production data into reservoir models
DESCRIPTION
Newton Rapshon Iterative SchemeTRANSCRIPT
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SummaryThis paper presents a new linearized iterative algorithm for build-ing reservoir models conditioned to multiphase production dataand geostatistical data. The significant feature of the proposedalgorithm is that the computation of the sensitivity coefficients ofproduction data, with respect to model parameters, can be avoided.This leads to dramatic reduction of computational cost. Instead ofgenerating the sensitivity matrix as required for the least-squaresalgorithms, the proposed approach relies on solving the inversionequations, which are derived from the necessary conditions for afunctional extremum.
It is proved that the proposed method requires considerably lesscomputational effort than the traditional algorithms such as theGauss-Newton method, the Levenberg-Marquardt method, and thegeneralized pulse-spectrum technique (GPST). This is because thecomputation of the sensitivity coefficients makes the traditionalalgorithms computationally intensive. However, for the proposedlinearized iterative scheme, the computational requirement onlydepends on the timesteps used in the reservoir simulator rather thanthe number of parameters or the number of observed data.
The linearized iteration scheme converges quickly because theinversion equations can be solved through the Newton-Raphsonmethod. At each iteration, the new approach requires solving thefinite difference equations and the linear adjoint equations onlyonce, respectively. Since the solver for the flow equations can beused to solve both the adjoint equations and the inversion equa-tions, the proposed algorithm can be easily applied to commercialreservoir simulators. In this paper, two numerical examples forincorporating water oil rate data into geostatistical models aregiven to prove the efficiency of the proposed algorithm.
IntroductionIntegrating production data into reservoir models is an inverseproblem and requires efficient optimization algorithms to mini-mize the least-squares objective function. In the past two decades,many optimization methods were proposed for the integration. Ingeneral, these optimization methods can be classified into threecategories: the gradient-based methods, the global optimizationmethods, and the sensitivity coefficient-based methods. As early asthe 1970s, the gradient-based methods were used in automatic history-matching problems. For single-phase flow, Chen et al.1 andChavent et al.2 proposed the optimal control method to calculatethe gradient of the objective function with respect to model param-eters and used the gradient-based optimization methods to mini-mize the least-squares objective function. Later, Wasserman et al.3
extended the optimal-control method to automatic history match-ing of a pseudosingle-phase reservoir. Watson et al.4 estimatedwater and oil relative permeabilities based on displacement exper-iment data. In the groundwater area, Carrera and Neuman5 used theoptimal-control method for solving the parameter identificationproblem of groundwater flow. Sun and Yeh6 have also applied sim-ilar procedures to estimate aquifer properties. Yeh7 presented aclear survey on parameter identification in the groundwater area.
Readers can refer to the latest review published by Ewing et al.8 onreservoir parameter estimation, including some discussion of optimization algorithms. In the mathematical community, Ito andKunisch9 and Kunisch and Tai10 used the system equations as constraints and introduced dual variables to study the parameter estimation problem.
For the gradient-based methods, the linear search for the opti-mal step size will dominate the inversion process. For the globaloptimization methods, low convergence rate is the major disadvantage. The numerous simulation runs for evaluating theobjective function become computationally expensive as the reser-voir model size increases. Normally, it is preferable to use the sensitivity coefficient-based methods such as the Gauss-Newtonmethod, the Levenberg-Marquardt method, and the least squaresQR decomposition (LSQR) method for minimization because oftheir fast convergence. However, if a large number of productiondata and reservoir parameters are involved, estimating the sensitivitycoefficients of production data with respect to the reservoir param-eters is time-consuming and costly. Reducing the cost for comput-ing the sensitivity coefficients can significantly reduce the compu-tational cost of the integration process. In previous research work,researchers have made great efforts to develop possible algorithmsto obtain the sensitivity coefficients. For single-phase flow,Carter's method is the most efficient for calculating the sensitivitycoefficients.11 Unfortunately, Carter's method cannot be applied tocalculate the sensitivity coefficients for multiphase flow. TheGPST method proposed by Tang et al.12 can only be used to calculate the sensitivity coefficients of pressure with respect to per-meability.13 Recently, the optimal control method to generate thesensitivity coefficients by Wu et al. is very efficient if the numberof observed data is less than the number of reservoir parameters.14
If reservoir models contain a few parameters, and the number ofreservoir parameters is less than the number of observed data, thegradient-simulator method is preferable. Obviously, neither theoptimal-control method nor the gradient-simulator method for calculating the sensitivity coefficients is efficient if the number ofobserved data and reservoir parameters are large.
In this paper, unlike previous research in which the primaryfocus is to develop the algorithm for estimating the sensitivitymatrix, a linearized iterative scheme for integrating production datainto geostatistical models is proposed. The feature of the proposedalgorithm is that the nonlinear inversion equations are derived fromthe necessary conditions for a functional extremum rather than thenormal equations formulated from the sensitivity matrix and, thus,the calculation for the sensitivity coefficients can be avoided. Forupdating reservoir parameters once, the computational requirementis to solve the adjoint equations n times, in which n is equal to thenumber of timesteps used in the reservoir simulator. The computa-tional cost of the proposed linearized iterative approach is onlydependent on the number of timesteps rather than the number ofobserved data or the number of model parameters.
The proposed linearized iteration method converges quicklybecause the inversion equations can be solved through the Newton-Raphson method. Moreover, the proposed iterative scheme offerstwo other advantages. First, the linearized iterative algorithm iseasier to implement, because the sensitivity coefficients are notrequired. Second, the adjoint equations are linear and have thesame coefficient matrix structure as the flow equations. No specif-ic solver for adjoint equations is required. One can apply the same
September 2001 SPE Journal 343
A Newton-Raphson Iterative Scheme forIntegrating Multiphase Production Data
Into Reservoir ModelsZhan Wu, SPE, Texas A&M U.
Copyright 2001 Society of Petroleum Engineers
This paper (SPE 74143) was revised for publication from paper SPE 62846, first presentedat the 2000 SPE/AAPG Western Regional Meeting, Long Beach, California, 1923 June.Original manuscript received for review 28 June 2000. Revised manuscript received 26 February 2001. Manuscript peer approved 26 March 2001.
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solver used in the reservoir simulator to solve the adjoint equa-tions. Actually, the solver can also be used to solve the inversionequations. However, it must be pointed out that one of the difficul-ties in solving the inversion equations is the ill-conditionedJacobian matrix. Theoretically, the covariance matrix can be usedto regularize the ill-posed problem, and therefore, many algorithmscan be used to solve the inversion equations. Currently, the pre-conditioned conjugate gradient approach or a partial singular valuedecomposition15,16 is still widely used for solving ill-posed sys-tems. In this paper we modify Moras approach16 to solving inver-sion equations.
There are three major parts in this paper. First, the partial dif-ferential equations for compressible two-phase flow in reservoirs,as well as the discrete formulation for the equations, are describedbriefly, and the inverse problem for integrating production datainto reservoir models is stated. Next, the adjoint equations aredeveloped by means of the optimal-control theory. Then, the inver-sion equations are derived according to the necessary conditionsfor a functional extremum. Finally, two numerical examples areincluded to demonstrate some of the features of the linearized iter-ative schemein particular, the convergence rate for integratingoil flow rate into geostatistical models.
Forward ModelThe two-phase (water/oil) flow equations in a reservoir with spa-tially dependent permeability and porosity in the flow domain, ,can be written as1719
and
The reservoir can be modeled as a rectangular domain with no flowboundary condition
The initial conditions are given by
The subscripts o and wthe oil and water phase. Sthe saturation,while viscosity. The relative permeabilities krw and krofunc-tions of water saturation, Bw and Boformation volume factors, andqw and qowater and oil flow rate, respectively. the boundary ofthe reservoir. For simplicity, the capillary pressure is ignored.
Eqs. 1 and 2 are a set of nonlinear-coupled partial differential equa-tions, which can be solved by use of finite difference methods. Whenusing the standard five-point scheme for the 2D reservoir, we maywrite the discrete formation of the partial differential equations as
where
and
denote the flow and accumulation terms, respectively. Here, nthetimestep, and loil phase or water phase.
If the permeability and porosity are given, the solution of equa-tions can be determined uniquely by applying the implicit-pressure, explicit-saturation (IMPES) method. The performance ofthe reservoir can be predicted accurately. In this study, the goal isto obtain more realistic reservoir models by minimizing the pro-duction mismatch while the geostatistical data are honored. Weneed to minimize the following objective function, which is given as
to obtain the maximum posteriori (MAP) estimation of reservoirmodels. CDthe diagonal covariance matrix for measured data,while CMthe covariance matrix of parameter field;
d contains the
calculated data. Finally, dobsa vector of the observed data, mthe
mean of reservoir parameters, and ma reservoir parameter vectorto be estimated. The first term in Eq. 8 represents the weighted sumof squares of the differences between the observed data and the cal-culated data. The second term denotes the prior information, whichcontains the mean, variance, and covariance of reservoir parame-ters. Oliver,20 Chu et al.,21 Reynolds et al.,22 and He et al.23 havepublished a series of papers dealing with the incorporation of production data into geostatistical data using the Bayesian estima-tion24 and Tarantolas inversion theory.25 More details on the theory basis can be found in their papers.
If the Gauss-Newton method is used, the MAP solution forminimizing the objective function Jo can be obtained by solvingthe following linear equations:
where Gthe sensitivity matrix. These relationships are known asthe inversion equations or the normal equations. Note that theinversion equations are explicit functions of the sensitivity matrix.That is, in order to invert the parameters m, the sensitivity coeffi-cient matrix must be evaluated in advance. As mentioned earlier,since the governing equations are nonlinear and the number ofgridblocks is large, calculating the sensitivity matrix is not an easytask. This forces us to develop new inversion equations to replaceEq. 9 without introducing the sensitivity matrix G.
In the next section, the linearized iterative scheme is proposedbased on the classical optimal control theory. The process of inte-grating production data into geostatistical data can be consideredas a distributed parameter optimal control problem in which thestate variables (pressure and saturation) are functions of space andtime, and the control variables (permeability and porosity) arefunctions of space only. However, the theory for dealing with thedistributed-parameter optimal-control problem is very tedious.Without making approximations, it is difficult to directly study thenonlinear distributed parameter optimal-control problem (i.e., thenonlinear partial differential equations for multiphase flow). Ingeneral, one can discretize the partial differential equations in bothspace and time domains. The distributed parameter system isapproximated by a lumped parameter system. Then the lumpedoptimal control theory can be used to analyze the distributedparameter system. Such an approximate approach is not very rig-orous in theory but very practical. For a more complete discussionof optimal control theory, readers can refer to many books andpapers on this subject.2631 In this work, the partial differentialequations for the two-phase flow are discretized in both space andtime domains. The linearized iterative scheme is described by alge-braic equations, and the matrix notation is applied throughout.
Inversion EquationsOn the basis of optimal control theory, the flow equations may beconsidered as constraints while the objective function J is mini-mized. By using the Lagrange multiplier, one can adjoin the equalityconstraints (two phase flow equations) to the objective function J0 bymeans of M dimensional adjoint variables, or the time-dependent
Lagrange multipliers . Thus,
the constraint optimal control problem becomes an unconstraintoptimal control problem. The objective function can be written as
1 1 1 1
,1 , 1 ,
Tn n n n
l l l M l M
+ + + + =
. . . . . . (9)1 1 1 1obs( ) ( ) ( ),
T
D M D MG C G C m C d d C m m + = +
. . . . . (8)1 1obs obs( ) ( ) ( ) ( ),T T
o D MJ d d C d d m m C m m = +
1 1 1 1
,1 , 1 ,
Tn n n n
l l l M l MX x x x+ + + +
=
1 1 1 1
,1 , 1 ,
Tn n n n
l l l M l MF f f f+ + + +
=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)1 1 ,n n nl l lF X X+ +=
. . . . . . . . . . . . . . . . (6)( , , ,0)w wiS x y z S= ( , , )x y z
. . . . . . . . . . . . . . . (5)0( , , ,0)p x y z p= ( , , )x y z
on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)0p
n
=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)1w oS S+ =
. . . . . . . . . . . . . (2)( )ro oo oo o o
k k Sp g z q
B t B
+ = +
. . . . . . . . . . . . (1)( ) ,rw wo ww w w
k k Sp g z q
B t B
+ = +
344 September 2001 SPE Journal
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where Nthe total number of timesteps for solving flow equations;the superscript Tthe transpose of a matrix. The objective functionJ can be expanded by the Taylor series, and the linear term isexpressed as
where variations p and Sw can be interpreted as the perturbationof pressure and saturation due to the perturbations of permeability,k, and porosity, .
After using the integration by parts, we can further rewrite Eq. 11as follows:
The necessary condition for a minimum is that the first variation ofthe objective function is equal to zero for arbitrary variations p,Sw (i.e., the following equation must hold):
In order to satisfy Eq. 13, the variations P and Sw must vanish.This leads to the discretized adjoint equations
and
for n1,2,,N1. Eqs. 13 through 15 are called the first-order nec-essary conditions for the optimal control problem defined in Eq. 10.
It must be pointed out that Eqs. 14 and 15 are a set of ordinaryequations. The initial conditions must be required for solving theEqs. 14 and 15. Fortunately, we simply specify the initial condition as
and the adjoint equations can be solved backward in time.Otherwise, a two-point boundary-value problem must be solved,which is much more costly than solving the initial-value problem.By applying Eqs. 14 and 15 to Eq. 12, the first-order variation ofthe objective function becomes
We can readily obtain the gradient expressions of the objectivewith respect to model parameters from Eq. 17.
where
Here, we use the fact p00 and Sw00 because the initial valuesof pressure and saturation are constant. Eqs. 18 and 19 provide thegradient of the objective function with respect to the permeabilityand porosity, and one can use the gradient-based methods to mini-mize the function J. However, the gradient-based methods requireestimating optimal step length, which is computationally costly.Instead of using the gradient-based approaches, we set the gradientequal to zero. This yields the following nonlinear equations withrespect to permeability and porosity:
, . . . . . . . . . . . . . . . . . . (21)11
1 0
, 0
0
TnN
nll
l o w n
F J
k k
++
= =
+ =
. . . . . . . . . . . . . . . . . . . . . . (20)10 0, ( )T
M
J JC m m
k
=
.
, . . . . . (19)11
1 1 0
, 0
T Tn nL
n nl ll l
l o w n
J X X J
++ +
= =
= +
, . . . . . . . . . . . . . . . . (18)11
1 0
, 0
TnN
nll
l o w n
J F J
k k k
++
= =
= +
. . . . . . . . . . . . . . . . . . . . . . . . . . (17) 1( )n
n T ll
X
+ +
1 11 1( ) ( )
n nn T n Tl ll l
F Xk
k
+ ++ + +
10 0
0
, 0
TTN
l o w n
J JJ J k
k
= =
+ +
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)1 1 0,N No w = =
. . . . . . . . . . . . . . . . . . . . . . . (15)
1 0
,
,
T T Tn n n
n n nl l ll l ln n n n
l o w w w w w
F X X J
S S S S
+
=
+ =
. . . . . . . . . . . . . . . . . . . . . . . (14)
,1 0,
T T Tn n n
n n nl l ll l ln n n n
l o w
F X X J
p p p p
+
=
+ =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)0.J =
. . . . . . . . . . . . . . . . . . . . . . . (12)
0 0 0 00 0 0 0
0 0
( ) ( )( ) ( ) .T Tl l l ll l w
w
F X F Ap S
p S
+ +
( ) ( )( ) ( )
N N N NN T N N T Nl l l l
l l wN N
w
F A F Xp S
p S
+ +
1( )n
n T ll
X
+ +
1 11 1( ) ( )
n nn T n Tl ll l
F Xk
k
+ ++ + +
1 1( ) ( )n n
n T n n T nl ll l wn n
w
X Xp S
p S
+ + + +
( ) ( )n l
n T n n T ll nl l wn n
w
X Xp S
p S
( ) ( )n n
n T n n T nl ll l wn n
w
F Fp S
p S
+ +
0 0
TTJ J
kk
+ +
10 0
, 0
TTN
n n
wn nl o w n w
J JJ p S
p S
= =
= +
. . . . . . . . (11)1
1 1( ) ( ) ,n n
n T n Tl ll l
X X
++ + +
11 1( ) ( )
n nn T l n Tl ll w ll
w
X FS k
S k
++ + + +
11 1 1
1( ) ( )
n nn T n n T nl ll w ln n
w
X XS p
S p
++ + +
+
+
1 11 1 1 1
1 1( ) ( )
n nn T n n T nl ll w ln n
w
F XS p
S p
+ ++ + + +
+ +
+
11 10 0
1( )
TT nn T nll n
J J Fk p
k p
++ +
+
+ + +
11 10 0
0 1 1, 0
TTN
n n
wn nl o w n w
J JJ J p S
p S
+ +
+ += =
+ +
. . . . . . . . . . . . (10)1
1 1 1
0
, 0
( ) [ ]N
n T n n n
l l l l
l o w n
J J F X X
+ + +
= =
= + + ,
September 2001 SPE Journal 345
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The above equations are the framework of linearized iterativescheme. In fact, the Gauss-Newton method and the Levenberg-Marquardt method are also developed based on the assumption thatthe first derivative is equal to zero. The only difference is that theproposed iterative method does not need to estimate the sensitivitycoefficients explicitly for developing the inversion equations.
For solving the nonlinear inversion Eqs. 21 and 22, one can usethe Taylor series to expand the nonlinear inversion equations
Neglecting the higher order terms and substituting Eqs. 21 and 22in Eq. 23, the change of the reservoir parameters can be estimatedby solving the following linearized system
where the Jacobian matrix, which can be obtained by
taking the partial derivatives of Eqs. 21 and 22 with respect to per-meability and porosity. It is not difficult to obtain the second orderpartial derivative expressions (see Appendix B):
The iterative procedure for the nonlinear Eq. 24 refers to theNewton-Raphson method. A more compact form can be obtainedby defining the Jacobian matrix as follows
and
A
The nonlinear inversion equations can be rewritten as
At each iteration, Eq. 31 can be solved to obtain the increment inparameters m and the model parameters can be updated at (k1)th iteration according to
The new parameters mk1 can be substituted into the flow equa-tions to obtain the calculated production data.
The Newton-Raphson method has high convergence rate forwell-posed problems, provided the initial values of the parametersare close enough to the solution. If the Jacobian matrix is ill-condi-tioned, regularization is required. Mathematically, the inversematrix of the covariance matrix CM
1 is not only a weak constraintterm for the reservoir parameters but also a regularization term forc-ing the Jacobian matrix to be nonsingular. Therefore, many solverscan be used to solve the inversion equations. In practice, the conju-gate gradient methods are popular choices for solving the ill-posedproblem. Here, we use the preconditioned conjugate gradientmethod proposed by Mora.16 In Appendix A, the main procedure ofthe modified preconditioning conjugate gradient method is given.
Even though the CM1 matrix is introduced in Eq. 31, the unique
solution of inverse problem might not be obtained. This is becausethe nonlinearity of the flow equations and the least-squares objec-tive function causes the objective function to be flat, or becausemany relative minima exist in the vicinity of the minimum. In suchcases, it is impossible to obtain the minimum except when the ini-tial guesses for the parameters are close enough to the true model.Nevertheless, for the MAP estimation, we always use the mean ofthe parameters as an initial model. In most cases, the mean is notan accurate initial guess. As a result, a large initial data mismatchresults in an overcorrection of the models, and a physically mean-ingful parameter field may not be generated after just one iteration.In order to generate smoother parameter fields when the mismatchis large, in the previous work,14 we artificially used large varianceof observed data at the early iterations. In this work, we adopt theLevenberg-Marquardt idea (i.e., add a term in the main diagonal ofthe Jacobian matrix). The modified inversion equation is given by
Note that the same solver used for solving the flow equations canbe applied to solve the inversion equations. Other efficient solverscan also be used to solve Eq. 33 because the matrix A is sparse andsymmetric. In such a case, the inverse of the covariance term mustbe calculated approximately because of the difficulty of computingthe inverse of the covariance matrix. In the next section, we willshow that satisfying inversion results can be obtained withoutdoing the inversion for the covariance matrix CM .
The main steps for the linearized iterative scheme are summa-rized as follows.1. Solve the flow equation to obtain the production data at each
well using the implicit scheme or IMPES scheme, and store thepressure and saturation data at all gridblocks on hard disk.
2. Take the partial derivatives of the flow equations with respectto the pressure and saturation of each gridblock to formulate thecoefficient matrix of adjoint equations.
3. Solve the adjoint equations backward in time to obtain theadjoint variables.
4. Set the gradient of the objective function, with respect to themodel parameters, to zero to obtain the nonlinear equations onthe permeability and porosity.
5. Take the first-order partial derivatives of the nonlinear inver-sion equations with respect to permeability and porosity to for-mulate the Jacobian matrix.
6. Solve the linearized inversion equations to obtain the param-eter increment, m, and update the model parameters.m k1m k1m .
7. Substitute m k1 into the flow equations and solve the flowequations. Evaluate the objective function. Check the conver-gence criterion. Stop if the criterion is satisfied. Otherwise,return to Step 2.
. . . . . . . . . . . . . . . . . . . . . . . . (33)1( )MJ
A C I mm
+ + =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (32)1k km m m+ = +
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (31)1( )mJ
A C mm
+ =
. . . . . . . . . . . . . . . . . . . . . . . (30)
11 1
20, 0
11 1 1
20, 0
( ) 0
0 ( ) ( )
Nn T n
l l
l w n
Nn T n n T n
l l l l
l w n
Fk
X X
+ +
= =
+ + +
= =
, . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)
2 2
0
2
1
2 2
0
2
M
J J
k kC
J J
k
=
. . . . . . . . . . . . . . . . . . . . . . . . . . (28)2 21
0
, 0
N
l o w n
J J
k k
= =
=
, . . . . . . . . . . . . . . . . . . . . . . . . . . (27)2 21
0
, 0
N
l o w n
J J
k k
= =
=
. . . . . . . . . . . . . . . . . . . . . . . (26)
,2 1 21
1 1 0
2 2, 0
T Tn nN
n nl ll l
l o w n
J X X J
++ +
= =
= +
, . . . . . . . . . (25)2 1 21
1 0
2 2, 0
TnN
nll
l o w n
J F J
k k k k
++
= =
= +
2
2
J
m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)
12
2,
J Jm
m m
=
. . . . . . (23)0 2
0 ( )( )J J J m J
m m mm m m m m
= + = + + +
. . . . . . (22)11
1 1 0
, 0
0
T Tn nN
n nl ll l
l o w n
X X J
++ +
= =
+ =
346 September 2001 SPE Journal
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Numerical Examples In the following two examples, we assume that the permeabilitydistribution of a reservoir is log-normal, with log variance of 0.40,and the log-mean of permeability is equal to 4.0 . The mean ofporosity is 0.20, with a prior variance of porosity equal to 0.0015.The covariance matrix can be generated from the variogram.19 Thelength of each cell is 60 ft. Fluid properties are shown in Table 1.Fig. 1 shows the gridblocks for the 2D reservoir and the well loca-tions. The reservoir consists of 625 gridblocks, and the number ofparameters to be estimated is 1,250 (permeability and porosity).The total flow rate for each production well is fixed at 250 B/D,and the injection rate at each well is fixed at 700.0 B/D. Theobserved data (oil rate) are generated from the true permeabilityand porosity fields by running the simulator 600 days. The totalmeasurement time for the observed data is between 135 and 255days and the maximum time interval of measurement data is 5 days. Thirty-six oil rates are taken as the observed data for eachwell, and the total of 334 observed data are integrated. We assumethe standard deviation of the observed data is one percentage of theoil rate.
In the first example, the means for porosity and permeabilityare selected as the initial values for the linearized iterative scheme.Figs. 2 and 3 show the true porosity and permeability fields,respectively, which are generated from the Cholesky decomposi-tion of the prior covariance matrix. The MAP estimates of porosityand permeability fields are shown in Figs. 4 and 5, respectively.We see that the MAP estimates capture the major features of thetrue cases and display their features, but the MAP estimates of bothpermeability and porosity are much smoother than the true cases.Similar results are also obtained by use of the Gauss-Newtonmethod.14 It is important to note that the Gauss-Newton methodrequires generating the sensitivity coefficients, while the Newton-Raphson linearized iterative scheme only requires solving theadjoint equations once for each iteration.
In the first example, the preconditioned conjugate gradientmethod is used to solve the inversion equations proposed byMora.16 The drawback of Moras procedure is that the inverse forcovariance matrix must be estimated. In general, the size of theparameter covariance is too large to be inverted efficiently. In
Fig. 2True porosity field. Fig. 3True permeability field.
Fig. 1Simulation gridblocks and well locations.
Producers
Injectors
1 7 13 19 25
1
7
13
19
25
0.13 0.16 0.19 0.23 0.26 0.30
1 7 13 19 25
1
7
13
19
25
2.70 3.24 3.77 4.31 4.84 5.38
TABLE 1RESERVOIR PARAMETERS
Parameter Value
Initial pressure, p 4000.00 psi
Residual oil saturation, Sor 0.20
Initial water saturation, Swc 0.16
Oil compressibility, Co 0.000001 psi-1
Water compressibility, Cw 0.0000004 psi-1
Water viscosity, w 1.00 cpOil viscosity, o 0.82 cpWellbore radius, rw 0.30 ft
Reservoir thickness, h 20.00 ft
Water volume formation factor, Bw 1.00 RB/STB
Oil volume formation factor, Bo 1.27 RB/STB
Number of production wells 9
Number of injection wells 4
September 2001 SPE Journal 347
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order to overcome this difficulty, an approximation is made forinverting the covariance matrix. Only the main diagonal elementsare kept, and nondiagonal elements are set to zero. Note that thisapproximation is made only when the inverse of the covariance isrequired. Otherwise, the true covariance matrix should be alwaysused. In the second example, the mean is still used as an initialguess for the iteration. Figs. 6 and 7 show the MAP estimates ofthe porosity and the log-permeability fields. Comparing Figs. 6and 7 with Figs. 4 and 5, we can see that the MAP solutionsobtaining by use of the preconditioned conjugate method and themodified preconditioned gradient method are quite similar. The
approximation for the inverse of the covariance matrix is validand will not affect the final inversion results. Mora thought that aslong as a suitable preconditioning technique is used, good inver-sion results could be obtained.
It was found that the uncertainty level of the permeability wasreduced significantly when the production data were honored.However, our recent results,14 as well as those of Landa et al.,32
showed that matching the water/oil ratio can slightly reduce theuncertainty. From the two examples, we can see that the productiondata may reflect the permeability characteristic, provided sufficientobserved data are incorporated into the objective function.
Fig. 6MAP estimate of porosity conditioned to oil flow rate. Fig. 7MAP estimate of permeability conditioned to oil flow rate.
Fig. 4MAP estimate of porosity conditioned to oil rate. Fig. 5MAP estimate of permeability conditioned to oil flow rate.
1 7 13 19 25
1
7
13
19
25
0.13 0.16 0.20 0.23 0.27 0.30
1 7 13 19 25
1
7
13
19
25
0.13 0.16 0.20 0.23 0.27 0.30
1 7 13 19 25
1
7
13
19
25
2.70 3.24 3.77 4.31 4.84 5.38
1 7 13 19 25
1
7
13
19
25
0.13 0.16 0.20 0.23 0.27 0.30
348 September 2001 SPE Journal
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Fig. 8 shows the objective function behavior at each iteration.We can observe that the convergence rate of the preconditionedconjugate gradient method is fast. In addition to the covariancematrix, the other new preconditioning techniques should be devel-oped to speed up the convergence rate of the proposed iterativescheme. However, the new preconditioning technique has not beenimplemented yet.
Fig. 9 shows the final match of oil flow rate at Well 1. It can beseen that the agreement between the observed data and the calculateddata from the MAP estimates are consistently matched very well.
ConclusionsIn this paper, a Newton-Raphson iterative approach for integratingproduction data into geostatistical models is proposed. This algo-rithm does not need to calculate and store the sensitivity matrix,because the least-squares inversion equations are derived from thenecessary conditions of a functional extremum rather than the sen-sitivity coefficients. This results in tremendous computational timeand storage savings over the traditional approaches. For each iter-ation, the new algorithm requires only one reservoir simulation runand solving the linear adjoint equations and the nonlinear inversionequations once, respectively. The fact that a single solver is usedto solve the flow equations as well as the adjoint equations makesthe new algorithm readily applicable to commercial simulators.Moreover, a modified preconditioned conjugate gradient is imple-mented to solve the ill-posed inversion equations. It is shown thathigh-resolution models can be obtained without inverting theparameter covariance matrix. This new method can be extended tosolve large-scale integration problems.
Discussion We are concerned about the existence and uniqueness of the solu-tion in the Newton-Raphson iterative scheme. If the initial valuesare not sufficiently close to the true model, the solution may not beobtained, even though the covariance matrix can be used as a reg-ularization term to remedy such a problem. In most cases, an addi-tional regularization term is still required in additional to thecovariance matrix. However, the determination of the optimal reg-ularization term is difficult, as no general theories exist.
We are also concerned about the convergence rate of the pro-posed linearized iterative scheme. It can be seen from Eq. 11 thatthe objective function is expanded by the Taylor series, and onlythe linear term is taken. Theoretically speaking, the new approachis still the first-order convergence-rate method, or the so-calledpseudosecond-order convergence-rate method. In order to furtherimprove the convergence rate, the second-order term of the Taylorseries should be taken into account. If so, it will make the iterativeprocedure more complex.
The efficiency of solving the adjoint equations plays an impor-tant role in improving the efficiency of the proposed algorithm. Asmentioned earlier, the coefficient matrix of the adjoint equations isthe same as that of the flow equations. In principle, the same solvercan be used to solve the adjoint equations. However, the coefficient
matrix of adjoint equations is nonsymmetric. The specific precon-ditioning techniques should be used to improve the convergencerate if iterative methods are used to solve the adjoint equations. Bynow, we use the direct and iterative methods to solve the adjointsystem, and they work very well.
Solving the adjoint equations backward in time means that thepressure and saturation data of all timesteps at each gridblock mustbe stored. This is a drawback of the proposed approach. In gener-al, the history data can be stored on the hard disk, and no extracomputer memory will be taken. Nevertheless, reading the datafrom data file may take some time. This is a tradeoff for storing thedata in the computer memory. In the near future, this disadvantageshould be overcome.
A single solver can be used to solve the flow equations, theadjoint equations, and the inversion equations, and hence the pro-posed linearized iterative method can be readily extended to anyreservoir simulator. The extra computational work for the proposedalgorithm is to calculate the first-order partial derivatives of theflow equations with respect to pressure, saturation, permeability,and porosity, as well as the second partial derivatives of flow equa-tions with respect to permeability and porosity. Actually, the first-order derivatives of the flow equations with respect to pressure andsaturation will be done if the flow equations are solved by meansof an implicit scheme.
The preconditioning operator is an important factor for the algorithm. The covariance matrix was chosen as a preconditioningmatrix because of no extra cost to the algorithm. In additional tothe covariance matrix, many techniques can be developed to gen-erate preconditioning matrices to speed up the convergence rate ofthe proposed iterative linearized method.
In this paper, two synthetic cases for integrating oil flow rateinto geostatistical models are given. The efficiency of the newalgorithm will be studied by using more cases.
NomenclatureA matrixB formation volume factorc compressibilityC covariance matrixd vector of dataf flow term
Fm flux term for phase mG sensitivity matrixh reservoir thicknessI identity matrixJ objective functionk permeabilitym vector of model parametersm mean of reservoir parametersN index of final timestep
Fig. 8Objective function with iteration. Fig. 9Oil flow-rate match at Well 1 for the first example.
0
50
100
150
200
250
300
0 50 100 150 200 250 300
Time, days
Observed data
1st iteration
30th iteration
Oil
Flo
w R
ate
, B
/D
100
1,000
10,000
100,000
0 5 10 15 20 25 30 35
Iteration Number
Permeability estimation
Obje
ctive F
unction
Porosity estimation
September 2001 SPE Journal 349
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p pressureQ source/sink term r radiusS saturationx accumulation termX accumulation terms boundary of the reservoirl vector of adjoint variables viscosity density porosity flow domain increment of reservoir parameter
SubscriptsD datal phase, oil or water
M modelobs observedor residual oilw water phase
wc initial watero oil phase
Superscriptsn the nth timestepT transpose
AcknowledgmentsI would like to thank Mr. T. Mathisens and Dr. K.N. Kulkarni fortheir comments and their review of the manuscript.
References1. Chen, W.H. et al.: A New Algorithm for Automatic History
Matching, SPEJ (December 1974) 593; Trans., AIME, 257.2. Chavent, G.M., Dupuy, M., and Lemonnier, P.: History Matching by
Use of Optimal Theory, SPEJ (February 1975) 74; Trans., AIME, 259. 3. Wasserman, M.L., Emanuel, A.S., and Seinfeld, J.H.: Practical
Application of Optimal-Control Theory to History-MatchingMultiphase Simulator Models, SPEJ (August 1975) 347; Trans.,AIME, 259.
4. Watson, A.T. et al.: History Matching in Two-Phase PetroleumReservoirs, SPEJ (December 1980) 521.
5. Carrera, J. and Neuman, S.P.: Estimation of Aquifer Parameters underTransient and Steady State Conditions: 1 Maximum LikelihoodMethod Incorporating Prior Information, Water Resour. Res. (1986)22, 199.
6. Sun, N.Z. and Yeh, W.G.: Coupled Inverse Problem in GroundwaterModeling Sensitivity Analysis and Parameter Identification, WaterResour. Res. (1990) 26, 2507.
7. Yeh, W.G.: Review of Parameter Identification Procedures inGroundwater Hydrology, Water Resour. Res. (1986) 22, 95.
8. Ewing, R.E. et al.: Estimating Parameters in Scientific Computation,IEEE Computational Science Engineering (1994) 1, 19.
9. Ito, K. and Kunisch, K.: The augmented lagrangian method for param-eter estimation in elliptic systems, SIM J. Control Optim. (1990) 28,113.
10. Kunisch, K. and Tai, X.-C.: Sequential and parallel splitting methodsfor bilinear control problems in Hilbert spaces, SIM J. Numer. Anal.(1997) 34, 91.
11. Carter, R.D. et al.: Performance Matching with Constraints, SPEJ(April 1974) 187; Trans., AIME, 257.
12. Tang,Y.N. et al.: Generalized Pulse-Spectrum Technique for 2-D and 2-PhaseHistoryMatching,AppliedNumericalMathematics (1989)5,529.
13. Chu, L., Reynolds, A.C., and Oliver, D.S.: Computation of SensitivityCoefficients for Conditioning the Permeability Field to Well-testPressure Data, In Situ (1995) 19, 179.
14. Wu, Z., Reynolds, A.C., and Oliver, D.S.: Conditioning GeostatisticalModels to Two-Phase Production Data, SPEJ (June 1999) 142.
15. Jackson, D.D. Interpretation of Inaccurate, Insufficient, andInconsistent Data, Geophys. J. R. Astr. Soc. (1972) 28, 97.
16. Mora, P.: Elastic Wave-field Inversion of Reflection and TransmissionData, Geophysics (1988) 53, 750.
17. Peaceman, D.W.: Fundamentals of Numerical Reservoir, Elsevier, NewYork City (1977).
18. Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Elsevier,London (1979).
19. Mattax, C.C. and Dalton, R.L.: Reservoir Simulation, MonographSeries, SPE, Richardson, Texas (1990) 13.
20. Oliver, D.S.: Incorporation of Transient Pressure Data into ReservoirCharacterization, In Situ (1994) 18, 243.
21. Chu, L., Reynolds, A.C., and Oliver, D.S.: Reservoir DescriptionFrom Static and Well-Test Data Using Efficient Gradient Methods,paper SPE 29999 presented at the 1995 SPE International Meeting onPetroleum Engineering, Beijing, 1417 November.
22. Reynolds A.C. et al.: Reducing Uncertainty in GeostatisticalDescription With Well-Testing Pressure Data, Proc., FourthInternational Reservoir Characterization Technical Conference,Houston, 24 March (1997) 443462.
23. He, N., Reynolds, A.C., and Oliver, D.S.: Three-DimensionalReservoir Description From Multiwell Pressure Data and PriorInformation, SPEJ (September 1997) 312.
24. Gavalas, G.R., Shah, P.C., and Seinfeld, J.H.: Reservoir HistoryMatching by Bayesian Estimation, SPEJ (December 1976) 337;Trans., AIME, 261.
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27. Lions, J.L.: Optimal Control of Systems Governed by PartialDifferential Equations, Springer, New York City (1971).
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Appendix AThis appendix presents a brief procedure of the modified precon-ditioned conjugate-gradient method for the nonlinear inversionequations. For a more complete account, readers can refer to Ref. 16.Here, we present the main steps.
For k1, maximum iteration number,
. . . . . . . . . optimal step-length (A-6)~
1
T
k kk
T T
k k k M k
v g
v Av v C v=
+
. . . . . . . conjugate gradient (A-5)11
( )Tk k kk k kT
k k
p g gv p v
p g
= +
. . . . . . . . . . . . . . . . . . . . . . preconditioned (A-4)k M kp C g=
. . . . . . . . . . . . . . . . . . . . . . . gradient (A-3),T
k
J Jg
k
=
. . . . . . objective function (A-2)~
11 ( ) ( )2
T
Mm m C m m+
1
0 obs obs
1( ) ( )
2
T
DJ d d C d d=
obskd d d =
, k km m m =
. . . . . . . . . . . . . . . . . . . . . . . . calculated data (A-1)cald d=
350 September 2001 SPE Journal
-
Because computing the inverse matrix of the covariance matrix CMis costly, in this work we modify the preconditioned conjugate gra-dient method. Instead of computing the inverse matrix CM , the non-diagonal elements are set to zero when CM
1 is required. Let
and the inverse matrix for~
CM can be obtained readily.
Appendix BThe proposed iterative scheme requires generating the second-order derivatives of the flow equations with respect to permeabili-ty and porosity. However, the second-order partial derivatives ofthe flow equations with respect to porosity do not exist. A trans-formation must be made.
If we define
the second-order derivatives of objective function with respect tocan be estimated
In theory, the first term of the right hand side of Eq. (B-2) can becalculated
Here, the optimal step length for the MAP estimates of the porosityis as follows
where
SI Metric Conversion Factorsft 3.048* E 01 m
cp 1.0* E 03 Paspsi 6.894 757 E 00 kPa
*Conversion factor is exact.
Zhan Wu is a postdoctoral research associate in the PetroleumEngineering Dept. of Texas A&M U. e-mail: [email protected]. His research interests include reservoir simulation,reservoir modeling, and reservoir engineering. He holds an MSdegree from the Southwest Petroleum Inst. in China and a PhDdegree from the U. of Tulsa, both in petroleum engineering.
SPEJ
. . . . . . . . . (B-5)11
1 1
, 0
T Tn nN
n nl ll l
l o w n
X Xb
++ +
= =
=
. . . . . . . . . . . . . (B-4)~ ~
1 1 ,T T T Tk k k M k k k k M kb Ab b C b b Ab v C v + = +
. . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)2
0 0
2
J J =
. . . . . . . . . . . . . . . . . . . . . . (B-2)
111 1
, 0
.
T Tn nN
n nl ll l
l o w n
X X
++ +
= =
+
2 2
0
2 2
J J
=
, . . . . . . . . . . . . . . . . . . . . . . . (B-1)2 2 21 2T
M =
. . . . . . . . . . . . . . (A-9)~
2 2 2
1 21/ 1/ 1/M MC diag =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-8)1.k k= +
. . . . . . . . . . update model parameters (A-7)1k k k km m v+ =
September 2001 SPE Journal 351