SPE 138864-PP · Web viewSPE 138864-PP A Study of the Impact of 4D-Seismic Data on TSVD-Based Schemes for History Matching F. Dickstein, P. Goldfeld, G.T. Pfeiffer, Universidade Federal

SPE 138864-PP

A Study of the Impact of 4D-Seismic Data on TSVD-Based Schemes for History MatchingF. Dickstein, P. Goldfeld, G.T. Pfeiffer, Universidade Federal do Rio de Janeiro, E.P.S. Amorim, R.W. dos Santos, Universidade Federal de Juiz de Fora, S.G. Gómez, Universidad Nacional Autónoma de México

Copyright 2010, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Latin American & Caribbean Petroleum Engineering Conference held in Lima, Peru, 1–3 December 2010.

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AbstractWe discuss a TSVD (truncated singular value decomposition) optimization scheme for history matching. We are particularly interested in studying the performance of TSVD under integration of P-wave velocity data. We compare the structure of the main singular value triplets with or without time-lapse seismics. We then analyze the impact of introducing the seismics in a synthetic two-phase model problem. We observe in our experiments a strong regularization of the inversion algorithm due to the seismics. At the same time, we obtain much better reservoir descriptions when 4D seismics is available.

IntroductionThis work concerns automatic history matching (AHM) for oil reservoir problems. Over the last decades this has been a very active area of research and considerable progress has been achieved in the design of robust algorithms for AHM. A comprehensive view of the development in this area can be found in the book by Oliver et al. (2005). For a recent review on the subject, see (Oliver and Chen 2010). In spite of the progress accomplished, there are still many aspects of AHM that need further investigation. In fact, history matching is a very delicate question, since one is in general trying to recover information out of scarce data, turning it into an ill-conditioned problem There are two natural ways to tackle this difficulty. First, by incorporating more data when available. A second way is to reduce the number of model parameters to be recovered. This is usually done by constraining the search of a solution to a low-dimensional subspace of the parameter space. This procedure, known as reduced parameterization, has been considered by many authors, with different strategies. The zonation method, see (Jacquard and Jain 1965) and (Jahns 1966), has been the first proposed scheme in that direction. The reservoir is subdivided into a small number of subregions, each one taken as homogeneous. The zonation is performed in an ad-hoc way before the matching (optimization) procedure starts, and remains fixed thereafter. The gradzone scheme proposed by Bissel (1994) improved upon the zonation method. Here, the zonation changes along the steps of the optimization procedure, according to the information contained in the current sensitivity matrix.

More precisely, the zonation is based on the first eigenvectors of , where is the Jacobian of the parameter to data function. Rodrigues (2005) has noticed that these eigenvectors are in fact the right singular vectors of . Using this observation, he has proposed a truncated singular value decomposition (TSVD) scheme, where the reduced parameter subspace is spanned by the first right singular vectors. Note that when is a matrix and is a vector in , the solution of the least square problem

involves the pseudo-inverse , which is given by the singular value decomposition (SVD) of . It is also well known that the truncated singular value decomposition (TSVD) is an efficient tool to numerically solve quadratic least square problem, see Hansen (1998). Therefore, it is natural to use TSVD for more general nonlinear optimization problems.

In fact, a nice theoretical support for using TSVD in history matching is presented in Tavakoli and Reynolds (2009). Using a Bayesian approach, the authors show that the reduction in uncertainty obtained from the knowledge of observed data can be estimated from the singular values of , suggesting that TSVD is an appropriated tool for reparameterization. The authors also present some numerical results in which a TSVD scheme perform efficiently when compared to a limited-memory Broyden-

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Fletcher-Goldfarb-Shanno (LBFGS) algorithm. Moreover, they propose a coupled LBFGS-TSVD scheme which outperforms LBFGS, in the sense that both schemes have comparable results, but TSVD-LBFGS is faster than LBFGS.

Most recently, various authors have investigated the use of 4D seismics in history matching. Since time-lapse seismics gives information over the whole reservoir, one might expect that this knowledge would complement the information provided by the spatially localized well data. Landa and Horne (1997) consider reservoirs that can be described by a small number of parameters and study the relative influence of the various types of data in the matching process. Gosselin et al. treat a real field case and conclude that the acquisition of 4D-seismics is particularly relevant for prediction. Emerick et al. (2007) point out that the introduction of seismic data reduces uncertainty in the parameter determination. Dong and Oliver (2005), using a LBFGS scheme, compare reservoir description using well data and well plus P-wave impedance data. They examine the impact of seismics when poor information from wells are available., in a synthetic and in a semi-synthetic cases. They do not recover some of the features of the true field. Nevertheless, the inclusion of seismics allows for a much better characterization.

In this work we use a TSVD scheme to integrate well and 4D seismic data. We model the 4D seismics using the P-wave velocity and Gassmann equations. Permeability is the only geophysical property we determine, in a 2D two-phase problem. Nevertheless, we believe that the features we describe will be present in more general situations.

We first discuss the structure of the singular value triplets produced by TSVD. Comparing the singular value distribution in the cases W (well data) and W+S (well plus 4D seismic data)., we see that the singular value decay of W+S is strongly attenuated. As a consequence, the problem is better conditioned and a large number of singular vectors may be used to produce better results. Surprisingly, though, in the case W+S the decay is accentuated when finer grids are considered. The opposite occurs for W. We don't have an explanation for this fact.

We next examine the structure of the right singular vectors., which allows for a better understanding of the behavior of the TSVD scheme. We remark that in the W+S case some unexpected features appear in the vectors configuration, and we give an explanation for that.

We finally study the performance of the TSVD scheme in the W and W+S cases in a model problem. In our experiments, the inclusion of seismics seems to provide enough information so as to fully characterize the reservoir, even in the absence of a priori information. At the same time it introduces a strong regularization into the inverse problem being solved. As a consequence, the optimization algorithm becomes much more stable. An important issue is the determination of the dimension of the reparameterized subspace, i.e., the number of right singular vectors to be considered. We compare the results of using a fixed number of singular vectors with a strategy of increasing the subspace dimension as proposed in (Tavakoli and Reynolds 2009). Our results indicate that this last approach in much more stable, especially in the absence of seismic data.

The rest of this paper is organized as follows. We first present the TSVD scheme and the treatment of the 4D seismics. We next describe the structure of the singular triplets. The performance of TSVD on our model problem is discussed subsequently. Some conclusions are presented in the last section. In the Appendix we prove some mathematical results concerning the solution of quadratic least square problems using SVD.

The TSVD schemeIn this section we describe our TSVD scheme. Let us denote by the set of model parameters to be determined and by the primary variables (in our case, saturation and pressure), so that . We call the set of observable data (in our case, bottomhole pressure, oil rate and P-wave velocity) and we suppose that is the observed data. If a priori information is available, a usual procedure is to minimize the objective function

00001()

where is the a priori model parameter, is the covariance matrix of data measurements errors and is the model covariance associated to the a priori information.

Equation 00001() can be viewed as a way of minimizing the mismatch between predicted and measured observations, honoring the a priori information. In a probabilistic framework, is associated to the probability density function (pdf) of conditioned to the observed data, and the minimization of 00001() gives the maximum a posteriori solution (MAP). We use a Gauss-Newton type iterative method to solve 00001(). This means that at -th step we replace by

, the linear approximation of around the current estimate . Calling , , and we are led to the least square problem

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00002()where

00003()

Consider , such that , . The normalized sensitivity matrix plays a fundamental role in the solution of 00002(). One can show, see the Appendix, that the corresponding minimum is given by

00004()where

is the singular value decomposition of normalized sensitivity matrix. Here, are orthogonal and is nonnegative and diagonal, being the dimension of the model parameter vector and the dimension of the observed data vector. However, for oil reservoir problems the computation of using 00004() would be usually too expensive, from one side, and unstable, from the other. A feasible alternative is to apply TSVD. It is shown in the Appendix that the truncation of 00004() given by

00005()

corresponds to the minimization of in the subspace spanned by , , , . This means that in 00005() gives the exact solution of 00004() restricted to the subspace spanned by the first RSVs. However, we found that updating

is not enough to obtain good convergence results. Instead, we perform an inexact line search in the direction to obtain as

00006()

In this way, the TSVD scheme goes as follows.

Initialize and compute , . Given , compute .

Compute the TSVD of . Compute given by 00005() Compute using 00006().

Here are some comments on the scheme. There is no uniqueness for the decompositions of and , one may consider, for example, Cholesky factorizations or square roots. The covariance matrix is usually considered to be diagonal, and then is easy to obtain. However, the computation of may be expensive. As for the SVD decomposition, we use the Lanczos method

(Golub and Van Loan 1989), which only requires the computation of the matrix-vector products and , where

. Thus, there is no need to assemble . On the other hand, the computation of involves . For a detailed

discussion of the resolution of , which is the adjoint (backward) linearized equation associated to the model problem being considered, see (Rodrigues 2005).

Before closing this section, let us make a last remark. Suppose there is no a priori information available, so that now reads

00007()

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We observe that now may have (infinitely) many minima and that the pseudo-inverse solution gives the one of least norm. Therefore, different scalar products will produce different pseudo-inverse solutions and a way of regularizing

the optimization problem is to consider alternative scalar products. Suppose that for some positive-definite matrix we use

00008()

as scalar product. We can then show that the corresponding TSVD approximate solution is given by

00009()

This formula should be compared with 00005(). It tells us that appropriate scalar products should produce analogous results as using a priori information and standard Euclidian norms.

The seismic model

Following (Dong and Oliver 2005), we use Gassmann equations to model the P-wave velocity as

where is the saturated rock density. is the bulk modulus and is the shear modulus. We don’t present here the formulas defining these quantities, which can be found in (Dong and Oliver 2005). We just mention that they depend explicitly on fluid saturation and pressure. Thus, it is simple to obtain the partial derivatives of with respect to the primary variables. In other words, the integration of seismics presents no major difficulty from the implementation point of view.

The synthetic problemIn the rest of this paper the reservoir will be given by a parallelepiped. We consider a 2D, two-phase water-oil 5-spot problem. Water is injected at the center of the reservoir, and oil is recovered through the four producers located at the corners. The fluids are supposed incompressible and immiscible, and no capillary effects are considered. The reservoir is isolated (no-flow boundary conditions), water injection rate at the injector well, and total production rate at the producers are prescribed.

In the history matching problem, we assume that permeability is the only parameter to be recovered. In fact, our mathematical model deals with log-permeabilities. The reservoir is assumed to be isotropic, so that one log-permeability value is attributed to each cell of the discretization grid. In this way, the dimension of the parameter space is equal to the number of grid blocks. Well history comprises bottomhole pressure (in all wells) and oil rate (in all producers) data. For the study of the influence of 4D seismics, we have considered the P-wave velocity as discussed previously.

Oil rate injection and total outflow rate at each producer are prescribed. The first breakthrough is around 65 days and the last one is around 175 days. Well data is recorded every 10 days and is acquired once, at day 250. The a priori covariance matrix

is taken to be , where is the (discrete) Laplacian operator. In this way, we are penalizing non-smooth fields.

TSVD structureIn this section we discuss the structure of the singular values (SVs) and of the right singular vectors (RSVs) of the model problem. We start by looking at the SV decay associated to a homogeneous field for two different grids: and , see Fig 1. This decay justifies the use of the TSVD scheme, since the most relevant information is concentrated in few RSV directions.

Consider the -th SV decay . Without seismics, , for the coarse grid, and for

the fine grid. When seismics is incorporated, and , respectively. We thus see that the SV initial decay is stronger for finer grids, when only well data is considered, , but the opposite holds when seismic data is integrated. When decay curves for the SVs associated with the nonhomogeneous field depicted in Fig. 2 are plotted, qualitatively equivalent results are found.

Let us now examine the nature of the RSVs. Fig. 3 shows the first 11 RSVs when no seismics is considered. The symmetry of the picture reflects the symmetry of the well locations, since the linearization is performed around a homogeneous reservoir and

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that the TSVD does not depend on the residual . We note that the first RSVs have their largest entries concentrated around the wells. This is due to the great sensitivity of the flow with respect to permeability in the vicinity of the wells. The main stream channels for the flow are also featured in these RSVs. Fig. 4 presents the first 11 RSVs when 4D seismics is integrated. The new feature is the appearance of horizontal and vertical channels. Surprising at first sight, this is actually the most efficient way to impact the saturation field, and hence . This is not what one wants to do if the residual associated to this region is small. This is precisely our case, since we don’t expect relevant flow in this region.

We consider now the RSVs (see Fig. 5) for the linearization around the permeability field depicted in Fig. 2. They now display some geometric features of the field (prominently the channel in the lower left). However, the previous remarks still apply: we observe concentration around some wells and vertical and horizontal channels associated with the seismics.

Numerical resultsIn this section we discuss some numerical experiments. In the results presented here we have considered the 2D two-phase 5-spot

problem previously described. The reservoir is , and the permeability map is as in Fig. 2. The permeability ranges from mD up to mD at the high permeability channel. Water injection rate and total flux at producers are prescribed. Bottomhole pressure and oil outflow is measured every 10 days, for 400 days. Pressure is measured at all wells and oil outflow is measured at each producer. The field corresponds to day 250. To minimize given by 00007(), we apply the TSVD scheme with , i.e., by taking 00008() as scalar product. As discussed before, this is essentially equivalent to considering the existence of a priori information associated to the covariance matrix . Concerning the number of singular triplets , we consider the case where is fixed a priori and the case where is initially set equal to 6 and increases by a fixed value at each iteration up to (we refer to the latter case as 6: : ).

We first consider the case without seismics. Fig. 6 shows the results from runs corresponding to fixed equals to 10 and 25, and to incremental equals to 2 and 3. In parenthesis, we present the norm of the error between the permeability field obtained and the true field, , and the quotient between the last and the initial values of the objective function. We see that the smoothing introduced by the Laplacian was not enough to stabilize the TSVD scheme for . Note that in spite of the poor representation of the reservoir, the matching is very good, see Fig. 7. Note also the regularizing effect of the increasing strategy. We would like to remark that Tavakoli and Reynolds (2009) report the need to introduce two regularizations in their TSVD scheme. They modify the pseudo-inverse solution in a way that is inspired by the Levenberg-Marquardt method. They also restrict in order to avoid large changes in the first steps. In this way, they are able to obtain stable solutions for fixed. They also propose the increasing strategy as a way of reducing computational costs. We would like to point out that the increasing strategy is also quite robust.

The strong smoothing effect when integrating seismics is presented in Fig. 8. The TSVD was stable even with the fixed strategy. We were able to obtain a very sharp representation of the truth using increasing up to 100. Fig. 9 and Fig. 10 display the oil rate and the P-wave velocity matchings. We observe very good agreements both for fixed =25 and for variable

.

ConclusionsWe considered a TSVD scheme to solve history matching problems. in the case where the history comprises pressure and oil flow rate at the wells and in the case where the P-wave velocity field is also available. A comparative analysis of the structure of the singular values triplets allows for a better understanding of the mechanisms driving the scheme.

Some numerical experiments for recovering a permeability field in a 2D two-phase model problem were performed. The integration of seismics has a strong regularizing effect on the optimization algorithm. It also yields better solutions, in particular it allows for a better resolution of the discontinuities of the true field. Another regularization effect is provided by the strategy of gradually increasing the dimension of the parameterizated TSVD subspace This is especially relevant in the case where there is no integration of 4D seismic data.

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Nomenclature= number of observations= number of parameters= log-permeability= true log-permeability field= number of singular triplets= incremental number= P-wave velocity vector= data covariance matrix= a priori covariance matrix= sensitivity matrix= objective function= objective function at -th iteration

= left orthogonal SVD matrix = right orthogonal SVD matrix

= truncated right orthogonalmatrix = singular value= singular value diagonal matrix

Abbreviations pdf = probability density function SV = singular value SVD = singular value decompositionTSVD = truncated singular value decomposition

AcknowledgementsWe are grateful to Petrobras for the support provided for the development of this work.

References

Bissell, R. 1994. Calculating optimal parameters for historymatching. Proc. 4th European Conference on the Mathematics of Oil Recovery.

Chen, Y., Oliver, D.S. 2010: Cross-covariances and localization for EnKF in multiphase flow data assimilation, Computat. Geosci. 14(5), 579--601.

Dong, Y., Oliver, D.S. 2005: Quantitative use of 4D seismicdata for reservoir description, SPE J. 10(1), 91--99.

Emerick, A.A. Moraes, R.J. and Rodrigues, J.R.P, 2007 History Matching 4D Seismic Data with Efficient Gradient Based Methods, (SPE 107179), Proc. SPE EUROPEC, London, UK, 2007.

Golub, G.H., van Loan, C.F., 1989: Matrix Computations, 2ndedn. The Johns Hopkins University Press, Baltimore.

Gosselin O., Aanonsen S.I., Aavatsmark I,, Cominelli A., Gonard R., Kolasinski R., Ferdinandi F., Kovacic L., Neylon K., 2003: History Matching Using Time-lapse Seismic, (SPE 84464-MS), Proc. SPE Annual Technical Conference and Exhibition.

Hansen, C. 1998: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation 4.

Jacquard, P., Jain, C. 1965: Permeability distribution from field pressure data, SPE J. 5(4), 281--294.

Jahns, H.O. 1966: A rapid method for obtaining a twodimensional reservoir description from well pressure responsedata. SPE J. 6(12), 315--327.

Landa, J.L., Horne, R.N. 1997: A procedure to integrate well test data, reservoir performance history and 4-D seismic informationinto a reservoir description (SPE-38653). Proc. 1997SPE Annual Technical Conference and Exhibition

Oliver D.S., Chen Y. 2010: Recent progress on reservoir history matching: a review, Computat. Geosci. DOI: 10.1007/s10596-010-9194-214.

Oliver, D.S., Reynolds, A.C., Liu, N. 2008: Inverse Theory for Petroleum Reservoir Characterization and History Matching,1st edn. Cambridge University Press, Cambridge.

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Rodrigues, J. R. P. 2005: Calculating derivatives for history matching in reservoir simulators (SPE-93445), Proc .SPE Reservoir Simulation Symposium.

Rodrigues, J. R. P. 2006: Calculating derivatives for automatic history matching, Computat. Geosci., 10, 119--136.

Tavakoli, R., Reynolds, A.C. 2009: History Matching With Parametrization Based on the SVD of a Dimensionless Sensitivity Matrix (SPE 118952), in Proc. SPE Reservoir Simulation Symposium.

AppendixConsider the matrices , , , let be a vector and be a vector. Assume that

, are positive-definite matrices, with decompositions , . We discuss here the minimization of

000010()We have the following result.

Proposition - has a unique minimum over the whole space, given by

000011()where

with , and is diagonal and nonnegative.

Furthermore, the minimization of restricted to the truncated subspace spanned by , , has also a unique solution, given by

000012()

where and are the submatrices formed by the first columns of and , respectively.

Proof.

We note that 000011() is a particular case of 000012(), for , so that it suffices to show 000012(). We write 000010() as

000013()

Taking it follows from that

Since is orthogonal, for all . Using this we write

We next note that, since the columns of are orthogonal, and , so that for all , . Using this we get

Hence,

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Thus,

so that

Therefore,

This ends the proof.

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Fig. 1: semi-log10 plot of singular value distribution.

Fig 3: right SVs for a homogeneous field, without seismics.

Fig. 2: true log-permeability field.

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Fig 4: right SVs for a homogeneous field, with seismics

Fig 5: right SVs for the true nonhomogeneous field, with seismics.

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true initial ( 0.33 / 1.0e+0 )

inc. 6:2:50( 0.15 / 8.9e-6 )

inc. 6:2:25( 0.17 / 1.6e-4 )

fixed 10 ( 0.24 / 4.8e-3 )

fixed 25 ( 0.43 / 5.7e-3 )

inc. 6:3:25 ( 0.21 / 7.0e-4 )

Fig. 6: log-permeability fields obtained without seismics and with Laplacian. “fixed n” stands for a fixed number n of s.v.’s “inc 6 m-n” stands for an incremental m from 6 up to n number of SVs . The numbers in parentheses are the relative errors between the displayed and the true fields.

Fig. 7: oil rate distribution at the well associated to the high permeability channel.

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true inc. 6:3:25 ( 0.17 / 4.6e-3 )

inc. 6:3:50( 0.15 / 1.3e-3 )

fixed 10 ( 0.18 / 5.2e-2 )

fixed 25 ( 0.20 / 6.2e-3 )

Fig. 8: log-permeability fields obtained with seismics and Laplacian. “fixed n” stands for a fixed number n of s.v.’s “inc 6 m-n” stands for an incremental m from 6 up to n number of SVs . The numbers in parentheses are the relative errors between the displayed and the true fields

Fig. 9: oil rate distribution at the well associated to the high permeability channel.

true initial fixed 25 inc. 6:3:25

Fig. 10: P-wave velocity fields

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