An IRBS for data inversion in reservoir modeling - School of

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Inverse Problems 25 (2009) 035006 (34pp)

doi:10.1088/0266-5611/25/3/035006

An iterative representer-based scheme for data inversion in reservoir modeling Marco A Iglesias and Clint Dawson Center for Subsurface Modeling-C0200, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712, USA E-mail: [email protected]

Received 30 November 2008 Published 15 January 2009 Online at stacks.iop.org/IP/25/035006 Abstract In this paper, we develop a mathematical framework for data inversion in reservoir models. A general formulation is presented for the identification of uncertain parameters in an abstract reservoir model described by a set of nonlinear equations. Given a finite number of measurements of the state and prior knowledge of the uncertain parameters, an iterative representer-based scheme (IRBS) is proposed to find improved parameters. In this approach, the representer method is used to solve a linear data assimilation problem at each iteration of the algorithm. We apply the theory of iterative regularization to establish conditions for which the IRBS will converge to a stable approximation of a solution to the parameter identification problem. These theoretical results are applied to the identification of the second-order coefficient of a forward model described by a parabolic boundary value problem. Numerical results are presented to show the capabilities of the IRBS for the reconstruction of hydraulic conductivity from the steady-state of groundwater flow, as well as the absolute permeability in the single-phase Darcy flow through porous media. (Some figures in this article are in colour only in the electronic version)

1. Introduction The main challenge of reservoir management is to achieve the maximum hydrocarbon recovery under the minimum operational costs. This goal cannot be accomplished without a reservoir simulator capable of predicting the reservoir performance. However, the reliability of reservoir models depends on the accuracy of the knowledge of the subsurface properties. These properties are highly heterogeneous and can be measured at few locations. Therefore, in order to obtain an accurate reservoir model, it is essential to develop techniques that allow us to characterize the petrophysical properties of the subsurface. 0266-5611/09/035006+34$30.00 © 2009 IOP Publishing Ltd Printed in the UK

1

Inverse Problems 25 (2009) 035006

M A Iglesias and C Dawson

Automatic history matching is one of the aspects involved in the characterization of the petrophysical properties of a reservoir. The aim of history matching is to obtain estimates of the petrophysical properties, so that the model prediction fits the observed production data. Traditionally, data acquisition for history matching was available by well-logging techniques which were expensive and required well intervention. However, after 1997 [15], the development of smart-well technology has dramatically changed the restrictive scenario for data acquisition. With the aid of down-hole permanent sensors, it is possible to monitor production data in almost real time. These data may be used for history matching in order to obtain a better characterization of the reservoir which in turn can be used to optimize recovery. Unfortunately, classical history matching techniques are not able to process large sets of data under reasonable computational cost. For this reason, the reservoir community has started to implement techniques that have been previously used in other areas. Such is the case in physical oceanography and meteorology [6, 8, 13, 30, 34], where several data assimilation technologies have been developed to overcome analogous problems to those faced by the reservoir community. Within the broad methods for data assimilation, we distinguish between two main approaches that have been applied to the reservoir modeling application: the Monte Carlotype or probabilistic framework [12] and the variational approach [6]. The former consists of generating an ensemble of different possible realizations of the state variable with given prior distribution. The model is run for each member of the ensemble up to the first measurement time. Then, each state variable is updated with standard formulae of Kalman filtering theory [22] to produce a posterior distribution which is used to continue the process. In the Kalman filter-type methods, the model simulator is used as a black box without the need of an adjoint code. This feature offers a significative advantage in history matching for which several applications of the ensemble Kalman filter (EnKF) have been conducted [9, 14, 17, 18, 24–27, 31]. However, there are still open questions arising from the EnKF implementation. Unphysical solutions, overestimation and poor uncertainty quantification have been reported with the EnKF [17, 18]. Despite attempts to remediate these undesired outcomes, the nonlinearity of reservoir models clearly goes beyond the capabilities of the EnKF. In the variational approach, the solution is sought as a minimizer of a weighted leastsquares functional that penalizes the misfit between model-based and real measurements. The corresponding minimization involves the implementation of an optimization technique which typically requires the development of an adjoint code for the efficient computation of the derivatives of the cost functional. It is worth mentioning that many variational approaches for history matching have been studied during the last few years. However, those approaches are suboptimal and computationally expensive because of the highly nonlinear and large-scale reservoir models. For this reason, variational ‘data assimilation’ techniques have received attention for reservoir and groundwater applications [2–4, 21, 29, 33]. In linear variational ‘data assimilation’, an inexpensive minimization of a weighted least-squares cost functional can be conducted by the so-called representer method [6]. Assuming that the measurement space is finite dimensional, the representer method provides the solution to the Euler–Lagrange equations (E–L) resulting from the variational formulation. For nonlinear problems, the representer method cannot be implemented directly. Nevertheless, some representer-based techniques have been explored to compute an approximation to the solution of the E–L equations [3, 29]. However, these approaches do not necessarily yield a direction of descent of the cost functional. Therefore, an additional technique such as line search is needed to accelerate or even obtain convergence [3, 29, 33]. Unfortunately, these additional implementations may compromise the efficiency required for reservoir applications. 2



In this paper, we present a novel implementation of the representer method as an iterative regularization scheme. The problem is posed in terms of a parameter identification problem where the objective is to find a parameter such that the model output fits the data within some error. In this formulation, a solution (identified parameters) is the limit of a sequence of solutions to linearized inverse problems. Each of these inverse problems is the minimization of a weighted least-squares functional that incorporates prior knowledge of the reservoir. In addition, we applied standard techniques from iterative regularization theory [10, 19] to obtain convergence results for the proposed scheme. Both analytical and numerical results show that this methodology is efficient, robust and therefore promising for the estimation of petrophysical properties in reservoir models. The robustness of the presented methodology is ensured by convergence results that were first developed by Hanke [19] in a more general framework. However, in our approach, we consider an implicit system (see equation (2)) instead of the parameter-to-observation map of [19]. The reason for this selection comes from the fact that the reservoir model is commonly described by a set of PDEs for which energy inequalities may be available. Second, we always assume that the observation space is finite dimensional so that the representer method can be applied. In contrast to [19], with the aforementioned property we do not need to interpolate the data which always come as a finite-dimensional set. Third, in our formulation we incorporate prior knowledge which is relevant to the application. Despite some subtle differences between our methodology and the one presented in [19], we show that our approach is a particular case of Hanke’s work in the case of a fixed Levenberg–Marquardt parameter α = 1 and an appropriate selection of the parameter norm. The outline of the paper is the following. In section 2, with the aid of the standard theory of parameter identification [10], we introduce an abstract formulation for parameter identification in reservoir models. In section 2.6, an iterative representer-based scheme (IRBS) is proposed for the computation of a solution to the parameter identification problem. Since it is assumed that the observation space is finite dimensional, we apply the representer method that we derive in section 2.7. In section 2.8, we discuss the required hypotheses for convergence of the IRBS. In section 3, we study an application of the IRBS to the identification of the second-order coefficient in a parabolic boundary value problem. For this case, the implementation of the IRBS is discussed in section 3.1. Finally, in section 4 we show some numerical experiments to determine the capability of the IRBS for the reconstruction of subsurface properties. 2. Data inversion in reservoir models In order to illustrate our approach, we first present a model problem that will be used in the subsequent sections for the validation of IRBS. We consider single-phase Darcy flow through porous media which is described by the following equation [5]: cφ

1 ∂p − ∇ · eY ∇p = f, ∂t μ

(1)

where c is the compressibility of the fluid, φ is the porosity, μ is the viscosity of the fluid and f is an injection/production rate. K = eY is the absolute permeability and p is the pressure. When c, μ, φ, K are given and some initial and boundary conditions are prescribed, then (1) can be solved uniquely for p in the appropriate spaces [23]. In this model, φ and K are the petrophysical properties associated with the reservoir. In practical applications, these parameters are ‘uncertain’ in the sense that only prior knowledge is available. In general, this knowledge does not capture the characteristics required for an accurate description of the reservoir. Therefore, if those parameters were utilized in (1), the solution would be different 3



from the measurements collected at the wells. Then, we need to address the following inverse problem. Given measurements collected at the well and prior knowledge of the petrophysical properties find improved parameters such that the model matches the measurements from the reservoir. 2.1. Preliminaries and notation

N 2 1/2 For any vector v ∈ RN we denote by v = the Euclidian norm. For i=1 |vi | Av N×N A ∈ R we also denote by A = supv=0 v . For any symmetric positive-definite matrix C ∈ RN×N , we denote by vC = (v T Cv)1/2 the corresponding induced norm in RN . Given a Hilbert space H, we denote the closed ball centered at k of radius r by B(r, k). Furthermore, the inner product of H is denoted by ·, ·H . Given ⊂ RN and 0 < T < ∞, we define = ∂, T = × (0, T ) and T = × (0, T ). We consider [23] the usual α 2 1/2 , and Sobolev spaces H 2 () with the norm given by uH 2 () = |α|2 D u 1/2 T . Additionally, H 1 (0, T ; L2 ()) with the norm uH 1 (0,T ;L2 ()) = 0 u(t)2L2 () dt we define the space H 2,1 (T ) = L2 (0, T ; H 2 ()) ∩ H 1 (0, T ; L2 ()) with the norm 1/2 T uH 2,1 (T ) = 0 u(t)2H 2 () dt + u2H 1 (0,T ;L2 ()) . Finally, we define H 1/2,1/4 (T ) = T L2 (0, T ; H 1/2 ()) ∩ H 1/4 (0, T ; L2 ()) with the norm uH 1/2,1/4 (T ) = 0 u(t)2H 1/2 () dt + 1/2 where H 1/2 () and H 1/4 (0, T ; L2 ()) are interpolation spaces defined u2H 1/4 (0,T ;L2 ()) in the sense of definition 2.1 in [23]. 2.2. The forward model The objective of this section is to develop a general mathematical framework for the estimation of uncertain parameters in a reservoir model described by an abstract nonlinear system in Hilbert spaces. Let us represent the reservoir by an open bounded and convex set ⊂ Rl with l = 2, 3 with C 1 boundary. Given some 0 < T < ∞, suppose that [0, T ] is the time of interest for the analysis of the reservoir dynamics. Assume that the reservoir has N uncertain parameters. The corresponding parameter space is defined by K = N i=1 Ki where each Ki is a linear subspace either of L2 () or L2 (T ). Let S and Z be arbitrary subspaces of L2 (T ). The set S is the space of all possible states of the reservoir. We now assume that the dynamics of the reservoir can be described, over a finite interval of time [0, T ], by a set of (forward) model equations G(k, s) = 0,

(2)

where G : K × S → Z is a nonlinear operator. For a fixed parameter k ∈ K, an element of the state space s(x, t) ∈ S that satisfies (2) is the corresponding state of the reservoir which simulates the reservoir dynamics over the interval of time. Parameters which are not considered uncertain in the sense described above are assumed to be contained in the functional relation chosen for the operator G. The subspaces S , K and Z , in addition to the functional relation for G, must be defined for the particular reservoir model of interest. 2.3. Measurements and prior knowledge Each measurement is treated independently at each location and time of collection. Let us η η suppose that we are given d η = d1 , . . . , dM a vector of M measurements of the state of the η reservoir, such that each dm was collected at (xm , tm ). In general, measurements are corrupted 4



by noise. In other words, the vector of measurements d η is only an approximation to the measurements of the state, that is d η − dC −1 η,

(3)

where d = [d1 , . . . , dM ] is the vector of perfect (noise-free) measurements, and the norm in (3) is weighted by the inverse of a positive-definite covariance matrix C. The observation process responsible for a perfect measurement dm is represented, on the space S , by the action (Lm (s) = dm ) of a linear functional Lm : S → R. Furthermore, prior knowledge information for each uncertain parameter ki is required to be given as a prior mean k i and the corresponding prior error covariance function Cki . In summary, the following information must be provided: η η (D1) d η = d1 , . . . , dM a vector of M (possibly corrupted by noise) measurements of the state of the reservoir at {(xm , tm )}M m=1 . (D2) Prior measurements (error) covariance matrix C (positive definite). (D3) Prior knowledge of parameters k = [k 1 , . . . , k N ]. (D4) Parameters prior error covariance operators {Ck1 , . . . , CkN }. (D5) Expressions for the linear functionals L = [L1 , . . . , LM ] representing the process from which d η = [d1 , . . . , dM ] was obtained. 2.4. Fundamental definitions Inspired by the standard theory of parameter identification [10, 19], we now formulate Definition 2.1. Assume that the state of the system is described by (2) and the measurement process is represented by L = [L1 , . . . , LM ] as described above. Given a vector d = [d1 , . . . , dM ] of exact measurements (noise free) of the state of the system, a solution to the parameter identification problem (PIP) is an element k ∈ K, such that G(k, s) = 0,

and

L(s) = d

(4)

for some s ∈ S . Note from the previous definition that no assumption has been made on the uniqueness of the solution to the PIP. Furthermore, as we indicated earlier, usually the vector of measurements d is not available but only a noisy version d η given by (3). In this case, the computation of a solution to the PIP will be understood as a stable approximation in the limit η → 0. Suppose for a moment that we are given a set of perfect (noise-free) measurements. If in addition we assume that the model dynamics and measurement process are represented correctly by the mathematical models under consideration, then, it is reasonable to think that the measurements were obtained from the ‘true state’ sT of the system which satisfies the model dynamics (2) for some ‘true parameter’ kT . For this reason we postulate the following: Attainability condition of data. There exists at least one solution kT ∈ K to the PIP. We refer to kT as the ‘true parameter’. For example, suppose we are interested in the identification of K in model (1) given pressure data from some well locations (assuming that c, φ and μ are perfectly known). It is clear that there may be several permeability configurations that will generate the same pressure response which is measured at the wells. However, the reservoir has one and only one permeability field which is responsible for the generation of the collected measurements. 5



2.5. Mathematical assumptions For the subsequent analysis consider the following assumptions on the (forward model) operator G: (F1) G : K × S → Z is continuously Frechet differentiable. (F2) For each k ∈ K, there exists a unique s ∈ S with G(k, s) = 0. ˜ s˜ ) = 0, we assume Ds G(k, ˜ s˜ ) : S → Z is a ˜ s˜ ) ∈ K × S , that satisfies G(k, (F3) For all (k, linear isomorphism. In addition, for every r > 0 and k∗ ∈ K we assume that there exists a constant C1 such that for every k ∈ B(r, k∗ ) and s ∈ S that satisfy G(k, s) = 0, the solution δs to Dk G(k, s)δk + Ds G(k, s)δs = 0

(5)

δsS C1 δkK

(6)

satisfies ˜ for every δk ∈ K. (Note that C1 is independent of k ∈ B(r, k)). (F4) Let us denote Gr the residual of G defined by the following relation, ˜ s˜ )[k − k] ˜ + Ds G(k, ˜ s˜ )[s − s˜ ] + Gr (k − k, ˜ s − s˜ ). ˜ s˜ ) + Dk G(k, G(k, s) = G(k, ˆ then Given kˆ fixed. There exist C2 > 0 and δ > 0 such that if k, k˜ ∈ B(δ, k) ˜ K L(s) − L(˜s ) L(sr ) C2 k − k where sr is the solution to ˜ s˜ )sr = Gr (k − k, ˜ s − s˜ ), Ds G(k, ˜ s˜ ) = 0 = G(k, s). s and s˜ are the (unique) solutions to G(k,

(7) (8) (9)

More general assumptions can be found in [19]. We now state some conditions on the assimilation information. By definition of the covariance operator, Cki : Ki → Ki is positive definite. Let us define the formal inverse [32]: Cki (ξ, χ )Ck−1 (χ , ξ ) dχ = δ(ξ − ξ ) (10) i

where is either or T . It is not difficult to see that φ(ξ )Ck−1 (ξ, ξ )ψ(ξ ) dξ dξ φ, ψCki ≡ i

(11)

defines an inner product on Ki for each i = 1, . . . , N. For this inner product, we consider the following assumption. Assumption (A1). For i = 1, . . . , N the space Ki is a Hilbert space with the induced norm ·Cki . Then, from standard functional analysis K = N i=1 Ki is a Hilbert space with the norm induced by N φ, ψK ≡ φi , ψi Cki .

(12)

i=1

In the context of reservoir modeling (model (1) for example), the prior error covariance reflects prior knowledge of the spatial variability of petrophysical properties. In real applications, the covariance may be determined from well logs, core analysis, seismic data and the geology of the formation. 6



2.6. An iterative representer-based scheme In order to find an approximation to a solution of the PIP (definition (2.1)), we propose the following algorithm. Assume we are given k 0 = k, an estimate of the error level η, prior error covariances C, {Cki }N i=1 and some τ > 1. For n = 1, . . . (1) Solution to the forward model. We compute sFn the solution to the forward model (13) G k n−1 , sFn = 0.

η n

n−1

(2) Stopping rule (discrepancy principle). If d − L sF C −1 τ η, stop. Output: k . (3) Parameter update by means of data assimilation with representers. Define k n as the solution to the linearized data assimilation problem J (s, k) = [d η − L(s)]T C −1 [d η − L(s)] + k − k n−1 2K → min

(14)

subject to

LG(k, s) = Dk G k n−1 , sFn [k − k n−1 ] + Ds G k n−1 , sFn s − sFn = 0.

(15)

In the following sections, we develop sufficient conditions for which this algorithm converges to a stable approximation of the solution to the PIP. 2.7. Step (3): solution to the linearized problem by the representer method Now we prove that problems (14) and (15) admit a unique solution. Moreover, an explicit formula for k will be provided in terms of ‘representers’. Note that (15) is nothing but the linearization of the reservoir model around k n−1 , sFn . The first term of (14) is a weighted least-squares functional that penalizes the misfit between real and predicted measurement. The second term accounts for the error between prior and estimated parameters. 2.7.1. The ‘representers’. We define the following variables: k = k − k n−1 and s = s − sFn . Then, problems (14) and (15) are equivalent to the minimization of T J n (s, k) = d η − L sFn − L(s) C −1 d η − L sFn − L(s) + k2K (16) subject to the linearized constraint Dk G k n−1 , sFn k + Ds G k n−1 , sFn s = 0.

(17)

We now state the following Lemma 2.1. Assume (A1) and (F1)–(F3) hold. Let r > 0 and k∗ ∈ K arbitrary. Take C1 given by (F3) for B(r, k∗ ). If k n−1 ∈ B(r, k∗ ), then, for each m ∈ {1, . . . , M}, there exists a ‘representer’ γmn ∈ K such that n γm , k K = Lm (s), (18) for all k ∈ K, where s is the solution to (17). Furthermore, the‘representer matrix’ defined by [R n ]ij = γin , γjn K (19) satisfies R n C12 L2

(20)

with L = supi Li . 7



Proof. By the invertibility of Ds G k n−1 , sFn we may consider the linear operator W n : K → S such that W n (k) = s where s is the solution to (17). Since k n−1 ∈ B(r, k∗ ), by hypothesis (F3) there exists C1 such that W n (k)S = sS C1 kK

(21)

for every k ∈ K. Then, by linearity and continuity of Li ◦ W we are allowed to apply the Riez representation theorem. Thus, for each i = 1, . . . , M, we assume the existence of γin such that n γi , k K = [Li ◦ W n ](k) = Li (s), (22) n

where s and k are related by (17). Then, we have proved (18). On the other hand, from definitions (19), (21) and (22) it follows that

|[R n ]ij | γ n γ n Li Lj W n 2 C 2 L2 . (23) i

K

j

K

1

which proves (20).

The following theorem establishes local existence and uniqueness of the nth step of the IRBS. Theorem 2.1. Assume (A1) and (F1)–(F3) hold. Let r > 0 and k∗ ∈ K arbitrary. Take C1 given by (F3) for B(r, k∗ ). If k n−1 ∈ B(r, k∗ ) and {Lm |W n (K) }M m=1 is linearly independent, then R n is positive definite and k(ξ ) = k n−1 (ξ ) + [γ n (ξ )]T [R n + C]−1 d η − L sFn (24) n n n n is the unique minimizer of J (14) subject to (15) in K, where γ (ξ ) = γ1 (ξ ), . . . , γM (ξ ) .

M Proof. From lemma 2.1 we assume the existence of the representers γmn m=1 . Clearly, (19) implies that R n is positive semidefinite. Moreover, if x T R n x = 0 for some x = (x1 , . . . , xM ) ∈ RM , then, from (19) and the linearity of the inner product it follows that

2 M

M M M

n n

xj Rj,m xm = xj γjn , xm γmn =

x γ (25) j j = 0.

j =1 m,j j =1 m=1 K n M From (18) it follows that Therefore, j =1 xj γj , k K = 0 for all k ∈ K. M n x L (s) = 0 for all s ∈ W ( K ). Since by hypothesis {Lm |W n (K) }M j =1 j j m=1 is linearly n independent, we conclude that x = 0 and so R is positive definite. Now we consider the following decomposition of K: K = span{γ n } + span{γ n }⊥ where, M M n n . Then, for every k ∈ K, span{γ } ≡ k ∈ K|k = M m=1 βm γm for some {βi }i=1 ∈ R k =

M

βm γmn + b

(26)

m=1

where

n γi , b K = 0

for all i = 1, . . . , M. If we substitute (26) into (22) and apply (27) we find n n n j βj γi , γj K = [Li ◦ W ](k) = Li (s). Therefore, from definition (19) R n β = L(s)

(27)

γin , k K

=

(28)

where L = [L1 , . . . , LM ] and β = [β1 . . . , βM ]. We now substitute (28) and (26) into (16) and (17). Then we use (27) and the definition of the representer matrix (19) to find that the minimization problem is posed in terms of T J (β, b) = d η − L sFn − R n β C −1 d η − L sFn − R n β + β T R n β + b2K (29) 8



over the set C = RM × b ∈ K γin , b = 0 . Since R n is positive definite, it is not difficult to see that J n is a strict convex functional. Additionally it is clear that C is a convex subset of RM × K. Then, it follows from standard optimization theory that any local minimum of J n is a unique global minimum over C . Therefore, we proceed to compute the first-order optimality condition which characterizes a global minimizer over C . Note that T 1 DJ n (β, b)[δβ, δb] = − d η − L sFn − R n β C −1 R n δβ + β T R n δβ + b, δbK . (30) 2 From (19) it is obvious that R n is positive definite. Since by definition C is also positive definite, then R n and R n + C are invertible. Therefore, DJ n = 0 implies (β, b) = ([R n + C]−1 d η − L sFn , 0). (31) We now use (31) in (26) which in turn, by the definition of k yields (24).

Remark 2.1. It is essential to note that the results from theorem 2.1 do not depend on the accuracy of the measurement vector d η . Expression (24) can be used with d η replaced by d (noise-free data). Analogously, the inverse of the covariance matrix C may be replaced with any positive-definite matrix. However, for practical applications, measurements will always be subject to errors and the error covariance plays an important role in the estimation. For this reason we have weighted the measurement misfit (see (14)) with the inverse of the covariance matrix, as is typically done in standard data assimilation [6], where J n may be derived from a Bayesian formulation for the linearized problem. Remark 2.2. The assumption of linear independence of the measurement functionals can be interpreted as a condition that prevents us from having redundant measurements. 2.8. Convergence properties The convergence results for the IRBS defined in section 2.6 can be summarized in the following theorem which can be proven by showing an equivalence to a particular case of the results presented in [19] (see also [10]). Theorem 2.2 (Convergence of the IRBS). Let d η be a vector of noisy measurements for η ∈ (0, η∗ ] where η∗ > 0 is the largest noise level given a priori. Let C ∈ RM×M be the (positive-definite) covariance matrix corresponding to d η∗ . Let {λi }M i=1 be the eigenvalues of C and we define α ≡ max λi / min λi . Suppose that d η − dC −1 η where d is the exact n n = 0. Let us assume be the solution to G kηn , sF,η data. Let kηn be defined by (24) and let sF,η (A1) and (F1)–(F4) and let k∗ be a solution to the PIP. For kˆ = k∗ , take C2 and δ > 0 as in (F4). Consider B(δ/2, k∗ ) and the corresponding constant C1 as in (F3). Let τ > 1/ρ with ρ defined by 1 ρ≡ 2 . (32) 2 C1 L C−1 + 1 If k 0 = k satisfies

k∗ − k K < min δ/2, 0

ρτ − 1 ρ , , α 1/2 C2 (1 + τ ) 2α 1/2 C2

(33)

then (discrepancy principle) there exists a finite integer n = n(η) such that d η −

n(η)

L sF C −1 τ η and d η − L sFm C −1 > τ η for m < n(η). Moreover, kηn(η) converges to a solution of the PIP as η → 0. Proof. See the appendix. 9



The previous theorem establishes conditions on the forward model, measurement functional and prior information so that the IRBS will converge to an approximate solution of the PIP (definition 1). 3. Application of the methodology for a system described by a parabolic boundary value problem In the previous section, we presented a variational formulation for parameter identification in a system that is described by (2) where the operator G satisfies hypotheses (F1)– (F4). In this section, we focus on a parabolic system described by the following operator G : C 1 () × H 2,1 (T ) → L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 (): ⎤ ⎡ ∂p − ∇ · eY ∇p − f ∂t (34) G(Y, p) = ⎣ −eY ∇p · n − BN ⎦ . p(x, 0) − I (x) As we indicated earlier this operator (see equation (1)) has important applications in subsurface flow modeling. Another interesting example can be obtained if eY is the hydraulic conductivity of an aquifer. Then, G(Y, p) = 0 may take the form of the PDE system that describes the groundwater flow. Therefore, the inverse estimation of Y is of great relevance for the subsurface community. For this reason we have chosen these problems in order to validate the IRBS. Before we discuss the numerical implementation of the IRBS for the aforementioned model, we first derive the conditions under which the operator G, the prior and measurement information satisfy the hypothesis required for convergence. We start by recalling the following theorem from standard PDE theory, Theorem 3.1. Let K ∈ C 1 () such that 0 < K∗ K(x) in . For any (f, BN , I ) ∈ L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 (), there exists a unique weak solution h ∈ H 2,1 (T ) to the following problem: ∂h − ∇ · K∇h = f ∂t ∇h · n = BN on h(x, 0) = I

in

× [0, T ],

∂ × [0, T ],

in .

(35) (36) (37)

Moreover, h satisfies hH 2,1 (T ) C[f L2 ((0,T );L2 ()) + BN H 1/2,1/4 (T ) + I H 1 () ],

(38)

where C is a constant that only depends on an upper bound for KC 1 () , K∗ , and T. Proof. See [23]. Let CY be a covariance operator such that Y Y ≡ Y CY−1 Y

(39)

is equivalent to a Sobolev norm H 3 () for N = 2, 3. Then, from the Sobolev imbedding theorem, H 3 () → C 1 (). Let us define the space Y to be the completion of C 1 () with the norm · Y . Then assumption (A1) is satisfied. Furthermore, the previous theorem can be applied for every Y ∈ Y . Therefore, we deduce the following. 10



Corollary 3.1. Let (f, BN , I ) ∈ L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 () be fixed. Take Yˆ ∈ Y arbitrary and r > 0. Consider B(r, Yˆ ) in Y . Then, there exists a constant C ∗ > 0 such that for every Y ∈ B(r, Yˆ ), the solution p to G(Y, p) = 0 satisfies pH 2,1 (T ) C ∗ [f L2 ((0,T );L2 ()) + BN H 1/2,1/4 (T ) + I H 1 () ]

(40)

where C ∗ depends only on T , , Yˆ and r. In addition, for any Y ∈ B(r, Yˆ ) we have that eY C 1 () ω

(41)

where ω is a constant that depends only on Yˆ and r. Proof. For all Y ∈ B(r, Yˆ ), we have that Y − Yˆ Y r. By the Sobolev imbedding we know that Y − Yˆ C 1 () r.

(42)

Trivially, |Y (x) − Yˆ (x)| r for all x ∈ . Then Y∗ ≡ −r + min Yˆ −r + Yˆ (x) Y (x) r + Yˆ (x) r + max Yˆ ≡ Y ∗ . (43)

A similar argument shows DY∗ ≡ −r + min Yˆ −r + Yˆ (x) Y (x) r + Yˆ (x) r + max Yˆ ≡ DY ∗ .

(44)

Note that the minimum and maximum above exist since Yˆ and Yˆ are continuous and defined on which is compact. Then, Y∗ Y (x) Y ∗ for all x ∈ and so 0 < eY∗ eY (x) eY

∗

(45)

for all x ∈ and for all Y ∈ B(r, Yˆ ). In addition, from (45) and (44), DY∗ eY∗ Y (x) eY (x) ∗ ∗ DY ∗ eY . Thus, |Y (x) eY (x) | min{−DY∗ eY∗ , DY ∗ eY } from which it follows ∗

∗

eY C 1 () = max[|eY (x) | + |Y (x) eY (x) |] eY + min{−DY∗ eY∗ , DY ∗ eY } ≡ ω,

(46)

where clearly ω depends only on Yˆ and r. Then, for every Y ∈ B(r, Yˆ ), ω is an upper bound of eY C 1 () and eY∗ as a lower bound of eY . Therefore, theorem 3.1 implies pH 2,1 (T ) C ∗ [f L2 ((0,T );L2 ()) + BN H 1/2,1/4 (T ) + I H 1 () ].

(47)

We now prove the following Lemma. Lemma 3.1. Consider B(r, Yˆ ) arbitrary and (f, BN , I ) ∈ L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 (). Let C ∗ be given by the previous corollary and Hm ∈ L2 (0, T ; L2 ()). Then, for all Y and Y˜ in B(r, Yˆ ), there exists a unique weak solution χ ∈ H 2,1 (T ) to the adjoint problem: ∂χ ˜ − ∇ · eY ∇χ = Hm (x, t), ∂t ∇χ · n = 0,

−

χ (x, T ) = 0.

(48) (49) (50)

Moreover, χ satisfies χ H 2,1 (T ) C ∗ Hm L2 ((0,T );L2 ()) .

(51) 11



Proof. For Y˜ ∈ B(r, Yˆ ), consider the following problem: ∂ξ ˜ − ∇ · eY ∇ξ = Hm (x, T − t) (52) ∂t with a homogenous Neummann boundary condition and a homogeneous initial condition. By hypothesis on Hm , corollary 3.1 implies that this problem has a solution ξ ∈ H 2,1 (T ). By the Sobolev imbedding, ξ(·, x) ∈ C(0, T ) so we can change and define χ (t, x) = ξ(T − t, x) which satisfies (48)–(50). As we will see in theorem 3.2, condition (F4) is the most difficult to prove. However, this condition is inherited from the exponential function. Therefore, we need the following basic result. Lemma 3.2. Let r > 0 such that r e−2r /2 and z ∈ R. For any w, u ∈ R such that |w − z| r and |u − z| r, then |ew − eu − eu (w − u)| e2r |u − w ew − eu |

(53)

|w − u| 2er−z |ew − eu |.

(54)

and

Proof. For any z ∈ R and r > 0, if |w − z| r and |u − z| r, by the mean value theorem it can be shown that |ew − eu | ez+r |w − u|.

(55)

Then,

1 u+(w−u)t u |e − e − e (w − u)| = [e − e ](w − u) dt 0 1 u+(w−u)t u z+r 2 |e − e (w − u)| dt e |u − w| w

u

u

0

1

t dt =

0

1 z+r e |u − w|2 2 (56)

From the previous expression it follows − 12 ez+r |u − w|2 −|ew − eu − eu (w − u)| |ew − eu | − |(w − u) eu |

(57)

from which we obtain |(w − u) eu | − 12 ez+r |u − w|2 |ew − eu |. Since u z − r, it follows that ez−r eu . Therefore, (58) becomes ez+r |w − u| e−2r − 12 |u − w| |ew − eu |. −2r

Note now that |u − w| 2r e 1 2

−2r

e

−2r

=e

(59)

where we have applied the hypothesis on r. Then,

−2r

− e 1 2

(58)

e−2r − 12 |u − w|

We substitute (60) into (59) to find ez+r |w − u| 12 e−2r |ew − eu |,

(60) (61)

which can be written as (54). This inequality is used in (56) to arrive at |ew − eu − eu (w − u)|

1 2

ez+r |u − w|2

ez+r |u − w|2er−z |ew − eu | e2r |u − w ew − eu |

which proves (53). 12

1 2

(62)



For any Y, Y˜ ∈ C 1 (), we define the residual ˜

˜

se [Y (x), Y˜ (x)] ≡ eY (x) − eY (x) − [Y (x) − Y˜ (x)] eY (x) ,

(63)

> ≡ {x ∈ Y (x) − Y˜ (x)| > 0}

(64)

the set and the function

⎧ ⎪ ⎨ se [Y (x), Y˜ (x)] ˜ W [Y, Y ](x) ≡ eY (x) − eY˜ (x) ⎪ ⎩0

if

x ∈ > ,

if

x ∈ − > .

(65)

From the previous lemma we now present the following proposition which is (F4) applied to eY . Proposition 3.1. For every Yˆ ∈ Y , W [Y, Y˜ ] ∈ H 1 () for all Y˜ , Y ∈ B(r, Yˆ ) and there exists C > 0, and r > 0 such that (66) W [Y, Y˜ ]H 1 () CY − Y˜ C 1 () . The constant C may depend only on , Yˆ and r. Proof. Let r > 0 be such that r < e−2r /2. For every Y˜ , Y ∈ B(r, Yˆ ), we have |Y˜ (x) − Yˆ (x)| Y˜ − Yˆ C 1 () r |Y (x) − Yˆ (x)| Y − Yˆ C 1 () r

(67)

for every x ∈ . We realize from lemma 3.2 that (53) holds independently of the choice of z. Therefore, for every x ∈ we apply the previous lemma for C1 = e2r , w = Y (x), z = Yˆ (x) and u = Y˜ (x) which implies ˜ |se [Y (x), Y˜ (x)]| C1 |Y (x) − Y˜ (x) eY (x) − eY (x) |.

(68)

Furthermore, from (67) and (54) it follows that ˆ

˜

|Y (x) − Y˜ (x)| 2er−Y (x) |eY (x) − eY (x) | for all x ∈ and all Y˜ , Y ∈ B(r, Yˆ ). Then, ˆ ˜ ˜ |Y (x) − Y˜ (x)| 2er−Y (x) |eY (x) − eY (x) | C2 |eY (x) − eY (x) |

(69) (70)

r−Yˆ (x)

|. where C2 = maxx∈ |2e Since Y − Y˜ ∈ C 1 (), it is easy to see that W [Y, Y˜ ] is continuously differentiable in > and in the interior of − > . Actually, a direct computation shows that ˜ ˜ eY (x) − eY (x) + [Y (x) − Y˜ (x)] eY (x) ˜ [∇Y (x) − ∇ Y˜ (x)] (71) ∇W [Y, Y˜ ](x) = eY (x) (eY (x) − eY˜ (x) )2 for all x ∈ > and ∇W [Y, Y˜ ](x) = 0 for all x ∈ − > . Then, from (69) and (70) we have that (72) |W [Y, Y˜ ](x)| C1 |Y (x) − Y˜ (x)| for all x ∈ and |Y − Y˜ | |∇Y (x) − ∇ Y˜ (x)| C1 C2 ω|∇Y (x) − ∇ Y˜ (x)| − eY˜ (x) | (73) ˜ ˜ for all x ∈ > and ∇W [Y, Y ](x) = 0 for all x ∈ − > . We now prove that W [Y, Y ] is Lipschitz continuous in . Then, since is convex, for all x, z ∈ > we have the following cases:

˜ |∇W [Y, Y˜ ](x)| C1 |eY (x) |

|eY (x)

13



(a) If the line segment Lx,z from x to z is contained in > , by the mean value theorem we have (74) |W [Y, Y˜ ](x) − W [Y, Y˜ ](z)| sup |∇W [Y, Y˜ ](x)x − z|. x∈>

(b) If the line segment Lx,z is not contained in > , then, there exists at least one x0 ∈ Lx,z such that x0 ∈ − > . In this case we have that W [Y, Y˜ ](x0 ) = 0 and |W [Y, Y˜ ](x) − W [Y, Y˜ ](z)| |W [Y, Y˜ ](x) − W [Y, Y˜ ](x0 )| + |W [Y, Y˜ ](x0 ) − W [Y, Y˜ ](z)| = |W [Y, Y˜ ](x)| + |W [Y, Y˜ ](z)| C1 |Y (x) − Y˜ (x)| + C1 |Y (z) − Y˜ (z)| C1 sup |∇Y (x) − ∇ Y˜ (x)|(|x − x0 | + |z − x0 |) x∈

C1 r(|x − x0 | + |z − x0 |) = C1 r|x − z|,

(75)

where we have used (72), the mean value theorem and the fact that x0 ∈ Lx,z . When x ∈ > and z ∈ −> , similar arguments show that |W [Y, Y˜ ](x)−W [Y, Y˜ ](z)| C1 r|x − z|. On the other hand, if x, z ∈ − > , then |W [Y, Y˜ ](x) − W [Y, Y˜ ](z)| = 0. In summary, for all z, x ∈ , |W [Y, Y˜ ](x) − W [Y, Y˜ ](z)| A|x − z| (76) where A = max{supx∈>0 |∇W [Y, Y˜ ](x)|, C1 r} which proves that W [Y, Y˜ ] is Lipschitz continuous in . Then, we apply [11] (section 5.8.2, theorem 4) to conclude that W [Y, Y˜ ] ∈ H 1 (). We now observe that the set ∂( − >) has measure zero in Rn , therefore, (72) and (73) imply that 2 2 2 ˜ ˜ ˜ |W [Y, Y ]| + |∇W [Y, Y ]| = |W [Y, Y ]| + |∇W [Y, Y˜ ]|2 CY − Y˜ 2C 1 ()

>

>

(77)

where C = C12 + ω2 C12 C22 ||.

We now use the previous results to prove the following lemma that will be needed to show that the operator G and the measurement functional defined below, satisfies (F4). Lemma 3.3. Let Yˆ ∈ C 1 (), (f, BN , I ) ∈ L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 () and Hm ∈ L2 (0, T ; L2 ()). Then, there exist r > 0 and D > 0 such that for all Y, Y˜ ∈ B(r, Yˆ ) there exists a unique weak solution v(·, t) ∈ u ∈ H 2 () : u = 0 to the following problem: −∇ · eY ∇v = −∇ · W eY ∇χ

in

−eY ∇v · n = 0 on ∂,

(78) (79)

for almost all t ∈ (0, T ), where W is defined by (65) and χ is the solution to (48)–(50). Moreover, v ∈ H 1 (0, T ; L2 ()), v(·, T ) = 0 and

∂v

DY − Y˜ C 1 () (80)

∂t 2 L ((0,T );L2 ()) where D may depend only on Hm , Yˆ , r, and T. Additionally, ∇v(·, t) − W ∇χ (·, t) = 0 a.e. in T . 14

(81)



Proof. Let r > 0 and C > 0 be given by proposition 3.1. For B(r, Yˆ ), let us define C ∗ and ω by corollary 5.2. We now prove the first part of the proposition. For each u ∈ H 2 let us consider the following problem: −∇ · eY ∇v = −∇ · W eY ∇u

in

(82)

−eY ∇v · n = −eY ∇u · n on ∂, which has a unique weak solution v ∈ {v ∈ H 2 () :

(83)

v = 0} such that

vH 2 () D1 ( − ∇ · (W e ∇u)L2 () + − W eY ∇u · nH 1/2 () ) Y

(84)

where D1 by the same arguments we used in corollary 3.1 can be chosen to be dependent only on , Yˆ and r > 0. From proposition 3.1 and the trace theorem we find that vH 2 () 2D1 D2 ωY − Y˜ C 1 () uH 2 () . (85) 2 2 Therefore, there is a linear operator g : H () → {v ∈ H : v = 0} such that g(u) = v where v is the solution to (82), (83). On the other hand, from a standard Sobolev imbedding we know that v ∈ C() with maxx∈ |v(x)| D3 vH 2 () . Then, for all x ∈ , gx (u) = v(x) defines a linear functional on H 2 (). From (86) it follows |v(x)| D3 vH 2 () 2D3 D2 D1 ωY − Y˜ C 1 () uH 2 () (86) which in turn implies that gx is bounded in H 2 () with gx 2D3 D2 D1 ωY − Y˜ C 1 ()

(87)

Additionally, since H 2 () is a linear subspace of L2 (), by the Hahn–Banach theorem, let g˜ x : L2 () → R be a continuous extension of gx . Moreover, by the Riez representation theorem, let us choose θx ∈ L2 () such θx u (88) v(x) = g˜ x (u) =

for all u ∈ L (). We now consider χ the solution to (48)–(50). From lemma 3.1 we know that χ ∈ H 2,1 (T ) and so χ (·, t) ∈ H 2 () a.e. in (0, T ) and ∇χ · n = 0 in the sense of traces. Then, for almost all t we repeat the previous argument to obtain that v(x, t) = g˜ x (χ (·, t)) = gx (χ (·, t)) = θx (y)χ (y, t) dy (89) 2

a.e. in (0, T ) where v(x, ·) is the solution to (78) and (79). Since χ (·, t) is strongly measurable, from (89) it follows that v(·, t) is also a strongly measurable function. In addition, |v(x, t)| |θx (y)χ (y, t)| dy θx L2 () χ (·, t)L2 ()

2D3 D2 D1 ωY − Y˜ C 1 () χ (·, t)L2 () ,

(90)

where we have used (87) and the fact that g˜ x = gx = θx . Therefore, vL2 ((0,T );L2 ()) 2D3 D2 D1 ωY − Y˜ C 1 () χ L2 ((0,T );L2 ()) < ∞. We now take φ ∈ C0∞ (0, T ) arbitrary and note that T T T ∂χ (y, t) dy (91) φ (t)v(x, t) dt = θx (y)φ (t)χ (y, t) dy = − θx (y)φ(t) ∂t 0 0 0 which implies that ∂v(x, t) ∂χ (y, t) θx (y) = dy (92) ∂t ∂t 15



in the sense of distributions. Following the same ideas as before we find

∂χ

∂v

˜ D3 D4 ωY − Y C 1 ()

∂t 2

∂t 2 2

L ((0,T );L2 ())

L ((0,T );L ())

2D3 D2 D1 ωY − Y˜ C 1 () χ H 2,1 (T )

(93)

and from (51) we have (80). Furthermore, since we have proved that v ∈ H (0, T ; L ()), from standard results we know that v ∈ C([0, T ]; L2 (). The same argument applies to χ and we may use (89) for t = T to obtain v(x, T ) = 0 a.e in . Additionally, for almost all t, let us now define 1

2

u(·, t) = eY (∇v(·, t) − W ∇χ (·, t)).

(94)

It is not difficult to see that u(·, t) ∈ [H ()] for almost all t. Moreover, for almost all t, u(·, t) satisfies ∇ · u = 0 a.e. and u · n = 0 in ∂ (in the sense of traces). Then, since is connected u(·, t) = 0 a.e. in for almost all t. 1

2

We finally arrive at the main result of this section. Theorem 3.2. Let (f, BN , I ) ∈ L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 (). m ∈ {1, . . . , M}, we define the measurement functional by T Lm (p) = Hm (x, t)p(x, t) dx dt 0

For each

(95)

with Hm ∈ L2 (0, T ; L2 (0, T )). Let L = {L1 , . . . , LM }. Assume that, for all Yˆ ∈ Y and r > 0, there exists C > 0 such that ˜ p˜ − pL2 ((0,T );L2 ()) CL(p − p)

(96)

˜ = 0, for all Y, Y˜ ∈ B(r, Yˆ ) where p and p˜ are the solutions to G(Y, p) = 0 and G(Y˜ , p) respectively. Then, the operator G : Y × H 2,1 (T ) → L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 () defined by (34) satisfies (F1)–(F4) Proof. (F1) G is continuously Frechet-differentiable. Moreover, ˜ ˜ ˜ DG(Y˜ , p)[δY, δp] = DY G(Y˜ , p)δY + Dp G(Y˜ , p)δp where

⎡

⎤ ˜ −∇ · eY δY ∇ p˜ ⎢ ⎥ ˜ DY G(Y˜ , p)δY = ⎣ −eY˜ δY ∇ p˜ · n ⎦, 0

(97)

⎡

⎤ ∂δp Y˜ − ∇ · e ∇δp ⎢ ∂t ⎥ ⎥ ˜ =⎢ Dp G(Y˜ , p)δp ⎣ −eY˜ ∇δp · n ⎦ δp(x, 0)

(98)

This result can be shown easily and so the proof is omitted. (F2) This is exactly theorem 3.1. (F3) From (98) and theorem 3.1 it follows that, for any (f, BC , I ) ∈ L2 (0, T ; L2 ()) × H 1/2,1/4 (T ) × H 1 (), problem ˜ Dp G(Y˜ , p)δp = (f, BN , I )T

(99)

has a unique solution δp which proves surjection. Uniqueness implies injectivity and so the isomorphism. Let us now consider any B(r, Yˆ ). We realize that (5) is equivalent to DY G(Y, p)δY + Dp G(Y, p)δp = 0 which implies 16



∂δp − ∇ · eY ∇δp = ∇ · eY δY ∇p, ∂t

(100)

eY ∇δp · n = −eY δY ∇p · n,

(101)

δp(x, 0) = 0.

(102)

Then, for Y ∈ B(r, Yˆ ), we apply (40) to find that δpH 2,1 (T ) C ∗ (∇ · eY δY ∇pL2 ((0,T );L2 ()) + eY δY ∇p · nH 1/2,1/4 (T ) ) C ∗ δY Y [eY Y pH 2,1 (T ) + BN H 1/2,1/4 (T ) ].

(103)

From (40) and (41) δpH 2,1 (T ) C ∗ δY Y [ω(r, Yˆ )C ∗ [f L2 ((0,T );L2 ()) + BN H 1/2,1/4 (T ) + I H 1 () ] + BN H 1/2,1/4 (T ) ]

(104)

from which it follows that δpH 2,1 (T ) C1 δY Y for C1 = C ∗ [ω(r, Yˆ )C ∗ [f L2 ((0,T );L2 ()) + BN H 1/2,1/4 (T ) + I H 1 () ] + BN H 1/2,1/4 (T ) ] (105) and so (F3) is proved. (F4) Let Yˆ ∈ Y arbitrary and let m ∈ {1, . . . , M}. Let δ˜ > 0 and D > 0 be given by lemma 3.3 ˜ Yˆ ) consider C ∗ given by corollary 3.1. Define and r > 0 such that (96) holds. For B(δ, ˜ ˆ ˜ 0 < δ < min(δ, r). Let Y, Y ∈ B(δ, Y ). It is not difficult to see that sr , defined in (9) is the solution to ⎤ ⎡ ⎤ ⎡ ∂sr Y˜ ˜ ] eY˜ ∇[p − p] ˜ − ∇ · e ∇s −∇ · s ∇p − ∇ · [Y − Y r e ⎥ ⎢ ⎥ ⎢ ∂t Y˜ ˜ (106) ⎣ −e ∇sr · n ⎦ = ⎣ −se ∇p · n − [Y − Y˜ ] eY ∇[p − p] ˜ · n⎦, 0

sr (x, 0)

˜ = 0 respectively. Let χ where p and p˜ are the solutions to G(Y, p) = 0 and G(Y˜ , p) be the solution to the adjoint problem (48)–(50) given by lemma 3.1. Multiply the first equation in (106) by χ , integrate by parts and apply initial, and boundary conditions on χ to obtain T T T ∂χ ˜ Y˜ ˜ · ∇χ . − − ∇ · e ∇χ sr = se ∇p · ∇χ + [Y − Y˜ ] eY ∇[p − p] ∂t 0 0 0 (107) From (65) it follows that T se ∇p · ∇χ =

0

T

˜

W [eY − eY ]∇p · ∇χ .

(108)

0

where from now on we define W = W [Y, Y˜ ]. We substitute (48) and (108) into (107) to find T T T ˜ ˜ ˜ · ∇χ . Hm sr = W [eY − eY ]∇p · ∇χ + [Y − Y˜ ] eY ∇[p − p]

0

0

0

(109) On the other hand, we observe that p − p˜ satisfies ˜ ∂(p − p) ˜ − ∇ · eY ∇p + ∇ · eY ∇ p˜ = 0, ∂t

(110) 17

Inverse Problems 25 (2009) 035006 ˜

M A Iglesias and C Dawson ˜

−eY ∇p · n + eY ∇ p˜ · n = 0

(111)

with homogeneous initial condition. Let v be the function defined by lemma 3.3. We now multiply (110) by v and integrate by parts to obtain that T ∂v Y Y˜ ˜ ˜ · ∇v = 0. −(p − p) + [e ∇p − e ∇ p] (112) ∂t 0 We rewrite (112) as T T ∂v Y˜ Y ˜ (p − p) + e ∇ p˜ · ∇v . e ∇p · ∇v = (113) ∂t 0 0 On the other hand, by definition of v (78) and (79) we know that T T Y W e ∇χ · ∇p = eY ∇v · ∇p. (114)

0

0

Then, using (113) and (114) we find T T T ˜ ˜ W [eY − eY ]∇χ · ∇p = eY ∇v · ∇p − W eY ∇χ · ∇p 0 0 0 T T ∂v ˜ ˜ ˜ (p − p) + eY ∇ p˜ · ∇v − = W eY ∇χ · ∇p ∂t 0 0 which can be written as T T T ∂v ˜ Y Y˜ ˜ − W [e − e ]∇χ · ∇p = (p − p) eY ∇ p˜ · [∇v − W ∇χ ] ∂t 0 0 0 T ˜ ˜ − W eY ∇χ · [∇p − ∇ p]

(115)

(116)

0

From lemma 3.3 we know that the second term on the right-hand side of the previous equation vanishes. Then, T T T ∂v ˜ ˜ ˜ ˜ + W [eY − eY ]∇χ · ∇p = (p − p) ∇ · [W eY ∇χ ][p − p]. (117) ∂t 0 0 0 Therefore, from (109) we have T T T ˜ ˜ ˜ · ∇χ Hm sr = W [eY − eY ]∇χ · ∇p + [Y − Y˜ ] eY ∇[p − p] 0 0 0 T ∂v ˜ ˜ ˜ + ∇ · ([W eY − [Y − Y˜ ] eY ]∇χ ) (p − p). = (118) ∂t 0 From proposition 3.1 and lemma 3.3, it is not difficult to see that T ∂v Y˜ Y˜ ˜ ˜ + ∇ · ([W e − [Y − Y ] e ]∇χ ) (p − p) ∂t 0

∂v

Y˜ Y˜

˜ ˜ L2 ((0,T );L2 ())

+ ∇ · ([W e − [Y − Y ] e ]∇χ )

p − p

2 ∂t L ((0,T );L2 ()) ˜ L2 ((0,T );L2 ()) , AY − Y˜ C 1 () p − p (119) where A is a constant that may depend only on Hm , Yˆ , δ, and T. Therefore, from (118) and (96) it follows T T T ˜ Y Y˜ ˜ · ∇χ Hm sr = W [e − e ]∇χ · ∇p + [Y − Y˜ ] eY ∇[p − p]

0

0

˜ AC2 Y − Y˜ C 1 () L(p) − L(p) 18

0

(120)



and using the same process with Hm replaced by −Hm we arrive at T ˜ |Lm (sr )| = Hm sr AC2 Y − Y˜ C 1 () L(p) − L(p).

(121)

0

We now use (121) and the fact that this result is valid for all m = {1, . . . , M}, to conclude that (F4) follows. For any Y ∈ Y let us define WY ≡ {δp ∈ H 2,1 (T ) : Dk G(Y, p)δY + Dp G(Y, p)δp = 0 for some δY ∈ Y },

(122)

where p is the solution to G(Y, p) = 0. Convergence of the IRBS for operator G and the measurement functional (95) is provided in the following corollary. Corollary 3.2. Assume the hypotheses of theorem 3.2 are satisfied. In addition, assume that ˜ ˆ for every Yˆ ∈ Y and r > 0, the set {Lm |WY˜ }M m=1 is linearly independent for all Y ∈ B(Y , r). Given τ > 1, if θ ≡ C −1 C12 L2 < τ − 1

(123)

and the initial guess (prior knowledge) k is close enough to a solution to the PIP, then the IRBS converges to a stable approximation to a solution to the PIP in the sense of theorem 2.2. Proof. Since (A1), (F1)–(F4) are satisfied, we can apply theorem 2.2.

Remark 3.1. From the definition of C1 (105) we point out that (123) balances relative magnitudes of the assimilation information and the forward model. More precisely, for a fixed ˆ ball B(r, Yˆ ), θ is a function of Yˆ , f, I, BC , eY , , T , L, C. In practice, (123) may be difficult or impossible to verify. Nevertheless, in section 6 the numerical experiments show that, for example, for different values τ > 1, the result of corollary 3.2 holds. Remark 3.2. Note that (96) is a restrictive assumption that cannot be in general satisfied for arbitrary elements p, p˜ ∈ L2 (T ). However, p and p˜ are the solutions to G(Y, p) = 0 ˜ = 0 respectively. Therefore, it can be shown that (96) can be satisfied when and G(Y˜ , p) p − p˜ belongs to the subspace generated by {Hm }M m=1 . Moreover, expression (96) holds if the mapping Y → Lm (p) is injective which in turn implies that the inverse problem is identifiable. Identifiability for operator G has been studied only for a particular case [16] and the general case is still an open problem. However, in the following section, we will show that convergence can be obtained for a measurement functional which approximates pointwise measurements at well locations. 3.1. Implementation For the implementation of the IRBS we need the explicit form of R n as well as γ n . To obtain closed form relations, we first recall that (see (section 2.7)) k n (24) was derived from the first-order optimality condition on J n . For the application under consideration, this can be obtained by considering the Lagrangian: n T −1 Q [Y, p, λ] = [d − L(p)] Cη [d − L(p)] + (Y (x) T λLG(p, Y ) (124) − Y n−1 (x))CY−1 (x, x )(Y (x ) − Y n−1 (x )) +

0

19


where

⎡ ∂p

− ∇ · (eY


n−1

∇p) − ∇ · (Y − Y n−1 ) eY

n−1

∇pFn − F

⎤

⎢ ∂t ⎥ ⎥ LG(p, Y ) ≡ ⎢ ⎣ eY n−1 ∇p · n + (Y − Y n−1 ) eY n−1 ∇pFn · n − BN ⎦ = 0 N p|t=0 − I

(125)

is the linearized constraint and pFn is the solution to G(Y n−1 , pFn ) = 0. Then, the first-order optimality condition D Qn [Y, λ] = 0 gives rise to the following Euler–Lagrange system. State: ∂pn n−1 n−1 − ∇ · (eY ∇pn ) − ∇ · (Y n − Y n−1 ) eY ∇pFn = F ∂t eY

n−1

∇pn · n + (Y n − Y n−1 ) eY

p = I, n

n−1

∇pFn · n = BN

× (0, T ],

on

N × (0, T ],

on

in × {0}.

(126) (127) (128)

Adjoint: ∂λn n−1 − ∇ · (eY ∇λn ) = (d − L(pn ))Cη−1 H ∂t in × {T }, λn = 0

−

Y n−1

e

∇λ · n = 0

Y n (x) = Y n−1 (x) −

T 0

CY (x, x )∇λn (x , t) · eY

n−1

(x )

(129) (130)

on N × (0, T ],

n

Parameter:

in × (0, T ],

(131)

∇pFn (x , t) dt dx .

(132)

where H = [H1 , . . . , HM ] and pFn is the solution to G Y n−1 , pFn = 0 (with G from (34)). The representers can be obtained by postulating that pn = pFn +

M

βmn rmn ,

m=1

λn =

M

Y = Y n−1 +

n βmn αm ,

m=1

M

βmn γmn

(133)

m=1

n for some functions rmn (x, t), αm (x, t) and γmn (x) and scalars βmn to be determined. When the previous expressions are substituted in the E–L system, we obtain that (133) solves the E–L system if and only if rmn is the solution to

n−1 ∂rmn n−1 − ∇ · eY ∇rmn − ∇ · γmn eY ∇pFn = 0, ∂t rmn = 0, in × {0} eY

n−1

∇rmn · n + γmn eY

n−1

∇pFn · n = 0

on

in × (0, T ],

(134) (135)

× (0, T ].

(136)

n is the solution to αm n ∂αm n−1 n ) = Hm − ∇ · (eY ∇αm ∂t n =0 in × {T } αm

−

20

in × (0, T ],

(137) (138)


eY

n−1


n ∇αm ·n=0

γmn satisfies γmn (x) = −

T

0

N × (0, T ].

on

n CY (x, x )∇αm (x , t) · eY

(139)

n−1

(x )

∇pFn (x , t) dt dx .

(140)

and the representer coefficients are given by (R n + Cη )β n = d − L pFn (141) with the matrix R n defined by Rijn = Li rjn . For this problem, we have directly derived the representer method in [21]. We now prove the following. Proposition 3.2. The representers are given by expression (140). Proof. Let us take δY ∈ Y arbitrary and consider p the solution to LG(p, Y = δY +Y n−1 ) = 0. We would like to prove (see section (section 2.7)) that for every m ∈ {1, . . . , M}, γm , δyY = Lm (δp) where δp = p − pFn . Multiplying (137) by δp, integrating by parts and using (138) and (139) as well as the definition of Lm yields T n ∂δp n−1 n − ∇ · (eY ∇δp) αm αm (t = 0)δp(t = 0) + ∂t 0 T n−1 n + eY ∇δp · nαm = Lm (δp) (142)

0

and from (125) we find T n n−1 ∇ · δY eY ∇pFn αm −

0

T

δY eY

0

n−1

n ∇pFn · nαm = Lm (δp).

(143)

Integrating by parts again yields T n−1 n − δy eY ∇pFn · ∇αm = Lm (δp).

(144)

On the other hand, we multiply (140) by CY−1 δY , integrate and use (10) to find T n−1 n γmn CY−1 δy = − ∇αm · eY ∇pFn δy.

(145)

Thus, combining (144) and (145) we obtain γm , δyY ≡ γmn CY−1 δy = Lm (δp)

(146)

0

0

for all δY and δp = p −pFn (where p satisfies (125)). Therefore, since the representer obtained by the Riez theorem is unique, then we have proved the result. 4. Numerical experiments In this section we present some numerical examples to show the capabilities of the IRBS. First, some aspects related to the implementation are discussed. 21



4.1. Discretization Note from the previous section that systems (129)–(139) need to be solved on each iteration. For the spatial discretization, a cell-centered finite differences (CCFD) scheme is utilized [1]. Time discretization is implemented with a backward Euler scheme. The coefficient of the second-order elliptic operator is assumed to be constant at each cell. Then, with the CCFD discretization, the coefficient at each cell edge is approximated by the harmonic average. For the present analysis we use covariance functions for which (140) becomes a convolution which can be computed efficiently by the FFT. 4.2. Measurement functional For each measurement location and time, we assume that the measurement functional is the average of the state variable, over a small cube around the measurement and time location. In other words, Lm is given by (95) with 1 if(x, y) ∈ R t − tm st , (147) Hm (x, y, t) = 0 else R = {(x, y) : |x − xm | sx , |y − ym | sy },

(148)

where sx and sy are defined so that R coincides with the cell containing (xm , ym ) and st is defined so that [tm , tm + st ) corresponds to the time interval containing tm . This definition of Hm serves as an approximation to pointwise measurements. 4.3. Prior error information and norms As indicated before, (A1) is an essential assumption for the previous analysis. More precisely, we need an equivalence between the norm induced by the inverse covariance operator and a Sobolev norm. Some equivalences of this type can be found in [28, 32, 35] for different types of covariance functions. However, the assumptions on the regularity of the coefficient of the second-order elliptic operator may not be preserved by the discretization scheme. In addition, direct computation of (39) may become computationally prohibited and so it is avoided in these experiments. Moreover, as we pointed out before, the permeability is assumed to be piecewise constant. For this reason, we relax the assumptions on the regularity for the parameter and the L2 ()-norm (l 2 norm for the discretized problem) is considered for convergence analysis in the parameter space. Additionally, we need to specify the prior error covariance CY , and the prior mean Y which is used as the first guess (Y 0 = Y ) in the IRBS. For the following examples, the experimental omnidirectional semivariogram [7] of the true parameter is computed and a variogram model is fitted. Then, CY is defined as the covariance of the associated variogram model of the true permeability. 4.4. Generation of synthetic data For the following set of experiments, a ‘true permeability’ (‘true hydraulic conductivity’) is prescribed. Then, we generate synthetic data d η by running the model simulator with the ‘true parameter’ and applying the measurement functional defined by (95) with (147). In order to avoid an inverse crime, these data are obtained from a finer grid than that utilized for the inverse computations. Moreover, Gaussian noise with zero mean and covariance C = σ 2 IM×M is added to the measurement. In our choice of C, IM×M is the M × M identity matrix where M is the number of measurements for the corresponding experiment. The true parameter is used only to compare with our final estimated parameter. 22



4.5. Stopping rule The stopping criteria for the algorithm presented in section 2 requires an estimate of the noise level. However, only the measurements (d η ) as well as the prior error statistics (σ ) may be available. To obtain an estimate of the error level in the case C = σ 2 I , we assume that d η = d + σ e where e is a Gaussian random variable with zero mean and variance equals one. Therefore, since d η − d = σ e, if we assume that |ej | 1, it follows d η − d2C −1 = 2 √ M 1 η = M |ej |2 M. Thus, d η − dC −1 M. With this argument we j σ 2 dj − dj j √ propose to use η = M as the noise level

for the √For the stopping criteria, we algorithm.

recall that the IRBS is terminated when d η − L sFn C −1 τ M for the first time. Having discussed some general considerations, we now introduce some examples. The objective of these experiments is to analyze the efficiency of the IRBS to obtain improved parameters while fitting the synthetic data. 4.6. The groundwater flow Groundwater flow can be modeled by the following equation [5]: ∂h (149) S − ∇ · eY ∇h = f, ∂t where S is specific storage, f accounts for sources, eY is the hydraulic conductivity and h is the piezometric head. When proper boundary conditions are imposed, (149) takes the form (up to known constants) of the operator (34). The parameter to be estimated is Y and all the other parameters are assumed to be perfectly known. 4.6.1. Experiment I. In order to validate the IRBS presented in the previous sections, we consider the same problem used in [19]. The forward model is the steady state of (149) over a square domain of dimension 6 km × 6 km. The boundary conditions are given by ux (6 km, y) = 0,

u(x, 0) = 100 m,

−1

−e ux (0, y) = 500 m km Y

3

−1

day ,

and a constant (in time) recharge is represented by ⎧ ⎪ ⎨0 f (x, y) = 0.137 × 10−3 m d−1 ⎪ ⎩ 0.274 × 10−3 m d−1

(150) uy (x, 6 km) = 0

(151)

0 < y < 4 km, 4 km < y < 5 km,

(152)

5 km < y < 6 km.

The grid used for the synthetic data generation and for the IRBS are 180 × 180 and 60 × 60 respectively. The measurement locations are shown in figure 1(a). For the prior mean of Y = ln K we select the natural logarithm of the initial guess used in [19]. Note from figure 1(b), that the true conductivity is anisotropic and therefore, the omnidirectional semivariogram is poor prior knowledge of the truth. It is the purpose of this example to illustrate the efficiency of our method when limited information is provided. The covariance associated with the variogram model fitted to the experimental semivariogram as well as other pertinent information are displayed in table 1. In Experiments IA and IB we follow [19] and test the scheme for two different signals-to-noise ratios (SNR). The estimated results are displayed in figure 1(c) and (d) where a good agreement to the true log hydraulic conductivity is obtained. Figure 1(e) shows the true and estimated hydraulic conductivity fields projected along the phantom shown in figure 1(a) (dotted red line). Convergence history for IA and IB are 23



6

5

15

50 5

4

Y(km)

50

15

3

2

150

1

0

0

1

2

3 X (km)

4

5

6

(a)

(b) 6

6

5.5

5.5 5

5

5

5

4.5

4.5

3 2.5

2

y (km)

y (km)

3.5 3

4

4

4

4

3.5 3

3 2.5

2 2

2 1.5

1

1.5

1

1

1 0

0 0

1

2

3 x (km)

4

5

6

0

1

2

3 x (km)

4

5

6

(d)

(c) 200 True Exp IA Exp IB

180 160

K (m/day)

140 120 100 80 60 40 20 0

0

1

2

3 4 Distance (km)

5

6

7

(e) Figure 1. Experiment I. Top left: aquifer configuration (observation wells are indicated by *). Top right and middle: log hydraulic conductivity fields [ln (md−1 )]. Bottom: hydraulic conductivity (m d−1 ) fields projected over the phantom. (a) Configuration, (b) true, (c) experiment IA, (d) experiment IB and (e) phantom.

displayed in table 2. Note that for smaller (SNR), the IRBS provides smaller error in the computed parameter. In contrast to the work in [19] where the hydraulic conductivity K is identified, in our approach the estimation is performed for Y = ln K. Although the latter 24



Table 1. Information for data assimilation. Experiment I. Experiment

IA

IB

Number of measurement locations τ ητ (m) σ (m) Y [ln md−1 ]

18 1.5 6.3640 1.8 ln 20

18 1.5 6.3640 0.18 ln 20

CY (ξ, ξ )[ln(m d−1 )]2

2.2 exp − ξ1−ξ km

Signal-to-noise ratio

102.27

!

2.2 exp − ξ1−ξ km

!

1034

Table 2. Numerical results. Experiment I. Experiment IA

Experiment IB

n

Y n−1 − Ytrue l 2

d η − L(pFn )C −1

Y n−1 − Ytrue l 2

d η − L(pFn )C −1

1 2 3 4 5

178.9116 114.0224 81.7970 70.4071 –

227.729 69.773 15.274 2.082 –

178.9116 116.4194 83.2218 69.2700 67.839

2280.319 692.873 138.325 13.538 0.381

approach imposes a stronger nonlinearity in the abstract formulation, the implementation does not require modification when negative values of the parameter occur. This may become an advantage in problems (see experiment III) where the true parameter changes by several orders of magnitude, and further modifications may be detrimental to the quality of the identification. We remark that the measurement data are assimilated only from the measurement locations. Unlike the standard parameter identification approaches, there is no need to interpolate the entire field of the state variable. Moreover, Hanke [19] reported that when the parameter identification problem is posed with a finite-dimensional measurement space, their method becomes considerably less efficient. On the other hand, the IRBS convergence is achieved in a few iterations with satisfactory results. Such behavior should not be surprising since the present approach takes into account prior information. 4.7. The single-phase Darcy flow through porous media We now study the reservoir model described by (1). 4.7.1. Experiment II. In experiment II we are interested in determining the efficiency for the estimation of the log permeability Y = ln K from pressure data collected at different configurations. The true log permeability (figure 2(a)) is generated stochastically with the routine SGSIM [7]. Table 3 shows the reservoir properties and table 4 presents prior error information. Table 5 provides the corresponding information for each run whose corresponding estimations are presented in figures 2(b)–(f). For experiments IIB, IIC and IID, the observation wells are equally spaced over the domain but they do not coincide with either the injection or the production wells. For experiments IIA and IIE measurements are collected at the injection and production wells. It comes as no surprise that the quality of the reconstruction depends on the number of measurements available. However, the measurement locations also affect 25



Table 3. Reservoir description. Experiments II and III. Experiment 3

Dimension (ft ) Grid blocks (for synthetic data) Grid blocks (for IRBS) Time interval [0, T ] (day) Time discretization dt (day) Porosity Water compressibility (psi−1 ) Rock compressibility (psi−1 ) Water viscosity (cp) Initial pressure (psi) Number injection wells Number production wells Shut-in time ts (day) Injection ratea (stb d−1 ) Production ratea (stb d−1 ) a

II

III

2000 × 2000 × 2 180 × 180 × 1 60 × 60 × 1 [0,1] 0.01 0.2 3.1 ×10−6 1.0×10−6 0.3 5000 2 6 0.75 2500 833.33

2000 × 4000 × 2 60 × 220 × 1 60 × 110 × 1 [0, 1] 0.01 0.2 3.1×10−6 1.0 ×10−6 0.3 10 000 4 5 0.75 1000 1000

Rate per well in the interval [0, ts ].

Table 4. Prior error information for experiment II. τ σ (psi) Y (ln md)

2.0 5.0 196.42

CY (ξ, ξ ) (ln(md))2

−ξ 0.95 exp − ξ400 ft

!

Table 5. Numerical results. Experiment II. Configuration

IIA

IIB

IIC

IID

IIE

Number of measured location Number of measured times τ η[psi] [ln(md)] Ytrue − Y ∗ a,b l2 Iterations

8 1 5.6569 68.74 5

16 1 8 63.75 3

25 1 10.00 58.08 4

36 1 12.00 47.27 4

8 2 8.00 59.84 6

a b

Y ∗ ≡ lim Y n . Y − Ytrue l 2 = 79.19.

the final outcome. It is also worth mentioning that the best estimate of log permeability was obtained with experiment IIE which corresponds to the configuration of maximum number of observation wells. Additionally, convergence is achieved within the first five iterations for all the different configurations. Clearly, non-uniqueness is always evident since for a small number of observations, there are many possible parameters that will fit the data. 4.7.2. Experiment III: SPE model. In experiment III we are interested in the estimation of a more realistic permeability field. The true permeability (figure 3(a)) is taken from the third layer of the absolute permeability data for the 10th SPE comparative project [20]. One of the interesting aspects of this permeability is that it has a variability of several orders of magnitude (10−3 –104 md). A heterogeneous field of this type is challenging even for the 26



2000 8

1800 1600

7

1400 6

y (ft)

1200 1000

5

800 4 600 3

400 200 0

2 0

500

(a)

1500

2000

(b)

2000

2000 8

1800 1600

8

1800 1600

7

1400

7

1400 6

1000

5

6

1200 y (ft)

1200 y (ft)

1000 x (ft)

800

1000

5

800 4

4

600

600 3

400 200 0

200

2 0

500

1000 x (ft)

1500

3

400

0

2000

2 0

500

(c)

1500

2000

(d)

2000

2000 8

1800 1600

8

1800 1600

7

1400

7

1400 6

1000

5

800

6

1200 y (ft)

1200 y (ft)

1000 x (ft)

1000

5

800 4

600 3

400 200 0

4 600

2 0

500

1000 x (ft)

(e)

1500

2000

3

400 200 0

2 0

500

1000 x (ft)

1500

2000

(f )

Figure 2. Log permeability fields [ln md−1 ]. (a) True; (b) experiment IIA; (c) experiment IIB; (d) experiment IIC; (e) experiment IID; (f) experiment IIE.

direct simulation of reservoir models. Any inverse approach that attempts its reconstruction should be robust enough to manage, not only the number of degrees of freedom, but also its lack of regularity. We test the methodology with the forward model given by (1), for the 27



4000

4000 10 3500

3000

3000

5

5

2500

y (ft)

2500

y (ft)

10

3500

2000

1500

2000

1500 0

0

1000

1000

500

500

0

0

500

1000

1500

0

2000

0

500

1000

x (ft)

1500

2000

x (ft)

(a)

(b) 4000 10 3500

3000

5

y (ft)

2500

2000

1500 0 1000

500

0

0

500

1000

1500

2000

x (ft)

(c)

Figure 3. Log permeability fields. (a) True; (b) experiment IIIA; (c) experiment IIIB [ln md].

estimation of the permeability from synthetic data (pressure). Measurements are collected at the injection and production wells, as wells as two additional observation wells. Table 3 shows the reservoir description and table 6 presents the information for the assimilation experiments. Figures 4(a) and (b) show the convergence history for both experiments. From figure 3(a) it is clear that the true log permeability is anisotropic. Although the isotropic covariance is only poor prior knowledge of the spatial variability, the estimated log permeability in figure 3(b) captures some of the main characteristics of the true log permeability field. In experiment IIIB we select, by inspection, a different error covariance function that reflects the higher spatial correlation in the x-direction. According to the definition of noise level (section 4.5), both experiments have the same stopping threshold. In fact, from figure 4(d) it can be observed that data have been fitted with about the same accuracy. However, in experiment IIIB, an improved estimation is achieved (figure 3(c)) because we considered ‘better’ prior knowledge (correlation lengths) than in experiment IIIA. 28



200

4.5 Exp IIIA

Exp IIIA Exp IIIB

Exp IIIB 4

195

3.5

3

n )|| F

true l

|| 2

190

185 2.5 180 2 175

170

1.5

1

2

3

4

5

6

1

7

1

2

3

Iterations

4

5

6

7

Iterations

Figure 4. Experiment III. (a) l 2 -error in Y = ln K; (b) log-measurement error log10 d − (pFn ).

Table 6. Data assimilation information. Experiment IIIA and IIIB. Experiment

IIIA

IIIB

Number of measurement locations Number of measurement times τ τ η (psi) σ (psi) Y (ln md)

12 2 2.0 9.798 7.5 24.85

12 2 2.0 9.798 7.5 24.85

a C (ξ, ξ )(ln md)2 Y aξ

−ξ 3.2 exp − ξ500ft

!

3.2 exp −

"

|x−x |2 (286ft)2

+

|y−y |2 (160ft)2

!a

= (x, y).

5. Summary and conclusions In this paper, we present the theory and application of an iterative representer-based scheme for parameter estimation in reservoir models. The objective of the proposed IRBS is to generate an estimate of the model parameters given measurements of the state variable (output), prior knowledge of the poorly known parameter as well as measurements error information. Since we assume that the measurement space is finite dimensional, the representer method is utilized to solve a linearized inverse problem on each iteration. Although there are other linear programming methods that can be used to solve the linearized problem, the representer method provides an efficient implementation that exploits the mathematical structure of the inverse problem. In the first part of this paper we develop the mathematical framework in order to prove that the IRBS converges to a solution to the parameter identification problem. Then, the IRBS is applied to a forward model described by a parabolic PDE and a covariance function whose inverse generates a weighted L2 -norm equivalent to a Sobolev norm. For this particular case, we developed the conditions to ensure local convergence. The convergence results predicted by the theory are confirmed with the numerical results. We discuss some examples for the estimation of log-permeability (log-hydraulic conductivity) given pressure (head) measurements in a single-phase Darcy flow (groundwater flow). Although the identification 29



problem is ill-posed, the iterative scheme has been proved to be robust enough to generate improved parameters while fitting the data. Stability was obtained when different signal-tonoise ratios are compared in experiment I. In contrast to the classical parameter identification approach, our approach assumes some prior knowledge of the uncertain parameter. However, our numerical results show that even with poor prior knowledge, the reconstructions capture the main heterogeneities of the fields. Moreover, when large problems and highly heterogenous fields are considered (experiment III), the IRBS generates improved parameters in a stable fashion. It is worth mentioning that the IRBS possesses an interesting flexibility for more general applications. The theory is applicable to the simultaneous estimation of reservoir properties. Furthermore, since timedependent parameters can also be estimated, ‘model error’ can be allowed in the formulation as in [6, 21, 33]. In addition, any reservoir model can be used with this theory as long as hypotheses (F1)–(F4) are satisfied. The numerical results are promising and indicate that the IRBS may become a tool for parameter estimation in reservoir modeling. With the integration of seismic data and smartwells technology, better prior knowledge and reliable measurements may result in an improved reservoir characterization. Future comparison with the state-of-the-art methodologies must be conducted. Acknowledgment This research was supported in part by NSF grant DMS-0411413. Appendix The proof of theorem 2.2 is very similar to the proof of theorem 2.2 [19]. Therefore, only some technical differences and equivalences are pointed out. First, we need the following Lemma A.1. Let d ∗ ∈ RM and C ∈ RM×M a positive-definite matrix. Suppose assumption (A1) and (F1)–(F3) hold. For any k∗ ∈ K and r > 0, let C1 be the constant in (F3) for the closed ball B(r, k∗ ). Define ρ as in (32) Suppose that for some n, the following inequality is satisfied

ρ

(A.1) d η − L(sr )C −1 d η − L sFn C −1 κ for some κ > 1, where sr is the solution to Dk G k n−1 , sFn [k∗ − k n−1 ] + Ds G k n−1 , sFn sr − sFn = 0. (A.2) If k n−1 ∈ B(r, k∗ ), then k n defined by (24) for d η = d ∗ satisfies

2 n−1 2 n 2 2 κ −1 ∗ d − L sFn C −1 . k∗ − k K − k∗ − k K > 2ρ κ

(A.3)

Proof. First we note that the assumptions on theorem 2.1 are satisfied. Then, k n is well defined in terms of (24). Let, s∗ be the solution to G(k∗ , s∗ ) = 0. Since G satisfies (F1)–(F3), by the implicit function theorem (IFT), there exists a neighborhood U of k∗ such that G(k, s(k)) = 0 for all k ∈ U , where s : U → S is a continuous function. Let us define the nonlinear operator F : K → RM by F (k) = (L ◦ s)(k), and consider the weighted norm in RM by C. From hypothesis (F2), F is continuous. Consider now T = F (k n−1 ). Note that if k n−1 ∈ B(r, k∗ ), by (F3) it follows that T LC1 and so T is locally bounded. Then F satisfies the hypotheses used in [19]. On the other hand, from the definition of sr (A.2) and the invertibility of 30



−1 Ds G k n−1 , sFn−1 it follows that sr = sFn − Ds G k n−1 , sFn−1 Dk G k n−1 , sFn−1 [k∗ − k n−1 ]. Therefore, (A.1) is equivalent to uη − F (k n−1 ) − F (k n−1 )[k∗ − k n−1 ]

= d η − L sFn − L ◦ Ds GDk G[k∗ − k n−1 ] C −1

ρ

ρ d η − L sFn C −1 = uη − F (k n−1 ) (A.4) κ κ which is condition (2.1) in [19]. Then we can apply the same argument that yields (2.16) in [19] to conclude that k∗ − k n−1 2K − k∗ − k n 2K

ρ

> 2 AC −1/2 d ∗ − L sFn AC −1/2 d ∗ − L sFn − d ∗ − L sFn C −1 κ (A.5) for A = [C −1/2 R n C −1/2 + I ]−1 . In contrast to the work of [19], the first term on the right-hand side of (14) is weighted by C −1 . For this reason C −1/2 appears in (A.5) after the appropriate factorization. From the definition of A, estimate (20) and (32) it follows that

−1/2 ∗

1

AC

d ∗ − L s n −1 d − L sFn F C −1 A

1

d ∗ − L s n −1 = ρ d ∗ − L s n −1 . (A.6) F F 2 C C C −1 C1 L2 + 1 This is substituted into (A.5) to obtain k∗ − k n−1 2K − k∗ − k n 2K

∗

∗ n

n

n

ρ

∗

d − L sF C −1 > 2ρ d − L sF C −1 ρ d − L sF C −1 − κ from which (A.3) follows.

Before we discuss the proof of theorem 2.2, we need the result for the idealized case of perfect measurements. Theorem A.1. Let d ∈ RM be the vector of exact measurements and C ∈ RM×M an arbitrary positive-definite matrix with eigenvalues {λi }M i=1 and α ≡ max λi / min λi . Let us assume (A1) and (F1)–(F4) and let k∗ be a solution to the PIP. For kˆ = k∗ , take C2 and δ > 0 as in (F4). Consider B(δ/2, k∗ ) and the corresponding constant C1 as in (F3). Let ρ be defined by (32). If k 0 = k (the initial guess) satisfies δ (A.7) , κ2 k∗ − kK < min 2 for some κ2
0 is fixed. Let sr1 be the solution to Ds G k 0 , sF1 sr1 − s∗ = Gr s∗ − sF1 , k∗ − k .

(A.8) 31



Since k∗ − kK δ/2 < δ, we apply (F4) (for kˆ = k = k∗ and k˜ = k) to obtain

L(s∗ ) − L s 1 C2 k∗ − kK L(s∗ ) − L s 1 . r F Then, it is not difficult to show that

L(s∗ ) − L s 1 −1 C −1 1/2 L(s∗ ) − L s 1

r r C

C −1 1/2 C2 k∗ − kK L(s∗ ) − L sF1

α 1/2 C2 k∗ − kK L(s∗ ) − L s 1 −1 . F

C

Therefore, the same steps of [19] can be used to find

η

d − L s 1 −1 ρ k∗ − k 0 K d η − L s 1 −1 r F C C κ

(A.9)

(A.10)

(A.11)

for κ=

ρτ . 1 + (1 + τ )C2 α 1/2 k∗ − k 0 K

(A.12)

Then, since k ∈ B(δ/2, k∗ ) we may apply lemma A.1 to obtain that kη1 , defined by (24), satisfies

2 0 2 1 2 2 κ −1 η

d − L sF1 C −1 (A.13) k∗ − k K − k∗ − kη K > 2ρ κ (we have added the subscript η to emphasize that kη1 depends on this case of d η ). From (A.12)

2 it is clear that κ > 1 which implies δ/2 > k∗ − k2K > k∗ − kη1 K . By applying this methodology inductively, we show that kηn given by (24) satisfies

k∗ − k n−1 2 > k∗ − k n 2 (A.14) η η K K

η

as long as τ η < d − L sFm C −1 and so the sequence kηn is monotone decreasing. Let us assume that for n = n(η),

η

d − L s n −1 τ η. (A.15) F C From (A.13) it follows that n(η)−1

k∗ − k n−1 2 − k∗ − k n 2 > 2ρ 2 κ − 1

d η − L s n 2 −1 . η η K F C K κ n=1 n=1

n(η)−1

(A.16)

Reducing some terms on the left-hand side and using the fact that τ η < d η − L sFn C −1 we find

2 2 n(η)−1 2 2 κ −1

τ 2 η2 (n(η) − 1). > 2ρ ∞ > k∗ − kK k∗ − kK − k∗ − kη K κ This inequality shows that, given η > 0 fixed, the discrepancy principle terminates the IRBS after a finite number n(η) of iterations. We now observe from (33) that theorem A.1 can be applied where now K n is the sequence given by (24) with d η replaced by d. We now point out to the fact that for theorem 2.2 in [19], the current estimate k n is related to the choice of the Levenberg–Marquardt parameter αk . In our approach this parameter is fixed (αk = 1) and the current estimate k n is computed from (24). However, with our selection of k n , the proof is 32



still valid since from (24) it is easy to see that kηn − K n = kηn−1 − K n−1 + [γ n (ξ )]T [R n + C]−1 [d η − d] n + [γ n (ξ )]T [R n + C]−1 L sFn − L sF,η ,

(A.17)

where sFn is the solution to G(K n−1 , sFn ) = 0. With an inductive argument and using the uniform bounds for γ n and R n (see (21)–(23)) it follows that kηn → K n as η → 0. Therefore, from (A.15) it follows that as η → 0, L sFn = d for n(δ) finite, which implies that kηn(η) converges to K n which in turn is a solution to the PIP. The proof when n → ∞ can be conducted as in [19]. References [1] Arbogast T, Wheeler M F and Yotov I 1997 Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences SIAM J. Numer. Anal. 34 828–52 [2] Baird J and Dawson C 2005 The representer method for data assimilation in single-phase Darcy flow in porous media Comput. Geosci. 9 (4) 247–71 (doi:10.1007/s10596-005-9006-2) [3] Baird J and Dawson C 2007 The representer method for two-phase flow in porous media Comput. Geosci. 11 (3) 235–48 (doi:10.1007/s10596-007-9048-8) [4] Baird J and Dawson C 2007 A posteriori error estimation of the representer method for single phase Darcy flow Comput. Methods Appl. Mech. Eng. 196 1623–32 [5] Bear J 1972 Dynamics of Fluids in Porous Media (New York: Dover) [6] Bennett A F 2002 Inverse Modeling of the Ocean and Atmosphere (Cambridge: Cambridge University Press) [7] Deutsch C V and Journel A G 1997 GSLIB: Geostatistical Software Library 2nd edn (New York: Oxford University Press) [8] Dimet F Le and Talagrand O 1986 Variational algorithms for analysis and assimilation of meteorological observations Tellus 38A 97–110 [9] Dong Y, Gu Y and Oliver D S 2006 Sequential assimilation of 4D seismic data for reservoir description using the Ensemble Kalman Filter J. Pet. Sci. Eng. 53 83–99 [10] Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems 1st edn (Dordrecht: Kluwer) [11] Evans L C 1998 Partial Differential Equations 1st edn (Providence, RI: American Mathematical Society) [12] Evensen G 1994 Sequential data assimilation with a non-linear quasi-geostrophic model using Monte-Carlo methods to forecast error statistics J. Geophys. Res. 99 10143–62 [13] Evensen G 2007 Data Assimilation 1st edn (Heidelberg: Springer) [14] Evensen G, Hove J, Meisingset H C, Reiso E and Seim K S 2007 Using the EnKF for assisted history matching of a North Sea reservoir model SPE Reservoir Simulation Symp. (Woodlands, TX, 26–28 February 2007, SPE 106184) [15] Gao C, Rajeswaran T and Nakagawa E 2007 A literature review on smart-well technology Proc. SPE Production and Operations Symp. (Oklahoma City, OK, 31 March–3 April 2007, SPE 106011) [16] Giudici M 1991 Identifiability of distributed physical parameters in diffusive-like systems Inverse Problems 7 231–45 [17] Gu Y and Oliver D S 2005 History matching of the PUNQ-S3 reservoir model using the Ensemble Kalman Filter SPE J. 10 51–65 [18] Gu Y and Oliver D S 2006 The ensemble Kalman filter for continuous updating of reservoir simulation models J. Energy Resour. Technol. 128 79–87 [19] Hanke M 1997 A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems Inverse Problems 13 79–95 [20] http://www.spe.org/csp/ (Spe 10th comparative project) [21] Iglesias M A and Dawson C 2007 The representer method for state and parameter estimation in single-phase Darcy flow Comput. Methods Appl. Mech. Eng. 196 4577–96 [22] Kalman R E 1960 A new approach to linear filtering and prediction problems Trans. ASME J. Basic Eng. 82 35–45 [23] Lions J L and Magenes E 1972 Non-Homogeneous Boundary Value Problems and Applications vol II (Berlin: Springer) [24] Liu N and Oliver D S 2005 Ensemble Kalman filter for automatic history matching of geologic facies J. Pet. Sci. Eng. 47 147–61 33



[25] Lorentzen R J, Naevdal G, Valls B and Berg A M 2005 Analysis of the ensemble Kalman filter for estimation of permeability and porosity in reservoir models SPE Annual Technical Conf. and Exhibition (Dallas, TX, 9–12 October 2005, SPE 96375) [26] Naevdal G, Johnsen L M, Aanonsen S I and Vefring E H 2005 Reservoir monitoring and continuous model updating using ensemble Kalman filter SPE J. 10 66–74 [27] Naevdal G, Mannseth T and Vefring E H 2002 Near-well reservoir monitoring through ensemble Kalman filter SEP/DOE Improved Oil Recovery Symp. (Tulsa, OK, April 2002, SPE 75235) [28] Oliver D S 1998 Calculation of the inverse of the covariance Math. Geol. 30 911–33 [29] Przybysz-Jarnut J K, Hanea R G, Jansen J D and Heemink A W 2007 Application of the representer method for parameter estimation in numerical reservoir models Comput. Geosci. 11 73–85 [30] Rosmond T and Xu L 2006 Development of NAVDAS-AR: non-linear formulation and outer loop test Tellus A 58 45–58 [31] Skjervheim J A, Evensen G, Aanonsen S I, Ruud B O and Johansen T A 2005 Incorporating 4d seismic data in reservoir simulation models using Ensemble kalman filter SPE Annual Technical Conf. and Exhibition (Dallas, TX, 9–12 October 2005, SPE 95789) [32] Tarantola A 1987 Inverse Problems Theory: Methods for Data Fitting and Model Parameter Estimation 1st edn (New York: Elsevier) [33] Valstar J R, McLaughlin D B, te Stroet C B M and van Geer F C 2004 A representer-based inverse method for groundwater flow and transport applications Water Resour. Res. 40 W05116 (doi:10.1029/2003WR002922) [34] Xu L and Daley R 2000 Towards a true 4-dimensional data assimilation algorithm: application of a cycling representer algorithm to a simple transport problem Tellus 52A 109–28 [35] Xu Q 2005 Representation of inverse covariance by differential operators Adv. Atmos. Sci. 22 181–98

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An IRBS for data inversion in reservoir modeling - School of

An IRBS for data inversion in reservoir modeling - School of

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