On inverse problems in secondary oil recovery

On inverse problems in secondary oil recovery Victor Isakov October 5, 2007 Department of M athematics and Statistics W ichita State U niversity W ichita, KS 67260 0033, U.S.A. e mail : Introduction victor.isakov@wichita.edu In oil production it is crucial to find out properties of the ground from various measurements of geophysical fields. In secondary oil recovery oil is extracted by pumping in water (through injecting wells) and creating pressure which pumps out oil through production wells. Water/oil pressure at wells is monitored providing with an information to determine two important characterictics of medium: permeability and porosity. These characteristics can indicate location of oil, and hence give valuable recommendations for drilling new wells and finding pumping regimes to optimize oil recovery. Looking for permeability and porosity can be viewed as an inverse problem for (system of) quasilinear partial differential equations describing fluids in porous media. This inverse problem is called history matching. Despite its obvious applied importance this inverse problem was studied only numerically, in most cases by using routine least squares minimization [2], [4], [27]. Due to large size of the problem and its nonconvexity these methods are not efficient and not reliable. In particular, there is no uniqueness and stability analysis. We review simplest models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational difficulties with both direct and inverse problems for this system mainly due to its degeneration. However, practical value of the problem justifies efforts producing any progress. At present, in inverse problems for elliptic and parabolic equations there is theoretical and numerical progress [6]. We believe that this progress can generate new efficient mathematical methods in oil recovery.

2 A general two phase model One of accepted mathematical models of filtration of oil and water through porous medium [] (see also [], [6], [0]), consists of two partial differential equations: div(k(α u α 2 γ J(S) α 3 g h)) = f in Ω () and φ t S div(kα 4 ( u 0.5γ J(S) ρ w g h)) = 0 in Ω (0, T ), (2) where u is (medium) pressure, k = k(x) is permeability, α = k w µ w + k o µ o, k w = k w (S), µ w are the relative permeability and viscosity of water, S is saturation of water, k o (S), µ o are relative permeability and viscosity of oil. Next, α 2 = 2 (k wµ w k o µ o ), γ is known function ( proportional to surface tension at water/oil interface), φ = φ(x) is porosity, J is known capillary pressure (Leverett function). Moreover, α 3 = k w ρ w µ w + k o ρ o µ o, where ρ w, ρ o are densities of water and oil, α 4 = k w µ w. Observe that k w, k o are known functions of S and hence α, α 2, α 3, α 4 are known functions of S. Finally, g is the gravity acceleration, h is the height function (h(x) = x 3 when n = 3), and f represents sources and sinks. We will impose the following practically feasible conditions: Ω is bounded domain in R n, n = 2, 3, its boundary Ω is a C 2 - surface, γ, k, φ L (Ω), δ < k, δ 2 < φ, δ 3 < γ on Ω where δ, δ 2, δ 3 are some positive constants, k o, k w, γ C (R), k w be nonincreasing and k o nondecreasing, k w (0) = 0, k o (0) =, J is increasing and J(s 0 ) = 0 for some s 0 (0, ). The partial differential equations (), (2) for u, S are supplemented by the initial conditions S = S(; 0) on Ω {0} (3) and the boundary value conditions α ν u α 2 γ ν J(S) α 3 g ν h = 0 on Ω (0, T ), S = S(; 0) on Ω (0, T ). (4) It is natural to assume that and to normalize pressure as follows Ω Ω f = 0, (5) u = 0. (6) Observe that the equation () for u is of elliptic type, while the equation (2) is a partial differential equation for S which is of parabolic type. The initial boundary value problem (), (2), (3), (4) is a system of quasi-linear equations with natural boundary conditions. For some results on (weak) solvability of this problem we refer to [], [6], and more recently to [7], [8]. A feature of the filtration system is degeneration at S = 0 which occurs near the domain filled by oil. This degeneration is accountable for several unusual phenomenae, 2

including finite speed of propagation, which is one of basic properties of solutions of hyperbolic equations and which is not possible for nongenerated (linear or nonlinear) elliptic and parabolic equations. As shown below, in some cases the equation (2) can be replaced by quasilinear first order equation for saturation S. This equation is hyperbolic and it exhibits shock solutions which are collapsing in finite time. In filtration theory this causes a phenomenon called fingering (developing of wet zones reminding shape of a hand with long fingers) and causing instability and blow-up of solutions. These reasons lead to serious difficulties in the theory of solvability of the boundary value problem (), (2), (3), (4): currently, only existence of weak solutions is established. We will briefly describe a typical result. The pair (u, S) is a weak solution to the initial boundary value problem (), (2), (3), (4) if u L (0, T ; H (Ω)), and S L (Ω (0, T )), J(S) L 2 (Ω (0, T )), 0 S on Ω (0, T ) Ω (0,t) Ω k(α u α 2 γ J(S) α 3 g h)) ϕ = Ω fϕ, ( φs t ϕ 2 + (kα 4 ( u 0.5γ J(S) ρ w g h)) ϕ 2 ) = φsϕ 2 ν t (Ω (0,t)) for all test functions ϕ H (Ω), ϕ 2 H (Ω (0, T )), t < T. Let us assume that f L ((0, T ), L 2 (Ω)), S(; 0) C([0, T ], H (Ω), 0 S(; 0), S(; 0) = 0 on Ω (0, T ) (no water on the outer boundary). Following arguments in [], [6], [8] one can show that there is a weak solution (u, S) to the initial boundary value problem (), (2), (3), (4). As in case of the Navier-Stokes system, uniqueness of a weak solution remains unknown, as well as existence of regular solutions. It seems that the main obstacle to completing theory of existence and uniqueness of solution of the initial boundary value problem (), (2), (3), (4) is its degeneration (of the parabolic equation (2) for S) at S = 0. In cases when the data exclude degeneration ( when one can show, say by using maximum principles, that for all solutions with this data ε 0 < S for some positive ε 0 ) global existence and uniqueness of solutions most likely can be derived from known theory of quasilinear elliptic and parabolic equations and systems [2], [22]. Maximum principles for the system of filtration theory can be found in []. In oil production the source function f can be modeled as the sum of point sources at x(m) ω, ω is a subdomain of Ω with intensities q m, so f = M q m δ( x(m)) (7) m= The system of equations (), (2) is too complicated, and satisfactory mathematical results for the direct initial boundary value problem are not available. 3

On the other hand, in practical situation there are simplifying assumptions. In particular, one neglects capillarity and gravity, to arrive at a version of Buckley- Leverett system div(kα u) = f, (8) φ t S div(kα 4 u) = 0 in Ω (0, T ). (9) Expressing div(k u) from the equation (8) one transforms the equation (9) into φ t S + α ( α 4 α ) k u S = α 4 α f. (0) with the initial and boundary value conditions ν u = 0 on Ω, S = S(; 0) on Ω (0, T ). () Even for the simplified system (8), (0) it is too hard to obtain theoretical results in direct and inverse problems. We write the system (8),(0) in the possible most convenient form where div(kα (S) u) = µ w f, (2) φ t S + α 5 (S)k u S = α 6 f, (3) α 5 (S) = k wk o k w k o µ o k w (S), α 6 (S) = (S). µ o k w + µ w k o µ o k w + µ w k o The functions k w (S), k o (S) are known from numerous experiments. In agreement with these experiments a good form of these functions is k w (S) = ( S 0. ) λ, when 0. < S, k w (S) = 0, when 0 < S 0., 0.9 k o (S) = k w ( S). (4) where λ = log0 log3 log9 log4. In many cases, µ o = 0.6 while µ w = [3]. Since typically there are hundreds of wells located in some region ω we can assume that f is a function in L 2 (Ω) supported in a subdomain ω Ω ω C 2. In practice pressure is measured at wells for a single given f or for many of them, provided one can change rates of pumping. It is of great importance to know two characteristics of the medium Ω, permeability k and porosity φ. These functions can be assumed to be piecewise constant, with constants reflecting properties of 20-30 typical materials forming soil. Complying with (5) and physical reality we partition ω into two open subsets ω + and ω with C 2 -boundaries so that 0 < f on ω + and f < 0 (and constant) on ω. Hence we formulate Inverse problem. Find k, φ entering the initial boundary value problem (), (2), (3), (4) ( or (8), (0), ()) from the additional data u = g 0 on ω (0, T ) (5) 4

given for one or many f on ω +. Even the direct problem for the system (), (2) is not well understood. So not surprisingly there are no theoretical results about uniqueness and stability of k, φ from the data (5). We will outline some linearization approach to the inverse problem (8), (0), (),(5) where one can at least claim uniqueness of k(x) for the data from many f. Let the initial data ε 0 be a small positive constant, f = f 0 + τf where τ is a small parameter. First we assume that f 0 = 0, S(; 0) = 0. By standard perturbation argument where... are terms bounded by Cτ 2. If f 0 = 0 we have u 0 = 0 and u = u 0 + τu +..., S = τs +... div(kα (0) u ) = µ w f, (6) with the boundary condition (4) for u. One can assume that k is known on ω +. Then the additional data (5) uniquely determine kα (0) ν u = g on ω +. (7) We will call a function k piecewise Lipschitz constant on Ω if there is a partition of Ω into finitely many Lipschitz subdomains such that k is constant on each subdomain. Theorem 2.. The map f u on ω + ( 0 < f on ω +, f < 0 on ω, f L 2 (ω) and satisfies (5)) uniquely determines piecewise Lipschitz constant k on Ω. Outline of proof. By using approximations one can show that for the solution u(; y +, y ) to the Neumann problem div(k u) = δ( y + ) δ( y ) on Ω, ν u = 0 on Ω the function u(; y +, y ) on ω + is uniquely determined. From the definition of a weak solution k u(; y +, y ) v = v(y + ) v(y ) Ω for any v C ( Ω). We can assume that k is a known constant k 0 in a connected neighborhood Ω(0) of ω +. Then by uniqueness of the continuation for elliptic equations u(; y +, y ) is given on Ω(0). Let Ω() be a connected component of Ω where k = k, k k 0 and which has a common piece of boundary Γ() with Ω(0). If v is given, then we are given Ω\ω + k u(; y +, y ) v. 5

Now let v(x) = x y +. Let γ be a (piecewise) analytic curve in Ω(0). Integrating by parts and using harmonicity of v outside ω + we conclude that we are given k 0 u(; y +, y ) ν v + (k k 0 )u(; y +, y ) ν v +... ω + Γ() where... denotes terms (integrals over surfaces of discotinuity of k away from Ω Γ() which are bounded when y + γ. Observe that the first term in the preceding equality is known when y + ω +. Hence I(y + ) = (k k 0 )u(; y +, y ) ν v +... Γ() is given when y + ω + and is obviously analytic with respect to y + Ω(0). Since k 0 k, as in [6], section 5.7, the first integral behaves as Clogd(y + ) where d(y + ) is the distance from y + to Γ(). Hence varying curves γ with starting points inside ω + and continuing I(y + ) along γ we can uniquely identify Γ() and k. We can repeat the same step and, moving subsequently from known discontinuity surfaces to adjacent ones, identify all discontinuity surfaces and piecewise k. Remark. Slightly changing this proof one can show that for uniqueness it is sufficient to have the data of Theorem 2. for one f on ω. The idea of using singular solutions of partial differential equations for identification of domains was proposed in [5]. Efficient numerical numerical algorithms based on this idea are given by Potthast [24]. For smooth k uniqueness can be shown if Ω = R 3, by using the Kelvin transform with the pole inside ω + and known results of Sylvester and Uhlmann for the inverse conductivity problem in bounded domains. If Ω is a ball in R 3, then we expect that uniqueness can be shown by the methods of the paper [7]. When Ω = R 2 by using inversion with respect to a point of ω + and recent results of Astala and Päivärinta [2] one obtain uniqueness of k L (Ω). Currently, there is a progress in the inverse conductivity problem with the partial boundary data [5], [9], however complete results with the data at a part of the boundary of a general domain Ω are still not available. For general f 0, S 0 and k = k 0 + τk +..., φ = φ 0 + τφ +... one similarly obtains div(k 0 α (S 0 ) u ) + div(k 0 α (S 0 )S u 0 ) = µ w f div(k α (S 0 ) u 0 ), φ 0 t S + α 5 (S 0 )k 0 S 0 u + α 5 (S 0 )k 0 u 0 S + (α 5(S 0 )k 0 u 0 S 0 α 6(S 0 )f 0 )S = ( t S 0 )φ α 5 (S 0 ) u 0 S 0 k + α 6 (S 0 )f. (8) These equations are augmented by zero initial and boundary value conditions () for u, S. The data of the linear inverse problem for first order corrections 6

k, φ are similar to (5). The term f can be interpreted as a new well, and solution of the linear inverse problem as updating old solution k 0, φ 0. Practical needs stimulate numerical solution of the inverse problem (in general and simplied formulations). Currently it is done mostly by regularized least squares matching [2], [4], [27], without analytic justification. Nonconvexity of minimization problems and absence of analytic theory result in poor resolution and low reliability of numerical methods. For this reason simplifications of the inverse problem which preserve its essential features are of obvious interest. Due to theoretical and numerical difficulties with the quasilinear degenerated system (2), (3) one uses the linear elliptic partial differential equation for the pressure u div(a(x) (u ρgh)) = f in Ω (9) with the natural boundary conditions ν (u ρgh) = 0 on Ω. (20) Here a(x) = k(x)µ(x), k is permeability and µ is some average viscosity. Of course, this average viscosity is not known. But if in some cases we assume µ to be known, then we arrive at the inverse conductivity problem where there is certain theoretical and numerical progress (as described above and in [6]). The model (9) is used in some cases when one can assume that compressibility of fluids is small. If f is given by (7) and n = 3, then as known from theory of elliptic equations where u(x) = M q m (K (x, x(m)) + K 2 (x, x(m)) +... m= K (x, x(m)) = 4πa(x) x x(m), K 2 (x, x(m)) = K (x, v)( v a(v) v 4πa(y) Ω v x(m) )dv, and... are bounded as well as their gradients with respect to x. Here Ω is some domain containing Ω. Integrating by parts it is not hard to show that K 2 (x, x(m)) = 6(π) 2 a(x(m)) ( loga(v) loga(y) ν(y) Ω x v v x(m) dγ(y)+ (loga(x(m) loga(v)) v Ω x v v dv). (2) v x(m) When u(x) is given for x ω and ω is a small neighborhood of x(m) (typically, a sphere) the most singular terms K, K, K 2 are uniquely determined by u, u. One can use the formula for K to find a near x(m). Moreover, one can use the second (less explicit) term K 2 of the expansion to get more detailed information about a near sources and sinks by solving linear integral equation 7

with respect to a(v). The first term on the right side of (2) can assumed to be known. Since we can take as Ω sufficiently large ball we can neglect the first term on the right side of (2) which will be small. Letting b(v) == (loga(v) loga(y)) 4πa(x(m)) we obtain for b the following integral equation b(v) v x v v v x(m) dv = U(x), x ω +, (22) 4π Ω where U(x) = K 2 (x, x(m)). It is not hard to show (integrating by parts when b is C smooth and compactly supported in Ω ) that U = div(b x(m) ) in Ω. Hence the integral equation (22) can be viewed as a linearization of the inverse conductivity problem [6], section 0.. Also it has similarities with the inverse gravimetrical problem [6], section 4.. Uniqueness of solution b of the integral equation (22) is not known (and it is not anticipated for general b), and stability is expected to be of logarithmic type (i.e. the corresponding linear inverse problem is exponentially ill-posed). 3 Muskat free boundary model In this model suggested in 930-s Muskat assumed that oil and water do not mix and occupy the domain Ω R n, n = 2, 3, so that domain occupied by water is Ω w and by oil is Ω o. As known, [6], div(a w (u w ρ w gh)) = f w in Ω w, div(a o (u o ρ o gh)) = f o in Ω o (23) where a j = σkk j µ j, u j = p j ρ j gh, j = o, w. σ(x) is the so-called section of field at x, σ = when n = 3, but for n = 2 it depends on reduction of the three-dimensional problem to the two-dimensional one. Here ρ j is density of the j th fluid. The boundary conditions at the oil-water interface Γ = Ω o Ω w are u o u w = 0 on Γ, a w (u w ρ w gh) = a o (u o ρ o gh) on Γ (24) and the standard boundary conditions are ν u j = 0 on Ω Ω j. (25) Given all coefficients and source terms, we have an elliptic free boundary problem (23), (24), (25). This free boundary problem is not overdetermined in R 2 and it is overdetermined in R 3. There are only few partial analytic results on this problem [6], [3], [26]. Observe that f w (rate of water injection) is given, 8

while f o (rate of oil production) may assumed to be constant determined from solvability condition (7). Letting τ to be the tangential component of we have from (24) τ u o = τ u w on Γ and hence ( k o µ o k w µ w ) τ u w = ( k oρ o µ o k wρ w µ w )g τ h on Γ so given Γ one can uniquely (modulo constant) determine u w on Γ. As above in the inverse problem one is looking for the coefficient k(x) from the additional data u w = g 0 in ω (26) where g 0 is a given function for one or several given f w. An advantage of the Muskat model for inverse problem is a possibility to use moving (with changing f w ) interfaces Γ to introduce into the inverse problem an additional parameter θ and hence some evolution with respect to this parameter. Now we will show that in a simple realistic case one has uniqueness in the inverse problem with the reduced (and most likely minimal) data. Inverse Problem 2. Find k from the data (26) given for a one-parametric family of domains Ω w and one f w. Now we will demonstrate uniqueness for a linearization of this inverse problem and analyze its stability. Let n = 2. We will neglect gravity force arriving at div(k u w ) = f 0 in Ω w. (27) Let ω be the unit disk {x : x <, Ω(θ) = { x < + θ}, Γ θ = Ω(θ). Moreover from the condition (24) as above we will have u w = const on Γ. We can assume that this constant is zero. We assume that k = + f (28) where f is small and equal to 0 in ω. We will assume that the data (26) are given for Ω w = Ω(θ), 0 < θ < T. By standard perturbation argument u = u 0 + u +... where u 0 corresponds to f = 0, u to first order perturbations (with respect to f), etc. Then We have and we are given u = div(f u 0 ) in Ω(θ) (29) u = 0 on Γ(θ), ν u = 0 on Γ(0), (30) u = g 0 on Γ(0), g 0 = g 00 + g 0 +... (3) We assume constant water pumping letting r u 0 (x; θ) = log + θ where r = x. (32) 9

Theorem 3.. A solution f L 2 (Ω(T )) to the linearized inverse problem (29), (30), (3), (32) is unique. This inverse problem is severely (exponentially) ill-posed. Proof. To find f we make use of its angular Fourier series f(x) = f k (r)e ikσ, x = r(cosσ, sinσ) and of solutions u k of an adjoint problem u r r k(x; θ) = (( + θ )k ( + θ ) k )e ikσ. Using that u are harmonic in Ω(θ) \ ω and u = 0 on Γ(θ) from the definition of a weak solution to (29) we have f ν u 0 u k + f u 0 u k = g 0 ν u k. Γ(0) Ω(θ) Γ(0) Using orthogonality of exponents and polar coordinates we get f u 0 u k = f r u 0 r u k = 2π +θ +θ 2π Ω(θ) Ω(θ) f k (r)r k + θ (( r r + θ )k + ( + θ ) k )rdr = kf k (r)(r k ( + θ) k + r k ( + θ) k )dr. Hence we have the following Volterra integral equation where +θ f k (r)(r k ( + θ) k + r k ( + θ) k )dr = F k (θ), (33) F k = F k + F k2, F k (θ) = 2π (( + θ)k + ( + θ) k ) 2π 0 g 0 (cosσ, sinσ; θ)e ikσ dσ, F k2 (θ) = k f k ()(( + θ) k ( + θ) k ). Observe that f k () are in a stable way determined by the data of the linearized inverse problem. Indeed, differentiating the both sides in (33) with respect to θ and letting θ = 0 we yield 2f k () = F k(0). (34) 0

Now stability of f(, ) in a Sobolev space follows from Parseval identity. To solve (33) for f k (θ) we introduce two functions Φ k (θ) = +θ From (33), (35) we have f k (r)r k dr, Φ 2k (θ) = +θ f k (r)r k dr. (35) Φ k(θ) = ( + θ) k f k ( + θ), Φ 2k(θ) = ( + θ) k f k ( + θ), ( + θ) k Φ k (θ) + ( + θ) k Φ 2k (θ) = F k (θ). (36) Expressing Φ 2k from the third equation (36), differentiating with respect to θ and using the second relation (36) we arrive at the linear ordinary differential equation with respect to Φ: (θ + ) 2k Φ k(θ) = ( + θ) 2k Φ k(θ) + 2k( + θ) 2k Φ k (θ) + (( + θ) k F k (θ)) (37) with the initial condition Φ k (0) = 0. (38) The initial condition (38) follows from (35). On the other hand, it is not hard to see that (37), (38), and the first relation (36) are equivalent to the Volterra equation (33). Solving (37) by using integrating factor we yield and using (38) we obtain ((θ + ) k Φ k (θ)) = (θ + ) k ((θ + ) k F k (θ)) Φ k (θ) = θ 2 (θ + )k (s + ) k ((s + ) k F k (s)) ds. Hence from (36) and (33) by elementary calculations k 2 θ 0 0 f k ( + θ) = ( + θ) k Φ k(θ) = (s + ) k (((s + ) k F k (s)) ds) + 2 (θ + )2k+ ((θ + ) k F k (θ)) = Due to (34) we have θ + 2 F k(θ) k2 2 θ 0 (s + ) F k (s)ds. (39) F k (θ) = F k (θ) + F k (0) (( + θ) k ( + θ) k ). 2

Using in addition the definition of F k in (33) by standard calculations we derive from (39) that F k(θ) = k(( + θ) k ( + θ) k ) 2π ((+θ) k +(+θ) k ) 2π 2π 0 2π 0 g 0 (cosσ, sinσ; θ)e ikσ dσ+ ( θ g 0 (cosσ, sinσ; θ)+ θ g 0 (cosσ, sinσ; 0))e ikσ dσ. This formula combined with (33) and (39) implies exponential ill-conditioning of the linearized inverse problem 2. We analyzed a simplest two-dimensional version of the linearized inverse problem which preserves some features of complete nonlinear three-dimensional problem. The Volterra equations (33) or corresponding ordinary differential equations (37) with the initial condition (38) There are no results on the full (nonlinear) inverse problem 2 when one dimensional family of domains growing with pumping time θ depends on the permeability. It will be interesting to study complete nonlinear case analytically and numerically and to adjust it to some realistic situation of oil recovery. 4 Compressible fluids For compressible fluids a simple model is the linear parabolic partial differential equation for the pressure with the initial data and the natural boundary condition a 0 (x) t u div(a(x) u) = f in Ω (0, T ) (40) u = u 0 on Ω {0} ν u = 0 (4) Here a 0 (x) = φ(x)c(x) where φ is the porosity of the medium and c(x) is compressibility of the fluid, and a(x) = µ k(x) where µ(x) is some average viscosity of fluids which is assumed to be known. We assume that φ, c L (Ω) and δ 0 < φ, δ 0 < c on Ω for some positive number δ 0. As above we can assume that f is a function in L 2 (Ω) supported in a subdomain ω Ω. We denote Ω ω = Ω \ ω. Inverse Problem 3. Find a 0, a in Ω ω from solution u to the problem given on ω (0, T ) for any function f L 2 (Ω (0, T )) which is zero on Ω ω (0, T ). Theorem 4.. Solution a 0, a to the Inverse Problem 3 is unique if either a) a 0, a C ( Ω) or b) a 0, a are Lipschitz piecewise constant functions. 2

We outline a proof of part a) based on the heat equation transform of parabolic problems to hyperbolic ones and application of results obtained by the boundary control method and a proof of part b) based on reduction to elliptic equations via stabilization method. First we will show that the data of Inverse Problem 3 uniquely determine the parabolic Dirichlet-to-Neumann map Λ p (Ω, ω). To define it we consider the parabolic initial boundary value problem a 0 (x) t w div(a(x) w) = 0 in Ω ω (0, T ), (42) with the initial data w = 0 on Ω ω {0} (43) and the boundary conditions w = g 0 on ω (0, T ), ν w = 0 on Ω (0, T ). (44) One can show that any g 0 C ( ω [0, T ]) can be approximated in L (0, T ; H 2 ( ω)) by solutions u of the problem (with various f L (0, T ; L 2 (ω))). By the conditions u on ω (0, T ) is given for any f. Hence ν u on ω (0, T ) is given as well. From standard estimates for solutions to parabolic initial boundary value problems it follows that convergence of u in L (0, T ; H 2 ( ω (0, T )) implies convergence of ν u in L (0, T ; H 2 ( ω (0, T )). Hence g 0 uniquely determines ν w on ω (0, T ), so we are given the parabolic Dirichlet-to-Neumann map Λ p (Ω, ω). To use results for hyperbolic equations we define w = a 2 v (45) which as known [6], section 5.2, transforms the partial differential equation for w into a (x) t v v + c(x)v = 0 in Ω ω (0, T ), a = a 2 a0, c = a 2 a 2, (46) v = 0 on Ω ω {0}, (47) u = g 0 on ω (0, T ), ν v = 0 on Ω (0, T ). (48) We remind the heat equation transform v of a function v : v(x, t) = (πt) 2 0 exp( τ 2 4t )v (x, τ)dτ. Let v solve the initial value mixed hyperbolic problem a (x) 2 t v v + c(x)v = 0 in Ω ω (0, T ), (49) v = t v = 0 on Ω ω {0}, (50) v = g 0 on ω (0, T ), ν v = 0 on Ω (0, T ). (5) 3

with g 0 C 2 0( ω (0, )) and g 0 being the heat equation transform of g 0. Available theory of the mixed hyperbolic problem (49), (50), (5) guarantees existence and uniqueness of the solution u with e τt v L (0, ; H 2 (Ω)), e τt 2 t v L (0, ; L 2 (Ω)) for some positive τ. For such functions the heat equation transform is well defined. We refer for its properties to [6], section 9.2. In particular, v solves the parabolic initial boundary value problem (46), (47), (48). Since for this parabolic equation the Dirichlet-to-Neumann is known, g 0 uniquely determines ν v on ω (0, T ). It is clear that ν v is the heat transform of ν v. Since the inverse heat transform is unique [5], section 9.2, ν v on ω (0, T ) is known (for any T ). From known results on inverse hyperbolic problems [4], [8] we derive that a, c are uniquely determined on Ω. Now from definition of c in (46) we have the elliptic partial differential equation a 2 ca 2 = 0 in Ω ω for a 2. Since a is known on ω we have the Cauchy data for a on ω. By uniqueness in the Cauchy problem for elliptic equations [5], section 3.3, a is unique. Since a, a are unique, from the definition of a in (46) a 0 is unique. This completes an outline of a proof in case a). Now we explain ideas of proof in case b). The substitution w = ve τt (52) transforms equation (42) into equation a 0 t v div(a v) + τa 0 v = 0 in Ω ω (0, T ). (53) Let g 0 = G 0 φ where G 0 is any function in C 2 ( Ω) and φ(t) C (R) satisfies the conditions: φ(t) = 0 on (, T 4 ) and φ(t) = eτt if T 2 < t. Since coefficients of equation (53) and the boundary data do not depend on t > T 2, solution to the initial boundary value problem is analytic with respect to t > T 2. So it is uniquely determined for all t > 0. Since equation (53) satisfies conditions of maximum principles and boundary data are time independent for large t by known stabilization results v(, t) v 0 () (Ω ω ) 0 as t, (54) where v 0 solves the following mixed boundary value problem div(a v 0 ) + τa 0 v 0 = 0 in Ω ω, v 0 = G 0 on ω, ν v 0 = 0 on Ω. (55) Since ν v(, t) on ω is given from (54) it follows that ν v 0 on ω is given as well (for any Dirichlet data G 0 ). Hence we are given a partial Dirichlet-to-Neumann map for all elliptic equations (55). Due to analyticity we can take τ = 0, then uniqueness of piecewise Lipschitz constant a follows from the generalization of results of of Kohn and Vogelius [20] given by Sever [25]. Similarly one can show uniqueness of a 0. The proof is complete. As follows from the proof of Theorem 4. and known results on the inverse conductivity problem and on inverse hyperbolic problems Inverse Problems 3 should be also severely (exponentially) ill-conditioned. 4

5 Conclusion We reviewed analytical results for inverse problems based on some models of filtration of water and oil through soil. In most cases we only outlined proofs which can be made complete. We hope that methods of proofs (especially formulas for the inverse Muskat model and use of the heat equation transform) can be used to design more efficient and reliable numerical algorithms. It seems that all possible formulations of inverse problems are severely ill-posed, which indicates that it is hard to expect fine numerical resolution. We observ that similar inverse problems arise in hydraulics [28]. The fitration system is a combination of a conservation-diffusion law (elliptic equation) with a drift (parabolic or first order equation). The same structure is a feature of partial differential equations models of semiconductors and ion channels. In all our consideration we decoupled system into scalar equations. It would be very interesting to find some approach to inverse problems for full drift-diffusion systems. As mentioned, the direct problem is a serious mathematical challenge which is not likely to be addressed in near future. So as in hydrodynamics it makes sense to simplify general system of filtration to particular interesting cases and study these cases analytically and numerically. Simulteneously, one can try to look at more difficult but more realistic model of three phases filtration (oil, water, and gas). Aknowledgement: The author is grateful to prof. H. W. Engl who during semester of Inverse Problems at IPAM initiated his work on inverse problems in oil recovery and writing of this review paper. Use of the Muskat problem in the radially symmetric case was discussed during a workshop at IPAM in September 2003 with Chris Farmer, Bill Lionheart and John Ockendon. This research was in part supported by the NSF through IPAM and by the NSF grants DMS 04-05976 and DMS 07-07734. References [] S. Antontsev, A. Kazhikhov, V. Monakhov. Boundary value problems in mechanics of nonhomogeneous fluids, North-Holland, 990. [2] K. Astala, L. Paivarinta. Calderon s inverse conductivity problem in the plane. Ann. Math.,63 (2006), 265-299. [3] J. Bear. Dynamics of fluids in porous media, Dover Publ., New York, 988. [4] M. I. Belishev. Boundary Control in Reconstruction of Manifolds and Metrics (the BC-method). Inverse Problems, 3 (997), R-R45. [5] A. L. Bukhgeim, G. Uhlmann. Recovering a potential from partial Cauchy data. Comm. Part. Diff. Equat., 27 (2002), 653-668. [6] G. Chavent, J. Jaffre. Mathematical models and finite elements for reservoir simulation. North-Holland, 986. 5

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