Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem
aa r X i v : . [ m a t h . O C ] J a n Necessary Optimality Condition for a DiscreteDead Oil Isotherm Optimal Control Problem
Moulay Rchid Sidi Ammi and Delfim F. M. Torres Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal [email protected] Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal [email protected]
Summary.
We obtain necessary optimality conditions for a semi-discretized opti-mal control problem for the classical system of nonlinear partial differential equationsmodelling the water-oil (isothermal dead-oil model).
Key words: extraction of hydrocarbons; dead oil isotherm problem; optimalityconditions.
We study an optimal control problem in the discrete case whose control systemis given by the following system of nonlinear partial differential equations, ∂ t u − ∆ϕ ( u ) = div ( g ( u ) ∇ p ) in Q T = Ω × (0 , T ) ,∂ t p − div ( d ( u ) ∇ p ) = f in Q T = Ω × (0 , T ) ,u | ∂Ω = 0 , u | t =0 = u ,p | ∂Ω = 0 , p | t =0 = p , (1)which result from a well established model for oil engineering within the frame-work of the mechanics of a continuous medium [3]. The domain Ω is an openbounded set in R with a sufficiently smooth boundary. Further hypotheseson the data of the problem will be specified later.At the time of the first run of a layer, the flow of the crude oil towardsthe surface is due to the energy stored in the gases under pressure in thenatural hydraulic system. To mitigate the consecutive decline of productionand the decomposition of the site, water injections are carried out, well beforethe normal exhaustion of the layer. The water is injected through wells withhigh pressure, by pumps specially drilled to this end. The pumps allow the dis-placement of the crude oil towards the wells of production. More precisely, theproblem consists in seeking the admissible control parameters which minimize Moulay Rchid Sidi Ammi and Delfim F. M. Torres a certain objective functional. In our problem, the main goal is to distributeproperly the wells in order to have the best extraction of the hydrocarbons.For this reason, we consider a cost functional containing different parametersarising in the process. To address the optimal control problem, we use theLagrangian method to derive an optimality system: from the cost function weintroduce a Lagrangian; then, we calculate the Gˆateaux derivative of the La-grangian with respect to its variables. This technique was used, in particular,by A. Masserey et al. for electromagnetic models of induction heating [1, 7],and by H.-C. Lee and T. Shilkin for the thermistor problem [5].We consider the following cost functional: J ( u, p, f ) = 12 k u − U k ,Q T + 12 k p − P k ,Q T + β k f k q q ,Q T + β k ∂ t f k ,Q T . (2)The control parameters are the reduced saturation of oil u , the pressure p ,and f . The coefficients β > β ≥ q >
1. The first two terms in (2) allow to minimize the differencebetween the reduced saturation of oil u , the global pressure p and the givendata U and P . The third and fourth terms are used to improve the quality ofexploitation of the crude oil. We take β = 0 just for the sake of simplicity. Itis important to emphasize that our choice of the cost function is not unique.One can always add additional terms of penalization to take into accountother properties which one may wish to control. Recently, we proved in [8]results of existence, uniqueness, and regularity of the optimal solutions to theproblem of minimizing (2) subject to (1), using the theory of parabolic prob-lems [4, 6]. Here, our goal is to obtain necessary optimality conditions whichmay be easily implemented on a computer. More precisely, we address theproblem of obtaining necessary optimality conditions for the semi-discretizedtime problem.In order to be able to solve problem (1)-(2) numerically, we use discretiza-tion of the problem in time by a method of finite differences. For a fixed real N , let τ = TN be the step of a uniform partition of the interval [0 , T ] and t n = nτ , n = 1 , . . . , N . We denote by u n an approximation of u . The discretecost functional is then defined as follows: J ( u n , p n , f n ) = τ N X n =1 Z Ω (cid:8) k u n − U k ,Ω + k p n − P k ,Ω + β k f n k q q ,Ω (cid:9) dx . (3)It is now possible to state our optimal control problem: find (¯ u n , ¯ p n , ¯ f n ) whichminimizes (3) among all functions ( u n , p n , f n ) satisfying u n +1 − u n τ − ∆ϕ ( u n ) = div ( g ( u n ) ∇ p ) in Ω , p n +1 − p n τ − div ( d ( u n ) ∇ p n ) = f n in Ω ,u | ∂Ω = 0 , u | t =0 = u ,p | ∂Ω = 0 , p | t =0 = p . (4) ecessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 3 The soughtafter necessary optimality conditions are proved in § Our main objective is to obtain necessary conditions for a triple (cid:0) ¯ u n , ¯ p n , ¯ f n (cid:1) to minimize (3) among all the functions ( u n , p n , f n ) verifying (4). In the sequelwe assume that ϕ , g and d are real valued functions, respectively of class C , C and C , satisfying:(H1) 0 < c ≤ d ( r ), ϕ ( r ) ≤ c ; | d ′ ( r ) | , | ϕ ′ ( r ) | , | ϕ ′′ ( r ) | ≤ c ∀ r ∈ R .(H2) u , p ∈ C (cid:0) ¯ Ω (cid:1) , and U, P ∈ L ( Ω ), where u , p , U, P : Ω → R , and u | ∂Ω = p | ∂Ω = 0.We consider the following spaces: W p ( Ω ) := { u ∈ L p ( Ω ) , ∇ u ∈ L p ( Ω ) } , endowed with the norm k u k W p ( Ω ) = k u k p,Ω + k∇ u k p,Ω ; W p ( Ω ) := (cid:8) u ∈ W p ( Ω ) , ∇ u ∈ L p ( Ω ) (cid:9) , with the norm k u k W p ( Ω ) = k u k W p ( Ω ) + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) p,Ω ; and the following notation: W := ◦ W q ( Ω ) ; Υ := L q ( Ω ) ; H := L q ( Ω ) × ◦ W − q q ( Ω ) . We define the following nonlinear operator corresponding to (4): F : W × W × Υ −→ H × H ( u n , p n , f n ) −→ F ( u n , p n , f n ) , where F ( u n , p n , f n ) = u n +1 − u n τ − ∆ϕ ( u n ) − div ( g ( u n ) ∇ p n ) , γ u n − u u n +1 − u n τ − div ( d ( u n ) ∇ p n ) − f n , γ p n − p ! ,γ being the trace operator γ u n = u | t =0 . Our hypotheses ensure that F iswell defined. Moulay Rchid Sidi Ammi and Delfim F. M. Torres
In addition to the hypotheses (H1) and (H2), let us suppose that(H3) | ϕ ′′′ | ≤ c .Then, the operator F is Gˆateaux differentiable and for all ( e, w, h ) ∈ W × W × Υ its derivative is given by δF ( u n , p n , f n )( e, w, h ) = dds F ( u n + se, p n + sw, f n + sh ) | s =0 = ( δF , δF ) = (cid:18) ξ , ξ ξ , ξ (cid:19) ,ξ = e − div ( ϕ ′ ( u n ) ∇ e ) − div ( ϕ ′′ ( u n ) e ∇ u n ) − div ( g ( u n ) ∇ w ) − div ( g ′ ( u n ) e ∇ p n ) , ξ = γ e , ξ = w − div ( d ( u n ) ∇ w ) − div ( d ′ ( u n ) e ∇ p n ) − h , ξ = γ w .Furthermore, for any optimal solution (cid:0) ¯ u n , ¯ p n , ¯ f n (cid:1) of the problem of mini-mizing (3) among all the functions ( u n , p n , f n ) satisfying (4) , the image of δF (cid:0) ¯ u n , ¯ p n , ¯ f n (cid:1) is equal to H × H . To prove Theorem 1 we make use of the following lemma.
Lemma 1.
The operator δF ( u n , p n , f n ) : W × W × Υ −→ H × H is linearand bounded.Proof (Lemma 1). For all ( e, w, h ) ∈ W × W × Υδ u n F ( u n , p n , f n )( e, w, h )= e − div ( ϕ ′ ( u n ) ∇ e ) − div ( ϕ ′′ ( u n ) e ∇ u n ) − div ( g ( u n ) ∇ w ) − div ( g ′ ( u n ) e ∇ p n )= e − ϕ ′ ( u n ) △ e − ϕ ′′ ( u n ) ∇ u n . ∇ e − ϕ ′′ ( u n ) e △ u n − ϕ ′′ ( u n ) ∇ e. ∇ u n − ϕ ′′′ ( u n ) e |∇ u n | − g ( u n ) △ w − g ′ ( u n ) ∇ u n . ∇ w − g ′ ( u n ) e △ p n − g ′ ( u n ) ∇ e. ∇ p n − g ′′ ( u n ) e ∇ u n . ∇ p n , where δ u n F is the Gˆateaux derivative of F with respect to u n . Using ourhypotheses we have k g ′′ ( u n ) e ∇ u n . ∇ p n k q,Ω ≤ k e k ∞ ,Ω k∇ u n . ∇ p n k q,Ω ≤ k e k ∞ ,Ω k∇ u n k q − q ,Ω k∇ p n k ,Ω ≤ c k u n k W k p n k W k e k W . Evaluating each term of δ u n F , we obtain k δ u n F ( u n , p n , f n )( e, w, h ) k q,Q T ≤ c ( k u n k W , k p n k W , k f n k Υ ) ( k e k W + k w k W + k h k Υ ) . (5) ecessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 5 In a similar way, we have for all ( e, w, h ) ∈ W × W × Υ that δ p n F ( u n , p n , f n )( e, w, h ) = w − div ( d ( u n ) ∇ w ) − div ( d ′ ( u n ) e ∇ p n ) − h = w − d ( u n ) △ w − d ′ ( u n ) ∇ u n . ∇ w − d ′ ( u n ) e △ p n − d ′ ( u n ) ∇ e. ∇ u n − d ′ ( u n ) e ∇ u n . ∇ p n − h , with δ p n F the Gˆateaux derivative of F with respect to p n . Then, using againour hypotheses, we obtain that k δ p n F ( u n , p n , f n )( e, w, h ) k q,Ω ≤ k w k q,Ω + k∇ w k q,Ω + c k△ w k q,Ω + c k∇ u n . ∇ w k q,Ω + c k e △ p n k q,Ω + c k∇ e. ∇ u n k q,Ω + c k e ∇ u n . ∇ p n k q,Ω + k h k q,Ω . (6)Applying similar arguments to all terms of (6), we then have k δ p n F ( u n , p n , f n )( e, w, h ) k q,Ω ≤ c ( k u n k W , k p n k W , k f n k Υ ) ( k e k W + k w k W + k h k Υ ) . (7)Consequently, by (5) and (7) we can write k δF ( u n , p n , f n )( e, w, h ) k H × H × Υ ≤ c ( k u n k W , k p n k W , k f n k Υ ) ( k e k W + k w k W + k h k Υ ) . ⊓⊔ Proof (Theorem 1).
In order to show that the image of δF ( u, p, f ) is equal to H × H , we need to prove that there exists ( e, w, h ) ∈ W × W × Υ such that e − div ( ϕ ′ ( u n ) ∇ e ) − div ( ϕ ′′ ( u n ) e ∇ u n ) − div ( g ( u n ) ∇ w ) − div ( g ′ ( u n ) e ∇ p n ) = α ,w − div ( d ( u n ) ∇ w ) − div ( d ′ ( u n ) e ∇ p n ) − h = β ,e | ∂Ω = 0 , e | t =0 = b ,w | ∂Ω = 0 , w | t =0 = a , (8)for any ( α, a ) and ( β, b ) ∈ H . Writing the system (8) for h = 0 as e − ϕ ′ ( u n ) △ e − ϕ ′′ ( u n ) ∇ u n . ∇ e − ϕ ′′ ( u n ) e △ u n − ϕ ′′′ ( u n ) e |∇ u n | , − g ( u n ) △ w − g ′ ( u n ) ∇ u n . ∇ w − g ′ ( u n ) e △ p n − g ′ ( u n ) ∇ p n . ∇ e − g ′′ ( u n ) e ∇ u n . ∇ p n = α ,w − d ( u n ) △ w − d ′ ( u n ) ∇ u n . ∇ w − d ′ ( u n ) e △ p n − d ′ ( u n ) ∇ u n . ∇ e − d ′ ( u n ) e ∇ u n . ∇ p n = β ,e | ∂Ω = 0 , e | t =0 = b ,w | ∂Ω = 0 , w | t =0 = a , (9) Moulay Rchid Sidi Ammi and Delfim F. M. Torres it follows from the regularity of the optimal solution that ϕ ′′ ( u n ) △ u n , ϕ ′′′ ( u n ) |∇ u n | , g ′ ( u n ) △ p n , g ′′ ( u n ) ∇ u n . ∇ p n , d ′ ( u n ) △ p n , and d ′ ( u n ) ∇ u n . ∇ p n belong to L q ( Ω ); ϕ ′′ ( u n ) ∇ u n , g ′ ( u n ) ∇ u n , g ′ ( u n ) ∇ p n , and d ′ ( u n ) ∇ u n be-long to L q ( Ω ). This ensures, in view of the results of [4, 6], existence of aunique solution of the system (9). Hence, there exists a ( e, w,
0) verifying (8).We conclude that the image of δF is equal to H × H . ⊓⊔ We consider the cost functional J : W × W × Υ → R (3) and the Lagrangian L defined by L ( u n , p n , f n , p , e , a, b ) = J ( u n , p n , f n ) + (cid:28) F ( u n , p n , f n ) , (cid:18) p ae , b (cid:19)(cid:29) , where the bracket h· , ·i denotes the duality between H and H ′ . Theorem 2.
Under hypotheses (H1)–(H3), if (cid:0) u n , p n , f n (cid:1) is an optimal solu-tion to the problem of minimizing (3) subject to (4) , then there exist functions ( e , p ) ∈ W ( Ω ) × W ( Ω ) satisfying the following conditions: e + div ( ϕ ′ ( u n ) ∇ e ) − d ′ ( u n ) ∇ p n . ∇ p − ϕ ′′ ( u n ) ∇ u n . ∇ e − g ′ ( u n ) ∇ p n . ∇ e = τ N X n =1 ( u n − U ) ,e | ∂Ω = 0 , e | t = T = 0 ,p + div ( d ( u n ) ∇ p ) + div ( g ( u n ) ∇ e ) = τ N X n =1 ( p n − P ) ,p | ∂Ω = 0 , p | t = T = 0 ,q β τ N X n =1 | f n | q − f n = p . (10) Proof.
Let (cid:0) u n , p n , f n (cid:1) be an optimal solution to the problem of minimizing(3) subject to (4). It is well known (cf. e.g. [2]) that there exist Lagrangemultipliers (cid:0) ( p , a ) , ( e , b ) (cid:1) ∈ H ′ × H ′ verifying δ ( u n ,p n ,f n ) L (cid:0) u n , p n , f n , p , e , a, b (cid:1) ( e, w, h ) = 0 ∀ ( e, w, h ) ∈ W × W × Υ, with δ ( u n ,p n ,f n ) L the Gˆateaux derivative of L with respect to ( u n , p n , f n ).This leads to the following system: ecessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 7 τ N X n =1 Z Ω (cid:0) ( u n − U ) e + ( p n − P ) w + q β | f n | q − f n h (cid:1) dx − Z Ω (cid:16) e − div ( ϕ ′ ( u n ) ∇ e ) − div ( ϕ ′′ ( u n ) e ∇ u n ) − div ( g ( u n ) ∇ w ) − div ( g ′ ( u n ) e ∇ p n ) (cid:17) e ! dx − Z Ω ( w − div ( d ( u n ) ∇ w ) − div ( d ′ ( u n ) e ∇ p n ) − h ) p dx −h γ e, a i + −h γ w, b i = 0 ∀ ( e, w, h ) ∈ W × W × Υ. The above system is equivalent to the following one: Z Ω τ N X n =1 ( u n − U ) e − div ( d ′ ( u n ) e ∇ p n ) p + e e − div ( ϕ ′ ( u n ) ∇ e ) e − div ( ϕ ′′ ( u n ) e ∇ u n ) e − div ( g ′ ( u n ) e ∇ p n ) e ! dx + Z Ω τ N X n =1 ( p n − P ) w + w p − div ( d ( u n ) ∇ w ) p − div ( g ( u n ) ∇ w ) e ! dx + Z Ω q β τ N X n =1 | f n | q − f n h − p h ! dx + h γ e, a i + h γ w, b i = 0 ∀ ( e, w, h ) ∈ W × W × Υ. (11)In others words, we have Z Ω τ N X n =1 ( u n − U ) + d ′ ( u ) ∇ p n . ∇ p − e − div ( ϕ ′ ( u n ) ∇ e )+ ϕ ′′ ( u n ) ∇ u n . ∇ e + g ′ ( u n ) ∇ p n . ∇ e ! e dx + Z Ω τ N X n =1 ( p n − P ) + p − div ( d ( u n ) ∇ p ) − div ( g ( u n ) ∇ e ) ! w dx + Z Ω q β τ N X n =1 | f n | q − f n h − p h ! dx + h γ e, a i + h γ w, b i = 0 ∀ ( e, w, h ) ∈ W × W × Υ. (12)Consider now the system Moulay Rchid Sidi Ammi and Delfim F. M. Torres e + div ( ϕ ′ ( u n ) ∇ e ) − d ′ ( u n ) ∇ p n . ∇ p − ϕ ′′ ( u n ) ∇ u n . ∇ e − g ′ ( u n ) ∇ p n . ∇ e = τ N X n =1 ( u n − U ) ,p + div ( d ( u n ) ∇ p ) + div ( g ( u n ) ∇ e ) = τ N X n =1 ( p n − P ) ,e | ∂Ω = p | ∂Ω = 0 , e | t = T = p | t = T = 0 , (13)with unknowns ( e , p ) which is uniquely solvable in W ( Ω ) × W ( Ω ) bythe theory of elliptic equations [4]. The problem of finding ( e, w ) ∈ W × W satisfying e − div ( ϕ ′ ( u n ) ∇ e ) − div ( ϕ ′′ ( u n ) e ∇ u n ) − div ( g ( u n ) ∇ w ) − div ( g ′ ( u n ) e ∇ p n ) = sign ( e − e ) ,w − div ( d ( u n ) ∇ w ) − div ( d ′ ( u n ) e ∇ p n ) = sign ( p − p ) ,γ e = γ w = 0 , (14)is also uniquely solvable on W q ( Ω ) × W q ( Ω ). Let us choose h = 0 in (12) andmultiply (13) by ( e, w ). Then, integrating by parts and making the differencewith (12) we obtain: Z Ω (cid:16) e − div ( ϕ ′ ( u n ) ∇ e ) − div ( ϕ ′′ ( u n ) e ∇ u n ) − div ( g ( u n ) ∇ w ) − div ( g ′ ( u n ) e ∇ p n ) (cid:17) ( e − e ) dx + Z Ω ( w − div ( d ( u n ) ∇ w ) − div ( d ′ ( u n ) e ∇ p n )) ( p − p ) dx + h γ e, γ e − a i + h γ w, γ p − b i = 0 ∀ ( e, w ) ∈ W × W. (15)Choosing ( e, w ) in (15) as the solution of the system (14), we have Z Ω sign ( e − e )( e − e ) dxdt + Z Ω sign ( p − p )( p − p ) dx = 0 . It follows that e = e and p = p . Coming back to (15), we obtain γ e = a and γ p = b . On the other hand, choosing ( e, w ) = (0 ,
0) in (12), we get Z Ω β τ N X n =1 | f n | q − f n − p ! h dx = 0 , ∀ h ∈ Υ. Then (10) follows, which concludes the proof of Theorem 2. ⊓⊔ We claim that the results we obtain here are useful for numerical im-plementations. This is still under investigation and will be addressed in aforthcoming publication. ecessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 9
Acknowledgments
The authors were supported by the
Portuguese Foundation for Science andTechnology (FCT) through the
Centre for Research on Optimization and Con-trol (CEOC) of the University of Aveiro, cofinanced by the European Commu-nity fund FEDER/POCI 2010. This work was developed under the post-docproject SFRH/BPD/20934/2004.
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