Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem

0.11 MB·English

by  Moulay Rchid Sidi Ammi

Necessary Optimality Condition for a Discrete
Dead Oil Isotherm Optimal Control Problem
8
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Moulay Rchid Sidi Ammi1 and Delfim F. M. Torres2
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1
Department of Mathematics, Universityof Aveiro, 3810-193 Aveiro, Portugal
n
[email protected]
a 2
Department of Mathematics, Universityof Aveiro, 3810-193 Aveiro, Portugal
J
[email protected]
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2
Summary. We obtain necessary optimality conditions for a semi-discretized opti-

] malcontrolproblemfortheclassicalsystemofnonlinearpartialdifferentialequations
C modelling thewater-oil (isothermal dead-oil model).
O
Key words: extraction of hydrocarbons; dead oil isotherm problem; optimality
.
h conditions.
t
a
m
1 Introduction
[

1
Westudyanoptimalcontrolprobleminthediscretecasewhosecontrolsystem
v
2 is given by the following system of nonlinear partial differential equations,
3
3 ∂ u−∆ϕ(u)=div(g(u)∇p) in Q =Ω×(0,T),
t T
4
∂ p−div(d(u)∇p)=f in Q =Ω×(0,T),
1.  t T (1)
0 u|∂Ω =0, u|t=0 =u0,
8 p|∂Ω =0, p|t=0 =p0,
:0 whichresultfromawellestablishedmodelforoilengineeringwithintheframe-
v
i work of the mechanics of a continuous medium [3]. The domain Ω is an open
X bounded set in R2 with a sufficiently smooth boundary. Further hypotheses
r on the data of the problem will be specified later.
a
At the time of the first run of a layer, the flow of the crude oil towards
the surface is due to the energy stored in the gases under pressure in the
natural hydraulic system. To mitigate the consecutive decline of production
andthedecompositionofthesite,waterinjectionsarecarriedout,wellbefore
the normal exhaustion of the layer. The water is injected through wells with
highpressure,bypumpsspeciallydrilledtothisend.Thepumpsallowthedis-
placementofthecrudeoiltowardsthewellsofproduction.Moreprecisely,the
problemconsistsinseekingtheadmissiblecontrolparameterswhichminimize
2 Moulay RchidSidi Ammi and Delfim F. M. Torres
a certain objective functional. In our problem, the main goal is to distribute
properly the wells in order to have the best extraction of the hydrocarbons.
For this reason,we consider a cost functional containing different parameters
arising in the process. To address the optimal control problem, we use the
Lagrangianmethodtoderiveanoptimalitysystem:fromthe costfunctionwe
introduce a Lagrangian;then, we calculate the Gˆateaux derivative of the La-
grangianwith respectto its variables.This technique wasused, 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:
1 1 β β
J(u,p,f)= ku−Uk2 + kp−Pk2 + 1 kfk2q0 + 2 k∂ fk2 .
2 2,QT 2 2,QT 2 2q0,QT 2 t 2,QT
(2)
The control parameters are the reduced saturation of oil u, the pressure p,
and f. The coefficients β > 0 and β ≥ 0 are two coefficients of penaliza-
1 2
tion, and q > 1. The first two terms in (2) allow to minimize the difference
0
between the reduced saturation of oil u, the global pressure p and the given
data U andP. The third and fourthterms are usedto improvethe quality of
exploitationof the crude oil.We takeβ =0 just for the sakeof simplicity. It
2
is important to emphasize that our choice of the cost function is not unique.
One can always add additional terms of penalization to take into account
other properties which one may wish to control. Recently, we proved in [8]
resultsofexistence,uniqueness,andregularityofthe optimalsolutionstothe
problem of minimizing (2) subject to (1), using the theory of parabolic prob-
lems [4, 6]. Here, our goal is to obtain necessary optimality conditions which
may be easily implemented on a computer. More precisely, we address the
problem of obtaining necessary optimality conditions for the semi-discretized
time problem.
Inordertobe abletosolveproblem(1)-(2)numerically,weusediscretiza-
tion of the problem in time by a method of finite differences. For a fixed real
N, let τ = T be the step of a uniform partition of the interval [0,T] and
N
t =nτ, n=1,...,N. We denote by un an approximationof u. The discrete
n
cost functional is then defined as follows:
N
τ
J(un,pn,fn)= kun−Uk2 +kpn−Pk2 +β kfnk2q0 dx.
2 2,Ω 2,Ω 1 2q0,Ω
n=1ZΩ
X (cid:8) (cid:9) (3)
Itisnowpossibletostateouroptimalcontrolproblem:find(u¯n,p¯n,f¯n)which
minimizes (3) among all functions (un,pn,fn) satisfying
un+1−un −∆ϕ(un)=div(g(un)∇p) in Ω,
τ
pn+1−pn −div(d(un)∇pn)=fn in Ω,
 τ (4)
u|∂Ω =0, u|t=0 =u0,
p| =0, p| =p .
∂Ω t=0 0

Necessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 3
The soughtafter necessary optimality conditions are proved in §3 under suit-
able hypotheses on the data of the problem.
2 Notation, hypotheses, and functional spaces
Our main objective is to obtain necessary conditions for a triple u¯n,p¯n,f¯n
tominimize(3)amongallthefunctions(un,pn,fn)verifying(4).Inthesequel
(cid:0) (cid:1)
we assume that ϕ, g and d are realvalued functions, respectively of class C3,
C2 and C1, satisfying:
(H1) 0<c ≤d(r), ϕ(r)≤c ; |d′(r)|, |ϕ′(r)|, |ϕ′′(r)|≤c ∀r ∈R.
1 2 3
(H2) u , p ∈ C2 Ω¯ , and U, P ∈ L2(Ω), where u , p , U, P : Ω → R, and
0 0 0 0
u | = p | =0.
0 ∂Ω 0 ∂Ω(cid:0) (cid:1)
We consider the following spaces:
W1(Ω):={u∈Lp(Ω), ∇u∈Lp(Ω)} ,
p
endowed with the norm kuk =kuk +k∇uk ;
W1(Ω) p,Ω p,Ω
p
W2(Ω):= u∈W1(Ω), ∇2u ∈Lp(Ω) ,
p p
withthenormkuk =ku(cid:8)k + ∇2u ;andth(cid:9)efollowingnotation:
Wp2(Ω) Wp1(Ω) p,Ω
(cid:13) (cid:13)
◦2(cid:13) (cid:13)
W :=W (Ω);
2q
Υ :=L2q(Ω);
◦2−1
H :=L2q(Ω)×W q (Ω).
2q
3 Main results
We define the following nonlinear operator corresponding to (4):
F :W ×W ×Υ −→H ×H
(un,pn,fn)−→F(un,pn,fn),
where
un+1−un −∆ϕ(un)−div(g(un)∇pn),γ un−u
F(un,pn,fn)= τ 0 0 ,
un+1τ−un −div(d(un)∇pn)−fn, γ0pn−p0!
γ being the trace operator γ un = u| . Our hypotheses ensure that F is
0 0 t=0
well defined.
4 Moulay RchidSidi Ammi and Delfim F. M. Torres
3.1 Gˆateaux differentiability
Theorem 1. Inaddition tothe hypotheses (H1) and (H2),let us suppose that
(H3) |ϕ′′′|≤c.
Then, the operator F is Gaˆteaux differentiable and for all (e,w,h) ∈ W ×
W ×Υ its derivative is given by
d
δF(un,pn,fn)(e,w,h)= F(un+se,pn+sw,fn+sh)|
ds s=0
ξ ,ξ
=(δF ,δF )= 1 2 ,
1 2 ξ ,ξ
3 4
(cid:18) (cid:19)
ξ =e−div(ϕ′(un)∇e)−div(ϕ′′(un)e∇un)−div(g(un)∇w)−div(g′(un)e∇pn),
1
ξ = γ e, ξ = w − div(d(un)∇w) − div(d′(un)e∇pn) − h, ξ = γ w.
2 0 3 4 0
Furthermore, for any optimal solution u¯n,p¯n,f¯n of the problem of mini-
mizing (3) among all the functions (un,pn,fn) satisfying (4), the image of
δF u¯n,p¯n,f¯n is equal to H ×H. (cid:0) (cid:1)
(cid:0)To prove T(cid:1)heorem 1 we make use of the following lemma.
Lemma 1.The operator δF(un,pn,fn) : W ×W ×Υ −→ H ×H is linear
and bounded.
Proof (Lemma 1). For all (e,w,h)∈W ×W ×Υ
δunF1(un,pn,fn)(e,w,h)
=e−div(ϕ′(un)∇e)−div(ϕ′′(un)e∇un)
−div(g(un)∇w)−div(g′(un)e∇pn)
=e−ϕ′(un)△e−ϕ′′(un)∇un.∇e−ϕ′′(un)e△un
−ϕ′′(un)∇e.∇un−ϕ′′′(un)e|∇un|2−g(un)△w−g′(un)∇un.∇w
−g′(un)e△pn−g′(un)∇e.∇pn−g′′(un)e∇un.∇pn,
where δunF is the Gˆateaux derivative of F with respect to un. Using our
hypotheses we have
kg′′(un)e∇un.∇pnk2q,Ω ≤kek∞,Ωk∇un.∇pnk2q,Ω
≤kek∞,Ωk∇unk24−qq,Ωk∇pnk4,Ω
≤ckunk kpnk kek .
W W W
Evaluating each term of δunF1, we obtain
kδunF1(un,pn,fn)(e,w,h)k2q,QT
≤c(kunk ,kpnk ,kfnk )(kek +kwk +khk ) . (5)
W W Υ W W Υ
Necessary 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
δpnF2(un,pn,fn)(e,w,h)=w−div(d(un)∇w)−div(d′(un)e∇pn)−h
=w−d(un)△w−d′(un)∇un.∇w−d′(un)e△pn
−d′(un)∇e.∇un−d′(un)e∇un.∇pn−h,
with δpnF the Gˆateaux derivative of F with respect to pn. Then, using again
our hypotheses, we obtain that
kδpnF2(un,pn,fn)(e,w,h)k2q,Ω ≤kwk2q,Ω +k∇wk2q,Ω +ck△wk2q,Ω
+ck∇un.∇wk +cke△pnk
2q,Ω 2q,Ω
+ck∇e.∇unk +cke∇un.∇pnk +khk . (6)
2q,Ω 2q,Ω 2q,Ω
Applying similar arguments to all terms of (6), we then have
kδpnF2(un,pn,fn)(e,w,h)k2q,Ω
≤c(kunk ,kpnk ,kfnk )(kek +kwk +khk ) . (7)
W W Υ W W Υ
Consequently, by (5) and (7) we can write
kδF(un,pn,fn)(e,w,h)kH×H×Υ
≤c(kunk ,kpnk ,kfnk )(kek +kwk +khk ) .
W W Υ W W Υ
⊓⊔
Proof (Theorem 1). In orderto show that the image of δF(u,p,f)is equalto
H ×H, we need to prove that there exists (e,w,h)∈W ×W ×Υ such that
e−div(ϕ′(un)∇e)−div(ϕ′′(un)e∇un)
−div(g(un)∇w)−div(g′(un)e∇pn)=α,
w−div(d(un)∇w)−div(d′(un)e∇pn)−h=β, (8)
e| =0, e| =b,
∂Ω t=0
w| =0, w| =a,
∂Ω t=0
for any (α,a) and (β,b)∈H. Writing the system (8) for h=0 as
e−ϕ′(un)△e−2ϕ′′(un)∇un.∇e−ϕ′′(un)e△un−ϕ′′′(un)e|∇un|2,
−g(un)△w−g′(un)∇un.∇w−g′(un)e△pn
−g′(un)∇pn.∇e−g′′(un)e∇un.∇pn =α,
w−d(un)△w−d′(un)∇un.∇w−d′(un)e△pn (9)
−d′(un)∇un.∇e−d′(un)e∇un.∇pn =β,
e| =0, e| =b,
∂Ω t=0
w| =0, w| =a,
∂Ω t=0
6 Moulay RchidSidi Ammi and Delfim F. M. Torres
it follows from the regularity of the optimal solution that ϕ′′(un)△un,
ϕ′′′(un)|∇un|2, g′(un)△pn, g′′(un)∇un.∇pn, d′(un)△pn,and d′(un)∇un.∇pn
belong to L2q0(Ω); ϕ′′(un)∇un, g′(un)∇un, g′(un)∇pn, and d′(un)∇un be-
long to L4q0(Ω). This ensures, in view of the results of [4, 6], existence of a
unique solutionofthe system (9).Hence, there exists a (e,w,0) verifying (8).
We conclude that the image of δF is equal to H ×H. ⊓⊔
3.2 Necessary optimality condition
We consider the cost functional J :W ×W ×Υ →R (3) and the Lagrangian
L defined by
p a
L(un,pn,fn,p ,e ,a,b)=J(un,pn,fn)+ F(un,pn,fn), 1 ,
1 1 e , b
1
(cid:28) (cid:18) (cid:19)(cid:29)
where the bracket h·,·i denotes the duality between H and H′.
Theorem 2. Underhypotheses (H1)–(H3), if un,pn,fn is an optimal solu-
tion to theproblem of minimizing (3) subject to (4), then there exist functions
(cid:0) (cid:1)
(e ,p )∈W2(Ω)×W2(Ω) satisfying the following conditions:
1 1 2 2
e +div(ϕ′(un)∇e )−d′(un)∇pn.∇p −ϕ′′(un)∇un.∇e
1 1 1 1
N
−g′(un)∇pn.∇e =τ (un−U),
1
n=1
X
e | =0, e | =0,
1 ∂Ω 1 t=T
N (10)
p +div(d(un)∇p )+div(g(un)∇e )=τ (pn−P),
1 1 1
n=1
X
p | =0, p | =0,
1 ∂Ω 1 t=T
N
q β τ |fn|2q0−2fn =p .
0 1 1
n=1
X
Proof. Let un,pn,fn be an optimal solution to the problem of minimizing
(3) subject to (4). It is well known (cf. e.g. [2]) that there exist Lagrange
multipliers (cid:0)(p ,a),(e(cid:1),b) ∈H′×H′ verifying
1 1
(cid:0) (cid:1)
δ L un,pn,fn,p ,e ,a,b (e,w,h)=0 ∀(e,w,h)∈W ×W ×Υ,
(un,pn,fn) 1 1
with δ (cid:0) L the Gˆateaux der(cid:1)ivative of L with respect to (un,pn,fn).
(un,pn,fn)
This leads to the following system:
Necessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 7
N
τ (un−U)e+(pn−P)w+q β |fn|2q0−2fnh dx
0 1
n=1ZΩ
X (cid:0) (cid:1)
− e−div(ϕ′(un)∇e)−div(ϕ′′(un)e∇un)
ZΩ (cid:16)
−div(g(un)∇w)−div(g′(un)e∇pn) e dx
1
!
(cid:17)
− (w−div(d(un)∇w)−div(d′(un)e∇pn)−h)p dx
1
ZΩ
−hγ e,ai+−hγ w,bi=0 ∀(e,w,h)∈W ×W ×Υ.
0 0
The above system is equivalent to the following one:
N
τ (un−U)e−div(d′(un)e∇pn)p +ee −div(ϕ′(un)∇e)e
1 1 1
ZΩ n=1
X
−div(ϕ′′(un)e∇un)e −div(g′(un)e∇pn)e dx
1 1
!
N
+ τ (pn−P)w+wp −div(d(un)∇w)p −div(g(un)∇w)e dx
1 1 1
ZΩ n=1 !
X
N
+ q β τ |fn|2q0−2fnh−p h dx
0 1 1
ZΩ n=1 !
X
+hγ e,ai+hγ w,bi=0 ∀(e,w,h)∈W ×W ×Υ.
0 0
(11)
In others words, we have
N
τ (un−U)+d′(u)∇pn.∇p −e −div(ϕ′(un)∇e )
1 1 1
ZΩ n=1
X
+ϕ′′(un)∇un.∇e +g′(un)∇pn.∇e edx
1 1
!
N (12)
+ τ (pn−P)+p −div(d(un)∇p )−div(g(un)∇e ) wdx
1 1 1
ZΩ n=1 !
X
N
+ q β τ |fn|2q0−2fnh−p h dx
0 1 1
ZΩ n=1 !
X
+hγ e,ai+hγ w,bi=0 ∀(e,w,h)∈W ×W ×Υ.
0 0
Consider now the system
8 Moulay RchidSidi Ammi and Delfim F. M. Torres
e +div(ϕ′(un)∇e )−d′(un)∇pn.∇p −ϕ′′(un)∇un.∇e
1 1 1 1
N
−g′(un)∇pn.∇e =τ (un−U),
1
n=1
X (13)
N
p +div(d(un)∇p )+div(g(un)∇e )=τ (pn−P),
1 1 1
n=1
X
e | = p | =0, e | = p | =0,
1 ∂Ω 1 ∂Ω 1 t=T 1 t=T
with unknowns (e ,p ) which is uniquely solvable in W2(Ω) × W2(Ω) by
1 1 2 2
the theory of elliptic equations [4]. The problem of finding (e,w) ∈ W ×W
satisfying
e−div(ϕ′(un)∇e)−div(ϕ′′(un)e∇un)−div(g(un)∇w)
−div(g′(un)e∇pn)=sign(e −e ),
1 1
(14)
w−div(d(un)∇w)−div(d′(un)e∇pn)=sign(p −p ),
1 1
γ e=γ w =0,
0 0
isalsouniquelysolvableonW2 (Ω)×W2(Ω).Letuschooseh=0in(12)and
2q 2q
multiply (13)by (e,w). Then, integrating by parts and making the difference
with (12) we obtain:
e−div(ϕ′(un)∇e)−div(ϕ′′(un)e∇un)−div(g(un)∇w)
ZΩ(cid:16)
−div(g′(un)e∇pn) (e −e )dx
1 1
(15)
(cid:17)
+ (w−div(d(un)∇w)−div(d′(un)e∇pn))(p −p )dx
1 1
ZΩ
+hγ e,γ e −ai+hγ w,γ p −bi=0 ∀(e,w)∈W ×W.
0 0 1 0 0 1
Choosing (e,w) in (15) as the solution of the system (14), we have
sign(e −e )(e −e )dxdt+ sign(p −p )(p −p )dx=0.
1 1 1 1 1 1 1 1
ZΩ ZΩ
It follows that e =e and p =p . Coming back to (15), we obtain γ e =a
1 1 1 1 0 1
and γ p =b. On the other hand, choosing (e,w)=(0,0) in (12), we get
0 1
N
β τ |fn|2q0−2fn−p hdx=0, ∀h∈Υ.
1 1
ZΩ n=1 !
X
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 a
forthcoming publication.
Necessary Optimality Condition for a Discrete Dead Oil Isotherm Problem 9
Acknowledgments
The authors were supported by the Portuguese Foundation for Science and
Technology (FCT)throughtheCentreforResearchonOptimizationandCon-
trol (CEOC)oftheUniversityofAveiro,cofinancedbytheEuropeanCommu-
nity fund FEDER/POCI 2010. This work was developed under the post-doc
project SFRH/BPD/20934/2004.
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