Generalized Forchheimer Equations for Porous Media.
Part V.
Luan Hoang†,∗ , Akif Ibragimov† , Thinh Kieu† and Zeev Sobol‡
† Department
of Mathematics and Statistics, Texas Tech University
Department, Swansea University
∗ http://www.math.ttu.edu/∼lhoang/
luan.hoang@ttu.edu
Project on Generalized Forchheimer Flows: also with Eugenio Aulisa (TTU), Lidia
Bloshanskaya (TTU), Tuoc Phan (Tennessee)
‡ Mathematics
Applied Mathematics Seminars
Texas Tech University
March 20, 2013
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Outline
1
Introduction
2
Bounds for the solutions
3
Dependence on the boundary data
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Introduction
Fluid flows in porous media with velocity u and pressure p:
Darcy’s Law:
αu = −∇p,
the “two term” law
αu + β|u| u = −∇p,
the “three term” law
Au + B |u| u + C|u|2 u = −∇p.
the “power” law
au + c n |u|n−1 u = −∇p,
Here α, β, a, c, n, A, B, and C are empirical positive constants.
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General Forchheimer equations
Generalizing the above equations as follows
g (|u|)u = −∇p.
Let G (s) = sg (s). Then G (|u|) = |∇p| ⇒ |u| = G −1 (|∇p|). Hence
u=−
∇p
g (G −1 (|∇p|))
where
K (ξ) = Kg (ξ) =
⇒ u = −K (|∇p|)∇p,
1
1
=
,
−1
g (s)
g (G (ξ))
sg (s) = ξ.
We derived a non-linear equation of Darcy type from Forchheimer
equations.
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IBVP for pressure
Let ρ be the density. Continuity equation
dρ
+ ∇ · (ρu) = 0.
dt
For slightly compressible fluid:
1
dρ
= ρ,
dp
κ
where κ 1. Then
dp
= κ∇ · K (|∇|)∇p + K (|∇p|)|∇p|2 .
dt
Since κ 1, we neglect the last terms, after scaling the time variable:
dp
= ∇ · K (|∇|)∇p .
dt
Consider the equation on a bounded domain U in R3 with boundary Γ.
Dirichlet boundary data:
u = ψ(x, t) is known on Γ.
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Historical remarks
Darcy-Dupuit: 1865
Forchheimer: 1901
Other nonlinear models: 1940s–1960s
Incompressible fluids: Payne, Straughan and collaborators since
1990’s, Celebi-Kalantarov-Ugurlu since 2005 (Brinkman-Forchheimer)
Derivation of non-Darcy, non-Forchheimer flows: Marusic-Paloka and
Mikelic 2009 (homogenization for Navier–Stokes equations), Balhoff
et. al. 2009 (computational)
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Works on generalized Forchheimer flows
A. Single-phase flows.
1990’s Numerical study
L2 -theory (for slightly compressible flows):
Aulisa-Bloshanskaya-Ibragimov-LH (2009), Ibragimov-LH: Dirichlet
B.C. (2011), Ibragimov-LH Flux B.C. (2012),
Aulisa-Bloshanskaya-Ibragimov total flux, productivity index (2011,
2012).
Lα -theory : Ibragimiv-Kieu-Sobol-LH (this talk).
L∞ -theory : Kieu-Phan-LH (in progress).
B. Multi-phase flows.
One-dimensional case: Ibragimov-Kieu-LH (2013).
Multi-dimensional case: Ibragimov-Kieu-LH (in preparation).
Note: there are more works on Forchheimer flows (2-terms or 3 terms).
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Class FP(N, α
~)
We introduce a class of “Forchheimer polynomials”
Definition
A function g (s) is said to be of class FP(N, α
~ ) if
g (s) = a0 s α0 + a1 s α1 + a2 s α2 + . . . + aN s αN =
N
X
aj s α j ,
j=0
where N > 0, 0 = α0 < α1 < α2 < . . . < αN , and a0 , aN > 0,
a1 , . . . , aN−1 ≥ 0.
Notation: αN = deg(g ), ~a = (a0 , a1 , . . . , aN ),
a=
αN
∈ (0, 1),
αN + 1
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b=
a
αN
=
∈ (0, 1).
2−a
αN + 2
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Degeneracy - The Degreee Condition
Lemma
Let g (s,~a) be in class FP(N, α
~ ). One has for any ξ ≥ 0 that
C1 (~a)
C2 (~a)
≤ K (ξ,~a) ≤
,
a
(1 + ξ)
(1 + ξ)a
C3 (~a)(ξ 2−a − 1) ≤ K (ξ,~a)ξ 2 ≤ C2 (~a)ξ 2−a .
Degree Condition (DC)
deg(g ) ≤
4
n(2 − a)
⇐⇒ 2 ≤ (2 − a)∗ =
.
n−2
n − (2 − a)
Under the (DC), the Sobolev space W 1,2−a (U) is continuously embedded
into L2 (U).
(NDC)
4
n(2 − a)
deg(g ) >
⇐⇒ 2 > (2 − a)∗ =
.
n−2
n − (2 − a)
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Definitions and notations
Let p̄(x, t) = p(x, t) − Ψ(x, t) where Ψ(x, t) is an extension of ψ(x, t)
from ∂U to U.
Z
2−a
Z
1
Z
r0 (1−a)
r0
2
r0
r0
G1 (t) =
|∇Ψ(x, t)| dx +
|Ψt (x, t)| dx
+
|Ψt (x, t)| dx
U
ZU
ZU
2
2
G2 (t) =
|∇Ψt (x, t)| dx +
|Ψt (x, t)| dx,
U
U
Z
G3 (t) = G1 (t) + G2 (t), G4 (t) = G3 (t) +
|Ψtt |2 dx,
U
where r0 =
n(2−a)
(2−a)(n+1)−n .
For any α > 0, we define
na
nαN
=
,
2−a
αN + 2
(
max{α, α∗ } in the NDC case
α
b = max{α, 2, α∗ } =
max{α, 2}
in the DC case.
α∗ =
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Estimates of solutions - I: for p
For α > 0 and t ≥ 0, we define
Z
Z
α(2−a)
A(α, t) =
|∇Ψ| 2 dx +
|Ψt |α dx,
U
Z
B(α, t) = e −c5 (α)t
t ≥ 0,
U
|p̄(x, 0)|α dx +
U
t
Z
e −c5 (α)(t−τ ) A(α, τ )dτ.
0
Theorem
Let α > 0. Then
Z
|p̄(x, t)|α dx ≤ C (1 + B(b
α, t))
U
for all t ≥ 0, and
Z
lim sup
t→∞
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U
|p̄|α dx ≤ C (1 + lim sup A(b
α, t)).
t→∞
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Ideas of proof
α
Let γ0 = α−a
. Multiplying equation of p̄ by |p̄|α−1 sgn(p̄), integrating over
U, and many other calculations give
Z
Z
1
Z
d
γ
α
α−2
2−a
|p̄|α dx 0 +Cε (1+A(α, t)).
dx+ε
|p| dx ≤ −C
|∇p̄| |p|
dt U
U
U
Poincaré-Sobolev inequality: for α ≥ max{2, α∗ }, applying the imbedding
α(2−a)
α−a
from the Sobolev space W̊ 1,2−a (U) into L α−a (U) to function |u| 2−a
gives
Z
Z
1
γ
2−a
α−2
|∇u| |u|
dx ≥ c2
|u|α dx 0 .
U
U
Using this,
Z
Z
1
Z
1
d
γ
γ
|p̄|α dx 0 + ε
|p̄|α dx 0 + Cε (1 + A(α, t)).
|p|α dx ≤ −C
dt U
U
U
Selecting ε sufficiently small, we obtain
Z
Z
1
d
γ
|p|α dx ≤ −C
|p̄|α dx 0 + C (A(α, t) + 1).
dt U
U
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Differential inequality
Then we use the following estimates.
Lemma
Let φ : [0, ∞) → [0, ∞) be continuous, strictly increasing. Suppose
y (t) ≥ 0 is a continuous function on [0, ∞) such that
y 0 ≤ −h(t)φ−1 (y (t)) + f (t),
t > 0,
where h(t) > 0, f (t)
≥ 0 for t ≥ 0 are continuous functions on [0, ∞).
f (t)
(i) If M(t) = Env h(t)
on [0, ∞) then
y (t) ≤ y (0) + φ(M(t)) for all t ≥ 0.
(ii) If
R∞
0
h(τ )dτ = ∞ then
f (t)
lim sup y (t) ≤ φ lim sup
.
t→∞
t→∞ h(t)
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Estimates of solutions - II: for ∇p
Improving L2 -theory. Previously,
Z
Z
Z
d
H(|∇p|)dx ≤ −d5
H(|∇p|)dx + C
p̄ 2 dx + CG3 (t),
dt U
U
U
where H(∇p) ≈ K (|∇p|)|∇p|2 ≈ |∇p|2−a ± 1.
Theorem
If t ≥ 0 then
Z
Z
b
2−a
−d5 t
|∇p(x, t)| dx ≤ C + Ce
|p̄(x, 0)|2 + |∇p(x, 0)|2−a dx
U
U
Z t
+C
e −d5 (t−τ ) (A(b
2, τ ) + G3 (τ ))dτ,
0
where d5 > 0. Consequently,
Z
lim sup
|∇p(x, t)|2−a dx ≤ C lim sup(1 + A(b
2, t) + G3 (t)).
t→∞
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U
t→∞
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Estimates of solutions - III: for pt
Theorem
If t ≥ t0 > 0 then
Z
Z
b
2−a
2
−d7 t
|∇p(x, t)|
+ |p̄t (x, t)| dx ≤ C + Ce
|p̄(x, 0)|2 dx
U
U
Z t0
o
nZ
+ Ce Ct0 t0−1 e −d7 t
G3 (τ )dτ
|p̄(x, 0)|2 + |∇p(x, 0)|2−a dx +
U
0
Z t
+C
e −d7 (t−τ ) (A(b
2, τ ) + G4 (τ ))dτ,
0
where d7 > 0. Consequently,
Z
|∇p(x, t)|2−a + |p̄t (x, t)|2 dx ≤ C lim sup(1 + A(b
2, t) + G4 (t)).
lim sup
t→∞
U
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t→∞
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Estimates of solutions - IV: for mixed terms
d
dt
Z
α
Z
|p| dx ≤ −C
|∇p̄|
U
Integrate in time or
2−a
α−2
|p|
dx+ε
Z
U
R
U
|p̄|α dx
1
γ0
+Cε (1+A(α, t)).
U
|∇p̄|2−a |p|α−2 dx ≤ α
R
U
|p̄t ||p|α−1 dx + . . .
Corollary
If α ≥ max{2, α∗ } and t > 0 then
Z tZ
|∇p(x, τ )|2−a |p̄(x, τ )|α−2 dx dτ
0
U
Z t
Z
α
≤C
|p̄(x, 0)| dx +
(1 + A(α, τ ))dτ ,
U
0
Z
2−a
α−2
|∇p(x, t)| |p̄(x, t)|
dx
U
Z
Z
2(α−1)
2
≤ C 1 + A(α, t) +
|p̄(x, t)|
dx +
|p̄t (x, t)| dx .
U
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U
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Dependence on the boundary data
• Let N ≥ 1, the exponent vector α
~ and coefficient vector ~a be fixed.
• Let pk = pk (x, t) (for i = 1, 2) be the solution with initial data pk (x, 0)
and boundary data ψk .
• Let Ψ1 , Ψ2 be extensions of ψ1 , ψ2 , and Φ = Ψ1 − Ψ2 .
• Let p̄k = pk − Ψk for k = 1, 2, and z = p1 − p2 , z̄ = p̄1 − p̄2 = z − Φ.
Then
(
∂z̄
∂t = ∇ · (K (|∇p1 |)∇p1 ) − ∇ · (K (|∇p2 |)∇p2 ) + Φt in U × (0, ∞),
z̄(x, t) = 0 on Γ × (0, ∞).
Goal. Estimating z̄(x, t) in terms of z̄(x, 0) and Φ.
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Main differential inequalities for z̄
Define N(λ, t) =
P2
k=1
R
λ
U |p̄k (x, t)| dx
1/λ
for λ > 0, and N(0, t) = 1.
Lemma
Let α ≥ 2. (i) In the DC case, for t > 0,
Z
hZ
i
a
d
α
|z̄(x, t)| dx ≤ −c6
|z̄(x, t)|α dx M1 (t)− 2−a + CF (α, t)D(α, t)
dt U
U
where
Z
M1 (t) = 1 +
|∇p1 (x, t)|2−a + |∇p2 (x, t)|2−a dx,
U
1−a
F (α, t) = 1 + N(α, t)α−1 + N(γ1 , t)α−2 M1 (t) 2−a ,
def
with γ1 = γ1 (α) == 2(α − 2)(2 − a), and
(
k∇Φ(·, t)kL2(2−a) + k∇Φ(·, t)k2Lα + kΦt (·, t)kLα
D(α, t) =
k∇Φ(·, t)kL2−a + k∇Φ(·, t)k2L2 + kΦt (·, t)kL2
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if α > 2,
if α = 2.
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Lemma (cont.)
(ii) In the NDC case,
Z
hZ
iθ
2−θ1
d
−
α
|z̄(x, t)| dx ≤ −c7
|z̄(x, t)|α dx M2 (α, t) θ1 +CF (α, t)D(α, t)
dt U
U
for all t > 0, where
Z
M2 (α, t) = 1 +
|∇p1 (x, t)|2−a + |∇p2 (x, t)|2−a
U
+ |p̄1 (x, t)|θ2 α + |p̄2 (x, t)|θ2 α dx.
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Monotonicity
Multiplying the equation of z̄ by |z̄|α−1 sgn(z̄), ...
Z
1 d
|z̄|α dx
α dt U
Z
= −(α − 1)
(K (|∇p1 |)∇p1 − K (|∇p2 |)∇p2 ) · (∇p1 − ∇p2 )|z̄|α−2 dx + . .
U
Notation. ∨ for max.
Lemma (Degenerate monotonicity)
For any y , y 0 ∈ Rn , one has
K (|y |)y − K (|y 0 |)y 0 · y − y 0 ≥ (1 − a)K (|y | ∨ |y 0 |)|y − y 0 |2 .
d
dt
Z
α
Z
|z̄| dx ≤ −C
U
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K (|∇p1 | ∨ |∇p2 |)|∇z̄|2 |z̄|α−2 dx + . . .
U
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Weighted Poincaré–Sobolev inequality
Lemma
Let ξ(x) ≥ 0 on U and u(x) = 0 on ∂U. Assume α ≥ 2.
(i) In the DC case,
Z
Z
hZ
ih
i a
2−a
|u|α dx ≤ c3
K (ξ)|∇u|2 |u|α−2 dx 1 +
ξ 2−a dx
.
U
U
U
(ii) In the NDC case, given two numbers θ and θ1 that satisfy
θ>
then
Z
α
|u| dx ≤ c4
U
where θ2 =
n
2
2n o
and
max
1,
≤ θ1 < 2 − a,
(2 − a)∗
nθ + 2
hZ
2
K (ξ)|∇u| |u|
α−2
dx
i1 h
θ
Z
1+
U
θ1 (θ−1)(2−a)
2(2−a−θ1 )
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ξ 2−a + |u|θ2 α dx
i 2−θ1
θθ1
,
U
> 0.
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Dependence - Ia. DC case.
Theorem
(i) Assume (DC) and α ≥ 2. Then
Z
α
|z̄(x, t)| dx ≤ e
−c8
Rt
0
−a
M 1 (τ ) 2−a dτ
Z
U
|z̄(x, 0)|α dx
U
t
Z
+C
e
−c8
Rt
τ
−a
M 1 (s) 2−a ds
F (α, τ )D(α, τ )dτ
0
R∞
a
for all t ≥ 0, where c8 = c8 (α) > 0. Moreover, if 0 M̃1 (t)− 2−a dt = ∞
then
Z
h
i
a
lim sup
|z̄(x, t)|α dx ≤ C lim sup F̃ (α, t)M̃1 (t) 2−a D(α, t) .
t→∞
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t→∞
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Dependence - Ib. NDC case.
Theorem (cont.)
(ii) Assume (NDC) and α ≥ α∗ . Then
Z
|z̄(x, t)|α dx ≤
U
Z
i 1
h
2−θ1
θ
θ1
α,
t)
D(α,
t)
|z̄(x, 0)|α dx+C Env F (α, t)M 2 (θd
2
U
for all t ≥ 0. Moreover, if
Z
0
−
M̃2 (t)
2−θ1
θ1
dt = ∞ then
α
|z̄(x, t)| dx ≤ C lim sup F̃ (α, t)M̃2 (θd
2 α, t)
lim sup
t→∞
R∞
U
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2−θ1
θ1
1
θ
D(α, t)
.
t→∞
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Dependence - Ic
Corollary
Let α ≥ max{2, α∗ }. Assume the functions Ψk (k = 1, 2) satisfy
sup {k∇Ψk (·, t)kL∞ , k(Ψk )t (·, t)kL∞ , k∇(Ψk )t (·, t)kL∞ } < ∞,
[0,∞)
lim k∇Φ(·, t)kLγ = lim kΦt (·, t)kLα = 0,
t→∞
t→∞
where γ = max{α, 2(2 − a)}. Then
Z
lim
|z̄(x, t)|α dx = 0.
t→∞ U
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Gradient
Lemma
In the NDC case, we have for all t > 0 that
hZ
2−a
|∇z|
dx
i
2
2−a
1
≤ CM1 (t) 2−a k∇Φ(·, t)kL2−a
U
a
1
+ CM1 (t) 2−a M3 (t) 2
hZ
|z̄|α∗ dx
i
1
α∗
,
U
where
Z
M3 (t) =
|(p̄1 )t (x, t)|2 + |(p̄2 )t (x, t)|2 dx +
U
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Z
|Φt (x, t)|2 dx.
U
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Dependence II. Finite time.
Theorem
Assume (NDC).
(i) Then
Z
|∇z(x, t)|2−a dx
2
2−a
1
≤ C M 1 (t) 2−a k∇Φ(·, t)kL2−a
U
a
1
+ C M 1 (t) 2−a M 3 (t) 2 D(t)
for all t ≥ 1, where
i 1
h
2−θ1
θα∗
θ
D(t) = kz̄(·, 0)kLα∗ + Env F (α∗ , t)M 2 (θd
.
2 α∗ , t) 1 D(α∗ , t)
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Dependence III. Asymptotics (a)
Theorem
(ii) Let η1 = 1 + Ā(µ1 ) + lim supt→∞ G4 (t). If η1 < ∞ then
lim sup
t→∞
Z
|∇z(x, t)|2−a dx
U
2
2−a
1
≤ C η12−a lim sup k∇Φ(·, t)kL2−a
t→∞
o 1
n
θα∗
+ C η1µ3 lim sup k∇Φ(·, t)kL2(2−a) + k∇Φ(·, t)k2Lα∗ + kΦt (·, t)kLα∗
.
t→∞
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Case η1 = ∞
Assume (NDC) and η1 = ∞. Define for t ≥ 0,

R t −d (t−τ )
7
1
+
Ā(µ
)
+
G4 (τ )dτ

1
0 e


µ1


µ
−2a

+ Ã(µ1 , t)
1) 1
1 + β̄(µ
R t −d (t−τ )
ω(t) =
+ 0e 7
(Ã(µ1 , τ ) + G4 (τ ))dτ

Rt



1 + Env Ã(µ1 , t) + 0 e −d7 (t−τ ) G4 (τ )dτ



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if Ā(µ1 ) < ∞,
if Ā(µ1 ) = ∞,
β̄(µ1 ) < ∞,
if Ā(µ1 ) = ∞,
β̄(µ1 ) = ∞.
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Dependence III. Asymptotics (b)
Theorem
R∞
Assume (NDC) and η1 = ∞. If
0
ω(t)−µ4 dt = ∞ and
def
η2 == lim sup[(ω(t)µ5 )0 ]+ < ∞
t→∞
then
lim sup
t→∞
Z
|∇z(x, t)|2−a dx
2
2−a
1
≤ C lim sup ω(t) 2−a k∇Φ(·, t)kL2−a
t→∞
U
h
+ C lim sup ω(t)µ6 D(t)
t→∞
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i
1
θα∗
1
θα∗
+ C η2
h
lim sup ω(t)µ7 D(t)
i
1
θ 2 α∗
.
t→∞
Generalized Forchheimer Equations (V)
TTU - Mar. 20, 2013
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Dependence on the Forchheimer polynomial
Collaborator Thinh Kieu will present this topic next week in this Applied
Math Seminar.
Luan Hoang - Texas Tech
Generalized Forchheimer Equations (V)
TTU - Mar. 20, 2013
30
THANK YOU FOR YOUR ATTENTION!
Luan Hoang - Texas Tech
Generalized Forchheimer Equations (V)
TTU - Mar. 20, 2013
31