On a general class of Forchheimer gas flows in porous media
Emine Celik
Joint with Luan Hoang and Thinh Kieu
March 15, 2016
The 40th SIAM-SEAS, Applied Mathematics
University of Georgia, Athens, GA
Outline
Fluid flows in porous media
Estimates of the Lebesgue norms
Maximum estimates
On a general class of Forchheimer gas flows in porous media
Fluid flows in porous media
Emine Celik | University of Georgia, Athens, GA
3/36
On a general class of Forchheimer gas flows in porous media
velocity: v(x, t) pressure: p(x, t) viscosity: µ constants: a, b, c, d
density: ρ(x, t) porosity: φ ∈ (0, 1)
permeability: k
k
• Darcy’s law (1856): v = − ∇p,
µ
• The Forchheimer two-term law (1901): av + b|v|v = −∇p,
• The Forchheimer three-term law (1901): av + b|v|v + c|v|2 v = −∇p,
• The Forchheimer power law (1930): av + d|v|m−1 v = −∇p, m ∈ [1, 2],
• Generalized Forchheimer equations:
N
X
ai |v|αi v = −∇p, ai > 0.
i=0
Emine Celik | University of Georgia, Athens, GA
4/36
On a general class of Forchheimer gas flows in porous media
Goal
• Explore Forchheimer flows of compressible fluids in porous media using analytical
techniques from PDE theory
!! dependence of coefficients ai ’s on the density ρ is essential.
Emine Celik | University of Georgia, Athens, GA
5/36
On a general class of Forchheimer gas flows in porous media
Dimensional relationships Muskat and Ward (1964):
−∇p = φ(v α k
α−3
2
ρα−1 µ2−α ).
• In particular the case α = 2 (Forchheimer, Muskat, Ward)
−∇p =
cF ρ
µ
v + √ |v|v,
k
k
cF > 0.
Emine Celik | University of Georgia, Athens, GA
6/36
On a general class of Forchheimer gas flows in porous media
• Adaptation of Generalized Forchheimer equation:
N
X
ai ραi |v|αi v = −∇p,
i=0
where N ≥ 1, α0 = 0 < α1 < . . . < αN are fixed real numbers, the coefficients
a0 , . . . , aN are positive.
• Integrate with gravity
N
X
ai ραi |v|αi v = −∇p + ρ~g ,
i=0
~g : constant gravitational field.
Emine Celik | University of Georgia, Athens, GA
7/36
On a general class of Forchheimer gas flows in porous media
Let g : R+ → R+ defined by
g(s) = a0 sα0 + a1 sα1 + · · · + aN sαN
for s ≥ 0,
with a0 , a1 , . . . , aN > 0.
⇒ g(ρ|v|)v = −∇p + ρ~g ,
Multiplying by ρ, we obtain
g(|ρv|)ρv = −ρ∇p + ρ2~g .
⇒ ρv = −K(|ρ∇p − ρ2~g |)(ρ∇p − ρ2~g ),
K : R+ → R+ is defined for ξ ≥ 0 by K(ξ) =
1
, with s(ξ) = s being the
g(s(ξ))
unique non-negative solution of sg(s) = ξ.
Emine Celik | University of Georgia, Athens, GA
8/36
On a general class of Forchheimer gas flows in porous media
Continuity equation:
φρt + div(ρv) = F,
• φ in (0, 1),
• source term F : rate of net mass production or loss due to any source and/or
sink in the media.
⇓
φρt = div(K(|ρ∇p − ρ2~g |)(ρ∇p − ρ2~g )) + F .
Emine Celik | University of Georgia, Athens, GA
9/36
On a general class of Forchheimer gas flows in porous media
Scenarios:
Isentropic gas flows.
p = c̄ργ
for some constants c̄, γ > 0.
γ: specific heat ratio.
⇒ ρ∇p = ∇
c̄γργ+1 γ+1
.
γ+1
γp γ
c̄γργ+1
• Let u =
= 1
.
γ+1
c̄ γ (γ + 1)
⇒ φc1/2 (uλ )t = ∇ · (K(|∇u − cu`~g |)(∇u − cu`~g )) + F ,
where
1
λ=
∈ (0, 1),
γ+1
` = 2λ
Emine Celik | University of Georgia, Athens, GA
and
c=
γ+1
c̄γ
`
.
10/36
On a general class of Forchheimer gas flows in porous media
Ideal gases.
p = c̄ρ for some constant c̄ > 0.
• special case of Isentropic gas flows with γ = 1, (u ∼ p2 )
√
2
2
2
⇒ φ √ (u1/2 )t = ∇ · (K(|∇u − u~g |)(∇u − u~g )) + F .
c̄
c̄
c̄
Slightly compressible fluids. The equation of state is
1 dρ
1
= = const. > 0.
ρ dp
κ
⇒ ρ∇p = κ∇ρ,
• Let u = κρ,
` = 2 and
c = 1/κ2 ,
⇒ φc1/2 (u)t = ∇ · (K(|∇u − cu`~g |)(∇u − cu`~g )) + F .
Emine Celik | University of Georgia, Athens, GA
11/36
On a general class of Forchheimer gas flows in porous media
⇓
(uλ )t = ∇ · (K(|∇u − cu`~g |)(∇u − cu`~g )) + F ,
with λ ∈ (0, 1], ` = 2λ, c > 0.
Boundary condition.
• Consider v · ~ν = ψ on ∂U, outward normal vector ~ν on the boundary.
ρv · ~ν = ψρ
⇓
−K(|∇u + cu`~g |)(∇u + cu`~g ) · ~ν = c1/2 ψuλ .
Emine Celik | University of Georgia, Athens, GA
12/36
On a general class of Forchheimer gas flows in porous media
General formulation and the initial boundary value problem.

∂(uλ )



 ∂t = ∇ · (K(|∇u + Z(u)|)(∇u + Z(u))) + f (x, t, u) on U × (0, ∞),
u(x, 0) = u0 (x)
on U,



K(|∇u + Z(u)|)(∇u + Z(u)) · ~ν = B(x, t, u)
on Γ × (0, ∞),
|Z(u)| ≤ d0 u`Z ,
Z(u): [0, ∞) → Rn ,
B(x, t, u): Γ × [0, ∞) × [0, ∞) → R,
B(x, t, u) ≤ ϕ1 (x, t) + ϕ2 (x, t)u`B ,
f (x, t, u): U × [0, ∞) × [0, ∞) → R,
f (x, t, u) ≤ f1 (x, t) + f2 (x, t)u`f
with constants d0 , `Z > 0, `f , `B ≥ 0, and functions ϕ1 , ϕ2 , f1 , f2 ≥ 0.
Emine Celik | University of Georgia, Athens, GA
13/36
On a general class of Forchheimer gas flows in porous media
Properties of K(ξ):
• Define
a=
αN
∈ (0, 1).
αN + 1
• ⇒ K(ξ) : [0, ∞) → (0, a10 ] is decreasing in ξ. ⇒ K(ξ) ≤
1
.
a0
d2
d1
≤ K(ξ) ≤
,
(1 + ξ)a
(1 + ξ)a
d3 (ξ 2−a − 1) ≤ K(ξ)ξ 2 ≤ d2 ξ 2−a ,
Emine Celik | University of Georgia, Athens, GA
14/36
On a general class of Forchheimer gas flows in porous media
U (bounded domain in Rn ) models a porous medium.
Γ = ∂U is C 2 boundary.
u(x, t) ≥ 0 is a solution of IBVP.
Define δ = 1 − λ ∈ [0, 1).
n(a − δ)
• Assume a > δ. Then α∗ =
>0
2−a
• Define
1−a
2−a
κB = 1 +
and κf = 1 +
.
n
n
• Let p1 , p2 , p3 , p4 be fixed numbers such that
•
•
•
•
1 < p1 , p2 < κB
and
1 < p3 , p4 < κf .
• For i = 1, 2, 3, 4 let qi be the conjugate exponent of pi .
Emine Celik | University of Georgia, Athens, GA
15/36
On a general class of Forchheimer gas flows in porous media
n
−p1 λ + a − δ p2 (−λ + `B ) + a − δ
η0 = max q1 λ, q2 (λ − `B ), n(`Z − 1),
,
,
κf − p1
κf − p2
−p1 λ + (a − δ) p2 (−λ + `B ) + a − δ
,
,
κB − p1
κB − p2
−p3 λ + a − δ p4 (−λ + `f ) + a − δ o
,
.
κf − p3
κf − p4
• Assume
n
na o
α ≥ max 2,
and
1−a
α > η0 .
Υ(t) = kϕ1 (t)kqL1q1 (Γ) +kϕ2 (t)kqL2q2 (Γ) +kf1 (t)kqL3q3 (U ) +kf2 (t)kqL4q4 (U )
Emine Celik | University of Georgia, Athens, GA
for t ≥ 0.
16/36
On a general class of Forchheimer gas flows in porous media
Estimates of the Lebesgue norms
Emine Celik | University of Georgia, Athens, GA
17/36
On a general class of Forchheimer gas flows in porous media
µ1 = `Z (2 − a) + α − λ − 1,
µ3 = p2 (α − λ + `B ),
µ5 = α +
µ4 = α +
µ2 = p1 (α − λ),
(2 − a)((p1 − 1)α − p1 λ) + a − δ
> α,
1−a
(2 − a)((p2 − 1)α + p2 (−λ + `B )) + a − δ
> α,
1−a
µ6 = p3 (α − λ)
and
µ7 = p4 (α − λ + `f ).
Emine Celik | University of Georgia, Athens, GA
18/36
On a general class of Forchheimer gas flows in porous media
Lemma
For t > 0, one has
d
dt
Z
U
u(x, t)α dx +
Z
|∇u(x, t)|2−a u(x, t)α−λ−1 dx
U
θ
α(1+ 2−a · 1−θ
)
n
≤ C0 (α) · ku(t)kα−λ−1
Lα (U ) + ku(t)kLα (U )
with
θ = θα :=
+ Υ(t) ,
max{µi : 1 ≤ i ≤ 7} − α
.
α(2 − a)/n + δ − a
Emine Celik | University of Georgia, Athens, GA
19/36
On a general class of Forchheimer gas flows in porous media
Theorem
If T > 0 satisfies
Z T
(1 + Υ(t))dt <
0
Z
(2−a)θ
− n(1−θ)
(2 − a)θ
· 1+
u0 (x)α dx
,
4nC0 (α)(1 − θ)
U
then for all t ∈ [0, T ]
Z
α
u (x, t)dx ≤
U
n
Z
1+
(2−a)θ
− n(1−θ)
u0 (x)α dx
U
− 4C0 (α)
(2 − a)θ
n(1 − θ)
Z
Emine Celik | University of Georgia, Athens, GA
t
(1 + Υ(τ ))dτ
o− n(1−θ)
(2−a)θ
.
0
20/36
On a general class of Forchheimer gas flows in porous media
Theorem (cont.)
In particular, if T > 0 satisfies
Z
T
0
(2−a)θ
θ(2 − a)(1 − 2− n(1−θ) ) 1+
(1 + Υ(t))dt ≤
4nC0 (α)(1 − θ)
Z
u0 (x)α dx
(2−a)θ
− n(1−θ)
,
U
then
Z
U
Z
u (x, t)dx ≤ 2 1 +
uα
0 (x)dx
α
for all
t ∈ [0, T ],
U
and
Z TZ
0
Z
n(1 − θ) |∇u|2−a uα−λ−1 dxdt ≤ 2 1 +
1+
u0 (x)α dx .
θ(2 − a)
U
U
Emine Celik | University of Georgia, Athens, GA
21/36
On a general class of Forchheimer gas flows in porous media
Maximum estimates
Emine Celik | University of Georgia, Athens, GA
22/36
On a general class of Forchheimer gas flows in porous media
Let
n (2 − a)p − 1 (2 − a)p − 1
o
1
2
,
, p3 , p4 ,
1−a
1−a
n p λ − a + δ p (λ − ` ) − a + δ o
1
2
B
η1 = max
,
,
(2 − a)p1 − 1
(2 − a)p2 − 1
n
a − δ + (2 − a)p2 `B o
η2 = max `Z (2 − a), `f ,
.
1−a
κ∗ = max
2−a
2−a
1−a
M0 = 1 + kϕ1 kL1−a
q1 (Γ ) + kϕ2 kLq2 (Γ ) + kf1 kLq3 (Q ) + kf2 kLq4 (Q ) ,
T
T
T
T
Emine Celik | University of Georgia, Athens, GA
23/36
On a general class of Forchheimer gas flows in porous media
• Positive powers ν5 and ν6 are defined by
n
p1 λ − a + δ p2 (λ − `B ) − a + δ o
ν5 = α − max λ + 1,
,
p1 (2 − a) − 1
p2 (2 − a) − 1
n
ν6 = α + max 0, `Z (2 − a) − λ − 1, `f − λ, `B − λ,
a − δ − p1 λ a − δ − p2 (λ − `B ) o
,
.
p1 (2 − a) − 1
p2 (2 − a) − 1
Emine Celik | University of Georgia, Athens, GA
24/36
On a general class of Forchheimer gas flows in porous media
Lemma (Caccioppoli Inequality)
Given κ̃ > κ∗ , suppose
α > max{2, α∗ , η1 }
If T > T2 > T1 ≥ 0 then
Z
Z
sup
uα (x, t)dx +
t∈[T2 ,T ]
U
≤ c6 (1 + T ) 1 +
T
T2
Z
and α ≥
η2
.
κ̃ − κ∗
|∇u(x, t)|2−a u(x, t)α−λ−1 dxdt
U
1
ν
ν
α2 M0 kukL5κ̃α (U ×(T1 ,T )) + kukL6κ̃α (U ×(T1 ,T )) ,
T2 − T1
where c6 ≥ 1 is independent of α, κ̃, T , T1 and T2 ,
Emine Celik | University of Georgia, Athens, GA
25/36
On a general class of Forchheimer gas flows in porous media
Lemma (Parabolic Sobolev inequality)
Assume 1 > a > δ ≥ 0, α ≥ 2 − δ
and
α > α∗ :=
If T > 0, then
1
Z T Z
κα
Z
1
κα
2−a κα
|u| dxdt
≤ (c5 α
)
0
Z
U
T
Z
Z
|u|α+δ−a dxdt
U
0
+
0
T
n(a − δ)
.
2−a
θ̃
Z
1−α θ̃
α+δ−a
· sup
,
|u(x, t)|α dx
|u|α+δ−2 |∇u|2−a dxdt
t∈[0,T ]
U
U
where c5 ≥ 1 is independent of α and T, and
θ̃ = θ̃α :=
1
1+
α(2−a)
n(α+δ−a)
,
κ = κ(α) := 1 +
Emine Celik | University of Georgia, Athens, GA
2−a a−δ
−
.
n
α
26/36
On a general class of Forchheimer gas flows in porous media
• Combining Caccioppoli and Parabolic Sobolev inequality and with some other
calculations, we have
α1
αν6
1
,
kukLκα (U ×(T2 ,T )) ≤ M1α kukνL5κ̃α (U ×(T1 ,T )) + kukLα+δ−a
κ̃α (U ×(T ,T ))
1
where
1
α2 M0
T2 − T1
α
α+δ−a
i
α2 M0
.
h
M1 = 8c5 α2−a (1 + |QT |)2 + c6 (1 + T ) 1 +
+ c6 (1 + T ) 1 +
1
T2 − T1
• Estimate M1 :
M1 ≤ 8c5 (1 + |U |)2 + 2c26 α6−a 1 +
Emine Celik | University of Georgia, Athens, GA
2
1
(1 + T )2 M20 .
T2 − T1
27/36
On a general class of Forchheimer gas flows in porous media
Lemma
If T > T2 > T1 ≥ 0 then
α1
αν6
1
,
kukLκα (U ×(T2 ,T )) ≤ Aαα kukνL5κ̃α (U ×(T1 ,T )) + kukLα+δ−a
κ̃α (U ×(T ,T ))
1
Aα = c7 (1 + T )2 1 +
2
1
α6−a M20
T2 − T1
with c7 ≥ 1 independent of α, κ̃, T , T1 and T2 .
• We will iterate this inequality (Moser).
Emine Celik | University of Georgia, Athens, GA
28/36
On a general class of Forchheimer gas flows in porous media
Iteration:
• For α = α0 , we have α0 > max{2, α∗ , η1 }
and
• Define for j ≥ 0, βj = κ̃j α0 .
∞
• κ̃ > 1 ⇒ {βj }∞
j=0 is increasing, so is {κ(βj )}j=0 .
a−δ
• κ(βj ) ≥ κ(α0 ) = κf −
≥ κ̃2 .
α0
• For j ≥ 0, let tj = σT (1 − 21j ).
• α = βj , T2 = tj+1 and T1 = tj , we have
1 β
r̃
kukLκ(βj )βj (U ×(tj+1 ,T )) ≤ Aβjj kukLjβj+1 (U ×(t
where r̃j = ν5 (βj ) and s̃j =
α0 ≥
η2
.
κ̃ − κ∗
s̃
j
+ kukLjβj+1 (U ×(t
,T ))
j ,T ))
β1
j
,
βj ν6
βj +δ−a .
Emine Celik | University of Georgia, Athens, GA
29/36
On a general class of Forchheimer gas flows in porous media
• Note that κ(βj )βj ≥ κ̃2 βj = βj+2 .
• Define for j ≥ 0 that Qj = U × (tj , T ) and Yj = kukLβj+1 (Qj ) .
1
• By Hölder’s inequality and |Qj+1 | βj+2
− κ(β1 )β
j
j
1
≤ (1 + |Q0 |) βj ,
1
Yj+1 = kukLβj+2 (Qj+1 ) ≤ (1 + |Q0 |) βj kukLκ(βj )βj (Qj+1 ) .
⇒
1
β
r̃
s̃
b j Y j +Y j
Yj+1 ≤ A
j
j
j
β1
j
∀j ≥ 0,
bj = (1 + |Q0 |)Aβ .
where A
j
bj .
•Estimate A
j+1 2
bj ≤ c7 (1 + |U |)β 6−a 1 + 2
A
(1 + T )3 M20 ≤ Aj+1
j
T,σ,α0 ,
σT
where
1 2
AT,σ,α0 = max 4κ̃6−a , 4c7 (1 + |U |)α06−a (1 +
) (1 + T )3 M20 ≥ 1.
σT
Emine Celik | University of Georgia, Athens, GA
30/36
On a general class of Forchheimer gas flows in porous media
⇒
j+1
β
r̃
s̃
j
Yj+1 ≤ AT,σ,α
Yj j + Yj j
0
Emine Celik | University of Georgia, Athens, GA
β1
j
.
31/36
On a general class of Forchheimer gas flows in porous media
Lemma
Let yj ≥ 0, κj > 0, sj ≥ rj > 0 and ωj ≥ 1 for all j ≥ 0. Suppose there is
ωj
r
s
1
A ≥ 1 such that yj+1 ≤ A κj (yj j + yj j ) κj
ᾱ :=
∞
X
ωj
j=0
κj
< ∞,
∀j ≥ 0. Assume
∞
∞
Y
rj Y sj
,
converge to positive numbers β̄, γ̄, resp.
κ
κ
j=0 j j=0 j
then
lim sup yj ≤ (2A)Gᾱ max{y0γ̄ , y0β̄ },
j→∞
with G = lim sup max{1, γm γm+1 . . . γn : 1 ≤ m ≤ n < j} .
j→∞
Emine Celik | University of Georgia, Athens, GA
32/36
On a general class of Forchheimer gas flows in porous media
Theorem
Let α0 be a positive number such that
α0 > max{2, α∗ , η1 }
and
α0 ≥ max
n
η2
a−δ o
,
.
κ̃ − κ∗ κf − κ̃2
Q∞ r̃
Q∞
Then there are positive constants C, µ̃ = j=0 βjj , ν̃ = j=0
P∞
Q∞
ω = k=1 βs̃kk j=0 j+1
βj such that if T > 0 and σ ∈ (0, 1) then
s̃j
βj
and
1 2ω
kukL∞ (U ×(σT,T )) ≤ C 1 +
(1 + T )3ω M2ω
0
σT
n
o
µ̃
· max kukLκ̃α0 (U ×(0,T )) , kukν̃Lκ̃α0 (U ×(0,T )) .
Emine Celik | University of Georgia, Athens, GA
33/36
On a general class of Forchheimer gas flows in porous media
Theorem
Let α0 satisfy
n
η0 o
α0 > max 2, α∗ , η1 ,
and
κ̃
α0 ≥ max
n
na
η2
a−δ o
,
,
.
(1 − a)κ̃ κ̃ − κ∗ κf − κ̃2
(i) Denote β1 = κ̃α0 . If T > 0 satisfies
Z
T
(1 + Υ(t))dt <
0
Z
(2−a)θ
− n(1−θ)
(2 − a)θ
· 1+
u0 (x)β1 dx
,
4nC0 (β1 )(1 − θ)
U
then for 0 < ε < min{1, T } and 0 ≤ t ≤ T , one has
kukL∞ (U ×(ε,T )) ≤ Cε−2ω (1 + T )3ω M2ω
0
n Z T
βµ̃ Z
1
· max
V(t)dt
,
0
Emine Celik | University of Georgia, Athens, GA
T
βν̃ o
1
V(t)dt
,
0
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On a general class of Forchheimer gas flows in porous media
Theorem (cont.)
V(t) =
Z
(2−a)θ
n Z
− n(1−θ)
o− n(1−θ)
4C0 (β1 )(2 − a)θ t
(2−a)θ
−
.
1+ u0 (x)β1 dx
Υ(τ )dτ
n(1 − θ)
U
0
(ii) If T > 0 satisfies
Z
T
(1+Υ(t))dt ≤
0
(2−a)θ
− n(1−θ)
Z
(2−a)θ
(2 − a)θ
·(1−2− n(1−θ) ) 1+ u0 (x)β1 dx
,
4nC0 (β1 )(1 − θ)
U
then for 0 < ε < min{1, T }, one has
ν̃
kukL∞ (U ×(ε,T )) ≤ Cε−2ω (1 + T )3ω+ β1 (1 + ku0 (x)kLβ1 (U ) )ν̃ M2ω
0 .
Above C is a positive constant independent of T and ε.
Emine Celik | University of Georgia, Athens, GA
35/36
Thank you!
Questions?