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Continuum Models of Two Phase Flow in Porous Media
Michael Shearer
North Carolina State UniversityDepartment of Mathematics
Research supported by NSF grants DMS 0968258 FRG, DMS 0636590 RTG
Shearer (NCSU) Flow in Porous Media 1 / 46
Acknowledgements
Students and former students
Rachel Levy (Harvey Mudd College)Kim Spayd (Gettysburg College)Melissa Strait (NC State University)
Collaborators
Ruben Juanes, Luis Cueto-Felgueroso (MIT Engineering)Zhenzheng Hu (NC State University)
Shearer (NCSU) Flow in Porous Media 2 / 46
Applications
Oil recovery: water flood
water injection
H O2
Water infiltration under gravity(DiCarlo 2004)
Carbon Sequestration in Deep SalineAquifers
Shearer (NCSU) Flow in Porous Media 3 / 46
Introduction I
Framework: Conservation of mass; Darcy’s law; incompressibility
Buckley-Leverett flux f (u) (1942) u is saturationnon-convex conservation law ut + f (u)x = 0
Modeling: pressure difference pc across interfaces: interfacial energy
Classical approach: pc = pe(u) equilibrium pressure; decreasing in uut + f (u)x = −(H(u)pe(u)x)x
Dynamic effects: Hassanizadeh and Gray (1991,1993):pc = pe(u)− τut
Netherlands group: undercompressive shocks; sharp shocks
Pop, van Duijn, Peletier, Cuesta, Fan (2006-2012)
Shearer (NCSU) Flow in Porous Media 4 / 46
Introduction II
2-d stability/instability for Lax shocks: hyperbolic/elliptic system
Phase field modeling: Juanes (2008), Witelski (1998), based onpolymer mixtures: introduce stored energy function with capillaryeffects
Capillary tube gas/liquid finger: degenerate fourth order PDE
Dynamical systems trick for traveling waves
Carbon sequestration: Huppert, Neufeld, Hesse, ... (2008-2011).Discontinuous dependence in conservation law to allow for depositionof CO2 bubbles as plume propagates in brine (deep saline aquifer)
Track interface; non-standard wave interactions
Shearer (NCSU) Flow in Porous Media 5 / 46
2-D Model
u(x , y , t) : saturation (vol. fraction) of water, (1− u) : oil saturationp(x , y , t) : pressure in water
Conservation of mass with Darcy’s law: velocity v = −λ(u)∇p :
φ∂u
∂t−∇ · (λ(u)∇p) = 0 φ = porosity, λ(u) =
Kk(u)
µ
K = absolute permeability, k(u) = relative permeability, µ = viscosity
Incompressibility: ∇ · vTotal = 0
∇ ·(
Λ(u)∇p + λoil(u)∇pc(u))
= 0
Λ = λwater + λoil , pc = poil − p : capillary pressure;
For simplicity, neglect gravity, set φ = 1.
Shearer (NCSU) Flow in Porous Media 6 / 46
One-Dimensional Equation for u(x , t) : ut − (λpx)x = 0
∇ · vTotal = 0 =⇒ vTotal = (vT , 0) = constant, after rescaling t
Thus, Λpx + (Λ− λ)pcx = −vT
Eliminate px : px = − vTΛ(u)
−(
1− λ(u)
Λ(u)
)pcx
Thenut + f (u)x = − (H(u)pc
x )x
f (u) = vT λ(u)
Λ(u), H(u) =
λ(u)
Λ(u)(Λ(u)− λ(u))
Shearer (NCSU) Flow in Porous Media 7 / 46
ut + f (u)x = − (H(u)pcx )x
f (u) = vT λ(u)
Λ(u), H(u) =
λ(u)
Λ(u)(Λ(u)− λ(u))
Fractional flow rate Dissipation coefficient
Recall: Λ(u) is total mobility: Λ(u)− λ(u) = λoil(1− u)λ, λoil are positive, increasing convex functions, with λ(0) = λoil(0) = 0.
0 0.5 10
0.2
0.4
0.6
0.8
1
u
f(u)
0 0.5 10
0.02
0.04
0.06
0.08
0.1
u
H(u)
Shearer (NCSU) Flow in Porous Media 8 / 46
ut + f (u)x = − (H(u)pcx )x , but what to do about pc?
Buckley-Leverett (1942): ignore it: pc = 0
scalar non-convex conservation law ut + f (u)x = 0
Lax-Oleinik construction: rarefaction-shock interface
Classical approach: equilibrium capillary pressure depends onsaturation: pc = pe(u), a decreasing function: d
dupe(u) < 0.
The PDE is parabolic but degenerate at u = 0, 1.
Max principle: 0 ≤ u ≤ 1 is positively invariant.
Lax-Oleinik solutions replaced by smooth approximation.
Shearer (NCSU) Flow in Porous Media 9 / 46
Hassanizadeh and Gray (1991, 1993): rate dependence in pc :
pc = pe(u)− τut
Modified Buckley-Leverett equation
ut + f (u)x = −(H(u) ∂
∂x pe(u))x
+ τ (H(u)utx)x
Equation is dissipative-dispersive. No maximum principle, but positiveinvariance of 0 ≤ u ≤ 1 (Yabin Fan thesis 2012)
Analogous equation: modified BBM-Burgers:
ut + (u3)x = αuxx − βuxxt
Traveling waves for undercompressive shocks (Spayd thesis, 2012)
Explicit construction similar to modified KdV-Burgers:
ut + (u3)x = αuxx − βuxxx
(Jacobs, McKinney, Shearer, 1995; LeFloch book, 2002)
Shearer (NCSU) Flow in Porous Media 10 / 46
Traveling Wave Solutions
u(ξ) = u(x − st)⇒ −su′ + (f (u))′ = [H(u)u′]′ − sτ [H(u)u′′]′
Integrating with boundary conditions u(±∞) = u± leads to secondorder ODE: −s(u − u−) + f (u)− f (u−) = H(u)u′ − sτH(u)u′′
Rescale u = u( x−st√sτ
) and write as first order system of ODEs:
u′ = v
v ′ =1√sτ
v +1
H(u)[s(u − u−)− f (u) + f (u−)]
Singular at u = 0, u = 1, where H(u) = 0
Shearer (NCSU) Flow in Porous Media 11 / 46
Saddle-Saddle Connections
Phase portraits of ODE system: up to three equilibria(ue , 0) : (u±, 0), (umid, 0), s = (f (ue)− f (u−))/(ue − u−)
Analyze phase plane with separation function R(u−, u+, τ)
0.4 0.6 0.8
−0.5
0
0.5
u
v=uv u
midu u-
+
R>0
0.4 0.6 0.8 1
−0.5
0
0.5
u
v
R<0
0.4 0.6 0.8 1
−0.5
0
0.5
u
v=uv
R = 0
Saddle-saddle connection: R = 0 :heteroclinic orbit from (u−, 0) to (u+, 0) undercompressive shock
Shearer (NCSU) Flow in Porous Media 12 / 46
Numerical procedure
For τ <∞, fix u− and determine u+ = uΣ(u−) for whichR(u−, u+, τ) = 0Monotonicity of uΣ(u−) from Melnikov integral: derivatives of R
For τ =∞, determine (u−, uΣ) by using exact integral:
12
d
duv 2 =
1
H(u)[s(u − u−)− f (u) + f (u−)] ≡ G (u; u−, u+)
Thus: connection when
∫ u+
u−
G (u; u−, u+) du = 0 : u+ = uΣ(u−)
Asymptotics in corners: e.g., u− → 0, u+ → 1 :
u+ = uΣ(u−) ∼ 1− uM−
Shearer (NCSU) Flow in Porous Media 13 / 46
Στ Curves
Fix τ > 0, plot (u−, u+) with saddle-saddle connection u− → u+
u−
u+
0 0.2 0.4 0.6 0.8 110
0.2
0.4
0.6
0.8
1
Σ0.1
Σ
Σ
1
∞
u−
u+
0 0.2 0.40.75
0.8
0.85
0.9
0.95
1
0.6
Σ0.1
c: R=0
a: R>0
b: R<0
τ=0.1
Shearer (NCSU) Flow in Porous Media 14 / 46
Scalar conservation law: ut + f (u)x = 0
Idealization: no capillary pressure; characteristic speed f ′(u)Scale invariant solutions: building blocks for solving initial value problem
Rarefactions
u(x , t) =
u− if x < f ′(u−)t
r( xt ) if f ′(u−)t ≤ x ≤ f ′(u+)t
u+ if x > f ′(u+)t
Shocks
u(x , t) =
{u− if x < st
u+ if x > st
Rankine-Hugoniot condition: shock speed s =f (u+)− f (u−)
u+ − u−
Shearer (NCSU) Flow in Porous Media 15 / 46
Admissible Shocks
f ′(u+) < s < f ′(u−) Lax shock is admissible if there is a travelingwave from u− = umid (unstable node) to u+ (saddle point)
s > f ′(u±) PLUS: shock has corresponding traveling wave:
undercompressive shock Σ.
Jacobs, McKinney, Shearer (1995), LeFloch Book (2002)
0 10
1
u
f(u)
s
u-u+
Lax
0 10
1
u
f(u)
u+
Σ
u-
Undercompressive
Shearer (NCSU) Flow in Porous Media 16 / 46
Buckley-Leverett solution 1942
Solve conservation law with initial jump from all water u− = 1 to alloil u+ = 0: water flooding:
ut + f (u)x = 0, u(x , 0) =
{1 if x < 0
0 if x > 0
0 10
1
u
f(u)
u
shock
rarefaction
*
Solution: rarefaction fromu− = 1 to u∗; Lax shock fromu∗ to u+ = 0 :rarefaction-shock.
Shearer (NCSU) Flow in Porous Media 17 / 46
The Riemann Problem: Classical Solution
Solve conservation law with initial jump from u` to ur
ut + f (u)x = 0, u(x , 0) =
{u` if x < 0
ur if x > 0
ul
ur
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
RS
RS R
R
S
SR: Rarefaction WaveS: Lax ShockRS : Rarefaction - Shock
Shearer (NCSU) Flow in Porous Media 18 / 46
The Riemann Problem; dynamic capillary pressure
(RP) : ut + f (u)x = 0, u(x , 0) =
{u` if x < 0
ur if x > 0
Solutions of (RP): leading order approximations to solutions of modifiedBuckley-Leverett equation
ul
ur
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
RΣ
RSRS
RΣ
R
R
S
S
SΣ
SΣ
R: Rarefaction WaveS: Admissible Lax ShockΣ : Undercompressive Shock
For every (u`, ur ) ∈ (0, 1), thereis a solution that stays in (0, 1)even though SΣ solutions arenonmonotonic.
Shearer (NCSU) Flow in Porous Media 19 / 46
PDE Simulations
Full PDE: ut + f (u)x = (H(u)ux)x + τ (H(u)utx)x
u` ∈ (0, ur ) : rarefaction wave
−2 0 2 40
0.2
0.4
0.6
0.8
1
x
u
u
ul
r
u` ∈ (ur , umid) : Lax shock
−2 0 2 40
0.2
0.4
0.6
0.8
1
x
u
u
u
l
r
Shearer (NCSU) Flow in Porous Media 20 / 46
PDE Simulations - Nonclassical Solutions
u` ∈ (umid , uΣ) :Lax shock u` to uΣ
undercompressive shockuΣ to ur
−2 0 2 40
0.2
0.4
0.6
0.8
1
x
u
u
u
u
l
Σ
r
u` ∈ (uΣ, 1) :rarefaction wave u` to uΣ
undercompressive shockuΣ to ur
−2 0 2 40
0.2
0.4
0.6
0.8
1
x
u
ur
ul
uΣ
Shearer (NCSU) Flow in Porous Media 21 / 46
‘Sharp’ Traveling waves TW u = u(x − st)
ut + f (u)x = −(
H(u)∂
∂xpe(u)
)x
+ τ (H(u)utx)x
−s(u − u−) + f (u)− f (u−) = −H(u)pe(u)′ − sτH(u)u′′, u(±∞) = u±
after one integration
Singular at u = 0, 1 Can have connections up to u = 1, or down to u = 0depending on relative permeability functions at u = 0, 1
Example: λ(u) = u3/2; λoil(u) = (1− u)3/2
Pop, Cuesta, van Duijn (2011): sharp shock TW with corner at u = 1
Shearer (NCSU) Flow in Porous Media 22 / 46
0 5 10 15 20 25 30 35
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1u
time = 0
time = 6
time = 3
time = 9
time = 12
x
0 0.2 0.4 0.6 0.8 1
0
1
2u = 1, u+= 0.025
sharp shock
u’
u0.8 0.85 0.9 0.95 1
0
0.5
1u = 0.95, u+= 1
u
u’
Lax shock profile
PDE Simula+on
Phase Portraits: Traveling waves
Rela+ve permeabili+es Integrable case: K(u) = u3/2
Shearer (NCSU) Flow in Porous Media 23 / 46
Planar Lax shock ut + f (u)x = 0
Interface x = st; velocity v± = −λ±∂xp±; mobility λ± = Kk(u±)/µ
Continuous pressure p = p±(x − st) = − v±λ±
z = − vT
Λ±z , z = x − st;
shock location: z = x − st = 0
u = up-
x = st
x
y
--- -
-u = u+p = (x - st) p+p = (x - st)
Shock speed
s =f (u+)− f (u−)
u+ − u−
Lax condition:
f ′(u+) < s < f ′(u−)
Characteristic speeds f ′(u±)
Shearer (NCSU) Flow in Porous Media 24 / 46
2-dimensional linearized stability of Lax shocks
2-d equations with pc ≡ 0 variables u, p saturation, pressure:
∂u
∂t−∇ · (λ(u)∇p) = 0
∇ · (Λ(u)∇p) = 0
(1)
Interface x = x(y , t), normal in t, x , y : (−xt , 1,−xy )Jump condition at shock: ([q] = q+ − q−)
−xt [u]− [λ(u)px ] + xy [λ(u)py ] = 0
−[Λ(u)px ] + xy [Λ(u)py ] = 0(2)
Base shock: u = u±, p = p±(x − st), x = st, constants u±, p±,V
s =f (u+)− f (u−)
u+ − u−, f (u) = vT λ(u)
Λ(u), p± = − vT
Λ(u±)
Shearer (NCSU) Flow in Porous Media 25 / 46
Equations ∂u∂t −∇ · (λ(u)∇p) = 0, ∇ · (Λ(u)∇p) = 0
∂u
∂t− λ′(u)∇u · ∇p − λ(u)∆p = 0
Λ′(u)∇u.∇p + Λ(u)∆p = 0
(1)
Linearize about shock on each side:u = u + U(x , y , t), p = p(x − st) + P(x , y , t)
Ut − λ′(u)pUx − λ(u)∆P = 0
Λ′(u)pUx + Λ(u)∆P = 0(2)
Thus, ∆P = −Λ′(u)
Λ(u)pUx . Subs. into Ut equation:
Ut − p
(λ′(u)− λ(u)
Λ′(u)
Λ(u)
)Ux = 0
Shearer (NCSU) Flow in Porous Media 26 / 46
CLAIM: Lax shock =⇒ U has to be zero.
Ut − p
(λ′(u)− λ(u)
Λ′(u)
Λ(u)
)Ux = 0, u = u±
Let U(x , y , t) = w(x − st)e iαyeσt . Then, with z = x − st,w ′ = dwdz :
σw − sw ′ − p λ(u)
(λ′(u)
λ(u)− Λ′(u)
Λ(u)
)w ′ = 0
But f (u) = vTλ(u)/Λ(u), p λ(u) = −f (u) so we have
σw = (s − f ′(u)) w ′, with solutions w(z) = a±eβ±z ,
σ = β±(s − f ′(u±))
For a Lax shock, f ′(u+) < s < f ′(u−).Thus, Re σ ≥ 0 implies w(z) does not decay at z = ±∞ unless w ≡ 0.
Consequently, U ≡ 0
Shearer (NCSU) Flow in Porous Media 27 / 46
2-d stability: perturb pressure in linearized equations
u = u±; p = p±z + P; P(z , y , t) = q±(z)e iαy+σt
z = x − Vt = ae iαy+σt , z = x − x
We seek σ = σ1α, α > 0. Linearized equations: ∆P = 0 :
q′′± − α2q± = 0 (1)
Decaying solutions: q± = b±e∓αz , ±z > 0,
Next: linearize jump conditions: two equations for three coefficients
a, b−, b+
Shearer (NCSU) Flow in Porous Media 28 / 46
Linearized jump conditions (no perturbation on u)
σa[u] + [λ(u)q′] = 0 (1)
[Λ(u)q′] = 0 (2)
Equation (2):
Λ(u+)b+ = −Λ(u−)b− (3)
Then (1) implies
b−Λ(u−)(f (u+)− f (u−)) = −σα
a(u+ − u−)vT
But (f (u+)− f (u−))/(u+ − u−) = s, the shock speed, so
b−Λ(u−)s = −aσ
αvT (4)
Shearer (NCSU) Flow in Porous Media 29 / 46
2-dimensional stability: third equation for a, b±
Third equation comes from continuity of pressure (linearized)at z = z(y , t) = ae iαy+σt
p = p±z + q±(z)e iαy+σt , q± = b±e∓αz (±z > 0)
Consequently,p+a + b+ = p−a + b− (5)
Thus, (p+ − p−)a = −σα
vT
sa
(1
Λ(u+)+
1
Λ(u−)
), (from (3,4))
Since p± = − vT
Λ(u±), we obtain, for a 6= 0 :
σ
α= s
Λ(u−)− Λ(u+)
Λ(u−) + Λ(u+)
Shearer (NCSU) Flow in Porous Media 30 / 46
Interpretation of stability condition: quadratic relativepermeabilities: k(u) = κu2
Lax shocks for u+ < u− ≤ u∗, f ′(u∗) = (f (u+)− f (u∗))/(u+ − u∗)
Stability boundary: u+ = −u− + 2MM+1
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
u−
u
u
+
US
I
*
M = 0.22M
M + 1=
1
3
Inflection point I of f (u) atuI = 0.2591
S: Stable Lax shocks
U: Unstable Lax shocks
Shearer (NCSU) Flow in Porous Media 31 / 46
Fingering Instability - Zhengzheng’s Simulations
Full parabolic/elliptic system with capillary pressure pc = −u + τut
Crank-Nicolson time-step; upwind advection; periodic side boundaryconditions, moving frame, plot middle contour u = 1
2 (u− + u+)
∆t = O(10−3), ∆x = ∆y = O(10−2)
Initial condition: u = randomly perturbed hyperbolic tangent
u− = 0.2, u+ = 0,M = 0.05
−3 −1.5 0 1.5 3−3
−1.5
0
1.5
3time = 0
−3 −1.5 0 1.5 3−3
−1.5
0
1.5
3time = 2
−3 −1.5 0 1.5 3−3
−1.5
0
1.5
3time = 4
Shearer (NCSU) Flow in Porous Media 32 / 46
Numerical Simulations - Stable case
M = 0.2
u− = 0.25, u+ = 0
Oil-water mixture displaces oil
Initial perturbation decays
Zoomed-in contour0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
u−
u+
US
I
time = 0.05
− 0 .05 0 0.05
−12
−6
0
6
12time = 1.25
−12
−6
0
6
12
− 0 .05 0 0.05
time = 20
−12
−6
0
6
12
−0.05 0 0.05
Shearer (NCSU) Flow in Porous Media 33 / 46
Numerical Simulations: Unstable case: Fingering Instability
M = 0.2
u− = 0.25, u+ = 0.15
weak Lax shock
Initial perturbation grows ⇒fingering instability
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
u−
u+
(u , u )i i
US
− 0 .05 0 0.05
−12
−6
0
6
12time = 0.05
− 0.05 0 0.05
−12
−6
0
6
12time = 1.5
− 0.05 0 0.05
time = 20
−12
−6
0
6
12
Shearer (NCSU) Flow in Porous Media 34 / 46
Conclusions: stability/fingering instability in 2-d
Analysis of hyperbolic/elliptic system linearized around 1-d shockLinear dependence of growth rate σ on transverse wave number αdistinguishes stable waves from unstable
Numerical simulations of full parabolic/elliptic system confirm results;(Riaz and Tchelepi (2006) also conducted numerical experiments)
Weak Lax shocks may be stable or unstable
Oil/water mixture displacing oil can be stable.
K. Spayd, M. Shearer and Z. Hu, Applicable Analysis, 2012.
Shearer (NCSU) Flow in Porous Media 35 / 46
Back to basics: capillary pressure in a capillary tube
x2r(x) 2R
γ = 1/Ca = σ/Uη
Ca =: capillary number
balance between surfacetension σ and viscosity η.
Saturation u : area fraction
u(x , t) = r(x)2/R2.
Write equation in form
ut + f (u)x = −γ∂x [f (u)λ(u)∂xψ]
f (u) = u/(u + (1− u)3)
Fractional flow rate for air/water(Bear 1972)
λ(u) = (1− u)3
ψ = δFδu , F : free energy
F = F 0(u) + κ(u)F 1(ux)
F 0 : bulk free energy
F 1(ux) : interfacial energy
κ(u) : interfacial energy weight
Shearer (NCSU) Flow in Porous Media 36 / 46
Phase field model: F = F 0(u) + κ(u)F 1(ux)
Cueto-Felgueroso and Juanes, PRL (2008, 2011)
Simple choices:
F 0(u) = −Σ
R(1− u)2u2 + 2
γwg cos θ
R(1− u)2
Double-well potential; tangent construction (like equal area rule)
Σ = γsw − γsg − γwg < 0 spreading parameter
θ : static contact angle Young-Laplace law at contact lineγsw − γsg = γwg cos θ, set γ = γwg
F 1(ux) = Γu2x
Γ = C (θ)γR
Shearer (NCSU) Flow in Porous Media 37 / 46
Phase field model: ut + f (u)x = −γ∂x [f (u)λ(u)∂xψ]
Variational derivative: ψ = δFδu
ψ = −C1u(1− u)(1− 2u) + C2
√κ(u)∂x(
√κ(u)∂xu), C1 > 0,C2 > 0
Chemical potential for polymer mixture:
de Gennes: J. Chem. Phys. (1980); Witelski, Appl. Math. Lett. (1998)
Benzi, Sbragaglia, Bernachi, Succi: Binary fluid mixtures, PRL (2011)
∂tu + ∂x f (u) = γ∂x [H(u)Q(u)∂xu]− γ∂x[H(u)∂xC2
√κ(u)∂x(
√κ(u)∂xu)
]Q(u) = C1(6u2 − 6u + 1), H(u) = f (u)λ(u)
Cahn-Hilliard type equation
Shearer (NCSU) Flow in Porous Media 38 / 46
Static solutions: bubbles and compactons
Now choose κ(u) so that a static bubble is an equilibrium solution
air �uid�uid
static air bubble
Ends of bubble are spherical caps (take contact angle θ = 0, or π) :
u(x) = r 2 = 1− x2, 0 ≤ x ≤ 1. Thus u′ = −2x = −2√
1− u, u′′ = −2.
Here, tube radius R = 1, θ = π, ′ = ∂x
∂x [f (u)λ(u)∂xψ] = 0
Consequently,
f (u)λ(u)ψ′ = c0 = 0, since f (0) = 0
Shearer (NCSU) Flow in Porous Media 39 / 46
ψ(u) = −C1G (u) + C2√κ∂x(√κ∂xu)); G (u) = u(1− u)(1− 2u)
f (u)λ(u)ψ′ = 0 =⇒ ψ = c1 = 0, since κ(0) = 0
Thus,
m(u)(m(u)u′)′ = Ku(1− u)(1− 2u), K = C2/C1,m(u) =√κ(u).
u(x) = 1− x2 =⇒ x =√
1− u : ODE for m(u) :
m(u)
(dm
du4(1− u)− 2m(u)
)= KG (u) = Ku(1− u)(1− 2u).
But κ(u) = m(u)2. Then
dκ
du(1− u)− κ =
K
2u(1− u)(1− 2u).
LHS = ddu ((1− u)κ), so (1− u)κ = K
2 ( 12 u2 − u3 + 1
2 u4). That is,
κ(u) =K
4(1− u)u2
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General contact angles θ : Coeff. of interfacial energy κ
Similar result for general contact angle θ, 0 < θ < π :
u = sec2 θ − x2/R2
κ(u) =K (θ)u2(1− u)2
4(sec2 θ − u), 0 ≤ u ≤ 1.
Idea works for more general constitutive laws
ψ(u) = −C1G (u) + C2√κ∂x(√κ∂xu)); G (u) bistable
d
du
(κ(u)(sec2 θ − u)
)=
K (θ)
2G (u)
κ(u) =K (θ)
∫G (u) du
4(sec2 θ − u), 0 ≤ u ≤ 1
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PDE simulations Rachel Levy
−10 −5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t=0 t=4 t=6
structure: rarefaction and leading (undercompressive) shock/TW fromplateau u = u− to u = 0 at x = st + x0 <∞.
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Traveling Waves
∂tu + ∂x f (u) = γ∂x [H(u)G (u)∂xu]− γC∂x [H(u)∂xm(u)∂x(m(u)∂xu)]
u = u(x − st), u(−∞) = u−, u(0) = 0, s = f (u−)/u−
−su + f (u) = γH(u)G (u)u′ − γCH(u)(m(u)(m(u)u′)′)′,
Note: constant of integration is zeroWrite as first order system, β = 1/(Cγ) (multiply 3rd eqn by m(u))
m(u)u′ = v
m(u)v ′ = w
m(u)w ′ = 1C G (u)v + β
m(u)
H(u)(su − f (u))
Eliminate m(u) to remove singularity at origin
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ODE system - nondegenerate vector field
Eliminate m(u) : m(u) ddξ = d
dη , u(ξ) = U(η), ...
U ′ = V
V ′ = W
W ′ = 1C G (U)V + β
m(U)
H(U)(sU − f (U))
(U,V ,W ) = (0, 0, 0) now a regular equilibrium
Seek solutions with −∞ < η <∞(U,V ,W )(−∞) = (u−, 0, 0), (u, v ,w)(∞) = (0, 0, 0)
Need real eigenvalues at origin: γ > γ∗ : Ca < Ca∗
Shooting: 1-d unstable manifold from u− to 2-d stable manifold atorigin: u− = u−(β).
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Recover original variable u(ξ)→ 0, ξ → 0−
Transform back: m(U(η)) dη = dξU(η) ∼ aeµη as η →∞, (µ < 0 : eigenvalue) implies
ξ = −∫ ∞η
U(y) dy = aeµη/µ
Thus, u(ξ) = U(η) ∼ µξ, as ξ → 0−, slope µ < 0.
0
0.1
0.2
0.3
0.4
0.5
0.6u
x
Traveling wave MATLAB plot
Shearer (NCSU) Flow in Porous Media 45 / 46
Summary
Many applications with two-phase porous media flow
Oil recovery context: oil/water interface; Lax shocks may be stable orunstable to 2-D perturbations; hyperbolic analysis gives condition forlinear stability
Yortsos and Hickernell (1989) traveling wave stability; most unstablewave number from dispersion relation tested by Zhengzheng Hu
Phase field analysis for flow in capillary tube; undercompressiveshocks, stationary compactons; degenerate diffusion traveling waves
Hassanizadeh-Gray modeldegenerate diffusion-dispersion; sharp shocks
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