Francisco J. Valdés-Parada
UAM-Iztapalapa, Mexico.
email: iqfv@xanum.uam.mx
Didier Lasseux
CNRS, Université de Bordeaux, France email: didier.lasseux@u-bordeaux.fr
Fabien Bellet
CNRS, Univ. Paris-Saclay, France
email: fabien.bellet@centralesupelec.fr
Introduction
• Unsteady flow in porous media is relevant in applications ranging from seismic waves to super fluid flow.
• This process is usually modeled using the heuristic momentum equation inferred from Darcy’s law:
ρ ∂ hvi
β∂t = −∇h p i
β− µεH
−1· hvi
β• Many analyses have been performed in the frequency but not in the time domain.
Pore-scale equations
∇ · v = 0 ρ Dv
Dt = − ∇ p ˜ + µ∇
2v − ∇hpi
β| {z }
sour c e
v = 0, at the solid-fluid interface v = v
0sour c e
|{z}
, when t = 0
ψ(r + l
i) = ψ(r), ψ = v, p, i ˜ = 1, 2, 3 h p ˜ i
β= 0
Formal solution
v = 1 µ
−
t0=t
Z
t0=0
∂ D
∂t
t−t0· ∇h p i
β t0d t
0+ m
0
˜
p = −
t0=t
Z
t0=0
∂ d
∂t
t−t0· ∇h p i
β t0d t
0+ n
0Upscaled model
∇ · hvi = 0
hvi = − 1 µ
t0=t
Z
t0=0
∂ H
t∂t · ∇hpi
βd t
0+ α where H
t= hDi and α = hm
0i/µ.
H21L-1923: Upscaling unsteady inertial flow in homogeneous porous media
Modeling macroscopic
unsteady interial flow in
porous media should take
into account the process history.
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Closure problems
Problem I
∇ · D = 0 ρ
µ
DD
Dt = −∇d + ∇
2D + I
D = 0, at the solid-fluid interface D = 0, t = 0
hdi
β= 0
ψ(r + l
i) = ψ(r), ψ = D, d, i = 1, 2, 3 Problem II
∇ · m
0= 0 ρ
µ
Dm
0Dt = −∇n
0+ ∇
2m
0m
0= 0, at the solid-fluid interface m
0= µv
0, t = 0
hn
0i
β= 0
ψ(r + l
i) = ψ(r), ψ = m
0, n
0, i = 1, 2, 3
Comparison with DNS
0 1 2 3 4 5
·10−2 0
0.5 1 1.5
2 ·10−3
