MULTIPHASE
FLOW
IN
MULTI-COMPONENT
POROUS VISCO-ELASTIC MEDIA
Evgeniy Tantserev, Christophe Y. Galerne, Yuri Y. Podladchikov
The Fourth Biot Conference on Poromechanics. 2009.
ABSTRACT
Modeling of multi-component multi-phase porous systems is fundamental to the
study of geological processes. Recipes of continuum mechanics and
thermodynamics are employed here in order to derive a closed system of equations.
We start from a set of balance equations for mass, momentum, energy, and entropy
written for a general multi-phase multi-component system. Based on a local
thermodynamic equilibrium assumption and non-negativity of entropy production,
we choose thermodynamically admissible expressions for fluxes and sources in the
set of balance equations and present a closed system of equations.
INTRODUCTION
Mechanical compaction is usually treated under isothermal [1] or isoentropic [2]
simplifying assumptions. The case of joint consideration of both mechanical
compaction and reactive porosity alteration requires somewhat greater than usual
care about thermodynamic consistency. There is no “conservation of porosity law”;
there are no equilibrium conditions for porosity similar to absence of thermal
gradients or continuity of stresses and chemical potentials. Porosity fields may
spontaneously develop jumps that do not have to disappear at equilibrium [3] and
may generate ever growing waves out of minuscule perturbations [4].
To simplify treatment of the chemical reactions, we assume local chemical
equilibrium reached in each point due to the high temperature of the system and
slow rate of percolation as it is assumed in an ideal chromatography theory, e.g. [5].
Our goal is to derive a thermodynamically admissible closed system of equations
describing the coupling of “mechanical” and “chemical” compaction, e.g.[5-7]. We
are restricting ourselves to a minimum set of the most essential processes. The
emphasis is not on generality or universality of the final system of equations but on
its transparency and thermodynamic consistency. For the mechanical interactions,
we include pore compressibility and viscosity to account for high temperature stress
__________
Tantserev, E., NTNU, S P Andersens Vei 15a, 7491 Trondheim, Norway
Galerne, C.Y., Podladchikov, Y.Y., Physics of Geological Processes, University of Oslo PO
Box 1048 Blindern 0316, Norway
relaxation and Darcian flow of the porous fluid. Due to their complexity, we
exclude the capillary effects. For the energy balance, we introduce a specific (per
unit mass) internal energy for each phase and assume local thermal equilibrium that
requires equal temperatures for all phases [8].
DIFFERENTIAL INTRA-PHASE BALANCE RELATIONSHIPS AND
INCREMENTAL LOCAL THERMODYNAMIC EQUILIBRIUM
ASSUMPTION
Eulerian balance of mass, momentum, energy and entropy are expressed by,
∂ ( ρ ⋅φ ⋅ A )
+ ∇i ⋅ ρ ⋅ φ ⋅ A ⋅ v i + q iA = QA ,
∂t
(
)
(1)
where the quantity A is a specific quantity (per unit mass), namely concentration
( ck , k=1..N), velocity ( vi ), total energy (E) and entropy (s). Symbols are further
deacribed in Table 1.
Summing up equations (1) for A = ck yields
∂ ( ρ ⋅φ )
+ ∇i ⋅ ( ρ ⋅ φ ⋅ vi ) = Qρ .
∂t
(2)
Using (2) to simplify non-divergent form of (1)-(2) after some algebra results in
φ⋅
Q dφ
1 dρ
⋅
= −φ ⋅∇ i ⋅ v i + ρ −
,
ρ dt
ρ dt
ρ ⋅φ ⋅
dA
= −∇i ⋅ qAi + Q% ,
dt
(3)
(4)
N
1
where Qρ = ∑ Qck , A = ck , vi , s, u ; u = E − vi ⋅ vi ; Q% = QA − A ⋅ Qρ ,
2
k =1
1
⎛
⎞
if A = ck , vi , s and Q% = vi ⋅∇ j ⋅ qvij + QE − vi ⋅ Qvi − ⎜ u − vi ⋅ v i ⎟ ⋅ Qρ , if A = u .
2
⎝
⎠
Local thermodynamic equilibrium equation follows representative volume
moving with the phase velocity:
τ dφ eq
d (1/ ρ ) N −1
dc
du
ds
.
=T ⋅ − P⋅
+ ∑ mk ⋅ k + φ
dt
dt
dt
dt ρ ⋅ φ dt
k =1
The last term accounts for poroelasticity.
(5)
DERIVATION OF EXPRESSION FOR ENTROPY PRODUCTION
Substituting (3)-(4) into (5) gives the expression for Qs = Qsintra + Qsinter .
Collecting similar terms and splitting obtained expression in two parts results in:
N
N −1
⎛
⎞
T ⋅ Qsintra = −∇i ⎜ qEi − T ⋅ qsi − ∑ µk ⋅ qci k − vi ⋅ qvij ⎟ − qsi ⋅∇ iT − ∑ qci k ⋅∇ i µ k −
k =1
k =1
⎝
⎠
− ( qvij − P ⋅ φ ⋅ δ ij ) ⋅∇i ⋅ v j ,
(6)
N
T ⋅ Qsinter = QE − vi ⋅ Qvi − ∑ µk ⋅ Qck + P ⋅
k =1
dφ
dφ eq
−τφ ⋅
.
dt
dt
No additional assumptions were made up to this point. Chemical potentials
µk in the final expressions for the entropy production are just shorthand notation
⎧m , k < N
1
P N −1
defined as µ k = u − vi ⋅ vi − T ⋅ s + − ∑ mn ⋅ cn + ⎨ k
,
2
ρ n =1
⎩ 0 ,k = N
internal mass flux is zero).
N
∑q
k =1
i
ck
= 0 (total
THERMODYNAMICALLY ADMISSIBLE FLUXES AND INTER-PHASE
INTERACTIONS
The following choice of intra-phase fluxes
N
qci k = −∑ M kn ⋅∇ i µ n ≈ − Dk ⋅∇ i µk ,
k =1
q = ( P ⋅ φ − η1 ⋅∇ i ⋅ vi ) ⋅ δ ij − η 2 ⋅ ( ∇ i ⋅ v j + ∇ j ⋅ vi ) ,
(7)
ij
v
qsi = −
λ
T
N
⋅∇ iT , qEi = T ⋅ qsi + ∑ µk ⋅ qci k + vi ⋅ qvij ,
k =1
results in non-negative intra-phase part of the entropy production, Qsintra ≥ 0 .These
expressions (first three) represent experimentally well verified Fick’s, Fourier and
Newton laws of mass diffusion, heat conduction and viscous rheology respectively.
Let us consider conservative exchange of mass, momentum, energy and entropy
between any two phases, called here fluid and solid. Conservations require:
Qck = Qck
fluid
= −Qck
solid
, Qvi = Qvi
Kinematics of the volume fractions:
fluid
= −Qvi
solid
, QE = QE
fluid
= −QE
solid
.
(8)
d sφ def dφ
=
dt
dt
− ⎡⎣vi ⎤⎦ ∇iφ f = −
fluid
dφ
dt
.
(9)
solid
Entropy production due to inter-phase exchange is:
N
⎡ vi ⎤
⎡ µk ⎤
⎡1⎤
inter
=
⋅
−
⋅
−
Q
Q
Q
∑
s
⎢⎣ T ⎥⎦ E ⎢ T ⎥ vi ∑ ⎢ T ⎥ ⋅ Qck +
⎦
phases
k =1 ⎣
⎣ ⎦
( P − τ φ ) ⋅ dφ eq .
P ⎛ dφ dφ eq ⎞
+ ∑ ⋅⎜
−
+
⎟ ∑
dt ⎠ phases T
dt
phases T ⎝ dt
(10)
We choose thermal equilibrium, chemical equilibrium, poro-elastic mechanical
equilibrium:
T =T
fluid
=T
solid
, µk = µk
fluid
= µk
solid
, k = 1..N , ⎡⎣ P − τ φ ⎤⎦ = 0 ,
(11)
entropy producing momentum exchange and inelastic porosity compaction,
respectively:
Qvi = ( P ⋅∇iφ + τ φ ⋅∇ iφ eq )
fluid
− α ⋅ ⎡⎣v i ⎤⎦ ,
d sφ d sφ eq [ P ]
−
=
.
η3
dt
dt
(12)
as admissible set of assumptions that guaranties non-negative entropy production,
∑ Qsinter ≥ 0 .
phases
CLOSED SYSTEM OF EQUATIONS
We consider multiphase multi-component system. All inertial effects are
neglected. For simplicity, we introduce only two velocity and two pressure fields,
one for all solid phases and another one for all fluid phases. Summing up
independent set of balance relationships over all phases yields
∂ρ
+ ∇i ⋅ ρ ⋅ vsi + ρ f ⋅ ( vif − vsi ) = 0
∂t
(
)
,
(13)
∂ck
+ ∇i ⋅ ck ⋅ vsi + ckf ⋅ ( vif − vsi ) = ∇ i ⋅ ( Dkeff ⋅∇i µk ) , k = 1..N − 1 ,
∂t
(14)
(
(15)
(
)
)
∇i ⋅ ( − P + η1 ⋅∇i ⋅ vsi ) + ∇ j ⋅ η2 ⋅ ( ∇i ⋅ vsj + ∇ j ⋅ vsi ) = 0 ,
∂s
+ T ⋅∇i ⋅ s ⋅ vsi + ss ⋅ ( vif − vsi ) = ∇i ⋅ ⎡⎣λ eff ∇iT ⎤⎦
∂t
2
⎛ N
P] ⎞
[
2
2
i 2
i
i 2
⎟.
+ ∑ ⎜ ∑ Dk ⋅ ∇i µk + η1 ⋅ ⎡⎣∇i ⋅ v ⎤⎦ + 2 ⋅η2 ⋅ ∑ eij + α ( v f − vs ) +
⎜
η3 ⎟⎠
phases k =1
i, j
⎝
(
T⋅
)
(16)
Summation of the momentum balances over fluid phases only yields Darcy’s law:
φ f2
φ f ⋅ ( v − v ) = − ⋅∇i ⋅ Pf .
α
i
f
(17)
i
s
Equations (13)-(17) constrain P , ck , vsi , T and v f i , respectively. Porosity
evolution equation is
d sφ f
=
dt
def
In these expressions A =
∑
d sφ eq
f
dt
+
(1 − φ )η
f
and A =
∑
φ lf ⋅A lf +
l , all fluid phases
.
(18)
3
∑
φ lf ⋅ ρ lf ⋅A lf +
l , all fluid phases
def
Pf − P
φsl ⋅ ρ sl ⋅ Als v, if A = ck , s
l , all solid phases
∑
φsl ⋅ A ls , if A = ρ , P .
l , all solid phases
Local thermodynamic equilibrium constrains provide closing relationships for
Pf and ckf , ρ and ρ f (equation of state), s and s f (caloric equation of state), mk
(solution models) and φ eq
(poroelasticity) close the system of equations consistent
f
with incremental local thermodynamic equilibrium assumption (5). We have
calibrated (see [9]) our representative smeared volume treatment of poroelasticity
by exact Gassman’s relationships [10] and obtained closing poroelastic rheological
relationships:
dPf
dt
= Kf ⋅
1 dρf
,
⋅
ρ f dt
⎛
1 dρ
1 dρf
≈ (1 − φ eq
⋅
Ks ⋅ ⋅ s − K f ⋅
f )⎜
⎜
ρ s dt
ρ f dt
dt
⎝
dPeff
−
dPeff
dt
≈ Kφ ⋅
dφ eq
f
dt
(19)
⎞
⎟⎟ ,
⎠
,
eq
1 1−φ f
1
.
where Peff = (1 − φ ) ⋅ ( Ps − Pf ) -effective pressure,
=
−
Kφ
K dry
Ks
eq
f
(20)
(21)
Table I. Notations
Symbols
i
N, ck, v
φ , P, T
ρ, u, E, s
Meanings
number of components, concentration of k-th component and
velocity in considered phase, respectively
volume fraction, pressure and temperature of the considered
phase, respectively
specific quantities (per unit mass): density, internal energy,
total energy and entropy of considered phase, respectively
eij
components of the solid strain rate
τφ
pore compressibility term
Q, q
source (exchange rate of mass, momentum, energy, entropy)
and internal flux, respectively.
η1 ,η2 ,η3 , Dk , λ , α , M kn , mk
non-negative experimental parameters
f, s, eq, eff,
inter, intra
[]
fluid and solid phases, equilibrium and effective, respectively
inter-phase and intra-phase, respectively
difference of the same quantity in the fluid and solid phases
,
quantity of the fluid phase or solid phase, respectively
fluid
solid
K f , K s , K dry , Kφ
bulk modulus of fluid, solid, dry solid skeleton and poroelastic
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