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Stability of vertical miscible displacements with developing density and viscosity gradients

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Abstract

Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement.

If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable.

A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.

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Abbreviations

a :

empirical constant in Equation (B3)

A :

dimensionless effective diffusion coefficient, defined by Equation (20)

b * :

gravity

B :

dimensionless parameter characterizing the effect of convection upon dispersion, defined by Equation (21)

B f :

some quantity associated with the fluid

C :

integration constant in Equation (50)

C mnp :

constant coefficients in Equation (51)

C(A):

volume fraction of species A

d * (A) :

mass density of pure species A

D :

density ratio, defined by Equation (59)

D(Aj) :

parameters upon which these functions depend indicated by Equation (8) (j = 1,2)

D (AB) (e) * :

effective dispersion tensor, defined by Equation (7)

D (e) (AB) * :

diffusion coefficient

E :

defined by Equation (43)

g * :

acceleration of gravity

G :

defined by Equation (22)

h :

reciprocal of the aspect ratio, which is the ratio of thickness to width and assumed to be less than one

I :

identity tensor that leaves vectors unchanged

j (A)(e) * :

effective mass flux vector with respect to \(\overline {V^* }\), represented by Equation (6)

k * :

permeability of the porous structure to the fluid

l 0 * :

characteristic dimension of the local pores

L * :

thickness of the reservoir

M :

mobility ratio, defined by Equation (57)

N pe :

local Peclet number, defined by Equation (9)

P * :

thermodynamic pressure

P :

defined by Equation (14)

r :

parameter in Equation (31)

R f :

region occupied by the fluid enclosed by S

S :

averaging surface

S (0) :

defined by Equation (42)

t * :

time

v * :

(mass-averaged) velocity of the fluid

υ c * :

critical velocity defined by Equation (61)

υ 0 * :

uniform magnitude of fluid velocity over the injection face

V (f) :

volume of R (f) enclosed by S

:

volume of the region enclosed by S

w :

defined by Equation (14)

ŵ :

defined by Equation (48)

X :

defined by Equation (B2)

Y(w) :

defined by Equation (B3)

z j * :

rectangular Cartesian coordinates (j= 1, 2, 3)

α :

dimensionless wave number in the z 2 direction

β :

dimensionless wave number in the z 3 direction

γ :

dimensionless wave number in the z 1 direction

δ :

defined by Equation (A2)

Δω :

defined by Equation (23)

ε :

perturbation parameter

λ mnp :

defined by Equation (54)

μ * :

viscosity of the fluid

μ (0) :

defined by Equation (35)

μ (0) :

defined by Equation (41)

μ 0 * :

viscosity of the displaced fluid

μ * :

viscosity of the displacing fluid

π :

3.141592...

ϱ * :

total mass density of the fluid

ϱ 0 * :

total mass density of the displaced fluid

ϱ 1 * :

defined by Equation (16)

ϱ * :

defined by Equation (60)

ϱ (A) * :

mass density of species A

τ mnp :

defined by Equation (55)

max(τ mnp ):

maximum value of τ mnp

φ * :

potential energy, defined by Equation (15)

ψ :

porosity

ω (A) :

mass fraction of species A, defined by Equation (10)

ω (A)0:

initial mass fraction of species A in the displaced fluid

ω (A)∞ :

final mass fraction of species A in the injection fluid

...(0) :

superscript denoting the stable solution

...(1) :

superscript denoting the first perturbation variable

...* :

superscript denoting the dimensional variable

div:

divergence operation

▽:

gradient operator

d V :

indicates that a volume integration is to be performed

dt′ :

indicates that a time integration is to be performed

\(\begin{array}{*{20}c} {\begin{array}{*{20}c} \sim \\ \ldots \\ \end{array} } \\ \end{array}\) :

indicates a variable defined by Equation (44)

\(\begin{array}{*{20}c} {\begin{array}{*{20}c} --- \\ \ldots \\ \end{array} } \\ \end{array}\) :

indicates a superficial average defined by Equation (1)

〈...〉:

indicates an intrinsic average defined by Equation (2)

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Chang, S., Slattery, J.C. Stability of vertical miscible displacements with developing density and viscosity gradients. Transp Porous Med 3, 277–297 (1988). https://doi.org/10.1007/BF00235332

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Key words

  • Stability
  • linear stability analysis
  • miscible displacement
  • aspect ratio
  • mobility control
  • dispersion