The effect of dispersion model on the linear stability of miscible displacement in porous media
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Abstract
Chang and Slattery (1986, 1988b) introduced a simplified model of dispersion that contains only two empirical parameters. The traditional model of dispersion (Nikolaevskii, 1959; Bear, 1961; Scheidegger, 1961; de Josselin de Jong and Bossen, 1961; Peaceman, 1966; Bear, 1972) has three empirical parameters, two of which can be measured in one-dimensional experiments while the third, the transverse dispersivity, must be measured in experiments in which a two-dimensional concentration profile develops. It is found that nearly the same linear stability behavior results from using either model.
Key words
Stability linear stability analysis miscible displacement aspect ratio mobility control dispersionNotation
- a1
dimensionless longitudinal dispersivity, defined by Equation (A3)
- at
dimensionless transverse dispersivity, defined by Equation (A4)
- A
dimensionless effective diffusion coefficient, defined by Equation (16)
- B
dimensionless parameter characterizing the effect of convection upon dispersion, defined by Equation (17)
- Cmnp
constant coefficients in Equation (A14)
- dp
particle diameter
- D
density ratio, defined by Equation (59) of Chang and Slattery (1988c)
- Dd
dimensionless effective diffusion coefficient, defined by Equation (A2)
- DL
dimensionless longitudinal dispersion coefficient, defined by Equation (A6)
- DT
dimensionless transverse dispersion coefficient, defined by Equation (A13)
- D(Aj)
parameters upon which these functions depend indicated by Equation (4) (j = 1,2
- D (AB)
effective dispersion tensor, defined by Equation (3)
- D*
diffusion coefficient
- G
defined by Equation (22) of Chang and Slattery (1988c)
- 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 v*, represented by Equation (2)
- k*
permeability of the porous structure to the fluid
- l0**
characteristic dimension of the local pores
- L*
thickness of the reservoir
- M
mobility ratio, defined by Equation (57) of Chang and Slattery (1988c)
- NPe
local Peclet number, defined by Equation (5)
- q(z1, t)
defined by Equation (A9)
- r
parameter in Equation (A8)
- S1(0)
defined by Equation (A11)
- t*
time
- Tt(0)
defined by Equation (A12)
- v*
(mass-averaged) velocity of the fluid
- v0*
uniform magnitude of fluid velocity over the injection face
- w
defined by Equation (14) of Chang and Slattery (1988c)
- zj*
rectangular Cartesian coordinates (j = 1, 2, 3)
Greek Letters
- αm
dimensionless wave number in the z1 direction
- βn
dimensionless wave number in the z2 direction
- γp
dimensionless wave number in the z3 direction
- δmmp
defined by Equation (A16)
- λmnp
defined by Equation (A15)
- μ(0)
defined by Equation (35) of Chang and Slattery (1988c)
- μ(0)
defined by Equation (41) of Chang and Slattery (1988c)
- ϱ*
total mass density of the fluid
- μ(A)*
mass density of species A
- σ
a measure of the inhomogeneity of the pack (Perkins and Johnston, 1963)
- τmnp
defined by Equation (A17) max(τmnp) maximum value of τmnp
- ψ
porosity
- ω(A)
mass fraction of species A, defined by Equation (6)
- 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
Other
- ...(0)
superscript denoting the stable solution
- ...(1)
superscript denoting the first perturbation variable
- ...*
superscript denoting the dimensional variable
- div
divergence operation
- Δ
gradient operator
- dt′
indicates that a time integration is to be performed
- ...
indicates a superficial average defined by Equation (1) of Chang and Slattery (1988c)
- 〈...〉
indicates an intrinsic average defined by Equation (2) of Chang and Slattery (1988c)
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References
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