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 ^{*} :

(massaveraged) 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)
References
Aleman, M. A., Ramamohan, T. R. and Slattery, J. C., 1988, A statistical structural model for unsteadystate displacement in porous media, to appear in Transport in Porous Media.
Bear, J., 1961, On the tensor form of dispersion in porous media, J. Geophys. Res. 66, 1185.
Bear, J., 1972, Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, New York.
Benham, A. L. and Olson, R. W., 1963, A model study of viscous fingering, Soc. Pe J.t. Eng. J. 3, 138.
Blackwell, R. J., Rayne, J. R., and Terry, W. M., 1959, Factors influencing the efficiency of miscible displacement, Trans. AIME 216, 1.
Brigham, W. E., Reed, P. W., and Dew, J. N., 1961, Experiments on mixing during miscible displacement in porous media, Soc. Petrol. Eng. J. 1, 1.
Chang, S.H. and Slattery, J. C., 1986, A linear stability analysis for miscible displacements, Transport in Porous Media 1, 179.
Chang, S.H. and Slattery, J. C., 1988, A new description for dispersion, to appear in Transport in Porous Media.
Clark, N. J., Shearin, H. M., Schultz, W. P., Garms, K., and Moore, J. L., 1958, Miscible drive  its theory and application, J. Petrol. Technol., 11 (June).
Craig, F. F. Jr., Sanderlin, J. L., Moore, D. W., and Geffen, T. M., 1957, A laboratory study of gravity segregation in frontal drives, Trans. AIME 210, 275.
Craig, F. F. Jr. and Owens, W. W., 1960, Miscible slug flooding  a review, J. Petrol. Technol., 11 (April).
Crane, F. E., Kendall, H. A., and Gardner, G. H. F., 1963, Some experiments on the flow of miscible fluids of unequal density through porous media, Soc. Petrol. Eng. J. 3, 277.
de Josselin de Jong, G. and Bossen, M. J., 1961, Discussion of paper by Jacob Bear, ‘On the tensor form of dispersion in porous media’, J. Geophys. Res. 66, 3623.
Dumore, J. M., 1964, Stability considerations in downward miscible displacements, Soc. Pet. Eng. J. 4, 356.
Garder, A. O. Jr., Peaceman, D. W., and Pozzi, A. L. Jr., 1964, Numerical calculation of multidimensional miscible displacement by the method of characteristics, Soc. Petrol. Eng. J. 4, 26.
Gardner, G. H. F., Downie, J., and Kendall, H. A., 1962, Gravity segregation of miscible fluids in linear models, Soc. Petrol. Eng. J. 2, 95.
Gardner, J. W. and Ypma, J. G. J. 1984, An investigation of phasebehavior/macroscopicbypassing interaction in CO_{2} flooding, Soc. Petrol. Eng. J. 24, 508.
Habermann, B., 1960, The efficiency of miscible displacement as a function of mobility ratio, Trans. AIME 219, 264.
Heller, J. P., 1966, Onset of instability patterns between miscible fluids in porous media, J. Appl. Phys. 37, 1566.
Hickernell, F. J. and Yortsos, Y. C., 1986, Linear stability of miscible displacement processes in porous media in the absence of dispersion, Stud. Appl. Math. 74, 93.
Hill, S., 1952, Channeling in packed columns, Chem. Eng. Sci. 1, 247.
Homsy, G. M., 1987, Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19, 271.
Jiang, T.S., Kim, M. H., Kremesec, V. J. Jr., and Slattery, J. C., 1987, The local volumeaveraged equations of motion for a suspension of nonneutrally buoyant spheres, Chem. Eng. Commun. 50, 1.
Kyle, C. R. and Perrine, R. L., 1965, Experimental studies of miscible displacement instability, Soc. Pet. Eng. J. 5, 189.
Lee, S.T., Li, K.M. G., and Culham, W. E., 1984, Stability analysis of miscible displacement processes, SPE/DOE 12631, Society of Petroleum Engineers, P. O. Box 833836, Richardson, TX 75083–3836.
Nikolaevskii, V. N., 1959, Convective diffusion in porous media, PMM, J. Appl. Math. Mech. (Engl. Transl.) 23, 1492.
Offeringa, J. and van der Poel, C., 1954, Displacement of oil from porous media by miscible liquids, Trans. AIME 201, 310.
Peaceman, D. W. and Rachford, H. H. Jr., 1962, Numerical calculation of multidimensional miscible displacement, Soc. Petrol. Eng. J. 2, 327.
Peaceman, D. W., 1966, Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement, Soc. Pet. Eng. J. 6, 213.
Perkins, T. K. and Johnston, O. C., 1963, A review of diffusion and dispersion in porous media, Soc. Pet. Eng. J. 3, 70.
Perrine, R. L., 1961a, Stability theory and its use to optimize solvent recovery of oil, Soc. Petrol. Eng. J. 1, 9.
Perrine, R. L., 1961b, The development of stability theory for miscible liquidliquid displacement, Soc. Petrol. Eng. J. 1, 17.
Peters, E. J., Broman, W. H. Jr., and Broman, J. A., 1984, A stability theory for miscible displacement, SPE 13167, Society of Petroleum Engineers, P.O. Box 833836, Richardson, TX 75083–3836.
Pozzi, A. L. and Blackwell, R. J., 1963, Design of laboratory models for study of miscible displacement, Soc. Petrol. Eng. J. 3, 28.
Scheidegger, A. E., 1961, General theory of dispersion in porous media, J. Geophys. Res. 66, 3273.
Schowalter, W. R., 1965, Stability criteria for miscible displacement of fluids from a porous medium, AIChE J. 11, 99.
Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media, AIChE J. 13, 1066.
Slattery, J. C., 1981, Momentum, Energy, and Mass Transfer in Continua, McGrawHill, New York (1972); second edition, Robert E. Krieger Publishing Co., Malabar, FL 32950.
Slobod, R. L., and Lestz, S. J., 1960, Use of a graded viscosity zone to reduce fingering in miscible phase displacements, Producers Monthly, 12 (August).
Slobod, R. L. and Thomas, R. A., 1963, Effect of transverse diffusion on fingering in misciblephase displacement, Soc. Petrol. Eng. J. 3, 9.
Slobod, R. L. and Howlett, W. E., 1964, The effects of gravity segregation in laboratory studies of miscible displacement in vertical unconsolidated porous media, Soc. Petrol. Eng. J. 4, 1.
Stalkup, F. I. Jr., 1983a, Status of miscible displacement, J. Pet. Technol. 35, 815.
Stalkup, F. I. Jr., 1983b, Miscible Displacement, Society of Petroleum Engineers Monograph Series 8, Dallas.
Wooding, R. A., 1962, The stability of an interface between miscible fluids in a porous medium, Z. Angew. Math. Phys. 13, 255.
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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