Linear Stability Analysis of Immiscible Displacement Including Continuously Changing Mobility and Capillary Effects: Part II— General Basic Flow Profiles
A. B. Huang, E. D. Chikhliwala, Yanis C. Yortsos · SPE Annual Technical Conference and Exhibition · 1984
ABSTRACT This paper reports on the continuation of previous work1 in the linear stability or immiscible, two-phase flow displacement processes in porous media that includes continuously changing mobility and capillary effects. In Part I simple Dasic-flow profiles that allow exact solutions to be obtained were investigated. This work uses the previous formulation in terms of eigenvalue problems to examine the stability characteristics of general basic flow profiles. hotivated by the connection between the present problem and the stability of shear layers in parallel flow2, the numerical method of "matched initial values" successfully used in the stuay of latter3 is adopted in this numerical investigation. First, the stability of non-capillary flows corresponding to a straight line fractional flow is examined. The mathematical equivalence of these flows to miscible Displacement in the absence of diffusion is established. It is shown that such displacement is neutrally stable to small disturbances of any wavelength provided that the total mobility is continuously increasing in the direction of flow. The disp1acement is unstable to small disturbances of any wavelength if the total mobility profile contains any segment of decreasing mobility. The numerical results are in agreement with the exact solutions of part I in the limits of small and large wavenumber. Next, the stability cf capillary fiov.-s for general basic flow profiles is examined. An analysis of the eigenvalue problem indicates the existence of a single stability curve parametrized by the viscosity ratio and relative permeability ano capillary pressure characteristics. Numerical investigations using power-law functional forms for the relative permeability show that the displacement is stable to any small disturbance provided that the viscosity ratio μoμw takes values below a critical value. The latter can toe evaluatec, fairly accurately both for arbitrarily fixed saturation end-points and the Buckley-Leverett problem, by the condition that the end-point total flow mobilities are equal. For values of the viscosity ratio above the critical, the numerical results show that the displacement is unstable to small disturbances of wavelength larger than a critical value, and stable otherwise. This effect is attributed to the stabilizing" action of capillarity. Values of wavelength corresponding to the highest rate of growth are numerically determined. It is found that stability is enhanced at lower values of the capillary number and the injection rate. Finally, a limited sensitivity study of the effect on stability of the functional forms of relative permeability ana capillary pressure is carried out.