Analytical Solutions for 1D Countercurrent Imbibition in Water-Wet Media
- Dimo Kashchiev (RERI) | Abbas Firoozabadi (RERI)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2003
- Document Type
- Journal Paper
- 401 - 408
- 2003. Society of Petroleum Engineers
- 5.3.1 Flow in Porous Media, 5.3.2 Multiphase Flow, 1.6.9 Coring, Fishing
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Analytical solutions for the initial stage of 1D countercurrent flow of water and oil in porous media are presented. Expressions are obtained for the time dependence of the water saturation profile and the oil recovered during spontaneous countercurrent imbibition in rod-like, cylindrical, and spherical cores, for which water is the wetting liquid. Some of the analytical solutions are found to be in good agreement with existing numerical solutions and available experimental data for oil recovery from cores with strong water wettability.
Capillary-driven fluid flow is often important in two-phase flow in fractured porous media and in layered media where individual layers are thin. The imbibition of water in the matrix block of a fractured reservoir when immersed in water is mainly by the capillary phenomena at water-wet conditions.1 In such cases, the parameters in the flow equations are complicated functions of saturation because of high nonlinearity arising from a realistic shape of the capillary-pressure curve. The common approach to the modeling of the process is the use of numerical techniques.
Analytical solutions to fluid-flow problems are desirable, because they allow a better understanding of the underlying physics and verification of numerical models. For capillary-driven flow, only a handful of authors have proposed analytical solutions of various degrees of complexity and with certain restrictive assumptions.
Yortsos and Fokas obtained an analytical solution for a 1D flow with account of capillary pressure; the relative permeabilities and capillary pressure were, however, severely restricted in functional form.2 Chen proposed combined analytical-numerical techniques for analysis of radial 1D flow.3 His work is based on the use of certain asymptotic conditions; it has a strong numerical component.
McWhorter and Sunada reported quasianalytical solutions for 1D linear and radial flow.4 Their work includes both countercurrent and cocurrent flow. These authors limited their solution to an infinite acting medium and assumed that the volume flux at the inlet is of the form At-1/2 where A is constant and t is time.
Pavone et al.5 also solved the 1D and 2D (gravity drainage) problem analytically; several assumptions were made by these authors to provide a closed-form solution. The assumptions included infinite gas mobility, linear liquid-phase relative permeability, and capillary-pressure dependence on saturation in the form of logarithmic function. As a result of these assumptions, the flow equations became linear.
In this paper, we provide approximate analytical solutions for the initial stage of linear, cylindrical, and spherical countercurrent flow of water and oil in a porous medium. We solve the flow equations without restricting the functional form of the relative permeabilities and the capillary pressure. We only assume that the imbibing and the displaced liquids are incompressible and that the porous medium is water-wet. These two assumptions have been made in the work of all authors referred to above.
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