Summary Flow experiments involving cocurrent and countercurrent spontaneous water/oil imbibition were performed on the same laterally coated sample of a natural porous medium with local saturation measurements and various boundary conditions. The experiments with countercurrent imbibition showed slower oil recovery, a smoother water/oil front, and slightly lower ultimate oil recovery than those with predominantly cocurrent imbibition. Numerical simulations revealed that the relative permeabilities that enabled good prediction of countercurrent oil recovery rate are about 30% less than the conventional cocurrent relative permeabilities at a given water saturation. Viscous coupling is assumed to permeabilities at a given water saturation. Viscous coupling is assumed to be the origin of this difference. A new formulation of Darcy equations that uses a matrix of mobilities was found to be in qualitative agreement with experimental results. Introduction Fractured reservoirs contain a substantial share of the world's oil reserves. Forecasts of the efficiency of a water-injection process for such reservoirs remain difficult because of poor knowledge of the different fracture networks and the individual production behavior of the matrix blocks in contact with water; i.e., each block produces its oil more or less independently from its neighbors produces its oil more or less independently from its neighbors under the combined effects of gravity and capillarity. Two-phase displacement of this kind is called spontaneous or free imbibition, and the mechanisms controlling such flow are analyzed in this study. Spontaneous imbibition involves both cocurrent and countercurrent flows in proportions that depend on the ratio of gravity to capillary forces and on the conditions applied at the boundaries of the block. The main concern has been finding a reliable procedure for scaling up laboratory imbibition tests performed on small cores. Experimental and numerical approaches have been considered. In the experimental approach, the scaling laws that apply to waterflooding were extended to spontaneous imbibition. Kyte proposed a centrifuging method for scaling up the effect of both gravity and capillary forces. Lefebvre du Prey, however, found that, when this centrifuging method was used to keep the ratio of gravity to capillary forces constant, a large discrepancy existed between the scaled-up recovery curves corresponding to different-sized blocks. The validity of standard macroscopic equations of two-phase displacements then became questionable because relative permeabilities are defined for fluids moving in the same direction and not permeabilities are defined for fluids moving in the same direction and not for countercurrent flows. Other possible origins of this discrepancy were suggested, such as imperfect knowledge of the boundary conditions and local heterogeneities of the porous medium that cannot be scaled up. Jacquin et al. therefore undertook a careful study of the mechanisms of spontaneous imbibition. Spontaneous imbibition tests (ID) on laterally coated sandstone samples 9.8 to 39 in. [25 to 100 cm] in length gave results in good agreement with conventional scaling laws. Therefore, the experimental scaling-up procedure requires numerous rules to be respected. The selection of the rock sample and the applied boundary conditions used for performing the imbibition test can strongly influence the results and prevent scaling up to reservoirblock sizes. The numerical method may be an alternative for solving this problem because the heterogeneities of the reservoir and various boundary conditions can easily be considered. It is necessary, however, to introduce the exact capillary pressure and relative permeability curves. Blair and Torsaeter and Silseth showed that these curves have an important impact on the oil recovery rate, with capillary pressures probably having a stronger effect than relative permeabilities. Some questions arise about using water/oil relative permeabilities, always determined from a cocurrent waterflood, to predict permeabilities, always determined from a cocurrent waterflood, to predict countercurrent imbibition flows. Unfortunately, there are very few experimental determinations of countercurrent relative permeabilities.

A macroscopic description of a two-phase flow in a porous medium is given by writing, firstly, mass and momentum-balance equations and, secondly, phenomenological equations derived from the theory of irreversible thermodynamic processes. The main results are as follows: (i) the law of capillary pressure is extended to dynamic conditions, (ii) an extended formulation of Darcy's law is established for each fluid phase and also for fluid/fluid interface which is considered as a phase of the system, and (iii) a coupling may appear between fluid phases.

This paper describes experimental studies of spontaneous imbibition of oil by water in a low-permeability outcrop chalk. At constant and high interfacial tension (IFT), the importance of capillary forces and the existence of a predominantly countercurrent mechanism were established. Additional experiments were performed to investigate the influence of length and of various boundary conditions. In another investigation, we modified the IFT at the sample boundary by using pairs of conjugate phases of the n-hexane/ethanol/brine ternary system. Final recovery increased when IFT was lowered. We give a numerical interpretation for this last result.

An extended formulation of Darcy's two-phase law is developed on the basis of Stokes' equations. It leads, through results borrowed from the thermodynamics of irreversible processes, to a matrix of relative permeabilities. Nondiagonal coefficients of this matrix are due to the viscous coupling exerted between fluid phases, while diagonal coefficients represent the contribution of both fluid phases to the total flow, as if they were alone. The coefficients of this matrix, contrary to standard relative permeabilities, do not depend on the boundary conditions imposed on two-phase flow in porous media, such as the flow rate. This formalism is validated by comparison with experimental results from tests of two-phase flow in a square cross-section capillary tube and in porous media. Coupling terms of the matrix are found to be nonnegligible compared to diagonal terms. Relationships between standard relative permeabilities and matrix coefficients are studied and lead to an experimental way to determine the new terms for two-phase flow in porous media.