Percolation Theory of Two-Phase Relative Permeability

Abstract

Summary This paper uses statistical concepts, some borrowed from the percolationtheory of disordered media, to develop a theory percolation theory ofdisordered media, to develop a theory of two-phase relative permeabilities inregimes in which one phase is strongly wetting. Inputs to the theory are anetwork model of pore space, the distributions of pore-throat radii, andfunctions that pore space, the distributions of pore-throat radii, andfunctions that relate pore-throat conductance and volume to its radius. Thetheory with a network model of pore space and several simple statisticaldistributions of pore dimensions, is found to predict all observed trends intwo-phase relative permeabilities of systems in which one phase is stronglywetting. Moreover, the inputs to the theory can be adjusted to fit experimentaldata closely. Introduction The slow concurrent flow of two immiscible fluids, such as oil and water, through a porous medium can be described by extending Darcy's law to apply toeach fluid. That is, the volumetric flux of each phase is proportional to thegradient of mechanical potential (pressure gradient and body force field)acting on Phase j and potential (pressure gradient and body force field) actingon Phase j and is inversely proportional to its viscosity. The coefficient ofproportionality, k, is the Darcy permeability to that phase. The relativeproportionality, k, is the Darcy permeability to that phase. The relativepermeability to Phase j is given by permeability to Phase j is given by (1) This definition seems to account for the fluid viscosity with muj and thepore space with the single-phase permeability, k, so that the relativepermeability might depend solely on the division of the PV between the twofluids-i.e., on their saturations, Sj. The situation is much more complicatedin reality. Measured relative permeabilities depend on the saturation historiesas well as the fluid saturations, the pore-space morphology, the fluids'wetting characteristics, the ratio of the fluid viscosities, and the capillarynumber of flow. The capillary number indicates the relative importance ofviscous stresses exerted by the fluids and the capillary pressures developed atinterfaces between them at the pore level. Moreover, the relative permeabilityto a phase typically becomes small or altogether negligible permeability to aphase typically becomes small or altogether negligible when its saturationfalls below a value that is distinctly above zero. The purpose of this paper isto develop earlier ideas of Larson et al. into a quantitative description ofmultiphase flow through permeable media has rested on experiment alone. Understanding of the underlying physical mechanisms and the roles of pore-spacegeometry and topology, however, is increasing, and a qualitative picture hasemerged. Each flowing phase occupies its own set of channels-i.e., a continuousconnected subspace of the pore space-and no movement of the fluid interfacesbetween the flowing phases usually occurs in steady two-phase flow. If onephase (say water) strongly wets the pore walls, then the set of channelsoccupied by the nonwetting phase (say oil) consists of sample-spanning clustersof pores that are filled by that phase, apart from thin films of low hydraulicconductivity and virtually isolated pendular bodies of the wetting phase. Ifthe two phases locally pendular bodies of the wetting phase. If the two phaseslocally exert negligible shear stress on each other, then the equations oflocal motion within the pore space (i.e., the Stokes equations) indicate thatthe relative permeability of a phase depends solely on the way it isdistributed within the pore space. The way a phase is distributed is controlledby the interplay of pore-space morphology, the capillary and normal viscousstresses on fluid interfaces, and the shapes and stability of the interfaces. The shapes actually taken by the interfaces depend on saturation and saturationhistory. History enters because there are multiple shapes that satisfy thedemands of the Stokes flow system, which is made nonlinear by the freeinterfaces. The relative importance of capillary and normal viscous stresses inlocating interfaces depends on the capillary number. If the two phases exertappreciable shear stress on each other, then the ratio of their viscositiesmust also affect their relative permeabilities. permeabilities. Most importantto the way a phase is distributed in the pore space is the morphology of thepore space. However highly irregular the pore space is, its connectivity, ortopology, can be represented by pore space is, its connectivity, or topology, can be represented by a network and its geometry can be represented to a firstapproximation by recording along the network the radius of the largest spherethat can be locally inscribed within the pore space. Other approximations alsocan be recorded along the network, for instance, the local resistance to flowthrough the pore space. Pore bodies and pore throats can be defined in terms ofappropriate maxima and minima of largest-inscribed-sphere radius. All thevolume of a pore body can be assigned to the corresponding network node;alternatively it can be apportioned among the network branches, which representthroats. Elementary branch points within a pore body can be fused into networknodes of coordination greater than three. All this has been worked out indetail. Thus, in principle any porous medium can be reduced to a topologicallyprinciple any porous medium can be reduced to a topologically equivalentnetwork and properties assigned to nodes and branches of that network. Insedimentary rocks and most other porous media, the coordination numbers anddimensional properties are chaotically distributed. The intuitive idea of porespace as a network led Fatt and others to model two-phase channel flow througha porous medium by (1) choosing a regularly coordinated network of finite size,(2) assigning pore radii to nodes and bonds according to some hypothesis aboutthe distribution of pore size and shape, (3) distributing two phases to a givensaturation level according to postulated mechanisms by which one phase wouldhave displaced the postulated mechanisms by which one phase would havedisplaced the other, (4) calculating flow rate through the channel network, and(5) evaluating the relative permeability from Eq. 1. This whole procedure canbe repeated for equivalent assignments of pore radii procedure can be repeatedfor equivalent assignments of pore radii and distributions of phases, andstatistics of relative permeability can he compiled. Another approach existsthat leads more directly to accurate estimates of relative permeability, thoughit sacrifices effects of spatial relationships among parts of the pore space. This approach uses the distributions of relevant features of pore morphology(i.e., no spatial correlations), still incorporates the mechanisms by which thephases compete for pore space, and relies on percolation theory. Larsonexplored this statistical approach's pertinence to relative permeability, butall the ingredients required were not clear at that time. The approach wassuccessful, however, in explaining the distribution of nonwetting phase inporous media and in accounting for capillary pressures measured when mercury isinjected and retracted from pore space. This paper brings percolation theoryfully to bear on relative permeability. percolation theory fully to bear onrelative permeability. In the next section, we introduce the concept of adisordered bond network as a model of pore space: all germane pore propertiesare assigned to bonds, which are elevated from throat cross sections to wholepore segments between branch points.