We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.
We have established in a simple and straightforward fashion that the analysis of quasistatic flow in fluid-saturated porous media due to Rice and Cleary is derivable from the low-frequency limit of Biot’s slow compressional/diffusive mode. The single material parameter of the problem, the diffusivity, is simply related to the bulk and shear moduli and permeability of the skeletal frame and to the viscous and elastic properties of the constitutive media. Since this common theory treats fluid and solid displacements on an equal footing, it is the most general linearized description of the problem; other treatments are special cases. These latter include the rigid frame approximation used in the petroleum industry and the weak frame approximation used by De Gennes to describe the motion of polymer gels.
The time behavior of the streaming potential is identical to that of differential fluid pressure when local laminar flow is maintained and electromagnetic relaxation is not important. The time evolution of the streaming potential waveform resulting from the application of a pressure pulse to one face of a cylindrical water-saturated porous structure is found to depend on the elastic properties of the fluid and rock frame and on the porosity and permeability of the core. The analysis of this quasi-static flow based on Biot's soil consolidation work [Biot, J. Appl. Phys. 12, 155 {1941)] and advanced by J.R. Rice and M.P. Cleary [Rev. Geophys. Space Phys. 14, 227 {1976)] is completely equivalent to that obtained from Biot's slow wave model [M.A. Biot, J. Acoust. Soc. Am. 28, 168 {1956)] in the limit of zero frequency. Both analyses result in a homogeneous diffusion equation in pore pressure (and streaming potential) where the diffusivity is expressed in terms of the bulk moduli of the fluid and solid constituents, bulk and shear moduli of the solid frame, fluid viscosity, porosity, and permeability. The predicted streaming potential transient response is in excellent agreement with experimental observations, permitting, for example, the extraction of sample permeability. Models assuming an incompressible frame are shown to be grossly inadequate in describing the temporal features of quasi-static flow.
The time behavior of the streaming potential is identical to that of differential pressure when local laminar flow is maintained and electromagnetic relaxation is not important. The time evolution of the streaming potential waveform resulting from the application of a pressure pulse to one face of a cylindrical water-saturated core is found to depend on the elastic properties of the fluid and rock frame and on the porosity and permeability of the core. The analysis of this quasi-static flow based on Biot's soil consolidation work (1941) is completely equivalent to that obtained from Biot's slow wave model (1956) in the limit of zero frequency. Both analyses result in a homogeneous diffusion equation in pore pressure (and streaming potential) where the diffusivity is expressed in terms of the bulk moduli of the fluid and solid constituents, bulk and shear moduli of the solid frame, fluid viscosity, porosity, and permeability. The predicted streaming potential transient response is in excellent agreement with experimental observations, permitting, for example, the extraction of core permeability. Models assuming an incompressible frame are shown to be grossly inadequate in describing the temporal features of quasi-static flow.
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