Capillary Pressure, Water Relative Permeability, Electrical Conductivity and Capillary Dispersion Coefficient of Fractal Porous Media at Low Wetting Phase Saturations

Summary

Transport properties are critical for many reservoir engineering calculations. This work describes recent progress in modeling transport properties of natural porous media at low saturations of a wetting phase, i.e. when total wetting phase saturation Sw is the sum of thin-films and pendular structures inventories. Capillary pressure Pc, wetting phase relative permeability krw, electrical conductivity s w, and capillary dispersion coefficient Dc have been observed to obey power laws in the wetting phase saturation. We relate power-law behavior at low wetting phase saturations, i.e. at high capillary pressures, to the thin-film physics of the wetting phase and the fractal character of the pore space of natural porous media. If wetting phase inventory is primarily pendular structures, and if thin films control the hydraulic resistance of wetting phase we deduce the power laws X=, with X=Pc, krw, sw and Dc, where for capillary pressure =-1/(3-D), for wetting phase relative permeability =3/m(3-D), for electrical conductivity =1/m(3-D), and for capillary dispersion coefficient =[3-m(4- D)/m(3-D), where m is the exponent in the relation of disjoining pressure and film thickness and D is the fractal dimension of the surface between the pore space and grains or matrix. Recent experimental work lends support to these scaling laws in the cases of natural sandstones and clayey soils. Recent displacement experiments show anomalously rapid spreading of wetting liquid during imbibition into a prewet porous medium. We explain this phenomenon, called hyperdispersion, as vicous flow along fractal pore walls in thin films of thickness h governed by disjoining forces and capillarity.

Introduction

The fact that capillary pressure, water relative permeability, electrical conductivity, and capillary dispersion coefficient sometimes follow power laws in water saturation is well known. The parameters of these power laws, however, have remained empirical, reducing the reliability of extrapolating scarce data. In this paper we describe recent work in which we show that applying Mandelbrot's fractal geometry1 and the physics of thin films in porous media suggests that the power-law exponents can be interpreted in terms of a fractal dimension which characterizes pore-wall roughness, and in terms of disjoining pressure which governs the thickness of liquid thin-films and thus the flow properties.

Pore Space Morphology

The geometry of pore walls in reservoir rocks and in soil strata varies from the smooth, crystalline surfaces of dolomites to the rough, pitted or clay-coated surfaces of sandstones and soils. Katz and Thompson, 2 using images from scanning electron microscopy, measured the number of features (asperities) versus size of the feature on several natural porous sandstone fracture surfaces and concluded that pore walls are surface fractals in a statistical sense on length scales between a minimum dimension l1 (of the order of 10 to 100A) and a maximum dimension l2 (of the order of 100µm). According to Katz and Thompson's argument the porosity f of fractal sandstone isEquation 1

where A is of the order of 1 and D is the fractal or Hausdorff dimension of pore walls. By plotting the number of geometric features versus size, they found values of D ranging from 2.57 to 2.87 from SEM studies of five natural sandstones.

If the volumes of the cavities formed by the various asperities of pore walls are fractally distributed, this has particular implications for the capillary pressure, surface wettability and transport properties of porous media.

Capillary Pressure

We assume one of the fluids strongly wets the porous medium. Thus, even at saturations so low that bulk wetting phase seemingly exists only as isolated regions or pendular structures, the wetting phase remains hydraulically connected through thin films. To achieve a given wetting phase saturation Sw, the pressures Pw in bulk wetting phase and Pnw in bulk nonwetting phase must satisfy the Young-Laplace (YL) equation for the capillary pressures Pc, i.e. PcPnw-P w=2H*gamma;, where H is the mean curvature and ? is the interfacial tension of the meniscus between wetting and nonwetting phases. The mean curvature is related to the principal radii of curvature R1 and R2 of the meniscus by the expression 2H=(1/R1+1/ R2). Fixing the mean curvature of the menisci between wetting and nonwetting phases fixes the saturation Sw.

Experimental Study of Cocurrent and Countercurrent Flows in Natural Porous Media

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.