ABSTRACT
The upper part of the earth’s crust is highly fractured on all scales ranging from microcracks to large scale joints and faults. Joints are well-known for their effects on the physical properties of a rock mass. Mechanical properties such as bulk elastic constants and shear strength are strongly affected by the presence of joints. Joints also control the hydraulic permeability of crystalline and tight sedimentary rocks. Fluid flow in fractured rocks is a subject of active research in petroleum engineering, engineering geology, and hydrology. Applications include oil and gas production from fractured reservoirs. reservoir stimulation by hydraulic fracturing, hazardous waste isolation, and geothermal energy extraction. For this reason considerable effort is being placed on characterization and modeling of fractures and fracture systems. Of central importance to all applications is the hydromechanical behavior of a single fracture.
In nearly all applications, fluid flow through a single fracture is assumed analogous to laminar flow between two perfectly smooth parallel plates. This leads to the so-called “cubic law,” where the volume flow rate varies as the cube of the separation between the plates. However, this parallel plate model can only be considered as a qualitative description of flow through real fractures. Real fracture surfaces are rough and contact each other at discrete points, leading to tortuous fluid paths and deviations from the cubic law.
Several approaches to develop an expression for fluid flow which explicitly accounts for the surface roughness have been made in the past. Various empirical flow laws have been presented that are based on experiments with idealized geometries such as parallel plates with sand glued to the walls to mimic small-scale roughness and parallel plates with various machining marks to mimic large-scale roughness. More theoretical approaches have focused on redefining the aperture term in the cubic law to account for the surface roughness and the resulting tortuosity of the fluid flow paths.
In this paper a more direct approach to the study of fluid flow through fractures is presented. Using the finite difference method, computer simulations, based on Reynolds’ equation, of laminar flow 4between rough surfaces have been carried out. The surfaces were modeled as fractals, which have been shown to be reasonable models of natural rock surfaces.
Lubrication engineers frequently use Reynolds’ equation as a basis for study of hydrodynamic lubrication of machine bearings. It is a simplification of the Navier-Stokes’ equations, which can be considered to describe the lowest order effects of surface roughness on flow between rough surfaces.
Recently it has been shown that linear profiles of natural rock surfaces have power-law power spectral density functions. This is a property of self-affine fractal distributions; hence, they are defined by two parameters, one of which is the fractal dimension. Qualitatively the fractal dimension is a jaggedness parameter, indicating the proportion of high frequency to low frequency roughness. In this study the fractal model was assumed to adequately describe the character o rock fracture surfaces. The concepts of fractals provides a convenient algorithm for generation of realistic numerical surfaces which can then be used in the solution of Reynolds’ equation.
To form a fracture, two surfaces with the same fractal dimension, but comprised of different sets of random numbers, were generated. Two surfaces were placed together at some fixed distance to form a rough-walled fracture. The resulting spatially varying aperture distribution was provided as input to Reynolds’ equation, which was solved by the finite difference method.
The solution of this equation is the local fluid pressure in the fracture plane, which is then used to calculate the local volume flow rate. Plots of the local flow rate vectors show that, when the surfaces touch, the flow is tortuous and tends to be channeled through high aperture regions around the contacts. When the surfaces are completely separated the effect of surface roughness is less noticeable. Finally, the total volume flow rate through the fracture was calculated. The total flow rate and the macroscopic pressure gradient were used in the cubic law to obtain an effective parallel plate or hydraulic aperture for the rough walled fracture. Quantitative measures of the validity of the cubic law were then obtained by comparing the hydraulic aperture to the mean aperture.
The results show marked deviations from the cubic law at small surface separations as measured by the ratio of the mean surface separation to the standard deviation of surface height. The first order, this dimensionless ratio controls the deviation from the cubic law and measures how far the surface asperities protrude into the fluid film. As this ratio decreases, the fracture deviates more from the idealized parallel plate geometry. Varying the fractal dimension has a second order effect on the total volume flow rate.
These simulations imply that the predictions of the cubic law become progressively worse as the surfaces of a rough-walled fracture are brought closer together. When the surfaces just touch, the actual flow rate is about 90% of that predicted by the cubic law. When the surfaces are pressed together with a contact area of about 20%, then the actual flow rate is 40–60% of that predicted by the cubic law. These results represent an upper bound to the agreement with the cubic law for fractures in the upper crust.
