The flow of a single-phase fluid through a rough-walled rock fracture is discussed within the context of fluid mechanics. The derivation of the ‘cubic law’ is given as the solution to the Navier-Stokes equations for flow between smooth, parallel plates - the only fracture geometry that is amenable to exact treatment. The various geometric and kinematic conditions that are necessary in order for the Navier-Stokes equations to be replaced by the more tractable lubrication or Hele-Shaw equations are studied and quantified. In general, this requires a sufficiently low flow rate, and some restrictions on the spatial rate of change of the aperture profile. Various analytical and numerical results are reviewed pertaining to the problem of relating the effective hydraulic aperture to the statistics of the aperture distribution. These studies all lead to the conclusion that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation. The tortuosity effect caused by regions where the rock walls are in contact with each other is studied using the Hele-Shaw equations, leading to a simple correction factor that depends on the area fraction occupied by the contact regions. Finally, the predicted hydraulic apertures are compared to measured values for eight data sets from the literature for which aperture and conductivity data were available on the same fracture. It is found that reasonably accurate predictions of hydraulic conductivity can be made based solely on the first two moments of the aperture distribution function, and the proportion of contact area.

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Hydraulic conductivity of rock fractures

Fluid flow through rock joints is commonly described by the parallel plate model where the volume flow rate varies as the cube of the joint aperture. However, deviations from this model are expected because real joint surfaces are rough and contact each other at discrete points. To examine this problem further, a computer simulation of flow between rough surfaces was done. Realistic rough surfaces were generated numerically using a fractal model of surface topography. Pairs of these surfaces were placed together to form a “joint” with a random aperture distribution. Reynolds equation, which describes laminar flow between slightly nonplanar and nonparallel surfaces, was solved on the two-dimensional aperture mesh by the finite-difference method. The solution is the local volume flow rate through the joint. This solution was used directly in the cubic law to get the so-called “hydraulic aperture.” For various surface roughnesses (fractal dimensions) the hydraulic aperture was compared to the mean separation of the surfaces. At large separations the surface topography has little effect. At small separations the flow is tortuous, tending to be channeled through high-aperture regions. The parameter most affecting fluid flow through rough joints is the ratio of the mean separation between the surfaces to the root-mean-square surface height. This parameter describes the distance the surface asperities protrude into the fluid and accounts for most of the disagreement with the parallel plate model. Variations in the fractal dimension produce only a second-order effect on the fluid flow. For the range of joint closures expected during elastic deformation these results show that the actual flow rate between rough surfaces is about 70–90% of that predicted by the parallel plate model.

Fluid flow through rock joints: The effect of surface roughness

https://cyberleninka.org/article/n/544115.pdf

Hydrodynamic lubrication of textured surfaces: A review of modeling techniques and key findings

A model is presented to study the potential use of micro-surface structure in the form of micro pores to improve tribological properties of reciprocating automotive components. The Reynolds equation and the equation of motion are solved simultaneously for a simplified “piston/cylinder” system with surface texturing. The solution provides the time behavior of both the clearance and the friction force between the “piston ring” and “cylinder liner” surfaces. It is shown that surface texturing can efficiently be used to maintain hydrodynamic effects even with nominally parallel surfaces. It is also shown that optimum surface texturing may substantially reduce the friction losses in reciprocating automotive components. Presented at the 56th Annual Meeting Orlando, Florida May 20–24, 2001

Friction-Reducing Surface-Texturing in Reciprocating Automotive Components

contacts is presented in this paper, using a new numerical approach that is simple and robust, capable of handling three-dimensional measured engineering rough surfaces moving at different rolling and sliding velocities. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed and boundary lubrication. In the hydrodynamically lubricated areas the Reynolds equation is used. In the asperity contact areas, where the film thickness is zero, the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. In order to save computing time, a multi-level integration method is used to calculate surface deformation. Sample cases under severe condition show that this approach is capable of analyzing different cases in a full range of l ratio, from infinitely large down to nearly zero (less than 0.03).

A Full Numerical Solution to the Mixed Lubrication in Point Contacts full numerical solution for the mixed elastohydrodynamic lubrication (EHL) in point

An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication

Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces

A numerical procedure for random generation of rough surfaces

The temperature distribution in spur gears operating in a state of thermal equilibrium is solved by using a finite element method. The effects of various dimensionless parameters on bulk temperature are shown. A table is provided which can be used to predict the bulk temperature on gear teeth, once the heat transfer coefficients and frictional heat input is estimated. Theoretical results for estimating heat transfer coefficients and frictional heat are also summarized. Presented as an American Society of Lubrication Engineers paper at the ASLE/ASME Lubrication Conference in Kansas City, Missouri, October 3–5, 1977

Prediction of the Bulk Temperature in Spur Gears Based on Finite Element Temperature Analysis

Closure to “Discussion of ‘Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces’” (1978, ASME J. Lubr. Technol., 101, pp. 229–230)

Nadir Patir

Papers

An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication

Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces

A numerical procedure for random generation of rough surfaces

Prediction of the Bulk Temperature in Spur Gears Based on Finite Element Temperature Analysis

Closure to “Discussion of ‘Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces’” (1978, ASME J. Lubr. Technol., 101, pp. 229–230)