Gudmundur S. Bodvarsson

Lubrication theory analysis of the permeability of rough-walled fractures

Abstract Lubrication theory is used to study the permeability of rough-walled rock fractures. In this approximation, which is valid for low Reynolds numbers and under certain restrictions on the magnitude of the roughness, the Navier-Stokes equations that govern fluid flow are reduced to the more tractable Reynolds equation. An idealized model of a fracture, in which the roughness follows a sinusoidal variation, is studied in detail. This fracture is considered to consist of a random mixture of elements in which the fluid flows either parallel or transverse to the sinusoidal bumps. The overall permeability is then found by a suitable averaging procedure. The results are similar to those found by other researchers from numerical analysis of the Reynolds equation, in that the ratio of the hydraulic aperture to the mean aperture correlates well with the ratio of the mean aperture to the standard deviation of the aperture. Higher-order approximations to the Navier-Stokes equations for flow between sinusoidal walls are then studied, and it is concluded that in order for the lubrication approximation to be valid, the fracture walls must be smooth over lengths on the order of one standard deviation of the aperture, which is much less restrictive a condition than had previously been thought to apply.

A numerical dual‐porosity model with semianalytical treatment of fracture/matrix flow

A new dual-porosity model is developed for single-phase fluid flow in fractured/porous media. Flow is assumed to take place through the fracture network and between the fractures and matrix blocks. The matrix blocks are treated in a lumped parameter manner, with a single average pressure used for each matrix block. Rather than assuming that fracture/matrix flux is proportional to the difference between the fracture pressure and matrix pressure at each point, as is done in the Warren-Root model, the authors use a nonlinear equation which more accurately models the flux over all time regimes, including both early and late times. This flux equation is compared with analytical solutions for spherical blocks with prescribed pressure variations on their boundaries. The nonlinear flux equation is also used as a source/sink term in the numerical simulator TOUGH. The modified code allows more accurate simulations than the conventional Warren-Root method, with a large savings (about 90%) in computational time compared to methods which explicitly discretize the matrix blocks. 33 refs., 7 figs.

Effective Transmissivity of Two-Dimensional Fracture Networks

Many of the sites that have been proposed as potential locations of underground radioactive waste repositories contain fractured rocks. The purpose of this paper is to describe a simple procedure for solving the problem of fluid flow in fractured rock masses, and to demonstrate its use in cases of both saturated and unsaturated flow.

A survey of geothermal reservoir properties

This paper presents results of a literature survey on thermal, hydrological and chemical characteristics of geothermal reservoirs. The data are presented in a table summarizing important fluid and rock parameters. The primary parameters of interest are the permeability, permeability-thickness, porosity, reservoir temperature and concentration of dissolved solids and non-condensible gases. Some preliminary correlations between these parameters are given.

A simple approximate solution for horizontal infiltration in a Brooks-Corey medium

A simple approximate solution is derived for the problem of one-dimensional absorption in a porous medium characterized by the Brooks-Corey equations. A piecewise-linear saturation profile, which satisfies flux continuity at the edge of the tension-saturated zone as well as an integrated form of the Richards equation, is assumed. The predicted sorptivity agrees very well with the results of numerical simulations.

Effective block size for imbibition or absorption in dual‐porosity media

A theoretical study is presented of liquid imbibition and solute absorption from a fracture network into a collection of matrix blocks of varying sizes and shapes. An individual irregularly-shaped matrix block can be modeled with reasonable accuracy using the results for a spherical matrix block, by defining an effective radius {bar a} = 3V/A, where V is the volume of the block and A is its surface area. In the early-time regime, a collection of spherical blocks of different sizes can be replaced by an equivalent spherical block with a radius of a{sub eq} = {sup {minus}1}, where the average is taken on a volumetrically-weighted basis. In the long-time limit, where no equivalent radius can rigorously be defined, asymptotic expressions are derived for the cumulative uptake as a function of the mean and the standard deviation of the radius distribution function, for both normal and lognormal radius distributions. 18 refs., 3 figs.

Coupled Reservoir-Wellbore Simulation of Geothermal Reservoir Behavior

Abstract The reservoir simulator TOUGH and the wellbore simulator WFSA have been coupled to model flow of geothermal brine in the reservoir as well as in the wellbore. An outline of the structure of the two computer codes is given, together with the relevant equations. A new module, COUPLE, has been written to serve as an interface between TOUGH and WFSA. Two sample problems are given to illustrate the use of the coupled codes. One of these problems compares the results of the new simulation method to those obtained by using the deliverability option in TOUGH. The coupled computing procedure is shown to simulate more accurately the behavior of a geothermal reservoir under exploitation.

Integral method solution for diffusion into a spherical block

Abstract An approximate analytical solution is derived for the problem of a Newtonian fluid infiltrating into a porous spherical block. The fluid in the block is initially at a constant pressure p i , and the pressure at the outer boundary is held at a constant value p o . Using the simple assumption of linear pressure profiles, the instantaneous and cumulative fluxes into the sphere are predicted with surprisingly high accuracy. The solution applies to all other physical processes governed by the same equation, such as heat conduction and chemical diffusion. The solution should be very useful for incorporation into double-porosity models for fractured reservoirs and aquifers.

Permeability of a fracture with cylindrical asperities

The permeability of a rock fracture that is modeled as two smooth, parallel faces propped open by randomly located, uniformly sized cyclindrical asperities, is investigated. The viscous resistance due to the asperities is accounted for by an in-plane permeability coefficient, and a Brinkman-type equation is used to find the velocity distribution across the thickness of the fracture. The resulting simple closed-form expression for the permeability of the fracture reduces to the known result for flow between parallel plates as the concentration of asperities approaches zero, and reduces to the known result for long, parallel cylinders as the distance between the plates goes to infinity. The results also compare very well with experimental results from the literature, obtained from a mechanical model with the same parallel-plate and cylindrical-post geometry.

An integral equation formulation for the unconfined flow of groundwater with variable inlet conditions

We combine an integral equation formulation with a hodograph transformation to solve self-similar problems describing the unconfined flow of groundwater with variable inlet conditions. A class of new semi-analytical solutions is obtained for both rectilinear and radial flow geometries. The solutions are in general agreement with those derived by Barenblatt, although there are some discrepancies for the case of radial flow. The formulation presented provides additional analytical insight, and for computational purposes is simpler than Barenblatts. In addition, the method proposed can be successfully used for the solution of a host of other nonlinear problems that admit self-similarity.