Paul A. Witherspoon

Numerical modeling of steam injection for the removal of nonaqueous phase liquids from the subsurface. 1. Numerical formulation

A multidimensional integral finite difference numerical simulator is developed for modeling the steam displacement of nonaqueous phase liquid (NAPL) contaminants in shallow subsurface systems. This code, named STMVOC, considers three flowing phases, gas, aqueous, and NAPL; and three mass components, air, water, and an organic chemical. Interphase mass transfer of the components between any of the phases is calculated by assuming local chemical equilibrium between the phases, and adsorption of the chemical to the soil is included. Heat transfer occurs due to conduction and multiphase convection and includes latent heat effects. A general equation of state is implemented in the code for calculating the thermophysical properties of the NAPL/chemical. This equation of state is primarily based on corresponding states methods of property estimation using a chemicals critical constants. The necessary constants are readily available for several hundred hazardous organic liquid chemicals. In part 2 (Falta et al., this issue), the code is used to simulate two one-dimensional laboratory steam injection experiments and to examine the effect of NAPL properties on the steam displacement process.

Numerical modeling of steam injection for the removal of nonaqueous phase liquids from the subsurface. 2. Code validation and application

The multiphase steam injection simulator developed in part 1 (Falta et al., this issue) is used to simulate two laboratory column steam displacement experiments. In the first simulation, steam is injected into a clean, water-saturated column, while in the second simulation, steam is injected into a column containing both water and separate phase trichloroethylene. In both cases, the numerical results are in good quantitative agreement with the experimental data. Based on the assumption of local chemical equilibrium between the phases, a simple criterion is derived for determining the major mechanism of nonaqueous phase liquid (NAPL)/chemical transport during the steam displacement process. Several one-dimensional simulations of the steam displacement of high-boiling-point NAPLs are discussed. These results are consistent with theoretical predictions and indicate that steam may efficiently displace organic liquids having boiling points substantially greater than that of water.

A finite-element method for coupled stress and fluid flow analysis in fractured rock masses

A variational principle is used in conjunction with the finite-element method to solve the nonlinear coupled field equations of the initial boundary value problem of flow in deformable fractured rock masses. This results in a powerful method for modeling of coupled stress and fluid flow behavior of rocks. Both stress and deformation history for both solid and liquid phases, for arbitrary boundary conditions and within complex geometrical configuration, can be determined. Direct application is to fluid flow problems in hydraulically fractured reservoirs and naturally fractured rocks.

The fractal dimension of pores in sedimentary rocks and its influence on permeability

Abstract Perimeter-area power-law relationship of pores in five sedimentary rocks are estimated from scanning electron micrographs of thin sections. These relationships for the pores of four sandstones were found to lie between 1.43 and 1.49, while that of an Indiana limestone was found to be 1.67. We show how the perimeter-area power-law relationship of pores, along with a pore-size distribution, can be used to estimate the hydraulic permeability. A discussion is given of how the fractal dimension of the pore perimeter derived by Mandelbrot for islands whose boundaries are fractal: P = ϵ D A D/2 , where ϵ is some constant that depends on the length of the measuring grid size and D is the fractal dimension of the pore perimeter, influences permeability.

Theoretical and field studies of coupled hydromechanical behaviour of fractured rocks—1. Development and verification of a numerical simulator

Abstract Analyses of hydromechanical behaviour of fractured rocks requires the use of numerical methods, such as the ROCMAS code developed at LBL. We have developed a new version of this simulator which uses mixed Newton-Raphson linearization within an incremental configuration. This code, that is named ROCMAS II, was verified against: (1) a semi-analytic solution; and (2) against its predecessor ROCMAS which uses a direct iteration linearization scheme. This new coupled phenomenological model has been used successfully in a validation study of a field experiment which is reported in Part 2 [Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.29, 411–419 (1992)]. The verified ROCMAS II with its new feature such as an incremental set-up, the strain-softening and dilating shear, and hyperbolic closure joint model and the new linearization scheme, allows more realistic simulations of a host of rock mechanical problems in saturated rocks. Furthermore, the ROCMAS II set-up provides a basis upon which procedures for general treatment of various kinds of material non-linearity can be built.

Mechanical and hydraulic properties of rocks related to induced seismicity

Abstract Witherspoon, P.A. and Gale, J.E., 1977. Mechanical and hydraulic properties of rocks related to induced seismicity. Eng. Geol., 11(1): 23–55. The mechanical and hydraulic properties of fractured rocks are considered with regard to the role they play in induced seismicity. In many cases, the mechanical properties of fractures determine the stability of a rock mass. The problems of sampling and testing these rock discontinuities and interpreting their non-linear behavior are reviewed. Stick slip has been proposed as the failure mechanism in earthquake events. Because of the complex interactions that are inherent in the mechanical behavior of fractured rocks, there seems to be no simple way to combine the deformation characteristics of several sets of fractures when there are significant perturbations of existing conditions. Thus, the more important fractures must be treated as individual components in the rock mass. In considering the hydraulic properties, it has been customary to treat a fracture as a parallel-plate conduit and a number of mathematical models of fracture systems have adopted this approach. Non-steady flow in fractured systems has usually been based on a two-porosity model, which assumes the primary (intergranular) porosity contributes only to storage and the secondary (fracture) porosity contributes only to the overall conductivity. Using such a model, it has been found that the time required to achieve quasi-steady state flow in a fractured reservoir is one or two orders of magnitude greater than it is in a homogeneous system. In essentially all of this work, the assumption has generally been made that the fractures are rigid. However, it is clear from a review of the mechanical and hydraulic properties that not only are fractures easily deformed but they constitute the main flow paths in many rock masses. This means that one must consider the interaction of mechanical and hydraulic effects. A considerable amount of laboratory and field data is now available that clearly demonstrates this stress-flow behavior. Two approaches have been used in attempting to numerically model such behavior: (1) continuum models, and (2) discrete models. The continuum approach only needs information as to average values of fracture spacing and material properties. But because of the inherent complexity of fractured rock masses and the corresponding decrease in symmetry, it is difficult to develop an equivalent continuum that will simulate the behavior of the entire system. The discrete approach, on the other hand, requires details of the fracture geometry and material properties of both fractures and rock matrix. The difficulty in obtaining such information has been considered a serious limitation of discrete models, but improved borehole techniques can enable one to obtain the necessary data, at least in shallow systems. The possibility of extending these methods to deeper fracture systems needs more investigation. Such data must be considered when deciding whether to use a continuum or discrete model to represent the interaction of rock and fluid forces in a fractured rock system, especially with regard to the problem of induced seismicity. When one is attempting to alter the pressure distribution in a fault zone by injection or withdrawal of fluids, the extent to which this can be achieved will be controlled in large measure by the behavior of the fractures that communicate with the borehole. Since this is essentially a point phenomenon, i.e., the changes will propagate from a relatively small region around the borehole, the use of a discrete model would appear to be preferable.

Large-scale hydraulic conductivity measurements in fractured granite

The large-scale hydraulic conductivity experiment at Stripa, Sweden, was an attempt to produce a macromeasurement of the average hydraulic conductivity of approximately 200,000 m/sup 3/ of low-permeability fractured granite. Groundwater seepage into a 33 m long, 5 m dia drift was measured as the net moisture pickup of a ventilation system. Water pressures were monitored at 90 locations in the rock mass. The experiment was designed to treat the rock as a porous medium. Analysis of test results indicates a behavior approximating radial flow in a porous medium. Tests made at three different drift air temperatures yielded very similar results. Computations indicate that the average hydraulic conductivity of the monitored rock mass, exclusive of a zone of lower conductivity immediately surrounding the drift, is approximately 9.8 x 10/sup -11/ m/sec. 7 references, 8 figures.

Characterization of leaky faults: Study of water flow in aquifer-fault-aquifer systems

Leaky faults provide important flow paths for fluids to move underground. It is often necessary to characterize such faults in engineering projects such as deep well injection of waste liquids, underground natural gas storage, and radioactive waste isolation. To provide this characterization, analytical solutions are presented for groundwater flow through saturated aquifer-fault-aquifer systems assuming that both the aquifers and the fault are homogeneous and that the fault has an insignificant effect on aquifer hydraulic properties. Three different conditions are considered: (1) drawdown in the unpumped aquifer is negligibly small; (2) drawdown in the unpumped aquifer is significant, and the two aquifers have the same diffusivity; and (3) drawdown in the unpumped aquifer is significant, and the two aquifers have different diffusivities. Methods are presented to determine the fault transmissivity from pumping test data. [References: 12]

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.

CHARACTERIZATION OF LEAKY FAULTS : STUDY OF AIR FLOW IN FAULTED VADOSE ZONES

Characterization of leaky faults in vadose zones of large thickness is very important for engineering problems such as high-level nuclear waste disposal. This paper presents analytical solutions for air flow in faulted vadose zones of large or small thicknesses. The focus is on those sites where the fault zone is more permeable than its adjacent rock matrix. On the basis of the assumptions of a two-dimensional air flow in the matrix and a one-dimensional air flow in the fault zone, analytical solutions are presented for a sinusoidal atmospheric pressure variation. A procedure is then provided for extending the application of the solutions to an arbitrary atmospheric pressure variation. The solutions can help determine air permeability of leaky faults in the vadose zone.