TOUGH2 is a numerical simulator for nonisothermal flows of multicomponent, multiphase fluids in one, two, and three-dimensional porous and fractured media. The chief applications for which TOUGH2 is designed are in geothermal reservoir engineering, nuclear waste disposal, environmental assessment and remediation, and unsaturated and saturated zone hydrology. TOUGH2 was first released to the public in 1991; the 1991 code was updated in 1994 when a set of preconditioned conjugate gradient solvers was added to allow a more efficient solution of large problems. The current Version 2.0 features several new fluid property modules and offers enhanced process modeling capabilities, such as coupled reservoir-wellbore flow, precipitation and dissolution effects, and multiphase diffusion. Numerous improvements in previously released modules have been made and new user features have been added, such as enhanced linear equation solvers, and writing of graphics files. The T2VOC module for three-phase flows of water, air and a volatile organic chemical (VOC), and the T2DM module for hydrodynamic dispersion in 2-D flow systems have been integrated into the overall structure of the code and are included in the Version 2.0 package. Data inputs are upwardly compatible with the previous version. Coding changes were generally kept to a minimum, and were only made as needed to achieve the additional functionalities desired. TOUGH2 is written in standard FORTRAN77 and can be run on any platform, such as workstations, PCs, Macintosh, mainframe and supercomputers, for which appropriate FORTRAN compilers are available. This report is a self-contained guide to application of TOUGH2 to subsurface flow problems. It gives a technical description of the TOUGH2 code, including a discussion of the physical processes modeled, and the mathematical and numerical methods used. Illustrative sample problems are presented along with detailed instructions for preparing input data.

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TOUGH2 User's Guide Version 2

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].

Reliability Engineering and System Safety

Characterizing flow and transport in fractured geological media: A review

A one-dimensional dual-porosity model has been developed for the purpose of studying variably saturated water flow and solute transport in structured soils or fractured rocks. The model involves two overlaying continua at the macroscopic level: a macropore or fracture pore system and a less permeable matrix pore system. Water in both pore systems is assumed to be mobile. Variably saturated water flow in the matrix as well as in the fracture pore system is described with the Richards' equation, and solute transport is described with the convection-dispersion equation. Transfer of water and solutes between the two pore regions is simulated by means of first-order rate equations. The mass transfer term for solute transport includes both convective and diffusive components. The formulation leads to two coupled systems of nonlinear partial differential equations which were solved numerically using the Galerkin finite element method. Simulation results demonstrate the complicated nature of solute leaching in structured, unsaturated porous media during transient water flow. Sensitivity studies show the importance of having accurate estimates of the hydraulic conductivity near the surface of soil aggregates or rock matrix blocks. The proposed model is capable of simulating preferential flow situations using parameters which can be related to physical and chemical properties of the medium.

A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media

On the basis of a review of recent theoretical and experimental studies of flow through fractured rocks, the authors have studied the fluid flow and solute transport in a tight fractured medium in terms of flow through channels of variable aperture. The channels are characterized by an aperture density distribution and a spatial correlation length. Aperture profiles along the channels are statistically generated and compared to laboratory measurements of fracture surfaces. Calculated tracer transport between two points in the fractured media is by way of a number of such channels. Tracer breakthrough curves display features that correspond well with recent data by Moreno et al., which lends support to the validity of our model. Calculated pressure profiles along the channels suggest possible measurements that may be useful in identifying the geometrical characteristics of the channels. Finally, predictions were made for tracer breakthrough curves in the case of single fractures under various degrees of normal stress. These suggest possible laboratory experiments which may be performed to validate this conceptual model.

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Channel model of flow through fractured media

LBL-12297 Preprint . To be published in the Journal of Geophysical Research HYDROMECHANICAL BEHAVIOR OF A DEFORMABLE ROCK FRACTURE SUBJECT TO NORMAL STRESS Y.W. Tsang and P.A. Witherspoon January 1981 TWO-WEEK lOAN COPY This is a library Circulating Copy which may be borrowed for two weeks .. For a personal retention copy, call Tech. Info. Diu is ion, Ext. 6782 Prepared for the U.S. Department of Energy under Contract W-7405-ENG-48

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Hydromechanical behavior of a deformable rock fracture subject to normal stress

Calculations for the flow and solute transport through a single rough-surfaced fracture were carried out. The fracture plane was discretized into a square mesh to which variable apertures were assigned. The spatially varying apertures of each single fracture were generated using geostatistical methods, based on a given aperture probability density distribution and a specified spatial correlation length. Constant head boundary conditions were assumed for the flow in the x direction of a single fracture with no flow boundaries in the y direction. The fluid potential at each node of the discretization mesh was computed and the steady state flow rates between all the nodes were obtained. Our calculations showed that fluid flow occurs predominantly in a few preferred paths. Hence, the large range of apertures in the single fracture gives rise to flow channeling. The solute transport was calculated using a particle tracking method. Both the spatial and time variations of tracer breakthrough results are presented. The spatial variation of tracer transport between a line of injection points and a line of observation points are displayed in contour plots which we labeled “transfer matrix.” Our results indicate that such plots can give information on the spatial correlation length of the heterogeneity in the fracture. The tracer breakthrough curve obtained from a line of point measurements is shown to be controlled by the aperture density distribution and is insensitive to statistical realization and spatial correlation length. These results suggest the importance of making line measurements in the laboratory and the field. Sensitivity of our results on parameter variations was also investigated.

Flow and tracer transport in a single fracture: A stochastic model and its relation to some field observations

Calculations to investigate the effect of path tortuosity and connectivity on fluid flow rate through a single rough fracture were carried out. The flow paths are represented by electrical resistors placed on a two-dimensional grid, and the resistances vary as the inverse of the fracture aperture cubed. The electric current through the circuit bears a one-to-one correspondence to the fluid flow rate. Both fracture apertures derived from measurements and from hypothetical analytic functions were used in a parameter study to investigate the dependence of tortuosity on fracture roughness characteristics. It was found that the more small apertures there are in the aperture distribution, the larger is the effect of tortuosity. When the fraction of contact area between the fracture surfaces rises above 30%, the aperture distributions are invariably large at small apertures, and the effect of fracture roughness and flow path tortuosity depresses flow rate from the value predicted by the parallel plate representation of a fracture by three or more orders of magnitude. The impact of these results on the calculations and measurements in fracture hydrology is discussed.

The Effect of Tortuosity on Fluid Flow Through a Single Fracture

LBL-27449 Preprint Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA I EARTH SCIENCES DIVISION Submitted to Water Resources Research On Two-Phase Relative Permeability and Capillary Pressure of Rough-Walled Rock Fractures K. Pruess and Y.W. Tsang September 1989 r n D :[ \JJn fDr+O I'D I'D v III rul:Z jj II! Q. UI r crn !lJ O r f.D r ru ...,J Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SFOOO98. . ru I::;: r.o. '..D

Y. W. Tsang

Papers

Channel model of flow through fractured media

Hydromechanical behavior of a deformable rock fracture subject to normal stress

Flow and tracer transport in a single fracture: A stochastic model and its relation to some field observations

The Effect of Tortuosity on Fluid Flow Through a Single Fracture

On Two-Phase Relative Permeability and Capillary Pressure of Rough-Walled Rock Fractures