Finite element modelling of temperature gradient driven rock alteration and mineralization in porous rock masses

We present finite element simulations of temperature gradient driven rock alteration and mineralization in fluid saturated porous rock masses. In particular, we explore the significance of production/annihilation terms in the mass balance equations and the dependence of the spatial patterns of rock alteration upon the ratio of the roll over time of large scale convection cells to the relaxation time of the chemical reactions. Special concepts such as the gradient reaction criterion or rock alteration index (RAI) are discussed in light of the present, more general theory. In order to validate the finite element simulation, we derive an analytical solution for the rock alteration index of a benchmark problem on a two-dimensional rectangular domain. Since the geometry and boundary conditions of the benchmark problem can be easily and exactly modelled, the analytical solution is also useful for validating other numerical methods, such as the finite difference method and the boundary element method, when they are used to dear with this kind of problem. Finally, the potential of the theory is illustrated by means of finite element studies related to coupled flow problems in materially homogeneous and inhomogeneous porous rock masses. (C) 1998 Elsevier Science S.A. All rights reserved.

Truncation errors in finite difference models for solute transport equation with first-order reaction

The truncation errors associated with finite difference solutions of the advection-dispersion equation with first-order reaction are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation and a temporally and spatially weighted parametric approach is used for differentiating among the various finite difference schemes. The numerical truncation errors are defined using Peclet and Courant numbers and a new Sink/Source dimensionless number. It is shown that all of the finite difference schemes suffer from truncation errors. Tn particular it is shown that the Crank-Nicolson approximation scheme does not have second order accuracy for this case. The effects of these truncation errors on the solution of an advection-dispersion equation with a first order reaction term are demonstrated by comparison with an analytical solution. The results show that these errors are not negligible and that correcting the finite difference scheme for them results in a more accurate solution. (C) 1999 Elsevier Science B.V. All rights reserved.

Finite element analysis of flow patterns near geological lenses in hydrodynamic and hydrothermal systems

We use theoretical and numerical methods to investigate the general pore-fluid flow patterns near geological lenses in hydrodynamic and hydrothermal systems respectively. Analytical solutions have been rigorously derived for the pore-fluid velocity, stream function and excess pore-fluid pressure near a circular lens in a hydrodynamic system. These analytical solutions provide not only a better understanding of the physics behind the problem, but also a valuable benchmark solution for validating any numerical method. Since a geological lens is surrounded by a medium of large extent in nature and the finite element method is efficient at modelling only media of finite size, the determination of the size of the computational domain of a finite element model, which is often overlooked by numerical analysts, is very important in order to ensure both the efficiency of the method and the accuracy of the numerical solution obtained. To highlight this issue, we use the derived analytical solutions to deduce a rigorous mathematical formula for designing the computational domain size of a finite element model. The proposed mathematical formula has indicated that, no matter how fine the mesh or how high the order of elements, the desired accuracy of a finite element solution for pore-fluid flow near a geological lens cannot be achieved unless the size of the finite element model is determined appropriately. Once the finite element computational model has been appropriately designed and validated in a hydrodynamic system, it is used to examine general pore-fluid flow patterns near geological lenses in hydrothermal systems. Some interesting conclusions on the behaviour of geological lenses in hydrodynamic and hydrothermal systems have been reached through the analytical and numerical analyses carried out in this paper.

Both CD4(+) and CD8(+) lymphocytes reduce the severity of tissue lesions in murine systemic candidiasis, and CD4(+) cells also demonstrate strain-specific immunopathological effects

The role of T lymphocytes in host responses to sublethal systemic infection with Candida albicans was evaluated by mAb depletion of CD4(+) and CD8(+) cells from BALB/c and CBA/CaH mice, which develop mild and severe tissue damage, respectively. Depletion of CD4(+) lymphocytes from BALB/c mice markedly increased tissue damage, but did not alter the course of infection. In CBA/CaH mice, depletion of CD4+ cells abrogated tissue destruction in both brain and kidney at day 4 after infection, and significantly decreased fungal colonization in the brain. However, the severity of tissue lesions increased relative to controls from day 8 onwards. A small increase in tissue damage was evident in both mouse strains after depletion of CD8(+) cells. There were no major differences between days 4 end 8 after infection in cDNA cytokine profiles of CD4(+) lymphocytes from either BALB/c or CBA/CaH mice. After passive transfer into infected syngeneic recipients, spleen cells from infected CBA/CaH mice markedly increased tissue damage when compared to controls, and also caused a significant increase in fungal colonization in the brain. A similar transfer in BALB/c mice increased the number of inflammatory cells in and around the lesions, but had no effect on the fungal burden in brain and kidney. The data demonstrate that both CD4(+) and CD8(+) lymphocytes contribute to the reduction of tissue damage after systemic infection with C. albicans, and that the development and expression of CD4(+) lymphocyte effector function is influenced by the genetic background of the mouse.

Finite element modelling of reactive mass transport problems in fluid-saturated porous media

We use the finite element method to solve reactive mass transport problems in fluid-saturated porous media. In particular, we discuss the mathematical expression of the chemical reaction terms involved in the mass transport equations for an isothermal, non-equilibrium chemical reaction. It has turned out that the Arrhenius law in chemistry is a good mathematical expression for such non-equilibrium chemical reactions especially from the computational point of view. Using the finite element method and the Arrhenius law, we investigate the distributions of PH (i.e. the concentration of H+) and the relevant reactive species in a groundwater system. Although the main focus of this study is on the contaminant transport problems in groundwater systems, the related numerical techniques and principles are equally applicable to the orebody formation problems in the geosciences. Copyright (C) 1999 John Wiley & Sons, Ltd.

The crack tip strain field of AISI 4340 - Part I - Measurement technique

This is the first paper in a study on the influence of the environment on the crack tip strain field for AISI 4340. A stressing stage for the environmental scanning electron microscope (ESEM) was constructed which was capable of applying loads up to 60 kN to fracture-mechanics samples. The measurement of the crack tip strain field required preparation (by electron lithography or chemical etching) of a system of reference points spaced at similar to 5 mu m intervals on the sample surface, loading the sample inside an electron microscope, image processing procedures to measure the displacement at each reference point and calculation of the strain field. Two algorithms to calculate strain were evaluated. Possible sources of errors were calculation errors due to the algorithm, errors inherent in the image processing procedure and errors due to the limited precision of the displacement measurements. Estimation of the contribution of each source of error was performed. The technique allows measurement of the crack tip strain field over an area of 50 x 40 mu m with a strain precision better than +/- 0.02 at distances larger than 5 mu m from the crack tip. (C) 1999 Kluwer Academic Publishers.

The crack tip strain field of AISI 4340 - Part II - Experimental measurements

Crack tip strain maps have been measured for AISI 4340 high strength steel. No significant creep was observed. The measured values of CTOD were greater than expected from the HRR model. Crack tip branching was observed in every experiment. The direction of crack branching was in the same direction as a major ridge'' of epsilon(yy) strain, which in turn was in the same direction as predicted by the HRR model. Furthermore, the measured magnitudes of the epsilon(y)y strain in this same direction were in general greater than the values predicted by the HRR model. This indicates more plasticity in the crack tip region than expected from the HRR model. This greater plasticity could be related to the larger than expected CTOD values. The following discrepancies between the measured strain fields for AISI 4340 and the HRR predictions are noteworthy: (1) The crack branching. (2) Values of CTOD significantly higher than predicted by HRR. (3) The major ridge'' of epsilon(yy) strain an angle of about 60 degrees with the direction of overall propagation of the fatigue precrack, in which the measured magnitudes of the epsilon(yy) strain were greater than the values predicted by the HRR model. (4) Asymmetric shape of the plastic zone as measured by the epsilon(yy) strain. (5) Values of shear strain gamma(xy) significantly higher than predicted by the HRR model. (C) 1999 Kluwer Academic Publishers.

The crack tip strain field of AISI 4340 - Part III - Hydrogen influence

This paper studied the influence of hydrogen and water vapour environments on the plastic behaviour in the vicinity of the crack tip for AISI 4340. Hydrogen and water vapour (at a pressure of 15 Torr) significantly increased the crack tip opening displacement. The crack tip strain distribution in 15 Torr hydrogen was significantly different to that measured in vacuum. In the presence of sufficient hydrogen, the plastic zone was larger, was elongated in the direction of crack propagation and moreover there was significant creep. These observations support the hydrogen enhanced localised plasticity model for hydrogen embrittlement in this steel. The strain distribution in the presence of water vapour also suggests that SCC in AISI 4340 occurs via the hydrogen enhanced localised plasticity mechanism. (C) 1999 Kluwer Academic Publishers.

Formal derivation of finite state machines for class testing

Previous work on generating state machines for the purpose of class testing has not been formally based. There has also been work on deriving state machines from formal specifications for testing non-object-oriented software. We build on this work by presenting a method for deriving a state machine for testing purposes from a formal specification of the class under test. We also show how the resulting state machine can be used as the basis for a test suite developed and executed using an existing framework for class testing. To derive the state machine, we identify the states and possible interactions of the operations of the class under test. The Test Template Framework is used to formally derive the states from the Object-Z specification of the class under test. The transitions of the finite state machine are calculated from the derived states and the class's operations. The formally derived finite state machine is transformed to a ClassBench testgraph, which is used as input to the ClassBench framework to test a C++ implementation of the class. The method is illustrated using a simple bounded queue example.

Finite element modelling of dissipative structures for nonequilibrium chemical reactions in fluid-saturated porous media

We use the finite element method to model and predict the dissipative structures of chemical species for a nonequilibrium chemical reaction system in a fluid-saturated porous medium. In particular, we explore the conditions under which dissipative structures of the species may exist in the Brusselator type of nonequilibrium chemical reaction. Since this is the first time the finite element method and related strategies have been used to study the chemical instability problems in a fluid-saturated porous medium, it is essential to validate the method and strategies before they are put into application. For this purpose, we have rigorously derived the analytical solutions for dissipative structures of chemical species in a benchmark problem, which geometrically is a square. Comparison of the numerical solutions with the analytical ones demonstrates that the proposed numerical method and strategy are robust enough to solve chemical instability problems in a fluid-saturated porous medium. Finally, the related numerical results from two application examples indicate that both the regime and the magnitude of pore-fluid flow have significant effects on the nature of the dissipative structures that developed for a nonequilibrium chemical reaction system in a fluid-saturated porous medium. The motivation for this study is that self-organization under conditions of pore-fluid flow in a porous medium is a potential mechanism of the orebody formation and mineralization in the upper crust of the Earth. (C) 2000 Elsevier Science S.A. All rights reserved.

Strain-dependent fluid flow defined through rock mass classification schemes

Strain-dependent hydraulic conductivities are uniquely defined by an environmental factor, representing applied normal and shear strains, combined with intrinsic material parameters representing mass and component deformation moduli, initial conductivities, and mass structure. The components representing mass moduli and structure are defined in terms of RQD (rock quality designation) and RMR (rock mass rating) to represent the response of a whole spectrum of rock masses, varying from highly fractured (crushed) rock to intact rock. These two empirical parameters determine the hydraulic response of a fractured medium to the induced-deformations The constitutive relations are verified against available published data and applied to study one-dimensional, strain-dependent fluid flow. Analytical results indicate that both normal and shear strains exert a significant influence on the processes of fluid flow and that the magnitude of this influence is regulated by the values of RQD and RMR.

Finite element analysis of heat transfer and mineralization in layered hydrothermal systems with upward throughflow

We use the finite element method to solve the coupled problem between convective pore-fluid flow, heat transfer and mineralization in layered hydrothermal systems with upward throughflow. In particular, we present the improved rock alteration index (IRAI) concept for predicting the most probable precipitation and dissolution regions of gold (Au) minerals in the systems. To validate the numerical method used in the computation, analytical solutions to a benchmark problem have been derived. After the numerical method is validated, it is used to investigate the pattern of pore-fluid Aom, the distribution of temperature and the mineralization pattern of gold minerals in a layered hydrothermal system with upward throughflow. The related numerical results have demonstrated that the present concept of IRAI is useful and applicable for predicting the most probable precipitation and dissolution regions of gold (Au) minerals in hydrothermal systems. (C) 2000 Elsevier Science S.A. All rights reserved.

Numerical modelling of single-layer folding: clarification of an issue regarding the possible effect of computer codes and the influence of initial irregularities

The influence of initial perturbation geometry and material propel-ties on final fold geometry has been investigated using finite-difference (FLAC) and finite-element (MARC) numerical models. Previous studies using these two different codes reported very different folding behaviour although the material properties, boundary conditions and initial perturbation geometries were similar. The current results establish that the discrepancy was not due to the different computer codes but due to the different strain rates employed in the two previous studies (i.e. 10(-6) s(-1) in the FLAC models and 10(-14) s(-1) in the MARC models). As a result, different parts of the elasto-viscous rheological field were bring investigated. For the same material properties, strain rate and boundary conditions, the present results using the two different codes are consistent. A transition in Folding behaviour, from a situation where the geometry of initial perturbation determines final fold shape to a situation where material properties control the final geometry, is produced using both models. This transition takes place with increasing strain rate, decreasing elastic moduli or increasing viscosity (reflecting in each case the increasing influence of the elastic component in the Maxwell elastoviscous rheology). The transition described here is mechanically feasible but is associated with very high stresses in the competent layer (on the order of GPa), which is improbable under natural conditions. (C) 2000 Elsevier Science Ltd. All rights reserved.

Finite element modelling of three-dimensional convection problems in fluid-saturated porous media heated from below

We use the finite element method to model three-dimensional convective pore-fluid flow in fluid-saturated porous media when they are heated from below. In particular, we employ the particle-tracking technique to mimic the trajectories of particles in three-dimensional fluid flow problems. The related numerical results demonstrated that: (1) The progressive asymptotic approach procedure, which was previously developed for the finite element modelling of two-dimensional convective pore-fluid flow problems, is equally applicable to the finite element modelling of three-dimensional convective pore-fluid flow in fluid-saturated porous media heated from below. (2) The perturbation of gravity at different planes has a significant effect on the pattern of three-dimensional convective pore-fluid flow and therefore, may influence the pattern of orebody formation and mineralization in three-dimensional hydrothermal systems. Copyright (C) 2001 John Wiley & Sons, Ltd.

Finite element modeling of fluid-rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins

In order to use the finite element method for solving fluid-rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins effectively and efficiently, we have presented, in this paper, the new concept and numerical algorithms to deal with the fundamental issues associated with the fluid-rock interaction problems. These fundamental issues are often overlooked by some purely numerical modelers. (1) Since the fluid-rock interaction problem involves heterogeneous chemical reactions between reactive aqueous chemical species in the pore-fluid and solid minerals in the rock masses, it is necessary to develop the new concept of the generalized concentration of a solid mineral, so that two types of reactive mass transport equations, namely, the conventional mass transport equation for the aqueous chemical species in the pore-fluid and the degenerated mass transport equation for the solid minerals in the rock mass, can be solved simultaneously in computation. (2) Since the reaction area between the pore-fluid and mineral surfaces is basically a function of the generalized concentration of the solid mineral, there is a definite need to appropriately consider the dependence of the dissolution rate of a dissolving mineral on its generalized concentration in the numerical analysis. (3) Considering the direct consequence of the porosity evolution with time in the transient analysis of fluid-rock interaction problems; we have proposed the term splitting algorithm and the concept of the equivalent source/sink terms in mass transport equations so that the problem of variable mesh Peclet number and Courant number has been successfully converted into the problem of constant mesh Peclet and Courant numbers. The numerical results from an application example have demonstrated the usefulness of the proposed concepts and the robustness of the proposed numerical algorithms in dealing with fluid-rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins. (C) 2001 Elsevier Science B.V. All rights reserved.