SUMMARYThe chemical-dissolution front propagation problem exists ubiquitously in many scientific and engineering fields. To solve this problem, it is necessary to deal with a coupled system between porosity, porefluid pressure and reactive chemical-species transport in fluid-saturated porous media. Because there was confusion between the average linear velocity and the Darcy velocity in the previous study, the governing equations and related solutions of the problem are re-derived to correct this confusion in this paper. Owing to the morphological instability of a chemical-dissolution front, a numerical procedure, which is a combination of the finite element and finite difference methods, is also proposed to solve this problem. In order to verify the proposed numerical procedure, a set of analytical solutions has been derived for a benchmark problem under a special condition where the ratio of the equilibrium concentration to the solid molar density of the concerned chemical species is very small. Not only can the derived analytical solutions be used to verify any numerical method before it is used to solve this kind of chemical-dissolution front propagation problem but they can also be used to understand the fundamental mechanisms behind the morphological instability of a chemical-dissolution front during its propagation within fluid-saturated porous media. The related numerical examples have demonstrated the usefulness and applicability of the proposed numerical procedure for dealing with the chemical-dissolution front instability problem within a fluid-saturated porous medium.

Various Green’s functions occurring in Poisson potential field theory can be used to construct non‐orthogonal, non‐compact, continuous wavelets. Such a construction leads to relations between the horizontal derivatives of geophysical field measurements at all heights, and the wavelet transform of the zero height field. The resulting theory lends itself to a number of applications in the processing of potential field data. Some simple, synthetic examples in two dimensions illustrate one inversion approach based upon the maxima of the wavelet transform (multiscale edges). These examples are presented to illustrate, by way of explicit demonstration, the information content of the multiscale edges. We do not suggest that the methods used in these examples be taken literally as a practical algorithm or inversion technique. Rather, we feel that the real thrust of the method is towards physically based, spatially local filtering of geophysical data images using Green’s function wavelets, or compact approximations thereto. To illustrate our first steps in this direction, we present some preliminary results of a 3‐D analysis of an aeromagnetic survey.

In this article, the effect of reactive surface areas associated with different particle shapes on the reactive infiltration instability in a fluid-saturated porous medium is investigated through analytically deriving the dimensionless pore-fluid pressure-gradient of a coupled system between porosity, pore-fluid flow and reactive chemical-species transport within two idealized porous media consisting of spherical and cubic grains respectively. Compared with the critical dimensionless pore-fluid pressure-gradient of the coupled system, the derived dimensionless pore-fluid pressure-gradient can be used to assess the instability of a chemical dissolution front within the fluid-saturated porous medium. The related theoretical analysis has demonstrated that (1) since the shape coefficient of spherical grains is greater than that of cubic grains, the chemical system consisting of spherical grains is more unstable than that consisting of cubic grains, and (2) the instability likelihood of a natural porous medium, which is comprised of irregular grains, is smaller than that of an idealized porous medium, which is comprised of regular spherical grains. To simulate the complicated morphological evolution of a chemical dissolution front in the case of the chemical dissolution system becoming supercritical, a numerical procedure is proposed for solving this kind of problem. The related numerical results have demonstrated that the reactive surface area associated with different particle shapes can have a significant influence on the morphological evolution of an unstable chemical-dissolution front within fluid-saturated porous rocks.

This article is concerned with chemical reactions that occur between two interacting parallel fluid flows using mixing in vertical faults as an example. Mineral precipitation associated with fluid flow in permeable fault zones results in mineralization and chemical reaction (alteration) patterns, which in turn are strongly dependent on interactions between solute advection (controlled by fluid flow rates), solute diffusion/dispersion and chemical kinetics. These interactions can be understood by simultaneously considering two dimensionless numbers, the Damkö hler number and the Z-number. The Damkö hler number expresses the interaction between solute advection (flow rate) and chemical kinetics, while the Z-number expresses the interaction between solute diffusion/dispersion and chemical kinetics. Based on the Damkö hler and Z-numbers, two chemical equilibrium length-scales are defined, dominated by either solute advection or by solute diffusion/dispersion. For a permeable vertical fault zone and for a given solute diffusion/dispersion coefficient, there exist three possible types of chemical reaction patterns, depending on both the flow rate and the chemical reaction rate. These three types are: (i) those dominated by solute diffusion and dispersion resulting in precipitation at the lower tip of a vertical fault and as a thin sliver within the fault, (ii) those dominated by solute advection resulting in precipitation at or above the upper tip of the fault, and (iii) those in which advection and diffusion/dispersion play similar roles resulting in wide mineralization within the fault. Theoretical analysis indicates that there exists both an optimal flow rate and an optimal chemical reaction rate, such that chemical equilibrium following focusing and mixing of two fluids may be attained within the fault zone (i.e. type 3). However, for rapid and parallel flows, such as those resulting from a lithostatic pressure gradient, it is difficult for a chemical reaction to reach equilibrium within the fault zone, if the two fluids are not well mixed before entering the fault zone. Numerical examples are given to illustrate the three possible types of chemical reaction patterns.

SUMMARYTo extend the application range of the distinct element method from a laboratory scale into a large scale such as a geological scale, we need to deal with an upscale issue associated with simulating spontaneous crack generation problems in large-scale quasi-static systems. Toward this direction, three important simulation issues, which may affect the quality of the particle simulation results of a quasi-static system, have been addressed in details in this paper. The first simulation issue is how to determine the particle-scale mechanical properties of a particle from the measured macroscopic mechanical properties of rocks. The second simulation issue is that the fictitious time, rather than the physical time, is used in the particle simulation of a quasi-static problem. The third simulation issue is that the conventional loading procedure used in the distinct element method is conceptually inaccurate, at least from the force propagation point of view. A new loading procedure is proposed to solve the conceptual problem resulting from the third simulation issue. The proposed loading procedure is comprised of two main types of periods, a loading period and a frozen period. Using the proposed loading procedure, the parameter selection problem stemming from the first issue can be somewhat solved. Since the second issue is an inherent one, it is strongly recommended that a particle-size sensitivity analysis of at least two different models, which have the same geometry but different smallest particle sizes, be carried out to confirm the particle simulation result of a large-scale quasi-static system. The related simulation results have demonstrated the usefulness and correctness of the proposed loading procedure for dealing with spontaneous crack generation problems in large-scale quasi-static geological systems.

This paper is concerned with the morphological evolution of three-dimensional chemical dissolution fronts that occur in fluid-saturated porous media. A fully coupled system between porosity, pore-fluid flow and reactive chemical species transport is considered to describe this phenomenon. Using the newly presented concept of the generalized dimensionless pore fluid pressure-gradient, which can be used to represent the interaction between solute advection, solute diffusion, chemical kinetics and the shape factor of the soluble mineral, a theoretical criterion has been established to assess the likelihood of instability at a chemical dissolution front in the reactive transport system. To simulate the chemical dissolution front evolution in a three-dimensional fluid-saturated porous medium, a numerical procedure combining both the finite difference method and the finite element method has been proposed. As the problem belongs to a complex system science problem, a small randomly generated perturbation of porosity is added to the initial porosity of a three-dimensional homogeneous domain to trigger instability of a planar chemical dissolution front during its propagation within the fluid-saturated porous medium. To test the correctness and accuracy of the proposed numerical procedure, a three-dimensional benchmark problem has been constructed and the related analytical solution has been derived. This enables using the proposed numerical procedure for simulating the morphological evolution of a three-dimensional chemical dissolution front from a stable, planar state into an unstable, fingering state. The related numerical results demonstrate that the proposed numerical procedure is useful for, and capable of, simulating the morphological instability of a threedimensional chemical dissolution front within a fluid-saturated porous medium.

SUMMARYNon-equilibrium redox chemical reactions of high orders are ubiquitous in fluid-saturated porous rocks within the crust of the Earth. The numerical modelling of such high-order chemical reactions becomes a challenging problem because these chemical reactions are not only produced strong non-linear source/sink terms for reactive transport equations, but also often coupled with the fluids mixing, heat transfer and reactive mass transport processes. In order to solve this problem effectively and efficiently, it is desirable to reduce the total number of reactive transport equations with strong nonlinear source/sink terms to a minimum in a computational model. For this purpose, the concept of the chemical reaction rate invariant is used to develop a numerical procedure for dealing with fluids mixing, heat transfer and non-equilibrium redox chemical reactions in fluid-saturated porous rocks. Using the proposed concept and numerical procedure, only one reactive transport equation, which is used to describe the distribution of the chemical product and has a strong non-linear source/sink term, needs to be solved for each of the non-equilibrium redox chemical reactions. The original reactive transport equations of the chemical reactants with strong non-linear source/sink terms are turned into the conventional mass transport equations of the chemical reaction rate invariants without any non-linear source/sink terms. A testing example, for some aspects of which the analytical solutions are available, is used to validate the proposed numerical procedure. The related numerical solutions have demonstrated that (1) the proposed numerical procedure is useful and applicable for dealing with the coupled problem between fluids mixing, heat transfer and non-equilibrium redox chemical reactions of high orders in fluid-saturated porous rocks; (2) the interaction between the solute diffusion, solute advection and chemical kinetics is an important mechanism to control distribution patterns of chemical products in an ore-forming process; and (3) if the pore-fluid pressure gradient is lithostatic, it is difficult for the chemical equilibrium to be attained within permeable cracks and geological faults within the crust of the Earth.

S U M M A R YWith the aim of designing signal processing tools that act locally in space upon specific features of a signal, we compare two algorithms to remove or isolate individual anomalies in potential field profiles. The first method, based on multiscale edge analysis, leaves other features in the signal relatively untouched. A second method, based on iterative lateral continuation and subtraction of anomalies, accounts for the influence of adjacent anomalies on one another. This allows a potential field profile to be transformed into a number of single anomaly signals. Each single anomaly can then be individually processed, which considerably simplifies applications such as inversion and signal processing.