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Through using rigorously mathematical deductions, this article derives analytical solutions for chemical dissolution‐front instability (CDFI) problems, in which radially divergent flow is involved within fluid‐saturated porous media. Since the acid injection‐well to be considered is a circle of nonzero radius in the horizontal plane, a polar coordinate system is used to describe the coupled mathematical governing equations of the CDFI problem. To facilitate deriving analytical solutions, an appropriate mathematical transform is used to convert the coupled mathematical governing equations from a dimensional form into a dimensionless one. This enables the generalized linear stability approach to be utilized for deriving both analytical base and perturbation solutions for the CDFI problem involving radially divergent flow within a fluid‐saturated porous medium. Consequently, a theoretical criterion, which is usable for assessing the CDFI in the chemical dissolution system (CDS), is further established from the corresponding analytical perturbation solutions. The related theoretical results have demonstrated that: (1) the proposed theoretical criterion is useful and applicable for assessing the CDFI associated with radially divergent flow within fluid‐saturated porous media; (2) the permeability ratio between the downstream and upstream regions can affect significantly the critical Peclet number of a CDS associated with radially divergent flow; (3) the initial porosity can have remarkable effects on the critical Peclet number of a CDS associated with radially divergent flow; and (4) the difference between the final and initial porosities can also affect significantly the critical Peclet number of a CDS associated with radially divergent flow in the fluid‐saturated porous medium. [ABSTRACT FROM AUTHOR]

This paper deals with three fundamental issues associated with theoretical analyses of reactive infiltration instability (RII) problems in fluid‐saturated porous media. The first fundamental issue is to determine the spatial shapes of chemical dissolution‐fronts in the limit case of the mineral dissolution ratio approaching zero in both conventional time (daily life time) and unconventional time (abstract time) domains. The chemical dissolution‐front is commonly represented by the spatial shape of the porosity profile. The second fundamental issue is to conduct theoretical analyses of the RII problems associated with the mineral dissolution ratio approaching zero through directly solving dimensional mathematical governing equations in the conventional time domain. The third fundamental issue is to carry out theoretical analyses of the RII problems associated with the mineral dissolution ratio approaching zero through solving dimensionless mathematical governing equations in the unconventional time domain. Through purely mathematical deductions, it has been proven that: (1) the spatially sharp shape of a chemical dissolution‐front, which is theoretically predicted either at a much smaller timescale than the dissolution timescale associated with the daily life time or at a much larger timescale than the dissolution timescale associated with the daily life time, can be observed in the daily life time domain; (2) the theoretical instability criterion of an RII problem can be established directly at three different length‐scales in the daily life time domain; and (3) although the spatial shape of a chemical dissolution‐front at the dissolution timescale associated with the daily life time cannot be observed in the daily life time domain, the theoretical instability criterion of an RII problem at the dissolution timescale associated with the daily life time can be established when a physically consistent mathematical transform is used in the theoretical analysis. [ABSTRACT FROM AUTHOR]

Purpose: The objective of this paper is to establish a solution strategy for obtaining dual solutions, namely trivial (conventional) and nontrivial (unconventional) solutions, of coupled pore-fluid flow and chemical dissolution problems in heterogeneous porous media. Design/methodology/approach: Through applying a perturbation to the pore-fluid velocity, original governing partial differential equations of a coupled pore-fluid flow and chemical dissolution problem in heterogeneous porous media are transformed into perturbed ones, which are then solved by using the semi-analytical finite element method. Through switching off and on the applied perturbation terms in the resulting perturbed governing partial differential equations, both the trivial and nontrivial solutions can be obtained for the original governing partial differential equations of the coupled pore-fluid flow and chemical dissolution problem in fluid-saturated heterogeneous porous media. Findings: When a coupled pore-fluid flow and chemical dissolution system is in a stable state, the trivial and nontrivial solutions of the system are identical. However, if a coupled pore-fluid flow and chemical dissolution system is in an unstable state, then the trivial and nontrivial solutions of the system are totally different. This recognition can be equally used to judge whether a coupled pore-fluid flow and chemical dissolution system involving heterogeneous porous media is in a stable state or in an unstable state. The proposed solution strategy can produce dual solutions for simulating coupled pore-fluid flow and chemical dissolution problems in fluid-saturated heterogeneous porous media. Originality/value: A solution strategy is proposed to obtain the nontrivial solution, which is often overlooked in the computational simulation of coupled pore-fluid flow and chemical dissolution problems in fluid-saturated heterogeneous porous media. The proposed solution strategy provides a useful way for understanding the underlying dynamic mechanisms of the chemical damage effect associated with the stability of structures that are built on soil foundations. [ABSTRACT FROM AUTHOR]

This paper deals with how to implement perturbations in the computational simulations of chemical dissolution‐front instability (CDFI) problems in fluid‐saturated porous media. On the basis of theoretical analysis, it is found that the application of a perturbation to the chemical dissolution front is equivalent to the application of an alternative perturbation to the dimensionless pore‐fluid normal velocity (relative to the planar chemical dissolution front) in the chemical dissolution zone, where the chemical dissolution front is located. This avoids the difficulty to find the spatial coordinates of the chemical dissolution front locations in the computational simulations of CDFI problems. Based on this new finding, a novel algorithm for implementing perturbations in the computational simulations of CDFI problems is proposed. The key point of the proposed algorithm is that the perturbed pore‐fluid normal velocity (relative to the planar chemical dissolution front) is used to directly replace the original pore‐fluid normal velocity (relative to the planar chemical dissolution front) in the related mathematical governing equations (MGEs), so that the proposed algorithm works for the porosity‐velocity‐concentration scheme when it is used to solve CDFI problems in fluid‐saturated porous media. In addition, the related theoretical analysis in this study has answered the previous unanswered question why the application of a perturbation to porosity works in the porosity‐pressure‐concentration scheme but does not work in the porosity‐velocity‐concentration scheme for solving the same CDFI problems in fluid‐saturated porous media. Through solving two illustrative examples with two different distributions of initial porosity, in which one is homogeneous and another is heterogeneous in the chemical dissolution system, the validity and usefulness of the proposed algorithm for implementing perturbations in the computational simulations of CDFI problems have been demonstrated. [ABSTRACT FROM AUTHOR]

This article presents an accurate porosity‐velocity‐concentration approach, in which porosity, pore‐fluid velocity and the concentration of dissolvable substances in the pore fluid are selected as four primary unknown variables for solving reactive mass transport problems involving chemical dissolution in fluid‐saturated porous media with arbitrarily initial porosity distributions. The first advantage of using the proposed approach is that since pore‐fluid velocity, instead of pore‐fluid pressure, is selected as the primary unknown variable to describe the pore‐fluid flow process, the pore‐fluid velocity obtained from the proposed approach is more accurate than that obtained from the numerical simulation, in which pore‐fluid pressure is selected as the primary unknown variable to describe the pore‐fluid flow. The second advantage of using the proposed approach is that because the property matrices of a four‐node rectangular element are precisely calculated in a purely mathematical way, the overall accuracy of numerical solutions can be ensured. After the proposed approach is verified by a benchmark problem, it has been applied for solving reactive mass transport problems involving chemical dissolution in fluid‐saturated porous media with three different kinds of initial porosity distributions. It has been demonstrated that: (1) the proposed approach can produce highly‐accurate numerical solutions for solving reactive mass transport problems involving chemical dissolution in fluid‐saturated porous media with arbitrarily initial porosity distributions; (2) the initial porosity distribution in a porous medium can have remarkable effects on the reactive mass transport process in the porous medium; (3) the porosity and dimensionless concentration fronts propagate from the entrance to the exit of the problem domain, which is identical to the pore‐fluid flow direction. [ABSTRACT FROM AUTHOR]

Reaction-infiltration instability, in which chemical reactions can dissolve minerals and therefore create preferential pore-fluid flow channels in fluid-saturated rocks, may play an important role in controlling groundwater quality in groundwater hydrology. Although this topic has been studied for many years, there is a recent debate, which says that the use of large-density asymptotics in the previous studies is invalid. However, there is a crucial conceptual mistake in this debate, which leads to results and conclusions that are inconsistent with the fundamental laws of physics. It is well known that in terms of distance, time and velocity, there are only two independent variables. But they are treated as three independent variables, a procedure that is the main source of the physically unrealistic results and conclusions in the debate. In this paper, we will discuss the results and conclusions related to the debate, with emphasis on the issues leading to the corresponding errors. In particular, we demonstrate that there is an unappreciated constraint condition between the dimensional/dimensionless distance, time and velocity in the debate. By using this constraint condition, it can be confirmed that as the ratio of the reactant concentration in the incoming fluid stream to the mineral concentration approaches zero, the dimensionless transport parameter, H , automatically approaches infinity. Therefore, it is further confirmed that the previous work conducted by Chadam and others remains valid. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the theoretical aspects of chemical dissolution-front instability problems in two-dimensional fluid-saturated porous media under non-isothermal conditions. In the case of the mineral dissolution ratio (that is defined as the ratio of the dissolved-mineral equilibrium concentration in the pore fluid to the molar concentration of the dissolvable mineral in the solid matrix of the fluid-saturated porous medium) approaching zero, the corresponding critical condition has been mathematically derived when temperature variation effects are considered. As a complementary tool, the computational simulation method is used to simulate the morphological evolution of chemical dissolution fronts in two-dimensional fluid-saturated porous media under non-isothermal conditions. The related theoretical and numerical results have demonstrated that: (i) a temperature increase in a non-isothermal chemical dissolution system can have some influence on the propagation speed of the planar chemical dissolution front in the system. Generally, the chemical dissolution front in the non-isothermal chemical dissolution system propagates slower than that in the counterpart isothermal chemical dissolution system when the temperature of the non-isothermal chemical dissolution system is higher than that of the counterpart isothermal chemical dissolution system; (ii) a temperature increase in the non-isothermal chemical dissolution system can stabilize the chemical dissolution front propagating in the system, because it can cause a decrease in the Zhao number of the system but does not affect the critical Zhao number of the system; and (iii) the temperature gradient in the upstream direction of a chemical dissolution front is smaller than that in the downstream direction of the chemical dissolution front when the non-isothermal chemical dissolution system is supercritical. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Numerical modeling has now become an indispensable tool for investigating the fundamental mechanisms of toxic nonaqueous phase liquid (NAPL) removal from contaminated groundwater systems. Because the domain of a contaminated groundwater system may involve irregular shapes in geometry, it is necessary to use general quadrilateral elements, in which two neighbor sides are no longer perpendicular to each other. This can cause numerical errors on the computational simulation results due to mesh discretization effect. After the dimensionless governing equations of NAPL dissolution problems are briefly described, the propagation theory of the mesh discretization error associated with a NAPL dissolution system is first presented for a rectangular domain and then extended to a trapezoidal domain. This leads to the establishment of the finger-amplitude growing theory that is associated with both the corner effect that takes place just at the entrance of the flow in a trapezoidal domain and the mesh discretization effect that occurs in the whole NAPL dissolution system of the trapezoidal domain. This theory can be used to make the approximate error estimation of the corresponding computational simulation results. The related theoretical analysis and numerical results have demonstrated the following: (1) both the corner effect and the mesh discretization effect can be quantitatively viewed as a kind of small perturbation, which can grow in unstable NAPL dissolution systems, so that they can have some considerable effects on the computational results of such systems; (2) the proposed finger-amplitude growing theory associated with the corner effect at the entrance of a trapezoidal domain is useful for correctly explaining why the finger at either the top or bottom boundary grows much faster than that within the interior of the trapezoidal domain; (3) the proposed finger-amplitude growing theory associated with the mesh discretization error in the NAPL dissolution system of a trapezoidal domain can be used for quantitatively assessing the correctness of computational simulations of NAPL dissolution front instability problems in trapezoidal domains, so that we can ensure that the computational simulation results are controlled by the physics of the NAPL dissolution system, rather than by the numerical artifacts. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper primarily deals with the computational aspects of chemical dissolution-front instability problems in two-dimensional fluid-saturated porous media under non-isothermal conditions. After the dimensionless governing partial differential equations of the non-isothermal chemical dissolution-front instability problem are briefly described, the formulation of a computational procedure, which contains a combination of using the finite difference and finite element method, is derived for simulating the morphological evolution of chemical dissolution fronts in the non-isothermal chemical dissolution system within two-dimensional fluid-saturated porous media. To ensure the correctness and accuracy of the numerical solutions, the proposed computational procedure is verified through comparing the numerical solutions with the analytical solutions for a benchmark problem. As an application example, the verified computational procedure is then used to simulate the morphological evolution of chemical dissolution fronts in the supercritical non-isothermal chemical dissolution system. The related numerical results have demonstrated the following: (1) the proposed computational procedure can produce accurate numerical solutions for the planar chemical dissolution-front propagation problem in the non-isothermal chemical dissolution system consisting of a fluid-saturated porous medium; (2) the Zhao number has a significant effect not only on the dimensionless propagation speed of the chemical dissolution front but also on the distribution patterns of the dimensionless temperature, dimensionless pore-fluid pressure, and dimensionless chemical-species concentration in a non-isothermal chemical dissolution system; (3) once the finger penetrates the whole computational domain, the dimensionless pore-fluid pressure decreases drastically in the non-isothermal chemical dissolution system. [ABSTRACT FROM AUTHOR]

Highlights: [•] To derive analytical solutions for NAPL dissolution problems with radial flow. [•] Three key factors can affect the propagating speed of an NAPL dissolution front. [•] Three key factors can also affect the location and breakthrough time of an NAPL dissolution front. [•] The Zhao number can be used to represent the overall hydrodynamic characteristic of a NAPL dissolution system. [•] Several key factors can also affect the distribution pattern and evolution of the NAPL dissolution front. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to investigate the effects of different domain shapes in general and trapezoidal domain shape in particular on the morphological evolution of nonaqueous phase liquid (NAPL) dissolution fronts in two-dimensional fluid-saturated porous media. After the governing equations of NAPL dissolution problems are briefly described, the numerical procedure consisting of a combination of the finite element and finite difference methods is used to solve these equations. The related numerical simulation results have demonstrated that: (1) domain shapes have a significant effect on both the propagating speed and the morphological evolution pattern of a NAPL dissolution front in the fluid-saturated porous medium; (2) an increase in the divergent angle of a trapezoidal domain can lead to a decrease in the propagating speed of the NAPL dissolution front; (3) the morphological evolution pattern of the NAPL dissolution front in a rectangular domain is remarkably different from that in a trapezoidal domain of a large divergent angle; (4) for a rectangular domain, the simplified dispersion model, which is commonly used in the theoretical analysis and numerical simulation, is valid for solving NAPL dissolution instability problems in fluid-saturated porous media; and (5) compared with diverging flow (when the trapezoidal domain is inclined outward), converging flow (when the trapezoidal domain is inclined inward) can enhance the growth of NAPL fingers, indicating that pump-and-treat systems by extracting contaminated groundwater might enhance NAPL dissolution fingering and lead to less uniform dissolution fronts. [Copyright &y& Elsevier]

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 © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Numerical methods are used to solve double diffusion driven reactive flow transport problems in deformable fluid-saturated porous media. In particular, the temperature dependent reaction rate in the non-equilibrium chemical reactions is considered. A general numerical solution method, which is a combination of the finite difference method in FLAC and the finite element method in FIDAP, to solve the fully coupled problem involving material deformation, pore-fluid flow, heat transfer and species transport/chemical reactions in deformable fluid-saturated porous media has been developed. The coupled problem is divided into two sub-problems which are solved interactively until the convergence requirement is met. Owing to the approximate nature of the numerical method, it is essential to justify the numerical solutions through some kind of theoretical analysis. This has been highlighted in this paper. The related numerical results, which are justified by the theoretical analysis, have demonstrated that the proposed solution method is useful for and applicable to a wide range of fully coupled problems in the field of science and engineering. [ABSTRACT FROM AUTHOR]

• To consider a reactive mass transport problem of two different porosity regions in porous media. • To derive theoretical solutions for the acid concentration in the left region of constant porosity. • To derive numerical solutions for the acid concentration in the right region of variable porosity. Reactive mass transport is a common phenomenon associated with groundwater pollution in the field of groundwater hydrology. For a reactive mass transport problem involving two different porosity regions, in which the porosity of the fluid-saturated porous medium has a constant distribution in the left region and an exponential distribution in the right region, theoretical solutions for both dimensional and dimensionless acid concentrations have been derived mathematically in the left region by using the interface condition substitution strategy. The fundamental idea behind this strategy is that the boundary condition at the entrance of the left region and the mathematical governing equation of the right region at the interface location between the left and right regions can be used to determine two independent constants involved in the theoretical solutions for the dimensional and dimensionless acid concentrations in the left region of the reactive mass transport system. Since both the acid concentration and its first derivative are known at the left boundary (i.e. the interface location) of the right region, it is possible to derive analytical expressions for the numerical solution (e.g. the finite element solution) in the right region by using the point-by-point marching strategy. The related theoretical results have demonstrated that the porosity distribution pattern in the right region of the system can have a significant influence on the dimensionless acid concentration distribution in the whole system, so that it affects not only the chemical dissolution front instability, but also the reactive mass transport process in an underground water system. [ABSTRACT FROM AUTHOR]