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The use of the asymptotic limit can greatly simplify the theoretical analysis of chemical dissolution front instabilities in fluid-saturated rocks and therefore make it possible to obtain mathematical solutions, which often play a crucial role in understanding the propagation behavior of chemical dissolution fronts in chemical dissolution systems. However, there has been a debate in recent years that the asymptotic limit of the acid dissolution capacity (i.e., the acid dissolution capacity number approaching zero) alone cannot lead to a sharp dissolution front of the Stefan type in the acidization dissolution system, in which the dissolvable minerals of carbonate rocks are chemically dissolved by the injected acid flow. The acid dissolution capacity number is commonly defined as the ratio of the volume of the carbonate rock dissolved by an acid to that of the acid. In this paper, we use four different proof methods, including (i) direct use of the fundamental concepts; (ii) use of the mathematical governing equations of an acidization dissolution system; (iii) use of the different time scaling approach; and (iv) use of a moving coordinate system approach, to demonstrate that the asymptotic limit of the acid dissolution capacity can indeed lead to sharp dissolution fronts of the Stefan type in acidization dissolution systems on a much larger time scale (than the dissolution time scale). Our new finding is that on the reaction time scale, the condition of the conventional time derivative of porosity approaching zero alone can ensure that the acidization dissolution front has a sharp shape of the Stefan type. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Acid dissolution capability plays a considerable role in controlling the propagation of an acid-dissolution front in the carbonate rocks that are saturated by pore fluids. This capability can be represented by a dimensionless number, known as the acid dissolution capability number, by which we mean the quotient of the volume of an acid-dissolved carbonate rock divided by that of the acid itself. This paper aims primarily to investigate why and how the acid dissolution capacity can affect the behaviors of the acid-dissolution front propagation in the carbonate rocks that are saturated by pore fluids. If the acid dissolution capacity number is a nonzero finite number, as in a general case, then the computational simulation method needs to be employed to get numerical solutions for the acid-dissolution system. The relevant computational simulation results have demonstrated that: (1) with an increase in the value of the Zhao number (namely another dimensionless number), which is used to denote the dynamic characteristics of an acid-dissolution system, the acid-dissolution front becomes more unstable in the corresponding supercritical acid-dissolution system. (2) When the acid dissolution capacity number is small enough, the propagating speed of a planar acid-dissolution front in the corresponding subcritical acid-dissolution system is linearly dependent on the acid dissolution capacity number, indicating that the smaller the acid dissolution capacity, the slower the propagating speed of a planar acid-dissolution front in the corresponding subcritical acid-dissolution system. (3) With a decrease in the acid dissolution capacity number, the acid-dissolution front can exhibit more unstable behavior in the corresponding supercritical acid-dissolution system. [ABSTRACT FROM AUTHOR]

Because dissolution of rocks may create and enhance groundwater flow channels, the chemical dissolution-front instability (CDFI) can control the quality of groundwater. This paper presents the theoretical analyses of porosity-permeability relationship effects on the CDFI in water-saturated porous rocks. Since the CDFI in a water-rock reaction system can be assessed by comparing the comprehensive dimensionless dynamic characteristic (CDDC) number with the corresponding critical CDDC number of the geochemical dissolution system, it is necessary to investigate theoretically how different porosity-permeability relationships can affect the CDDC number and critical CDDC number of a water-rock reaction system. With the commonly-used Kozeny–Carman (KC) formula taken as a reference porosity-permeability formula, the permeability variation indicator (PVI), which is defined as the ratio of the permeability obtained from any porosity-permeability formula to that obtained from the KC formula, is proposed to reflect the effect of the porosity-permeability formula on the CDFI in a water-rock reaction system. The theoretical results demonstrated that: (1) since the porosity-permeability formula with a higher PVI can result in a stronger Darcy flow velocity, it may have a significant influence on the CDFI in the water-rock reaction system. (2) With an increase in the PVI of a porosity-permeability formula, there is a decrease in the critical CDDC number of the water-rock reaction system. This means that the porous rock with a higher PVI can enable the CDFI to take place much easier in the water-rock reaction system. (3) The use of the porosity-permeability formula with a higher PVI can also cause an increase in both the dimensionless growth rate of a perturbation and the propagation speed of the chemical dissolution front in the water-rock reaction system. [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]

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]

This paper deals with the theoretical aspects of chemical-dissolution front instability problems in two-dimensional fluid-saturated porous media including medium anisotropic effects. Since a general anisotropic medium can be described as an orthotropic medium in the corresponding principal directions, a two-dimensional orthotropic porous medium is considered to derive the analytical solution for the critical condition, which is used to judge whether or not the chemical dissolution front can become unstable during its propagation. 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 medium permeability anisotropic 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 including medium anisotropic effects. The related theoretical and numerical results demonstrated that: (1) a decrease in the medium anisotropic permeability factor (or ratio), which is defined as the ratio of the principal permeability in the transversal direction to that in the longitudinal direction parallel to the pore-fluid inflow direction, can stabilize the chemical dissolution front so that it becomes more difficult for a planar chemical-dissolution front to evolve into different morphologies in the chemical dissolution system; (2) the medium anisotropic permeability ratio can have significant effects on the morphological evolution of the chemical dissolution front. When the Zhao number of the chemical dissolution system is greater than its critical value, the greater the medium anisotropic permeability ratio, the faster the irregular chemical-dissolution front grows. [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]

This paper presents a numerical procedure for computational modeling of moving interfaces between fluid and porous medium domains. To avoid the direct description of the interface boundary conditions at the interface between the fluid and porous medium domains, Darcy's law is used to simulate fluid flows in both the fluid and porous medium domains. For the purpose of effectively simulating the fluid flow using Darcy's law, the artificial permeability of the fluid domain is used to establish a permeability ratio between the artificial permeability in the fluid domain and the real permeability in the porous medium domain. Using the proposed permeability ratio, the ratio of fluid pressure gradient in the porous medium domain to that in the fluid domain can be appropriately simulated. To verify the proposed numerical procedure, analytical solutions have been derived for a benchmark problem, which can be simulated using the proposed numerical procedure. Comparison of the numerical solutions with the derived analytical solutions has demonstrated the correctness and accuracy of the proposed numerical procedure. Through applying the proposed numerical procedure to several examples associated with the fluid-porous medium interface propagation problems, the related numerical solutions have demonstrated that: (1) the proposed numerical procedure is capable of simulating the morphological instability of the fluid-porous medium interface in a coupled fluid flow-chemical dissolution system when the system is in a supercritical state; and (2) the permeability ratio can have a considerable effect on the evolved morphologies of the fluid-porous medium interface in the coupled fluid flow-chemical dissolution system. [ABSTRACT FROM AUTHOR]

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]

Highlights • Asymptotic limit of the acid dissolution capacity can lead to sharp dissolution fronts. • Main critique made in the comment is rendered invalid by a physical inconsistence. • There are two significantly-different approaches for assessing the dissolution front instability. Abstract In this reply we further demonstrate and confirm that the main conclusion (namely, that the asymptotic limit of the acid dissolution capacity can lead to sharp dissolution fronts) in our recent publication in this journal is correct and valid. The basis of the incorrect critique in the comment is whether or not a physically-consistent velocity scale should be used in deriving the dimensionless partial differential equations of the reactive infiltration instability problem in fluid-saturated rocks. In addition, there is confusion in the comment between the conclusions drawn from results obtained using different time scales (such as a dissolution time scale and a much larger time scale than the dissolution time scale) in the process of describing the reactive infiltration instability problem, and using different mathematical transforms (such as a physically-consistent mathematical transform and a physically-inconsistent one) in the nondimensionalization of dimensional partial differential equations. [ABSTRACT FROM AUTHOR]