## 1. Introduction

[2] Numerical simulations are an important tool widely used to assess the mechanisms of reactive transport processes in geochemical environment. Particularly challenging from the computational point of view are the simulations of systems in which precipitation and/or dissolution reactions led to significant changes of porosity in the media. Strong porosity changes typically observed at interfaces of engineered technical barriers and natural rock formations with different geochemical characteristics, e.g., clay and cement interfaces [*Marty et al.*, 2009; *Kosakowski et al.*, 2009; *De Windt et al.*, 2004]. Mineral precipitation in aquifers and reservoirs change their transport properties significantly and may lead eventually to complete clogging of the pore space [*Saripalli et al.*, 2001]. The porosity reduction for example at claystone/concrete interfaces has been experimentally reported by *Read et al.* [2001] as well as from small scale diffusive experiment using cement paste [*Sarott et al.*, 1992]. Similar processes have been observed at the Maqarin site in Jordan, a natural analog for cement/clay interactions at which hyperalkaline groundwater interact with marl formations [*Smellie*, 1998].

[3] Several numerical computer codes dealing with precipitation-dissolution reactions and porosity changes in multidimensional reactive transport problems have been developed over last decades, for example HYTEC [*van der Lee and De Windt*, 2001; *van der Lee et al.*, 2003; *van der Lee*, 2005], TOUGHREACT [*Xu et al.*, 2004, 2006, 2011], PHREEQC [*Parkhurst and Appelo*, 1999], MIN3P [*Mayer et al.*, 2002], PHAST [*Parkhurst et al.*, 2002], CRUNCH [*Steefel*, 2001]. These codes and others have been continuously cross benchmarked [*De Windt et al.*, 2003; *Carrayrou*, 2010; *Carrayrou et al.*, 2010a, 2010b]. Although the agreement between codes is satisfactory, it is shown that the numerical methods used in solving the transport equations, i.e., finite differences, finite elements, mixed-hybrid finite elements, and the sequential coupling schemes, i.e., the sequential noniterative approach, the sequential iterative approach and the global implicit approach, may lead to systematic discrepancies. Moreover, because of their inability to describe subgrid pore space changes correctly, the numerical approaches predict discretization-dependent values of porosity changes and clogging times [*Marty et al.*, 2009]. In this context, analytical solutions become an essential tool to verify numerical solutions.

[4] Mass transport influenced by precipitation-dissolution reactions via changing porosity can be represented mathematically by a nonlinear system of partial differential equations. In contrast to conservative mass transport which is described by a linear system of partial differential equations, such systems are not trivial to solve. Analytical solutions for transport of reactive solutes involving precipitation-dissolution reactions are available in the case of constant porosity. *Lichtner et al.* [1986] proposed analytical and numerical solutions to the moving boundary problem resulting from reversible heterogeneous reactions and aqueous diffusion in a porous medium. *De Simoni et al.* [2005, 2007] presented a methodology to compute homogeneous and heterogeneous analytical reaction rates directly under instantaneous equilibrium. *Donado et al.* [2009] extended this approach to the multirate mass transfer equation. Analytical expressions for solute concentrations and reaction rates associated with a multispecies reactive transport problem in the presence of both kinetic and equilibrium chemical reactions are developed and presented by *Sánchez-Vila et al.* [2010].

[5] However, analytical solutions for problems coupled with porosity changes are very few. The only available works on this topic are the two recent works of *Lagneau and van der Lee* [2010] and *Hayek et al.* [2011]. *Lagneau and van der Lee* [2010] proposed an analytical solution for a one-dimensional system containing one species and one mineral. This analytical solution was obtained on the basis of Fourier series and was used to verify numerical solution obtained from the numerical computer code HYTEC. They obtained good agreement between the analytical and the numerical solutions for small porosity changes far away from pore space clogging. However, this analytical solution cannot be applied to systems with strong porosity changes, in which a complete clogging of the pore space takes place. Recently, *Hayek et al.* [2011] presented new analytical solutions for one-dimensional diffusion problems coupled with one precipitation-dissolution reaction and strong feedback of porosity change for a system contained two aqueous species where the concentration of one species is fixed in time. Their solutions describe a system that contains two aqueous species where only one species is mobile. These solutions were obtained by using the original method of simplest equation [*Kudryashov*, 2005a, 2005b]. The proposed solutions are exact and do not contain any approximation. They can be used to describe systems with strong porosity changes and especially systems which reach clogging.

[6] In this paper, a general methodology is developed to derive analytical solutions for one-, two-, and three-dimensional diffusive transport coupled with precipitation-dissolution reactions and porosity changes. These solutions can be used to simulate systems containing multiple aqueous species and a single solid phase. To our knowledge, the analytical solutions proposed here are the first dealing with transport of reactive species coupled with precipitation-dissolution reactions and porosity changes in several dimensions. The method of the simplest equation [*Kudryashov*, 2005a, 2005b] used by *Hayek et al.* [2011] to derive the analytical solutions for one-dimensional problems was applied here to solve the nonlinear transport equations in two and three dimensions. The obtained analytical solutions describe spatial and temporal evolutions of solute concentrations, porosity, and mineral distribution for a set of initial and boundary conditions. The main purpose of the derived solutions is in verification numerical reactive transport codes for systems with strong porosity changes.