Natural convection and the evolution of a reactive porous medium

We describe a mathematical model of buoyancy-driven flow and solute transport in a saturated porous medium, the porosity and permeability of which evolve through precipitation and dissolution as a mineral is lost or gained from the pore fluid. Imposing a vertically varying equilibrium solubility creates a density gradient which can drive convective circulation. We characterise the onset of convection using linear stability analysis, and explore the further development of the coupled reaction–convection system numerically. At low Rayleigh numbers, the effect of the reaction–permeability feedback is shown to be destabilising through a novel reaction–diffusion mechanism; at higher Rayleigh numbers, the precipitation and dissolution have a stabilising effect. Over longer time scales, reaction–permeability feedback triggers secondary instabilities in quasi-steady convective circulation, leading to rapid reversals in the direction of circulation. Over very long time scales, characteristic patterns of porosity emerge, including horizontal layering as well as the development of vertical chimneys of enhanced porosity. We discuss the implications of these findings for more comprehensive models of reactive convection in porous media

Instabilities in the dissolution of a porous matrix

A reactive fluid dissolving the surrounding rock matrix can trigger an instability in the dissolution front, leading to spontaneous formation of pronounced channels or wormholes. Theoretical investigations of this instability have typically focused on a steadily propagating dissolution front that separates regions of high and low porosity. In this paper we show that this is not the only possible dissolutional instability in porous rocks; there is another instability that operates instantaneously on any initial porosity field, including an entirely uniform one. The relative importance of the two mechanisms depends on the ratio of the porosity increase to the initial porosity. We show that the "inlet" instability is likely to be important in limestone formations where the initial porosity is small and there is the possibility of a large increase in permeability. In quartz-rich sandstones, where the proportion of easily soluble material (e.g. carbonate cements) is small, the instability in the steady-state equations is dominant.Comment: to be published in Geophysical Research Letter

Mixing across fluid interfaces compressed by convective flow in porous media

We study the mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyze the behavior of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyze a double-gyre synthetic velocity field. Second, a Rayleigh-B\'enard instability (the Horton-Rogers-Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh-Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated, and convection shutdown. The interface compression model correctly predicts the behavior of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behavior of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behavior between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns

Steadily translating parabolic dissolution fingers

Dissolution fingers (or wormholes) are formed during the dissolution of a porous rock as a result of nonlinear feedbacks between the flow, transport and chemical reactions at pore surfaces. We analyze the shapes and growth velocities of such fingers within the thin-front approximation, in which the reaction is assumed to take place instantaneously with the reactants fully consumed at the dissolution front. We concentrate on the case when the main flow is driven by the constant pressure gradient far from the finger, and the permeability contrast between the inside and the outside of the finger is finite. Using Ivantsov ansatz and conformal transformations we find the family of steadily translating fingers characterized by a parabolic shape. We derive the reactant concentration field and the pressure field inside and outside of the fingers and show that the flow within them is uniform. The advancement velocity of the finger is shown to be inversely proportional to its radius of curvature in the small P\'{e}clet number limit and constant for large P\'{e}clet numbers.Comment: to be published in SIAM J. Appl. Mat

Simulation of rock salt dissolution and its impact on land subsidence

Extensive land subsidence can occur due to subsurface dissolution of evaporites such as halite and gypsum. This paper explores techniques to simulate the salt dissolution forming an intrastratal karst, which is embedded in a sequence of carbonates, marls, anhydrite and gypsum. A numerical model is developed to simulate laminar flow in a subhorizontal void, which corresponds to an opening intrastratal karst. The numerical model is based on the laminar steady-state Stokes flow equation, and the advection dispersion transport equation coupled with the dissolution equation. The flow equation is solved using the nonconforming Crouzeix-Raviart (CR) finite element approximation for the Stokes equation. For the transport equation, a combination between discontinuous Galerkin method and multipoint flux approximation method is proposed. The numerical effect of the dissolution is considered by using a dynamic mesh variation that increases the size of the mesh based on the amount of dissolved salt. The numerical method is applied to a 2D geological cross section representing a Horst and Graben structure in the Tabular Jura of northwestern Switzerland. The model simulates salt dissolution within the geological section and predicts the amount of vertical dissolution as an indicator of potential subsidence that could occur. Simulation results showed that the highest dissolution amount is observed near the normal fault zones, and, therefore, the highest subsidence rates are expected above normal fault zones