In this paper, we discuss two possible modifications to a numerical solution method for a model of microbiologically induced calcite precipitation (MICP). MICP provides a means to seal cracks in the surfaces of geological structures. This process can be useful in the context of carbon sequestration where even minor fissures in the storage medium can compromise the isolation of the gas.
From a mathematical and computational point of view MICP has very interesting features which make it challenging for numerical solution methods to solve the system. First, MICP models are very complex. There are multiple sensitivities to every element of the system, and computational approximations to the solutions to any part of the system can effect the solution of this part either very strongly, or not at all. Additionally, the effects can propagate to different parts of the system.
Second, a subset of the chemical reactions occur much faster than the remainder of the system. A solution method must use a fine discretization parameter to capture these dynamics, while the system must be solved on a large time scale in order for the calcite precipitation to occur. An assumption that has been used to mitigate this problem is that the reactions occur fast enough that the ratio of products to reactants remains constant.
Our proposed modification is to combine the fast reactions with physical conservation laws, posing them as a system of differential algebraic equations.
The third challenge with MICP is that some reaction terms are discontinuous, thus traditional numerical error bounds may not apply. We propose to allow the precipitation rate to increase sharply on a neighborhood of the threshold instead of jumping from "off'" to "on."