Accurate Discretization of Vertically-averaged Models of CO₂ Plume Migration

When CO₂ is injected into a deep formation, it will migrate as a plume that moves progressively higher in the formation, displacing the resident brine. The invasion front is driven by gravity, and the upward movement of the plume is limited by a low-permeable caprock. Several authors have recently proposed to make a sharp-interface assumption and only describe the plume migration in a vertically-averaged sense. For inhomogeneous permeability, the plume migration is then described by a system of conservation laws with spatially discontinuous flux. If one disregards dissolution and residual trapping, the system reduces to a scalar conservation law with a spatially dependent flux function, which may exhibit different solutions depending on the entropy condition that is enforced to pick a unique solution. We propose a certain set of assumptions that lead to the so-called minimum-jump condition and derive the corresponding solutions to the Riemann problem. Solutions to this problem are fundamental when developing accurate Godunov-type schemes. Here, we take a slightly different approach and present an unconditionally stable front-tracking method, which is optimal for this type of problem. Moreover, we verify the well-known observation that a standard upstream mobility discretization can give wrong solutions in certain cases.