# A Scalar Conservation Law for Plume Migration in Carbon Sequestration

## TL;DR

This paper develops a new mathematical model using a quasi-linear hyperbolic PDE with dual flux curves to better understand and analyze the complex migration and trapping of CO2 in geological formations during sequestration.

## Contribution

It introduces a novel dual flux model with a new cross-hatch characteristic construction, extending the classical theory of scalar conservation laws with convex flux.

## Key findings

- Characterizes shock and rarefaction waves in the dual flux model
- Reveals differences from classical scalar conservation laws with convex flux
- Provides a generalized entropy condition for shocks in the dual flux context

## Abstract

A quasi-linear hyperbolic partial differential equation with a discontinuous flux models geologic carbon dioxide migration and storage. Dual flux curves characterize the model, giving rise to flux discontinuities. One convex flux describes the invasion of the plume into pore space, and the other captures the flow as the plume leaves carbon dioxide bubbles behind, which are then trapped in the pore space. We investigate the method of characteristics, the structure of shock and rarefaction waves, and the result of binary wave interactions. The dual flux property introduces unexpected differences between the structure of these solutions and those of a scalar conservation law with a convex flux. During our analysis, we introduce a new construction of cross-hatch characteristics in regions of the space-time plane where the solution is constant, and there are two characteristic speeds. This construction is used to generalize the notion of the Lax entropy condition for admissible shocks, and is crucial to continuing the propagation of a shock wave if its speed becomes characteristic.