A Nonhomogeneous Porous-Medium Equation for Field Scale CO$_2$ Plume Spreading

Abstract

We derive a nonlinear diffusion model for field scale CO$_2$ plume spreading from a Global Buckley--Leverett component balance. The reduced variable $u$ is the vertically averaged mobile gas phase CO$_2$ content normalized by its maximum column value; under vertical segregation, $u=h/H$, where $h$ is plume thickness and $H$ is aquifer thickness. The resulting equation is a nonhomogeneous porous medium type equation in which nonlinear lateral spreading is coupled to source/sink terms for injection, dissolution, mineral fixation, and retention. Using the nonlinear diffusivity $D_u(u)\simeq D_0u^{1-q}$, we analyze Barenblatt-type profiles with prescribed mobile mass and a capped plume constrained by $0\le u\le1$. The capped solution contains a ful-thickness core of radius $a(t)$ and a compact plume edge $R(t)$. Constant net mobile injection can sustain the core and gives square-root growth of $R(t)$, whereas shut-in or weak mobile addition causes the core to shrink and disappear. We compare these regimes with equivalent radii from time lapse seismic plume maps at Sleipner, Aquistore, and Weyburn--Midale. The data distinguish injection controlled growth, delayed layer filling, and tail dominated redistribution, but do not determine a unique nonlinear exponent. The model provides an analytical reference for interpreting plume footprint evolution while separating cumulative injected CO$_2$ from mobile gas phase CO$_2$.