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

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$_2$.