Adjoint-based inversion for porosity in shallow reservoirs using pseudo-transient solvers for non-linear hydro-mechanical processes

Abstract

Porous flow is of major importance in the shallow subsurface, since it directly impacts on reservoir-scale processes such as waste fluid sequestration or oil and gas exploration. Coupled and non-linear hydro-mechanical processes describe the motion of a low-viscous fluid interacting with a higher viscous porous rock matrix. This two-phase flow may trigger the initiation of solitary waves of porosity, further developing into vertical high-porosity pipes or chimneys. These preferred fluid escape features may lead to localised and fast vertical flow pathways potentially problematic in the case of for instance CO2 sequestration. Constraining the porosity and the non-linearly related permeability distribution in such environments is a major challenge. Although seismic imaging methods accurately localise the high-porosity chimneys in the inverted wave-speed field, the conversion to porosity is not straightforward. We develop an inversion framework to reconstruct the unknown porosity field using relevant observable quantities such as subsurface fluid fluxes. We introduce the adjoint framework for the two-phase flow equations, which allows for efficient calculations of the pointwise gradients of the flow solution with respect to the porosity. We then use the gradients in a gradient descent method to invert for the pointwise porosity. We solve the forward and the adjoint equations using an iterative matrix-free pseudo-transient approach with the finite difference method. The proposed parallel solving procedure executes optimally on the latest many-core hardware accelerators such as GPUs. Numerical results show that an inversion for porosity is challenging if data is sparse since the porosity is very locally sensitive to the fluid flux. We introduce the concepts of sensitivity kernels as employed in seismology for the set of two-phase equations and suggest this approach as a standard for future studies.