Leveraging Differentiable Programming in Julia: Forward and Inversion Modeling of Two-Phase Systems

From complex magmatic systems to geothermal reservoirs, fluid-rock dynamics have posed immense modeling hurdles in the Geosciences for decades. These systems are influenced by interactions between magmatic heat sources, fluid flow, host rock deformation, and chemical heterogeneities. To enhance our understanding and predictive capabilities of these intricate systems, we develop a novel forward and inverse modeling code designed to simulate fluid migration within a deforming, porous host-rock. We employ the Julia programming language, chosen for its differentiability and efficiency, facilitating the simple integration of various composable packages.

Forward simulations are performed in the advanced automatic differentiation (AD) framework of Julia, allowing for flexible adjustments of the underlying coupled system of equations. We utilize a staggered-grid, implicit finite-difference solver, along with the GeoParams.jl package to implement visco-elasto-plastic rheologies and solve the coupled fluid-rock interactions under non-linear Darcy and incompressible Stokes-flow regimes.

The AD framework of Julia allows for the application of the adjoint method in parameter sensitivity analysis, significantly reducing computational demands compared to traditional inversion techniques. This framework lays the foundation for adjoint inversions as the gradients calculated for the sensitivities are needed for gradient descent algorithms. The effectiveness of our framework is demonstrated through representative case studies, illustrating its applicability to understanding the dynamic behavior of two-phase systems influenced by both thermal and mechanical processes.