Permeability sensitivity functions and rapid simulation of multi-point pressure measurements using perturbation theory

Abstract

We introduce a perturbation method for the forward modeling of transient diffusion phenomena in aquifers with pressure measurements acquired at arbitrary observation points. Our numerical approach requires only two simulations to compute the Permeability Sensitivity Function (PSF) at an observation point under multi-source and multi-rate conditions; one simulation captures the pressure solution due to the flow history of the sources, while the second simulation is required to extract the Green's function of the groundwater system. The PSF is calculated on the spatial-temporal domain by convolving the gradients of the pressure solution and of the Green's function. The first-order term in the perturbation expansion is obtained by weighting the spatial variations of permeability with a flow-history-dependent PSF.Our work confirms the flexibility and reliability of the method after successful validation with forward numerical simulations in cylindrical and Cartesian coordinates. Multidimensional synthetic studies including multi-well and formation testing conditions, were examined for diverse anisotropy-dominated fluid-flow regimes. Against perturbations of more than one order of magnitude in background permeability, results show that first-order approximations can be computed in tens of CPU seconds with relative errors in pressure of < 7%. Results indicate that sensitivity functions enable improved qualitative understanding of permeability-pressure correlations and local Darcy flow dynamics.