Functional parameterization for hydraulic conductivity inversion with uncertainty quantification

Functional inversion based on local approximate solutions (LAS) is developed for steady-state flow in heterogeneous aquifers. The method employs a set of LAS of flow to impose spatial continuity of hydraulic head and Darcy fluxes in the solution domain, which are conditioned to limited measurements. Hydraulic conductivity is first parameterized as piecewise continuous, which requires the addition of a smoothness constraint to reduce inversion artifacts. Alternatively, it is formulated as piecewise constant, for which the smoothness constraint is not required, but the data requirement is much higher. Success of the inversion with both parameterizations is demonstrated for both one-dimensional synthetic examples and an oil-field permeability profile. When measurement errors are increased, estimation becomes less accurate but the solution is stable, i.e., estimation errors remain bounded. Compared to piecewise constant parameterization, piecewise continuous parameterization leads to more stable and accurate inversion. Moreover, conductivity variation can also be captured at two spatial scales reflecting sub-facies smooth-varying heterogeneity as well as abrupt changes at facies boundaries. By combining inversion with geostatistical simulation, uncertainty in the estimated conductivity and the hydraulic head field can be quantified. For a given measurement dataset, inversion accuracy and estimation uncertainty with the piecewise continuous parameterization is not sensitive to increasing conductivity contrast.