Well-posed Bayesian geometric inverse problems arising in subsurface flow

Abstract

Abstract. In this paper, we consider the inverse problem of determining the permeability of the subsurface from hydraulic head measurements, within the framework of a steady Darcy model of groundwater flow. We study geometrically defined prior permeability fields, which admit layered, fault and channel structures, in order to mimic realistic subsurface features; within each layer we adopt either constant or continuous function representation of the permeability. This prior model leads to a parameter identification problem for a finite number of unknown parameters determining the geometry, together with either a finite number of permeability values (in the constant case) or a finite number of fields (in the continuous function case). We adopt a Bayesian framework showing existence and well-posedness of the posterior distribution. We also introduce novel Markov Chain-Monte Carlo (MCMC) methods, which exploit the different character of the geometric and permeability parameters, and build on recent advances in function space MCMC. These algorithms provide rigorous estimates of the permeability, as well as the uncertainty associated with it, and only require forward model evaluations. No adjoint solvers are required and hence the methodology is applicable to black-box forward models. We then use these methods to explore the posterior and to illustrate the methodology with numerical experiments.

On the Well-posedness of Bayesian Inverse Problems

“…In practical applications, allowing only continuous functions as diffusion coefficients may be too restrictive. Iglesias et al [22] consider more realistic geometric priors measures. In [22, Theorem 3.5], the authors show local Lipschitz continuity for some of those prior measures, but only Hölder continuity with coefficient γ = 0.5 for others.…”

“…Several authors have discussed, what we choose to call, (Lipschitz, Hellinger) well-posedness for a variety of Bayesian inverse problems. For example, elliptic partial differential equations [8,22], level-set inversion [23], Helmholtz source identification with Dirac sources [13], a Cahn-Hilliard model for tumour growth [24], hierarchical prior measures [25], stable priors in quasi-Banach spaces [36,37], convex and heavy-tailed priors [20,21]. Moreover, to show well-posedness, Stuart [34] has proposed a set of sufficient but not necessary assumptions.…”

“…Another pressing issue is the measurement of marginal changes in the posterior. Most authors have discussed Lipschitz continuity with respect to the Hellinger distance; exceptions are, e.g., the articles of Iglesias et al [22] and Sprungk [33]. While the Hellinger distance has useful properties, the actual choice of the metric should depend on the area of application.…”

Assisted History Matching for Fractured Reservoirs by Use of Hough-Transform-Based Parameterization

“…The probability distribution functions (PDFs) of the model parameters are described as weighted sums of Gaussian PDFs to be able to sample bimodal parameters and posteriors. Iglesias et al (2014) parameterized the simplified geometry of geological facies in layered and channelized reservoirs into a small number of geometric parameters, and sampled the posterior distributions of both the geometric parameters and permeabilities within each facies by Markov-chain Monte Carlo methods. Nejadi and Leung (2012) used discrete-fracture-network models and upscaling techniques to construct initial realizations of a dual-porosity model before history matching.…”

“…Both algorithms converged relatively quickly on multiple test cases, including the Brugge benchmark case (Peters et al 2010). Iglesias (2014) introduced an iterative regularizing-ensemble (Kalman) smoother to provide a robust ensemble approximation of the Bayesian posterior in ill-posed inverse problems.…”