Discrete Regularization for Calibration of Geologic Facies Against Dynamic Flow Data

Abstract

Subsurface flow model calibration involves many more unknowns than measurements, leading to ill‐posed problems with nonunique solutions. To alleviate nonuniqueness, the problem is regularized by constraining the solution space using prior knowledge. In certain sedimentary environments, such as fluvial systems, the contrast in hydraulic properties of different facies types tends to dominate the flow and transport behavior, making the effect of within facies heterogeneity less significant. Hence, flow model calibration in those formations reduces to delineating the spatial structure and connectivity of different lithofacies types and their boundaries. A major difficulty in calibrating such models is honoring the discrete, or piecewise constant, nature of facies distribution. The problem becomes more challenging when complex spatial connectivity patterns with higher‐order statistics are involved. This paper introduces a novel formulation for calibration of complex geologic facies by imposing appropriate constraints to recover plausible solutions that honor the spatial connectivity and discreteness of facies models. To incorporate prior connectivity patterns, plausible geologic features are learned from available training models. This is achieved by learning spatial patterns from training data, e.g., k‐SVD sparse learning or the traditional Principal Component Analysis. Discrete regularization is introduced as a penalty functions to impose solution discreteness while minimizing the mismatch between observed and predicted data. An efficient gradient‐based alternating directions algorithm is combined with variable splitting to minimize the resulting regularized nonlinear least squares objective function. Numerical results show that imposing learned facies connectivity and discreteness as regularization functions leads to geologically consistent solutions that improve facies calibration quality.

Delineating Facies Spatial Distribution by Integrating Ensemble Data Assimilationand Indicator Geostatistics With Level‐Set Transformation

“…Facies have a broad definition across different contexts of subsurface characterization, such as the most common lithofacies for lithologic features, hydrofacies for hydraulic properties used in hydrology model, and reactive facies for reaction potential (Bayer et al, ; Yabusaki et al, ). The facies approach is especially useful for systems where sharp contrasts in properties dominate the physical processes such as the flow and transport, while within‐facies heterogeneity is of secondary importance (Khaninezhad et al, ; Liu & Oliver, ; Park et al, ; Ronayne et al, ). The intrinsic scale of the facies is dependent on the properties of interest, which could vary from centimeters in laboratory to meters in field (Sassen et al, ).…”