Geological Field Restoration through the Lens of Image Inpainting

Abstract

We study an ill-posed problem of geological field reconstruction under limited observations. Engineers often have to deal with the problem of reconstructing the subsurface geological field from sparse measurements such as exploration well data. Inspired by image inpainting, we model this partially observed spatial field as a multidimensional tensor and recover missing values by enforcing a global low rank structure together with spatial smoothness. We solve the resulting optimization via the Alternating Direction Method of Multipliers. On the SPE10 model 2 benchmark, this deterministic approach yields consistently lower relative squared error than ordinary kriging across various sampling densities and produces visually coherent reconstructions.

Figures (19)

• Figure 1: 3D porosity field of SPE10 model 2 geological benchmark.
• Figure 2: Example of well data available during reconstruction process, $100$ wells.
• Figure 3: Reconstruction results of the porosity field from SPE10 model 2. Cross-section along the $z$-axis at $z = 12$ with $500$ wells. Left to right: (i) ground truth from SPE10 model 2; (ii) kriging; (iii) tensor completion; (iv) well mask.
• Figure 4: Reconstruction results of porosity field from the SPE10 model 2. Cross-section along $z$-axis at $z = 12$ with $100$ wells. From left to right: (i) ground truth data from SPE10 model2; (ii) reconstruction with kriging; (iii) reconstruction with tensor completion; (iv) well mask.
• Figure 5: Reconstruction results of porosity field from the SPE10 model 2. Cross-section along $z$-axis at $z = 12$ with $300$ wells. From left to right: (i) ground truth data from SPE10 model2; (ii) reconstruction with kriging; (iii) reconstruction with tensor completion; (iv) well mask.