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Curvelet-Regularized SPDE Inversion on Piecewise-Planar Fractures with Trace-Graph Coupling

J. J. Segura

TL;DR

This work addresses the interpolation/inversion of fracture-conditioned fields from sparse measurements by mapping 3D fracture traces onto per-plane charts and solving a convex objective that blends a grid SPDE/GMRF prior with a curvelet-based sparsity penalty. The main approach combines a Gaussian baseline on planes with a curvelet regularizer solved via ADMM, enabling efficient, anisotropic reconstruction of fracture-aligned features. A next layer introduces along-fracture coupling using a trace network graph Laplacian to promote connectivity-consistent estimates across plane intersections. The framework is validated on synthetic benchmarks, with reproducible code and a clear roadmap for extending to explicit graph-based regularization and physics coupling, offering a scalable, network-aware alternative to traditional geostatistical priors in fractured media.

Abstract

We formulate a sparse-to-dense reconstruction layer for fractured media in which sparse point measurements are mapped onto piecewise-planar fracture supports inferred from 3D trace polylines. Each plane is discretized in local coordinates and estimated via a convex objective that combines a grid SPDE/GMRF quadratic prior with an $\ell_1$ penalty on undecimated discrete curvelet coefficients, targeting anisotropic, fracture-aligned structure that is poorly represented by isotropic smoothness alone. We further define an along-fracture distance through trace-network geodesics and express connectivity-driven regularization as a quadratic form $z^\top P^\top L_G P z$, where $L_G$ is a graph Laplacian on the trace network and $P$ maps plane grids to graph nodes; plane intersections are handled by linear consistency constraints sampled along intersection lines. The resulting optimization admits efficient splitting: sparse linear solves for the quadratic block and coefficient-wise shrinkage for the curvelet block, with standard ADMM convergence under convexity. We specify reproducible synthetic benchmarks, baselines, ablations, and sensitivity studies that isolate directional sparsity and connectivity effects, and provide reference code to generate the figures and quantitative tables.

Curvelet-Regularized SPDE Inversion on Piecewise-Planar Fractures with Trace-Graph Coupling

TL;DR

This work addresses the interpolation/inversion of fracture-conditioned fields from sparse measurements by mapping 3D fracture traces onto per-plane charts and solving a convex objective that blends a grid SPDE/GMRF prior with a curvelet-based sparsity penalty. The main approach combines a Gaussian baseline on planes with a curvelet regularizer solved via ADMM, enabling efficient, anisotropic reconstruction of fracture-aligned features. A next layer introduces along-fracture coupling using a trace network graph Laplacian to promote connectivity-consistent estimates across plane intersections. The framework is validated on synthetic benchmarks, with reproducible code and a clear roadmap for extending to explicit graph-based regularization and physics coupling, offering a scalable, network-aware alternative to traditional geostatistical priors in fractured media.

Abstract

We formulate a sparse-to-dense reconstruction layer for fractured media in which sparse point measurements are mapped onto piecewise-planar fracture supports inferred from 3D trace polylines. Each plane is discretized in local coordinates and estimated via a convex objective that combines a grid SPDE/GMRF quadratic prior with an penalty on undecimated discrete curvelet coefficients, targeting anisotropic, fracture-aligned structure that is poorly represented by isotropic smoothness alone. We further define an along-fracture distance through trace-network geodesics and express connectivity-driven regularization as a quadratic form , where is a graph Laplacian on the trace network and maps plane grids to graph nodes; plane intersections are handled by linear consistency constraints sampled along intersection lines. The resulting optimization admits efficient splitting: sparse linear solves for the quadratic block and coefficient-wise shrinkage for the curvelet block, with standard ADMM convergence under convexity. We specify reproducible synthetic benchmarks, baselines, ablations, and sensitivity studies that isolate directional sparsity and connectivity effects, and provide reference code to generate the figures and quantitative tables.
Paper Structure (59 sections, 3 theorems, 15 equations, 2 figures, 2 tables)

This paper contains 59 sections, 3 theorems, 15 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and Motivation
  2. Scope statement.
  3. Related Work and Positioning
  4. Fractured media, DFNs, and connectivity-driven transport.
  5. SPDE/GMRF baselines.
  6. Directional multiscale transforms in inverse problems and geoscience.
  7. Graphs and network priors.
  8. Data Model
  9. Inputs
  10. 3D trace polylines.
  11. Sparse samples.
  12. Physical meaning of the latent field $z_p(u,v)$
  13. What does $z_p$ represent physically?
  14. Why along-fracture metrics matter.
  15. Notation summary
  16. ...and 44 more sections

Key Result

Theorem 1

Let $f$ be $C^2$ except for a discontinuity across a $C^2$ curve. Let $f_N$ be the best $N$-term approximation of $f$ in a curvelet tight frame. Then $\left\|f-f_N\right\|_2^2 \le C\, N^{-2}(\log N)^3$. In contrast, wavelet best $N$-term approximation is $O(N^{-1})$ for the same model.

Figures (2)

  • Figure 1: Representative outputs from the synthetic pipeline. Panels illustrate how the observation geometry (traces + sparse samples) and the reconstruction differ under GMRF smoothing versus curvelet-regularized recovery.
  • Figure 2: Sensitivity and resolution diagnostics for the synthetic benchmarks (Section \ref{['sec:experiments']}). Panel (a) supports the choice of $n=128$ as a cost--accuracy compromise; panel (b) illustrates the residual--sparsity trade-off for $\lambda$; panels (c--d) characterize robustness to $\kappa$ and to sampling density.

Theorems & Definitions (3)

  • Theorem 1: Curvelet approximation for cartoon-like functions (informal)
  • Theorem 2: ADMM convergence (standard)
  • Proposition 1: Dirichlet energy on a fracture network