From Interpolation to Physics: A Reaction–Diffusion Framework for Grid-Aware Stochastic Field Generation

Abstract Stochastic field generation plays a central role in modeling subsurface properties such as permeability, porosity, and lithofacies. The traditional approach in the petroleum industry—Kriging and its extension, Sequential Gaussian Simulation (SGS)—treats the field as a statistical interpolation problem based on limited data. As such, it often struggles to capture sharp transitions, long-range connectivity, and directional features—key geological heterogeneities that influence flow and arise from natural depositional and diagenetic processes. We propose a forward-modeling algorithm that mimics the formation of geological parameter fields by evolving a stochastic system over time. The method solves a reaction–diffusion equation with additive Gaussian noise. The reaction term incorporates constraints from geostatistical data and geologic structure, while the diffusion term promotes spatial coherence. Gaussian noise introduces heterogeneity and simulates the inherent randomness of natural systems. By calibrating noise strength and diffusion coefficients, we generate spatially correlated fields whose variogram and value distribution (histogram) emerge naturally and can be monitored dynamically. In contrast to traditional methods based on sequential simulation and Kriging-based interpolation, our approach embeds these geostatistical targets directly into the dynamics of a stochastic partial differential equation. Once the desired spatial properties are achieved, the simulation is stopped. One of the key advantages is that stochastic parameter fields are generated directly on simulation grids—such as corner-point grids or unstructured meshes—rather than mapped after grid creation. Since both reservoir simulation and reaction–diffusion are governed by evolution equations, the transition between parameter generation and flow simulation becomes seamless. This also enables local control of heterogeneity near faults, boundaries, and well zones by adjusting diffusion coefficients. The approach supports a new modeling paradigm: generate the simulation grid first, then evolve the parameter field within it, reversing the conventional workflow. We demonstrate the method through three numerical experiments. First, we demonstrate that the empirical variogram of the simulated field converges toward the prescribed target model as the field evolves. Second, we incorporate hard conditioning data using reaction terms that pull the solution toward specified values at target locations. Third, we embed sharp geologic features—such as streaks or barriers—by applying spatial masks that locally modify the field. While the first two cases use regular grids, the third applies the method on corner-point grids to ensure compatibility with standard reservoir modeling workflows. The results show that this reaction–diffusion framework offers a flexible, interpretable, and physics-consistent alternative to SGS, producing geologically realistic parameter fields directly on simulation grids.