A Unified Workflow for Analyzing Power-Law Pressure Derivatives in Heterogeneous and Unconventional Reservoirs

Abstract

Summary Conventional pressure-transient analysis (PTA) has historically relied on the assumption of reservoir homogeneity—a simplification rooted in the foundational work of Theis (1935) and Horner (1951). Although multilayer, double-porosity, and double-permeability models were later introduced to account for heterogeneity, each constituent medium is still assumed to be internally homogeneous. As a result, classical PTA predicts that once radial or pseudo-radial flow is established, the logarithmic pressure derivative should stabilize into a horizontal plateau. Extensive field observations contradict this expectation. Structurally complex sandstones, naturally fractured formations, and unconventional shale reservoirs frequently exhibit long-lasting power-law flow regimes, in which the pressure derivative traces a persistent straight line on log-log diagnostic plots—often for months or years. Recent studies (Chen and Raghavan 2015; Chu et al. 2019; Chen et al. 2025) show that this behavior is not anomalous but a fundamental signature of reservoir heterogeneity. This paper introduces a unified, computationally efficient interpretation workflow based on the time-fractional diffusion equation. The model characterizes power-law flow using two physically meaningful parameters: the sub-diffusion exponent α (0 < α ≤ 1), which quantifies the degree of heterogeneity, andthe effective permeability parameter k~α (or k˜αLF2 for fractured wells). Because α and k˜α are coupled through terms of the form Δt1−α and Δt1−α ln Δtα, direct analytical estimation becomes inaccurate if α is neglected. To address this challenge, we propose a practical two-step workflow: Estimate α directly from the slope of the apparent power-law straight line on the pressure-derivative diagnostic plot.Estimate the characteristic flow parameter, depending on the well type:Non-fractured wells: determine the effective permeability, k˜α, using a generalized Horner-type semilog regression.Fractured wells: determine the characteristic fracture parameter directly from the fractional analytical solution, without requiring any regression:if compound linear flow is established, calculate the effective permeability, k˜α,if fracture linear flow regime is established, calculate the fracture-linear-flow parameter, k˜αLF2,if bilinear flow regime is established, calculate the fracture conductivity, kFwF. This regression-based semilog formulation generalizes the classical Horner approach while explicitly incorporating fractional-order effects, eliminating the systematic errors that arise when heterogeneous reservoirs are interpreted using homogeneous radial-flow or linear-flow assumptions. The proposed methodology is especially valuable for unconventional reservoirs. Current industry practice often applies the linear-flow equation to estimate fracture parameters k˜αLF2 from square-root-of-time plots without verifying the actual slope on the diagnostic derivative plot. In many cases, the observed slope deviates significantly from the ideal ½ value—sometimes approaching ¼ or even ⅔—leading to unreliable permeability-fracture-length estimates. The proposed framework resolves this ambiguity by directly linking the observed power-law slope to the fractional exponent α and embedding it into the interpretation procedure. Field applications to sandstone, naturally fractured systems, and shale reservoirs demonstrate that the workflow yields stable, robust, and physically interpretable permeability estimates—even from short-duration tests. It preserves the operational simplicity of classical semilog analysis while providing a physically consistent framework suited to heterogeneous and unconventional systems.