Practical Applications of Space-Time Fractional-Diffusion in Naturally Fractured Reservoirs

Abstract This study introduces a radial space-time fractional-diffusion (ST-FD) model, now adapted for Laplace space. This model incorporates dual porosity factors and, when applied to well tests in naturally fractured reservoirs, addresses the effects of rock heterogeneity in fractured or fissured formations by solving a multi-index ST-FD equation through the Laplace transformation. For the first time, pressure distributions in cylindrical coordinates for heterogeneous fractured rocks with space and time-dependent properties are presented. These findings are novel and unreported in existing literature, highlighting the utility of this approach in analyzing fractured reservoirs. The method developed in this paper was applied to five example well tests from fractured carbonate fields in Northern Iraq, demonstrating the practical application of this methodology. Two key models commonly used in the industry to study naturally fractured reservoirs are the Warren and Root (1963) dual porosity model and de Swaan's (1976) model. The Warren and Root model assumes a pseudo-steady-state flow between the matrix and fractures, while de Swaan's model addresses transient flow between these components. When applied to pressure buildup responses in fractured reservoirs from Northern Iraq that exhibit a power-law pattern, these traditional models could lead to misinterpretations, attributing the behavior to boundary effects such as faults. In contrast, employing the Continuous Time Random Walk (CTRW) framework, specifically the space-time fractional-diffusion (ST-FD) model developed by Raghavan and Chen (2021, 2023), and now modified in this paper with solutions in Laplace space, provides a more accurate interpretation of these pressure responses. The analysis of real-world data using this modified ST-FD model reveals that the power-law behavior in pressure buildup responses is linked to the reservoir's complex heterogeneity as shown in five example well tests presented here. The effectiveness of the modified ST-FD model in Laplace space is validated, offering practical insights for well testing and accurate evaluations of reservoir recovery potential. Although the system is intricate at a fine scale, the results are applicable in the conventional sense at the Theis (1935) scale. Long-range connectivity and distorted flow paths that result in pauses, flow reversals, and related phenomena are quantified through three exponents and fractional derivatives. The findings highlight the influence of diverse heterogeneity indexes and introduce an effective workflow for interpretation and reservoir assessment. These insights significantly enhance comprehension of the reservoir's complex heterogeneity, leading to better estimations of Ultimate Recovery (EUR).