Practical Applications of Time Fractional Diffusion in Heterogeneous Sandstone Reservoirs

Abstract Transient pressure test analyses for sandstone reservoirs typically assume homogeneity. However, deviations in natural logarithmic derivatives, such as power-law trends, often challenge this assumption. Analysts frequently attribute these deviations to nearby boundaries or channel effects. Still, such interpretations usually suggest unrealistically close boundaries, sometimes within tens of feet, raising questions about their validity and physical feasibility. This study introduces a novel solution: the radial-dimensional time fractional-diffusion (T-FD) composite model. By leveraging this advanced approach, we aim to provide a more accurate framework for interpreting well test data in heterogeneous sandstone reservoirs. Our work resolves the counterintuitive implications of traditional analyses and introduces a systematic workflow for effectively exploring reservoir dynamics and managing the complexities of heterogeneous formations. This study examines pressure buildup responses in sandstone reservoirs from diverse geological settings, characterized by widespread power-law behavior. Traditional models often misinterpret these responses as boundary effects, leading to oversimplified or inaccurate analyses. Such misinterpretations can be particularly problematic during development planning, as they may influence decisions based on assumed boundary orientations—such as avoiding drilling additional wells beyond certain zones. However, in many cases, these boundaries cannot be reliably confirmed using geological data. This uncertainty poses significant challenges in formulating optimal development strategies and may result in costly—and potentially irreversible—financial losses. We employ the Continuous Time Random Walk (CTRW) framework to address these limitations and develop the T-FD model in Laplace space. This advanced approach offers deeper insight into the power-law dynamics characteristic of heterogeneous reservoirs, capturing the complex interactions within these systems more effectively than conventional methods. Field evaluations demonstrate that the T-FD model in Laplace space provides a more accurate representation of power-law behaviors in pressure buildup responses compared to traditional methods. A compelling case study illustrates the model's capabilities: during the drawdown phase, the data exhibited a slope greater than one-half, followed by a buildup period characterized by a positive power-law slope. Traditional methods, such as multi-zone composite or layered system models, failed to reconcile these behaviors, revealing their limitations. In contrast, the T-FD model effectively captures both drawdown and buildup responses, offering valuable insights into the reservoir's dynamic behavior. These findings underscore the importance of accounting for heterogeneity indices and their significant impact on reservoir performance. Additionally, this study offers a streamlined workflow for interpreting well test data and evaluating reservoir properties, enabling more reliable predictions of reservoir dynamics and estimated ultimate recovery (EUR). This study shows the prevalence of power-law behavior in pressure buildup responses in heterogeneous sandstone reservoirs. By introducing the T-FD model—a novel framework for modeling vertical wells in cylindrical composite reservoirs—we demonstrate its effectiveness in capturing the heterogeneous nature of these formations. Formulated in Laplace space, the model addresses key challenges in reservoir characterization and offers a systematic workflow for well testing. The insights from this research carry important implications for optimizing recovery, enhancing reservoir management, and advancing the development of heterogeneous sandstone reservoirs.