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Gas-liquid anomalous diffusion and its numerical modeling in unconventional reservoirs
Yildirim, Gizem Hazal
Yildirim, Gizem Hazal
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2024
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Abstract
The velocity field for flow in porous media may include jumps and retardations due to multi-scale heterogeneities of the domain. The concurrent flow of multiple phases, particularly that of liquid and gas, may further exacerbate the discontinuities of the velocity field causing sharp changes in capillary pressure and relative permeability behaviors. Flow over a discontinuous velocity field cannot be represented by the conventional normal-diffusion models.
This PhD study introduces a one-dimensional, three-phase, numerical model for anomalous-diffusion in porous media and discusses the impact of gas-liquid anomalous-diffusion on production from ultra-tight, nanoporous, fractured unconventional reservoirs. The objective of this research is not to contribute to the theoretical basis of anomalous- diffusion concept, but to extend its use to highly non-linear flow conditions caused by the existence of a mobile gas phase. The effect of a compressible gas phase on anomalous- diffusion in porous media has not been reported in the literature and constitutes the central new contribution of this research.
A multiple-fractured horizontal well surrounded by a stimulated reservoir volume is considered. Hydraulic fractures along the horizontal well are assumed to be identical and uniformly distributed. Further assuming negligible contribution from the outer reservoir and considering the symmetry of flow between hydraulic fractures, one-dimensional flow from the stimulated reservoir to a single hydraulic fracture is modeled. Fracture surface is assumed to be at a constant pressure and the bisector of the distance between two hydraulic fractures is a no-flow boundary.
A modified flux law for anomalous diffusion is extended to multiphase flow by the relative permeability formalism. Time- and space-fractional derivatives are expressed by Caputo’s representation and Muiro’s fractional finite difference approximation is used. A fully implicit computational scheme with Newton’s iteration is used to solve the resulting non-linear equations. The oil pressure, gas saturation, and water saturation are calculated simultaneously. A table-look-up approach with interpolation is used to update rock and fluid properties in each time step to overcome the stabilization problems due to the existence of gas.
Production, pressure, and saturation behaviors are presented for super-diffusion cases caused by the existence of natural or induced fractures in the stimulated reservoir volume and sub-diffusion due to localized retardations. The results signify deviations from normal diffusion behavior as the anomalous-diffusion effects increase particularly in the presence of the gas phase. It is shown that super-diffusion accelerates production and yields more efficient depletion of the reservoir. Sub-diffusion hinders the fluid transport, leading to less effective drainage.
The study emphasizes the importance of accounting for anomalous diffusion for multiphase flow in unconventional reservoirs. It provides guidelines to incorporate a gas phase in anomalous-diffusion models and demonstrates that the non-linearity caused by a highly compressible phase accentuates anomalous-diffusion and vice versa
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