Abstract
In this paper we focus on a general model to describe compressible and immiscible three-phase flow in porous media. The underlying idea is to replace Darcy’s law by more general momentum balance equations. In particular, we want to account for viscous coupling effects by introducing fluid-fluid interaction terms. In [Qiao, et al. (2018) Adv Water Resour 112: 170–188] a water-oil model based on the theory of mixtures was explored. It was demonstrated how the inclusion of viscous coupling effects could allow the model to better capture flow regimes which involve a combination of co-current and counter-current flow. In this work we extend the model in different aspects: (i) account for three phases (water,oil,gas) instead of two; (ii) deal with both the compressible and incompressible case; (iii) include viscous terms that represent frictional forces within the fluid (Brinkman type). A main objective of this work is to explore this three-phase model, which appears to be more realistic than standard formulation, in the context of petroleum related applications. We first provide development of stable numerical schemes in a one-dimensional setting which can be used to explore the generalized water-oil-gas model, both for the compressible and incompressible case. Then, several numerical examples with waterflooding in a gas reservoir and water alternating gas (WAG) experiments in an oil reservoir are investigated. Differences and similarities between the compressible and incompressible model are highlighted, and the fluid-fluid interaction effect is illustrated by comparison of results from the generalized model and a conventional model formulation.
Highlights
Viscous coupling GenerallyThe processes of multiphase flow in porous media occur in many subsurface systems and have found many applications of practical interest, such as hydrology, petroleum engineering, geothermal energy development and carbon storage (Bakhshian et al, 2019; Bakhshian and Hosseini, 2019; Wu, 2016)
The immiscible three-phase flow is always encountered in waterflooding for oil reservoirs with gas cap, in immiscible CO2 storage in depleted oil and gas reservoirs, and steam floods and water-alternating-gas (WAG) processes (Bentsen and Trivedi, 2012; Juanes, 2008)
We have presented a three-phase compressible and incompressible viscous model based on the mixture theory approach
Summary
The processes of multiphase flow in porous media occur in many subsurface systems and have found many applications of practical interest, such as hydrology, petroleum engineering, geothermal energy development and carbon storage (Bakhshian et al, 2019; Bakhshian and Hosseini, 2019; Wu, 2016). Using the same method, Li et al (2005) showed that their model was able to capture main experimental effects caused by viscous coupling They mentioned that the interfacial area between the fluids is a key variable for relative permeability functions for two immiscible fluids flow in porous media. Main observations from our numerical experiments with two and three-phase flow scenarios where the flow dynamics are generated by injection of water or gas in the center of the domain and production of fluids at the left and right boundary are: (i) The simulation cases involve competition between pressure driven co-current flow and countercurrent gravity driven flow; (ii) Both the incompressible and compressible discrete version of the model appear to have good stability properties. The details of the compressible and incompressible scheme are given in Appendix A–Appendix D
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