Simulation of two-phase flow in porous media with sharp material discontinuities
Introduction
Multiphase flow through porous soils, sedimentary rocks or volcaniclastics, forms an integral part of a wide range of processes occurring in different geologic settings such as infiltration of the vadose zone, geothermal circulation, biogenic methanogenesis, or secondary (hydrocarbon) migration. These processes occur on vastly different time and length scales. The length scale affects the balance between the driving forces: while capillary forces tend to dominate on the pore-scale, viscous and gravitational forces determine flow on the km-scale. Notwithstanding, dynamic, km-scale capillary barriers are well documented in the engineering literature (e.g., Watts (1987)) and are related to the interplay of geologic heterogeneity with wettability and saturation. This interplay gives rise to rich flow patterns and sharp saturation discontinuities collocated at material interfaces. As illustrated in a photograph of an outcrop of fractured sandstone that was infiltrated by water draining from overlying soil (Fig. 1).
Beyond variation of the external conditions, flow can become episodic and varying discontinuously over time. Thus, multiphase flow across a coarse-fine material interface can come to a halt when the non-wetting phase collects upstream of it, yet remaining at a pressure that is below the entry pressure of the fine-grained layer. This process has been termed vapour-lock, and Benzing and Shook (1996) studied its dynamics in the laboratory. They showed that, in the presence of gravity, the build-up of buoyant non-wetting phase below horizontal interfaces may eventually break the lock (dynamic seal), leading to a sudden fluid discharge that only ends when the accumulation has drained and the process can repeat. In addition to such relaxation oscillations, spontaneous imbibition of a fine-grained layer saturated with non-wetting phase will occur, when this layer is brought in contact with the wetting phase. As this happens, the spontaneous pressure drop that ensues when water enters the fine-grained layer, it causes rapid displacement of the non-wetting phase from the layer.
Both, the formation and breakdown of a vapour lock and the spontaneous imbibition into a water-wet non-wetting phase-saturated layer are material interface processes posing modelling challenges because they locally perturb system behavior in a nonlinear fashion. Such perturbations induce spatio-temporal discontinuities in pressure and saturation, requiring additional degrees of freedom to capture them: While a vapour lock exists at a capillary barrier, fluid pressure is decoupled across it and flows parallel to the interface can evolve independently on either side. This state will persist until the capillary entry pressure of barrier is exceeded, flow occurs, and the pressure discontinuity disappears again. It follows that a realistic model of this process will need to be able to deal with such temporal jump discontinuities in pressure and saturation as well as their toggle-switch control of fluid flow. Notably, geologic porous media are full of capillary interfaces with intricate shapes, difficult to resolve already at the level of geometric modelling. Capturing the dynamics of multiphase flow in such heterogeneous rock sequences provides a formidable modelling and upscaling challenge (Kueper et al. (1989), Dawe et al. (1992), Krause et al. (2009), and Shi et al. (2011)).
Introducing a new numeric method for the simulation of such interface dynamics is the subject of this article.
The described behavior of capillary interfaces has already been subjected of considerable analysis on the basis of which a new numeric model can be built. Van Duijn et al. (1995) proposed a semi-analytic model, including interface conditions to represent the effect of the capillary entry pressure. Correa and Firoozabadi (1996) studied gas-oil gravity drainage, concluding that capillary pressure contrasts promote instabilities in the system. Other studies have attempted to treat interface transfer numerically: classic finite difference approaches only consider transfer along the grid axes, and zero-dimensional transfer functions are used in the modelling of fracture-matrix transfer in dual continua models, see review in Gilman (2003). Finite volume (FV) and finite element (FE) methods and combinations thereof permit the use of unstructured meshes for the discrete representation of material interfaces (Cordes and Kinzelbach (1992) and Geiger et al. (2004)). Without specific provisions, these methods cannot describe jump discontinuities in pressure and/or saturation because, in the FEM, dependent variables are continuous across element interfaces. In the FVM, interfaces can only be resolved using transmissibility multipliers. Thus, dependent variable discontinuities are smeared across interfaces. To study this behavior, Helmig and Huber (1998) compared a fully upwind Galerkin method with standard Galerkin and Petrov-Galerkin methods: while the standard Galerkin method is not even locally conservative, the Petrov-Galerkin method can create nonphysical results. The fully upwind Galerkin method provides reasonable results, but the jump discontinuity of saturation at the material interface gives way to a sharp gradient. To resolve the jump discontinuity correctly, Bastian (1999) and Niessner et al. (2004) introduced additional degrees of freedom at the interface nodes. However, this approach does not guarantee mass conservation when flow occurs at an angle to the interface. Cancés et al. (2009) study the convergence of finite volume scheme which takes account the effect of capillary pressure discontinuity. Wolff et al. (2012) and Bazrafkan et al. (2014) use cell-centred finite volume method for two-phase flow, but discontinuities in capillary pressure are not represented. On cell-centred multi-point flux approximation framework (CVD-MPFA, Friis et al. (2008)), Ahmed et al. (2019) formulate capillary flux approximation which implies van Duijn - de Neef condition Van Duijn et al. (1995), Hoteit and Firoozabadi (2008) successfully combined the mixed finite element approach with the discontinuous Galerkin method (DGM), capturing the saturation discontinuity in a mass conservative scheme. Ern et al. (2010) and Bastian (2014) subsequently pioneered the fully Discontinous Galerkin method. In the DGM, each node carries multiple saturation values. As a result of this increase in the degrees of freedom, the method requires much more computational resources than competing finite element-finite volume methods (FEFVM) (Kim and Deo (1999), Geiger et al. (2004), and Nick and Matthäi (2007)). In these hybrid methods, the rock is represented by a finite element mesh, which conforms to material interfaces, see Fig. 2(a). Porosity, permeability and relative permeability are treated as piecewise constant within each element, while pressure and saturation are discretized as piecewise linear on the finite element nodes. Therefore, undesirable smearing effects occurs between heterogeneous regions. Nick and Matthäi (2011a) improve this FEFVM by introducing interface elements, see Fig. 2(b). Where nodes coincide with material interfaces, they are replicated as many times as different materials are joint with one another at this location. Thus, the mesh is split along internal boundaries and then reconnected by interface elements. These lightweight elements act as a proxy which controls the mesh connectivity. This node-centred finite volume finite element discretization with embedded discontinuities (DFEFVM), can capture both pressure and saturation discontinuities precisely. All cited studies show that when capillary entry pressure is not considered, the interface condition of Van Duijn et al. (1995) is not satisfied.
Here, we extend the DFEFVM by an interface coupling algorithm that manages to reproduce the complex toggle-switch behavior discussed above. It will be shown that as a consequence, pressure and saturation are no longer artificially continuous nor smeared across the interface, respectively. The scheme has been developed with large capillary pressure contrasts between layers in mind, and the coupling algorithm for the disjointed mesh satisfies the interface conditions without any flux averaging. Part of the concept is a local time-stepping control framework to manage, localize and capture quick changes of physical variables across the interface. We show that this scheme is locally conservative, and interface conditions are satisfied at all times.
This paper is organized as follows: we first review the interface conditions and governing equations for slightly compressible, immiscible two-phase flow in porous media with material interfaces in detail. Second, we introduce the new numerical treatment of interface fluxes and decoupling/coupling behavior. Third, a range of test cases is presented and discussed, for model verification and validation with results from a recent physical experiment involving the flow of supercritical CO2 through a laminated drill core that was monitored real-time with X-ray tomography.
Section snippets
Methodology
For simplicity, the new scheme is described here only for incompressible, immiscible two-phase flow in heterogeneous porous media.
Test cases and applications
We have applied the new interface coupling scheme to a series of one-, two-, and three-dimensional models. First, we verify the code by comparing the numerical result with the semi-analytical solution. Second, the infiltration of DNAPL into a high permeability lens with different entry pressure test cases is conducted. The purpose of the test case is to verify that the scheme handles the interface condition correctly for both equilibrium and non-equilibrium capillary pressure cases. Next, a
Discussion
The DFEFVM has great potential for simulating two-phase flow in porous media with discontinuities (Nick and Matthäi (2011b), Nick and Matthäi (2011a)). Because flow domains can be modelled accurately using a minimal number degree of freedoms. However, Nick and Matthäi (2011a) did not yet demonstrate the strong effects of these discontinuities, when capillary pressure involved. The current study achieves this by integrating the interface conditions of Van Duijn and De Neef (1998) within the
Conclusion
In this work, we present a capillary interface coupling scheme for immiscible two-phase flow and transport in heterogeneous porous media, taking account discontinuities of material properties. This scheme is based on IMPES, entails a global pressure formulation and extends the finite element - finite volume method with embedded discontinuities by coupling/decoupling of variables at material interfaces. This is achieved in three steps: (a) observe the physical state of flow at the material
Declaration of Competing Interests
None.
CRediT authorship contribution statement
L.K. Tran: Conceptualization, Methodology, Software, Writing - original draft. J.C. Kim: Software. S.K. Matthäi: Supervision, Software, Writing - review & editing.
Acknowledgments
The authors wish to acknowledge the financial assistance provided through the Australian National Low Emissions Coal Research and Development (ANLEC R&D) (7-1115-0258). ANLEC R&D is supported by COAL21 Ltd and the Australian Government through the Clean Energy Initiative.
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