Stable and Efficient Time Integration of a Dynamic Pore Network Model for Two-Phase Flow in Porous Media

Abstract

We study three different time integration methods for a dynamic pore network model for immiscible two-phase flow in porous media. Considered are two explicit methods, the forward Euler and midpoint methods, and a new semi-implicit method developed herein. The explicit methods are known to suffer from numerical instabilities at low capillary numbers. A new time-step criterion is suggested in order to stabilize them. Numerical experiments, including a Haines jump case, are performed and these demonstrate that stabilization is achieved. Further, the results from the Haines jump case are consistent with experimental observations. A performance analysis reveals that the semi-implicit method is able to perform stable simulations with much less computational effort than the explicit methods at low capillary numbers. The relative benefit of using the semi-implicit method increases with decreasing capillary number Ca, and at Ca ∼ 10 −8 the computational time needed is reduced by three orders of magnitude. This increased efficiency enables simulations in the low-capillary number regime that are unfeasible with explicit methods and the range of capillary numbers for which the pore network model is a tractable modeling alternative is thus greatly extended by the semi-implicit method. method [2, 3] to keep track of the fluid interface locations, have been used. The lattice-Boltzmann method is another popular choice, see e.g. [4]. These methods can provide detailed information on the flow in each pore. They are, however, computationally intensive and this restricts their use to relatively small systems.Pore network models have proven to be useful in order to reduce the computational cost [5], or enable the study of larger systems, while still retaining some pore-level detail. In these models, the pore space is partitioned into volume elements that are typically the size of a single pore or throat. The average flow properties in these elements are then considered, without taking into account the variation in flow properties within each element.Pore network models are typically classified as either quasi-static or dynamic. The quasi-static models are intended for situations where flow rates are low, and viscous pressure drops are neglected on the grounds that capillary forces are assumed to dominate at all times. In the quasi-static models by Lenormand et al. [6], Wilkinson and Willemsen [7] and Blunt [8], the displacement of one fluid by the other proceeds by the filling of one pore at the time, and the sequence of pore filling is determined by the capillary entry pressure alone.The dynamic models, on the other hand, account for the viscous pressure drops and thus capture the interaction between viscous and capillary forces. As three examples of such models, we mention those by Hammond and Unsal [5], Joekar-Niasar et al.[9] and Aker et al. [10]. A thorough review of dynamic pore network models was performed by Joekar-Niasar and Hassanizadeh [11].The pore network model we consider here is of the dynamic type that was first pre...

Pore Network Modeling of the Effects of Viscosity Ratio and Pressure Gradient on Steady-State Incompressible Two-Phase Flow in Porous Media

“…In this section, we briefly describe the pore network model used in this study. For a more detailed description of the model and the numerical methods used to solve it, the reader is referred to [13]. An in-depth discussion of a slightly different model, which is also of the Aker type [7], can be found in [14].…”

“…Close to the nodes, they are subject to additional models that account for interface interactions in the nodes. This is described in [13].…”

“…We present results from more than 6000 steady-state simulations, that cover a large range of viscosity ratios and capillary numbers. The chosen pore network model is also of the Aker type [7], specifically the variant described by Gjennestad et al [13]. Other variants of the Aker model can be found in [8][9][10]14].…”

“…First, it is dynamic and thus captures the effects of both viscous and capillary forces. Second, it can be solved in a numerically stable manner at arbitrarily low capillary numbers [13]. Third, it is possible to apply periodic boundary conditions, keeping the saturation constant and eliminating effects of saturation gradients.…”

“…In the simulations, we utilize a recent innovation in the numerical solution method [13] to perform numerically stable simulations at low and moderate capillary numbers. The new methodology has an important effect at capillary numbers below 10 −3 .…”