Homogenized Lattice Boltzmann Model

A new homogenization approach for the simulation of multi-phase flows in heterogeneous porous media is presented. It is based on the lattice Boltzmann method and combines the grayscale with the multi-component Shan-Chen method. Thus, it mimics fluid-fluid and solid-fluid interactions also within pores that are smaller than the numerical discretization. The model is successfully tested for a broad variety of single- and two-phase flow problems. Additionally, its application to multi-scale and multi-phase flow problems in porous media is demonstrated using the electrolyte filling process of realistic 3D lithium-ion battery electrode microstructures as an example. The new method shows advantages over comparable methods from literature. The interfacial tension and wetting conditions are independent and not affected by the homogenization. Moreover, all physical properties studied here are continuous even across interfaces of porous media. The method is consistent with the original multi-component Shan-Chen method. It is accurate, efficient, easy to implement, and can be applied to many research fields, especially where multi-phase fluid flow occurs in heterogeneous and multi-scale porous media. Keywords: two-phase flow, transport in porous media, Darcy, Brinkman, Buckley-Leverett, Washburn, Shan-Chen 

 

Introduction Fluid flow in porous media plays an important role in many technical and natural processes such as hydrogeology, reservoir and process engineering, electrochemical energy storage, or medical applications. Most of these examples involve complex flow phenomena such as transport of solutes, reactions, or the interaction of multiple phases or immiscible fluid components [1–6], structures that are heterogeneous regarding their chemical composition and wetting properties [2, 5, 7, 8], and pore sizes that range from nanometers to the macroscale [7–14]. Thus, and because most of the interesting physical phenomena happen on the pore scale, they are hard to study experimentally [2, 9, 14, 15]. Therefore, in the literature often direct numerical simulations and more specifically the lattice Boltzmann method (LBM) are used to conduct pore-scale simulations. LBM is a reliable tool for studying multi-scale and multi-physics transport processes within complex porous geometries [16, 17]. It has also been successfully applied to solve multiphase flows in high-resolution real-world image data of porous media samples that were recorded using X-ray micro-computed tomography (µ-CT) or focused ion beam scanning electron microscopy (FIB-SEM) [4, 7–9, 11, 12, 14, 17, 18].

 

Unfortunately, LBM is computationally expensive, especially when simultaneously simulating flow in structurally resolved pores at different length scales. Therefore, homogenization methods have been developed, where the detailed structure of pores at the smallest length scale is ignored and, instead, the flow is described by a Darcy-Brinkman-type approach. A volume average of the structurally resolved geometry is taken and its effects on fluid flow are mimicked as permeability-related parameter. These homogenization methods can be basically subdivided into two groups. Those are the Brinkman force-adjusted models (BF) [12, 15, 19–24], where a drag force is applied locally, and the grayscale models (GS) [24–33], where flow populations are partially bounced back to mimic flow resistance. Although the aforementioned homogenization methods have been heavily discussed and further developed for single-phase fluids [24, 27–29, 32, 33], this is not the case for multi-phase or multi-component fluids. Only a few methods combining GS and multiphase physics [34–36] as well as methods combining other homogenization approaches with multi-component physics [37, 38] have been reported recently. However, despite 2 the fact that the multi-component Shan-Chen method (MCSC) is most widely used for studying all kinds of immiscible fluids, only one homogenized method has been developed combining GS with MCSC [39]. This method is however not fully consistent with the original MCSC and shows deficiencies with respect to heterogeneous porous media.

 

Therefore, in the current paper, a new model is presented, that follows the approach of Pereira [39] and combines GS by Walsh et al. (GS-WBS) [28] with MCSC [40]. It is therefore called the homogenized multi-component Shan-Chen method (HMCSC) in the following. GS-WBS is chosen as it is known to recover Darcy–Brinkman flow, conserves mass, and allows an efficient computational parallelization as only local bounce-back operations are performed. MCSC uses a physically-based approach to model fluid-fluid and solid-fluid interactions without the need for interface tracking. It also achieves a good compromise between computational efficiency and physical reality, and thus is widely adopted for modeling immiscible fluids. The HMCSC inherits all positive features from the aforementioned models, but overcomes their deficiencies which are mainly related to the discontinuity of properties in heterogeneous porous media. For example, using HMCSC, the interfacial tension and the wetting properties are constant and not affected by the homogenization. Thus, especially the MCSC-related model parameters can be chosen consistently to the original MCSC and no further parametrization is required. Besides, the new method switches freely between free-flow and Darcy regime, and can be also applied to study single-phase flows. As part of this paper, the HMCSC was rigorously tested for a broad variety of singlephase and two-phase flow benchmark cases that are relevant in the context of porous media. Those were Stokes-Brinkman-Darcy flow under Couette and Poiseuille conditions, fluid flow in stratified heterogeneous porous media and partially porous channels, as well as steady bubble tests and Washburn-type capillary flow. It was also shown to predict Buckley-Leverett waterflooding to some extent (cf. Section S2 in the Supporting Information). The results were compared with analytical and semi-analytical solutions where available.

 

Finally, the HMCSC was applied to a two-phase flow issue of current research interest in the field of electrochemical energy storage: The electrolyte filling of lithiumion battery microstructures with partially permeable nanoporous components. However, other research fields where multi-phase fluid flow occurs in multi-scale porous media can 3 benefit from the new method, too. In the context of hydrology, geoscience and petroleum engineering, potential applications are the prediction of microbiologically affected groundwater flow [41, 42], geologic carbon storage or sequestration [43, 44], and the recovery of oil, dry natural gas, or shale gas from tight gas sandstones [14, 45], carbonates [14, 45] and shale formations [13], respectively. This paper is organized as follows. In Section 2, the HMCSC is described. In Section 3, it is tested for a broad variety of benchmark cases and the corresponding results and features of the method are discussed. In Section 4, the results of the electrolyte filling simulations in realistic and partially homogenized battery microstructures are presented. Finally, conclusions are drawn in Section 5.