# Modelling Imbibition Processes in Heterogeneous Porous Media

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## Abstract

Imbibition is a commonly encountered multiphase problem in various fields, and exact prediction of imbibition processes is a key issue for better understanding capillary flow in heterogeneous porous media. In this work, a numerical framework for describing imbibition processes in porous media with material heterogeneity is proposed to track the moving wetting front with the help of a partially saturated region at the front vicinity. A new interface treatment, named the interface integral method, is developed here, combined with which the proposed numerical model provides a complete framework for imbibition problems. After validation of the current model with existing experimental results of one-dimensional imbibition, simulations on a series of two-dimensional cases are analysed with the presences of multiple porous phases. The simulations presented here not only demonstrate the suitability of the numerical framework on complex domains but also present its feasibility and potential for further engineering applications involving imbibition in heterogeneous media.

## Keywords

Porous media Imbibition Heterogeneity Interface## 1 Introduction

Imbibition as a typical type of capillary flow in porous media is a ubiquitous physical phenomenon, which has a wide range of applications from daily commodities, e.g. napkins and baby diapers, to advanced engineering applications, such as paper-based chromatography (Block et al. 2016), microfluidics for medical diagnosis (Liu et al. 2015; Tang et al. 2017), energy-harvesting devices (Nguyen et al. 2014), fractured reservoir (Meng et al. 2017), and oil recovery (Morrow and Mason 2001; Rokhforouz and Akhlaghi Amiri 2017). The research on imbibition phenomena starts from the pioneer work by Lucas (1918) and Washburn (1921) who proposed an analytical model for capillary rise in tubes, known as the Lucas–Washburn equation. Imbibition in porous media shares the same physical principle as that in capillary tubes, i.e. they both are capillary pressure driven liquid flow in essence, though the phenomena are more complicated in porous media. Adequately, modelling imbibition processes in porous media is required, in particular when dealing with heterogeneous porous media (Reyssat et al. 2009), i.e. with spatial distribution and variability of effective properties (Durlofsky 1991; Warren and Price 1961).

Motivated by limitations of the Lucas–Washburn equation, a series of modifications have been proposed, to consider the inertia (Quéré 1997), gravity (Fries and Dreyer 2008), evaporation (Fries et al. 2008), and shape and tortuosity of the pore space (Cai et al. 2014; Cai and Yu 2011). These modified theoretical models replenish the deviation between experimental results and predictions of the original Lucas–Washburn equation, but the corresponding solutions are only available in one-dimensional cases. For higher-dimensional cases, a few analytical solutions for homogeneous porous media of limited geometrical shapes are provided for radial penetration (Conrath et al. 2010; Liu et al. 2016), fan-shape membrane (Mendez et al. 2009), variable-width paper strips (Elizalde et al. 2015), semi-infinite domain (Xiao et al. 2012), and fractal porous media (Jin et al. 2017; Kun-Can et al. 2017). Recently, the combined effects of geometry and evaporation and gravity are also investigated (Liu et al. 2018; Xiao et al. 2018). Moreover, employing the nonlinear Richard’s equation, Perez-Cruz et al. (2017) developed a two-dimensional imbibition model that is suitable for arbitrary geometries of homogeneous porous media, i.e. effective properties are spatially independent.

Imbibition in heterogeneous porous media is encountered commonly across different scales. For example, at the macroscale, layered soils are typical heterogeneous porous media, which are composed of several layers of sediments characterised by grain and pore sizes (Al-Maktoumi et al. 2015; Zhuang et al. 2017). Other media may exhibit hierarchical structure across different length scales, including fractured reservoirs and soils containing double porosity (Di Donato et al. 2007; Lewandowska et al. 2005). When observed at the mesoscale, concrete materials are typical examples, since the main compositions include aggregates, cements, and C–S–H phase, each with distinct permeability. The overall permeability of concretes is determined by this mesoscale structure, which responds for barrier performance, concrete durability (Hall 1994; Hanžič et al. 2010; Navarro et al. 2006), etc. However, earlier work on overall flow properties of heterogeneous porous media (Durlofsky 1991; Pettersen 1987; Warren and Price 1961) focuses on effective permeability instead of detailed imbibition processes, e.g. the evolution of liquid front, while recent work is mostly limited to layered configuration (Bal et al. 2011; Debbabi et al. 2017; Ern et al. 2010; Guerrero-Martínez et al. 2017; Helmig et al. 2007; Patel and Meher 2017; Reyssat et al. 2009; Schneider et al. 2018). Reyssat et al. (Reyssat et al. 2009) conducted a research on the imbibition process of layered granular media experimentally and theoretically. Guerrero-Martínez et al. (2017) carried out a series of numerical simulations on a three-layer porous media and adopted a hyperbolic tangent function to treat the interfaces approximately. In addition, at microscale, Spaid and Phelan (1998) simulated the multicomponent flow in heterogeneous porous media through modified lattice Boltzmann method. Thus, it is necessary to extend the continuum numerical framework to more complex domains, in particular, solving problems with the presence of interfaces among distinct types of porous media.

In this work, a new model for imbibition in heterogeneous porous media is proposed based on the nonlinear Richard’s equation, with consideration of spatial distribution of properties. In particular, we develop an efficient and exact method for interface treatment, combined with which the proposed model provides a complete numerical framework for imbibition processes in arbitrarily heterogeneous porous media, including graded and stepwise structures. The developed numerical scheme is verified through comparing with existing experimental results. Finally, imbibition processes in various two-dimensional complex domains are simulated to demonstrate the versatility and suitability of the developed numerical framework.

## 2 Modelling

### 2.1 Liquid Front Tracking

The right hand of Eq. (10) is divided into two terms, in which the first term, centred with \( \nabla S \), describes liquid transfer induced by saturation gradient, i.e. naturally it can be regarded as a diffusion process. Since \( P_{{{\text{c}},\hbox{min} }} \left( {\mathbf{x}} \right) \) is related to the property of matrix materials as shown in Eq. (9), the second term is an additional flux induced by the non-homogeneity of matrix materials, while it is spontaneously eliminated for homogenous matrix materials.

### 2.2 Interface Treatment

Depending on the actual spatial distribution of the material properties, Eq. (10) can be applied to provide the evolution law for the local degree of saturation. However, for some special cases with the presence of interfaces, e.g. in layered porous materials (Guerrero-Martínez et al. 2017; Reyssat et al. 2009), singularity needs to be handled in the numerical solutions. This singularity appears when differentiating a step function at the interface where the corresponding material property has a sharp change. Conventional solutions to deal with this type of problems include (1) smoothing the step function and (2) requiring special mesh treatment at the vicinity of the interface, which are difficult for higher-dimensional cases and likely result in mesh-dependent predictions for imbibition processes. In this work, based on the weak form of Eq. (10), we propose a new interface treatment approach, named the *interface integral method*, which is not only suitable for simple one-dimensional cases but also for the higher-dimensional interfaces of complex geometry.

### 2.3 Numerical Realisation

In this work, COMSOL Multiphysics^{®}, a partial differential equation solver based on finite element method, was adopted to obtain the numerical solution of Eq. (10). Special treatment has been introduced in WEAK CONTRIBUTION module to implement the weak form for interface regimes.

To summarise, we develop a complete and generalised model for imbibition processes, and this proposed model is applicable to various types of homogeneous and heterogeneous porous media when \( k_{\hbox{max} } \left( {\mathbf{x}} \right) \) and \( P_{{{\text{c}},\hbox{min} }} \left( {\mathbf{x}} \right) \) are defined accordingly. Moreover, an interface integral method is also proposed to consider the presence of interfaces in the heterogeneous porous media. In order to verify the practicability of the proposed model and present its potential applications, several numerical cases are provided in the following sections.

## 3 Numerical Validations and Examples

In this section, we validate the proposed imbibition model against available experimental data and further test more generalised numerical examples. First, in Sects. 3.1 and 3.2, the model is validated by two one-dimensional experimental cases from Reyssat et al. (2009), where the flow direction was predefined and aligned with the orientation of the changing properties. Subsequently, typical examples of two-dimensional cases are used to demonstrate the potential of this numerical scheme. Several parameters regarding material properties in Eq. (10) are shown as follows: \( \phi = 0.38 \), \( \gamma = 0.02 \) N/m, \( \mu = 0.01 \) Pa s, \( c^{*} = 0.02 \), and \( \lambda = 0.1 \). Here, these are typical for silicon oil, as an example, but other settings are also feasible.

### 3.1 Granular Media with Permeability Gradients

*x*-direction. From Eq. (7), we can have the relation for the grain size as \( d\left( x \right) = \sqrt {\frac{{180 \cdot \left( {1 - \phi } \right)^{2} \cdot \left( {k_{0} + \beta \cdot x} \right)}}{{\phi^{3} }}} \), and then substitute it into Eq. (9). The numerical solutions for Eq. (10) are presented in Fig. 3c and compared with corresponding experimental results (Reyssat et al. 2009).

As shown in Fig. 3c, the numerical results of the proposed model are in good agreement with experimental measurements of the liquid front for granular materials with different permeability gradients (\( \beta \)), and specifically the relative errors of these cases are less than 4%. Furthermore, the numerical solutions of Eq. (10) capture the intrinsic imbibition behaviour in gradually changed porous media, i.e. (1) a log–log linear relationship between the position of liquid front and imbibition time, and (2) a significantly higher advancing speed of the liquid front in granular medium with decreasing grain sizes than the one with increasing grain sizes along the flow direction. For the latter, this demonstrates the significance of the second term on the left hand side of Eq. (10) for graded porous media.

### 3.2 Two-Layer Granular Materials

It can be seen from Fig. 4c that simulation results also exhibit decent agreement with the experimentally measured liquid front motion and the maximum relative error of these cases is less than 14%. In Reyssat et al. (2009), the authors also provided a piece-wise theoretical solution for two-layered cases, in which the advancing speed of liquid front drops since the flux continuity condition at the interface was ignored, and thus violating the mass conservation law. However, for the current model, the continuity condition is satisfied at the interface automatically, via Eq. (10), so that both the position and advancing speed of the liquid front maintain continuous at the interface.

Through the numerical examples and their comparisons to the experimental data in previous and current sections, it is shown that the kinetics of the wetting front is associated with the actual spatial distributions of the measurable material properties, e.g. pore (grain) size. The numerical cases presented here not only verified the proposed simulation scheme including governing equations and interface treatments, but also lay the root for more complex applications since interface issues are ubiquitous in various fields.

### 3.3 Regions with Oblique Interfaces

Figure 5a, b shows different spatial arrangements of porous regions, PM1 with characteristic pore size \( d_{1} \) = 41 μm and PM2 with \( d_{2} \) = 196 μm, which generate different imbibition patterns. The boundary conditions were prescribed as follows: (1) Dirichlet boundary condition, \( S = 1 \), is applied at the bottom edge, and (2) zero-flux conditions are imposed on the other three edges. The gravity term is ignored in this example. First of all, the soaking speed of Case I (the smaller pore size region, PM1, on the top) is significantly faster than that of Case II (the larger pore size region, PM2, on the top) through comparing the time it takes to form final configures, i.e. the whole imbibition rate is determined by the local property of porous media close to the reservoir. As for the liquid front across the interface, in Case I, a “drag effect” was observed for the imbibition process when the liquid front invades from the region with a larger pore size (PM2) to PM1, and tangent lines of liquid fronts at the boundary of two regions form a convex angle (marked in yellow in Fig. 4). While reversely in Case II, the imbibition in PM1 region impedes that in PM2, and thus, a concave angle (marked in green) is formed at the interface, i.e. a “pull effect”.

### 3.4 Regions with Inclusions

## 4 Typical Heterogeneous Porous Media

*U*-shaped curve with a minimum value around \( \gamma = 1 \). When compared with the upper and lower bound predictions, the cases of pore size ratio smaller than one, \( D_{\text{eff}} \), tend to approximate the upper bound, while the lower bound gives a better estimation of \( D_{\text{eff}} \) when pore size ratio is larger than one. Thus, this work can be extended as a numerical tool for solving inverse problems and predicting the effective diffusivity of composite materials, utilising the spontaneous imbibition processes.

## 5 Discussion

Based on the proposed numerical framework, imbibition processes are effectively simulated in heterogeneous porous media ranging from one-dimensional to higher-dimensional cases. In this framework, the liquid front is captured by implicitly imposing a narrow transition zone, and a special interface treatment has also been proposed and implemented for cases with the presence of material interfaces.

Generally, for layered porous media, the interfaces cannot be ignored and it brings certain effects on the imbibition process, such as additional flow resistance shown in Sect. 3.2. Also, the configuration shown in Sect. 3.3 can be adopted to simulate the groundwater flow and the resultant effective stress states for certain slopes where the interfaces between layered soils may be non-parallel to the underwater level (Al-Maktoumi et al. 2015; Zhuang et al. 2017). In fact, the interface issue often occurs in various types of porous media. For concrete, as a typical example, aggregates, cements, and C–S–H phases have typical porous media interfaces due to their distinct permeability, and the numerical cases shown in Sect. 3.4 can work as a simplified REV model to analyse barrier performance of concrete structures (Hall 1994; Hanžič et al. 2010; Navarro et al. 2006). Besides interface issues, the porous media with gradually changed permeability also presents special flow pattern and liquid front dynamics, as shown in Sect. 3.1, which can guide the design of paper-based microfluidic chips (Block et al. 2016; Nguyen et al. 2014; Tang et al. 2017). The example presented in Sect. 4 is a typical heterogeneous porous medium, which can be used to analyse soil containing two sub-domains characterised by contrasted pore sizes (Lewandowska et al. 2005), and the present approach provides rich information from the microstructural heterogeneity. Based on the analysis in Sect. 4, a microstructure informed prediction is developed, i.e. the effective diffusive parameter of the whole heterogeneous structure can be estimated under the proposed framework, as an analogue to the spontaneous imbibition experiments of fibrous composites (Cai et al. 2012) and carbonate rocks (Alyafei and Blunt 2018).

## 6 Conclusion

In this work, we develop a comprehensive numerical framework to solve imbibition problems in heterogeneous porous media. Detailed wetting front dynamics has been captured by introducing the partially saturated transition zone at the front vicinity. To extend the method to more complex domains, special interface treatments, i.e. the interface integral method, have also been proposed and implemented. To validate the proposed numerical model, simulations are compared with experimental results of layered porous media and good agreements are achieved. The dynamics of wetting front is found to be associated with the spatial distributions of the material properties, e.g. pore size. Furthermore, combined with the proposed interface treatment, the model is capable of predicting imbibition processes in porous media with arbitrary topology. The proposed framework in this paper contributes to discover the underlying physics behind imbibition processes in heterogeneous materials and to improve quantitative and microstructure informed perditions for capillary flow-related phenomena in various fields.

## Notes

### Acknowledgements

This work was financially supported by Australian Research Council (Projects DP170102886) and The University of Sydney SOAR Fellowship.

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