Microscopic theory of capillary pressure hysteresis based on pore-space accessivity and radius-resolved saturation
Introduction
From rocks and wood to concrete and catalysts, porous materials vary widely in origin, properties, and applications. Despite their macroscopic appearance as solid objects, porous media are distinguished by their ability to contain fluids internally, owing to their heterogeneous microstructure – the solid matrix occupies only a portion of the macroscopic domain, while the complementary pore space is able to accommodate one or more fluid phases (Bear, 1972).
In the typical case of well-connected pores, diverse physical phenomena in porous media, such as fluid flow, heat and mass transfer, gas adsorption, and phase transformations, may be amenable to homogenized macroscopic descriptions (Torquato, 2013), although the exact connection with microscopic details of the porous medium is not always clear. Remarkably, any continuum model implicitly assumes that the overall effects of the often nontrivial pore-space morphologies (Sahimi, 2011) can be encapsulated in a small number of parameters, e.g., porosity, , tortuosity, , intrinsic permeability, , effective thermal conductivity, , etc., and the state of any fluid phase can be described by a small number of distributed state variables, e.g., pressure, p, saturation, s, etc. For certain simple physical processes, especially those involving a single fluid phase, simple continuum formulations generally work well, and a rigorous connection between the pore-scale and continuum-scale governing equations can be sought – examples include single-phase flow (Allaire, 1989, Hornung, 2012) and heat transfer (Loeb, 1954, Carson et al., 2005) – although estimating the transport coefficients from microscopic features of the porous medium is still an open research problem (Ranut, 2016, Pabst and Gregorova, 2017, Neithalath et al., 2010, Chareyre et al., 2012, Mostaghimi et al., 2013).
In comparison, it is considerably more challenging to develop continuum models for immiscible multiphase flow in porous media (including unsaturated flow and condensate transport) (Bear, 1972, Scheidegger, 1974, Richards, 1931, Buckley and Leverett, 1942, Childs and Collis-George, 1950, Klute, 1952, Tamon et al., 1981, Lee and Hwang, 1986, Jaguste and Bhatia, 1995, Do and Do, 2001) based on microscopic physics. While it is possible to upscale pore-scale equations by careful averaging (Durlofsky, 1998, Arbogast, 2000, Cushman et al., 2002, Wood, 2009, Li et al., 2006), the resulting model varies depending on the macroscopic state variables selected and the scaling laws assumed for the application considered, not to mention that the mechanisms for pore-scale fluid motions are highly complex and are still actively researched (Lenormand et al., 1983, Pak et al., 2015, Holtzman and Segre, 2015, Zhao et al., 2016). Thus, in formulating continuum models of multiphase flow in porous media, there exists a trade-off between mathematical simplicity and consideration of pore-scale physics. In conventional models that are widely accepted in practice, saturation is the primary state variable; it is used to compute capillary pressure (Brooks and Corey, 1964, van Genuchten, 1980) and relative permeabilities (Corey, 1954, Irmay, 1954, Brooks and Corey, 1964) via empirical constitutive relationships, which are able to fit typical experimental measurements by virtue of having several adjustable parameters (Pinder and Gray, 2008). By recognizing that pore-scale phenomena like viscous flow and capillary equilibrium must first and foremost depend on the microscopic dimensions of pores, some of the most popular continuum constitutive relationships in the literature have also incorporated the concept of a pore-size distribution (Thomson, 1872, Washburn, 1921, Burdine, 1953, Mualem, 1976), conceptualizing the pore space as interconnected pores of various sizes, either implicitly or explicitly (van Brakel, 1975, Quiblier, 1984), such as in the well-known “capillary bundle” model.
However, these conventional models suffer from hysteresis, meaning that the relationships between state variables are non-unique and history-dependent. This suggests that saturation alone cannot fully describe the microscopic state of fluid phases in real porous media, and additional macroscopic state variables are required to capture hysteresis. Many authors have considered extending the conventional constitutive relationship between capillary pressure and saturation, , by assuming that may also depends on the “rate of saturation”, (or at least its sign) (Barenblatt, 1971, Luckner et al., 1989, Hassanizadeh and Gray, 1990, Barenblatt et al., 2003, Juanes, 2008), thereby attributing hysteresis to nonequilibrium effects. Some authors have identified “specific interfacial area”, , or the interfacial area between fluid phases w and n per unit volume of the porous medium, as a physically relevant state variable on thermodynamic grounds, and have advocated for its inclusion in continuum models to reduce capillary pressure hysteresis (Kalaydjian, 1987, Hassanizadeh and Gray, 1990, Hassanizadeh and Gray, 1993, Beliaev and Hassanizadeh, 2001, Hassanizadeh et al., 2002). This hypothesis seems to hold in many but not all cases, as revealed by micromodel experiments (Cheng et al., 2004, Chen et al., 2007, Pyrak-Nolte et al., 2008, Karadimitriou et al., 2014), lattice-Boltzmann simulations (Porter et al., 2009), pore-network simulations (Reeves and Celia, 1996, Held and Celia, 2001, Joekar-Niasar et al., 2008, Joekar-Niasar and Hassanizadeh, 2012), and analysis (Helland et al., 2007), while the exact form of any new constitutive relationships required may not be completely clear. More recently, Hilfer put forth a new class of continuum models for two-phase flow in porous media (Hilfer, 1998, Hilfer, 2006, Doster et al., 2010) involving four fluid saturation variables, , as opposed the conventional two, , for the two fluid phases (in either case, all saturation variables must sum up to unity); we have and , where and correspond to “percolating regions” of the respective fluid phases, and and correspond to “non-percolating regions”. By differentiating between the contributions of percolating and non-percolating fluid “subphases”, Hilfer’s model naturally predicts hysteresis as a result of the dynamics of the newly introduced state variables. Because the model is not derived from the principles of microscopic physics, phenomenological assumptions are still required to, say, model the “mass transfer rates” between and , or and (namely, percolating fluid regions becoming non-percolating and vice versa), which result in model parameters that may be difficult to physically interpret, though potentially determinable from experiments.
It appears that the intrinsic complexity of multiphase flow in porous media would ensure that any continuum model to come in the foreseeable future will not match the performance of pore-scale methods (pore-network, Lattice-Boltzmann, phase-field, etc.; see this review (Meakin et al., 2009) for instance) in terms of either predictability or connection to first principles. On the one hand, conventional models are preferred for macroscopic simulations and used routinely in practical applications, where hysteresis is either described empirically or neglected altogether, although it is considered crucial for making certain types of predictions (e.g., Essaid et al., 1993). On the other hand, new continuum models, despite having rightly introduced new and physically meaningful state variables (, Hilfer’s ) so as to naturally predict hysteresis, deviate significantly from conventional models, and involve somewhat unintuitive constitutive laws with phenomenological constants that lack a clear connection to the pore-scale descriptions, possibly due to the emphasis placed on reproducing certain macroscopic observations.
In this work, we take the view that there is great value in identifying new physically meaningful concepts that are relevant to continuum modeling of multiphase flow in porous media. These concepts should be connected to essential aspects of pore-scale physics, yet intuitive enough for a wide range of continuum-scale applications – the pore-size distribution would fall within this category. In the short term, these concepts may be incorporated into conventional continuum models for incremental improvements, while in the long term, they may be subject to pore-scale investigations, and ultimately play a role in future continuum models of multiphase flow.
The first concept we shall propose is the “pore-space accessivity”, denoted by . Accessivity is a continuum-scale property of porous media that, in the simplest possible fashion, contrasts serial and parallel arrangements of different sized pores. This extends the capillary bundle model, which is known to be insufficient, but still routinely invoked because of its simplicity (Diamond, 2000, Hunt et al., 2013). The capillary bundle picture coincides with the limit in our framework.
The second concept we shall propose is the “radius-resolved saturation”, , where is the cumulative distribution function (CDF) of the pore-size distribution, where r denotes the pore radius. Radius-resolved saturation would replace saturation as a better continuum-scale descriptor for the distribution of fluid phases in the pore space, where “saturation” is now defined for pores of each particular size given by F, hence “radius-resolved”.
The paper is organized as follows. In Sections 2 Characterization of pore-space morphology, 3 Characterization of fluid distribution, we introduce the concepts of accessivity and radius-resolved saturation, and relate them to existing ideas in the literature. In Section 4, we present a simple statistical theory based on pore branching that leads to simple governing equations involving the proposed concepts. In Section 5, we present simple illustrative examples to highlight the usefulness and limitations of our theory. Finally, we discuss the broader utility of accessivity and radius-resolved saturation and identify outstanding questions and future research directions in Section 6, before concluding in Section 7.
Section snippets
Characterization of pore-space morphology
In this section, we consider macroscopic descriptors for the pore-space morphology of a porous medium. For that purpose, the pore space need not be filled with any particular fluid and can be left “empty” (or, alternatively and conceptually equivalently, filled uniformly with an inert fluid). We will briefly review some existing descriptors and introduce pore-space accessivity at the end of the section.
Characterization of fluid distribution
In this section, we will consider a porous medium whose pore space now hosts multiple fluids. The fluid phases may distribute in any arbitrary fashion that is consistent with pore-scale physics. Our goal is to characterize the distribution of fluid phases at the macroscopic scale. The concepts we propose here are general and can be readily extended to systems involving any number of immiscible fluids, though for simplicity, we shall confine our discussion to the case of two fluid phases, which
Statistical theory
So far, we have proposed accessivity, , as a continuum-scale descriptor for the connectivity of different sized pores, and radius-resolved saturation, , for the distribution of immiscible fluid phases in the pore space. We have qualitatively demonstrated that, in a porous medium with , serial connections between different sized pores may contribute to the accumulation of “metastable” fluids (i.e., wetting phase in “larger” pores, or nonwetting phase in “smaller” pores, relative to a
Illustrative examples
In this section, we present several exploratory case studies to illustrate the implications of our theory. We will examine the behaviors of the key formulae derived, including: rules for updating radius-resolved saturations during arbitrary drainage-imbibition cycles, Eqs. (54), (55); their simplified forms for primary drainage and imbibition, Eqs. (50), (51); and the resulting formulae for conventional saturations, (43) and (45), respectively. Of particular interest is the role of accessivity,
Further discussions
In this section, we will further expound the conceptual usefulness as well as limitations of our theory. We will discuss the broader utility of accessivity and radius-resolved saturation, including how they may be generalized.
Conclusions
We have proposed the pore-space accessivity, , as a continuum-scale parameter for describing the arrangement of different sized pores in porous media. Defined quantitatively based on the amount of hysteresis observed in macroscopic invasion percolation processes, can also be interpreted geometrically as the ratio of the length (or volume) scale for pore radius variation to the typical size of a pore instance (the pore space explored by an average meniscus in a control volume) observed at
Acknowledgement
The authors wish to acknowledge funding from Saudi Aramco, a Founding Member of the MIT Energy Initiative. We are also grateful to Samuel J. Cooper for useful discussions.
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