Kinematic closures for two-fluid-phase flow in porous media: interfacial state equations, the interfacial velocity term, and slow-relaxation effects on imbibition and nonwetting-phase entrapment

Gray, Dye, McClure, Pyrak-Nolte and Miller (2015) formulated a macroscale model for two-fluid-phase flow in porous media that couples conservation equations with kinematic equations for the fluid phases, the fluid-fluid interface, and the common curve (the line along which the two fluids and the solid phase meet), so that saturation, fluid pressures, capillary pressure, and specific interfacial area all evolve dynamically. Its implementation is blocked by three linked closure gaps, identified in the source and in later restatements by the same group: (1) state equations for the macroscale capillary pressure $p^c=p^c(s^w,\epsilon^{wn})$ and for the equilibrium specific interfacial area $\epsilon^{wn}_{\mathrm{eq}}(p^n,p^w,s^w)$ were unavailable, because interfacial relaxation toward equilibrium is extremely slow and interfacial curvatures and areas are hard to measure in three-dimensional porous media; (2) the interfacial velocity-like term $\mathbf{w}^{wn}$ in the interfacial-area evolution equation is difficult to approximate for general flow conditions, and its role when the interface undergoes net convection is unexplored; (3) the effect of this slow interfacial relaxation on imbibition displacement and nonwetting-phase entrapment is unknown, since most existing pressure-saturation data are not recorded at true equilibrium. Later work partially resolved component (1): a non-hysteretic geometric state function relating capillary pressure (mean curvature), saturation, interfacial area, and the Euler characteristic (a topological invariant of the fluid phase) has been formulated and validated on idealized and several natural systems, which also shows that the original two-variable form $p^c(s^w,\epsilon^{wn})$ is incomplete. Relaxation timescales of hours have been measured by fast X-ray tomography, and empirical interfacial-area-saturation relations conditional on relaxation history exist. However, general validated state equations for arbitrary media and flow conditions, an accepted closure and role for $\mathbf{w}^{wn}$ under net interfacial convection (recent implementation attempts remain contested), and a quantitative, predictive account of how slow relaxation biases imbibition displacement and nonwetting-phase entrapment estimates all remain open.

Background

In a porous medium saturated by two immiscible fluids, the wetting fluid ($w$, e.g. water) preferentially coats the solid while the nonwetting fluid ($n$, e.g. air or oil) occupies the pore centers. The macroscale state is usually described by the wetting-phase saturation $s^w$ (volume fraction of pore space occupied by the wetting fluid), the fluid pressures $p^w$ and $p^n$, and the capillary pressure $p^c=p^n-p^w$. For a century, constitutive behavior has been represented by an equilibrium capillary pressure-saturation curve $p^c(s^w)$, which is famously hysteretic: drainage and imbibition follow different curves, so the relation is not a state function of saturation alone. Hassanizadeh and Gray (1990) argued that interphase boundaries should enter the theory explicitly, motivating the description of the fluid-fluid interface by its specific interfacial area $\epsilon^{wn}$ (interfacial area per unit bulk volume of porous medium). Reeves and Celia (1996) computed a surface $p^c(s^w,\epsilon^{wn})$ from a pore-scale network model (a lattice of idealized pore bodies and throats representing the pore space), and dynamic pore-network modeling by Joekar-Niasar, Hassanizadeh and Dahle showed that interfacial area relaxes toward equilibrium slowly and that capillary pressure is a non-equilibrium quantity during flow, leading Joekar-Niasar and Hassanizadeh to call specific interfacial area the 'missing state variable' of two-phase flow equations. The thermodynamically constrained averaging theory (TCAT) of Gray, Miller and coworkers derives macroscale two-fluid models by rigorous averaging of microscale conservation and thermodynamic relations. Gray, Dye, McClure, Pyrak-Nolte and Miller (2015) extended this program with macroscale kinematic equations (equations for the motion and geometry of phases and interfaces derived from averaging theorems rather than conservation principles). Including conservation and kinematic equations for the interface itself yields a novel closed-form model in which $s^w$, $p^w$, $p^n$, $p^c$, and $\epsilon^{wn}$ all obey evolution equations; the interfacial-area evolution equation contains a macroscale interfacial velocity-like term $\mathbf{w}^{wn}$ (an averaged velocity of the $w$-$n$ interface). Using microfluidic experiments and lattice Boltzmann simulations (a mesoscopic simulation method that resolves fluid interfaces at the pore scale), they showed that relaxation of interface shape and distribution toward equilibrium is slow compared with the time typically allowed before pressure-saturation data are declared 'equilibrium', so most such data sample a still-evolving system. The authors therefore identified the closures needed to make the kinematic model usable. Subsequent work sharpened the state-variable question: McClure et al. (2018) formulated a geometric state function relating average mean curvature (equivalently capillary pressure), saturation, interfacial area, and the Euler characteristic $\chi$ (a topological invariant that counts, e.g. for the nonwetting phase, the number of connected blobs minus the number of loops/handles), and Miller et al. (2019) demonstrated a non-hysteretic capillary pressure state equation of this extended type on model systems. Synchrotron-based X-ray micro-computed tomography ($\mu$CT, which images fluid configurations in 3D inside real rock samples) now permits in-situ measurement of curvature, interfacial area, and topology under quasi- and non-equilibrium flow, so the ingredients needed to attack the closures exist.

Problem Statement

Close and validate the kinematic model for two-fluid-phase flow in porous media formulated by Gray et al. (2015) by resolving the three constitutive and kinematic gaps its authors identified. First, determine which state variables are required, and establish the macroscale state equations for the capillary pressure $p^c$ and the equilibrium specific interfacial area $\epsilon^{wn}_{\mathrm{eq}}$ as validated, parameterizable relations for general three-dimensional porous media. Second, construct an approximation for the macroscale fluid-fluid interfacial velocity vector $\mathbf{w}^{wn}$ appearing in the model's interfacial-area evolution equation - the equation containing the term $\nabla\cdot(\epsilon^{wn}\mathbf{w}^{wn})$, restated as equation (4) of Bruning & Miller (2019) - that is valid for general flow conditions, and determine the role this term plays when the fluid-fluid interface undergoes net convection rather than merely relaxing in place. Third, quantify the effect of the observed slow interfacial relaxation on imbibition displacement and on nonwetting-phase entrapment - in particular, how predictions of trapped nonwetting saturation and displacement patterns are biased when conventional pressure-saturation data, which are sampled before true equilibrium is reached, are treated as equilibrium data. These three components jointly support one objective: an implementable, thermodynamically and kinematically consistent two-fluid-phase model whose interfacial state evolves correctly and whose trapping predictions survive validation against independent experiments or high-resolution simulations. The model is closed when these state equations and closures are available, when each state equation's domain of validity has been established by transfer to real and idealized three-dimensional porous media beyond those used to construct it - rather than by fit to the construction set alone - and when the complete resultant macroscale model has been evaluated and validated against independent experiments, macroscale data sets, or high-resolution simulations.

References

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