Upscaling Multiphase Flow in Porous Media

Traditional two-phase flow models use an algebraic relationship between capillary pressure and saturation. This relationship is based on measurements made under static conditions. However, this static relationship is then used to model dynamic conditions, and evidence suggests that the assumption of equilibrium between capillary pressure and saturation may not be be justified. Extended capillary pressure-saturation relationships have been proposed that include an additional term accounting for dynamic effects. In the present work we study some of the underlying pore-scale physical mechanisms that give rise to this so-called dynamic effect. The study is carried out with the aid of a simple bundle-of-tubes model wherein the pore space of a porous medium is represented by a set of parallel tubes. We perform virtual two-phase flow experiments in which a wetting fluid is displaced by a non-wetting fluid. The dynamics of fluid-fluid interfaces are taken into account. From these experiments, we extract information about the overall system dynamics, and determine coefficients that are relevant to the dynamic capillary pressure description. We find dynamic coefficients in the range of 10² – 10³ kgm⁻¹ s⁻¹, which is in the lower range of experimental observations. We then analyze certain behavior of the system in terms of dimensionless groups, and we observe scale dependency in the dynamic coefficient. Based on these results, we then speculate about possible scale effects and the significance of the dynamic term.

We show how to predict flow properties for a variety of rocks using pore-scale modeling with geologically realistic networks. The pore space is represented by a topologically disordered lattice of pores connected by throats that have angular cross-sections. We successfully predict single-phase non-Newtonian rheology, and two and three-phase relative permeability for water-wet media. The pore size distribution of the network can be tuned to match capillary pressure data when a network representation of the system of interest is unavailable. The aim of this work is not simply to match experiments, but to use easily acquired data to estimate difficult to measure properties and to predict trends in data for different rock types or displacement sequences.

The basic aim of this work is to present a combination of techniques for the reconstruction of the porous structure and the study of transport properties in porous media. The disordered structure of porous systems like random sphere packing, Vycor glass and North Sea chalk, is represented by three-dimensional binary images. The random sphere pack is generated by a standard ballistic deposition procedure, while the chalk and the Vycor matrices by a stochastic reconstruction technique. The transport properties (Knudsen diffusivity, molecular diffusivity and permeability) of the resulting 3-dimensional binary domains are investigated through computer simulations. Furthermore, physically sound spatial distributions of two phases filling the pore space are determined by the use of a simulated annealing algorithm. The wetting and the non-wetting phases are initially randomly distributed in the pore space and trial-and-error swaps are performed in order to attain the global minimum of the total interfacial energy. The effective diffusivities of the resulting domains are then computed and a parametric study with respect to the pore volume fraction occupied by each phase is performed. Reasonable agreement with available data is obtained in the single- and multi-phase transport cases.

In this paper we present numerical results obtained with a pore-network model for the drying of porous media that accounts for various processes at the pore scale. These include mass transfer by advection and diffusion in the gas phase, viscous flow in the liquid and gas phases and capillary effects at the liquid-gas interface. We extend our work by studying the effect of capillarity-induced flow in macroscopic liquid films that form at the pore walls as the liquid-gas interface recedes. A mathematical model that accounts for the effect of films on the drying rates and phase distribution patterns is presented. It is shown that film flow is a major transport mechanism in the drying of porous materials, its effect being dominant when capillarity controls the process, which is the case in typical applications.

It is known that the classical capillary pressure-saturation relationship may be deficient under non-equilibrium conditions when large saturation changes may occur. An extended relationship has been proposed in the literature which correlates the rate of change of saturation to the difference between the phase pressures and the equilibrium capillary pressure. This linear relationship contains a damping coefficient, τ, that may be a function of saturation. The extended relationship is examined at the macro-scale through simulations using the two-phase simulator MUFTE-UG. In these simulations, it is assumed that the traditional equilibrium relationship between the water saturation and the difference in fluid pressures holds locally. Steady-state and dynamic “numerical experiments” are performed where a non-wetting phase displaces a wetting phase in homogeneous and heterogeneous domains with varying boundary conditions, domain size, and soil parameters. From these simulations the damping coefficient τ can be identified as a (non-linear) function of the water saturation. It is shown that the value of τ increases with an increased domain size and/or with decreased intrinsic permeability. Also, the value of τ for a domain with a spatially correlated random distribution of intrinsic permeability is compared to a homogeneous domain with equivalent permeability; they are shown to be almost equal.

Several alternative mathematical models for describing water flow in unsaturated porous media are presented. These models are based on an equation for conservation of mass of water, and a generalized linear law for water flux (Darcy's law) containing a term called the dynamic capillary pressure. The distinct form of each alternative model is based on the specific form of expression used to describe the dynamic capillary pressure. The conventional representation arises when this pressure is set equal to the equilibrium pressure given by the capillary pressure — saturation function for unsaturated porous media, and this conventional approach leads to the Richards equation. Other models are derived by representing the dynamic capillary pressure by a rheological relationship stating that the pressure is not given directly by the capillary pressure — saturation function. Two forms of rheological relationship are considered in this manuscript, a very general non-equilibrium relation, and a more specific relation expressed by a first-order kinetic equation referred to as a relaxation relation. For the general non-equilibrium relation the system of governing equations is called the general Non-Equilibrium Richards Equation (NERE), and for the case of the relaxation relation the system is called the Relaxation Non-Equilibrium Richards Equation (RNERE). Each of the alternative models was analyzed for flow characteristics under gravity-dominant conditions by using a traveling wave transformation for the model equations, and more importantly the flow described by each model was analyzed for linear stability. It is shown that when a flow field is perturbed by infinitesimal disturbances, the RE is unconditionally stable, while both the NERE and the RNERE are conditionally stable. The stability analysis for the NERE was limited to disturbances in the very low frequency range because of the general form of the NERE model. This analysis resulted in what we call a low-frequency criterion (LFC) for stability. This LFC is also shown to apply to the stability of the RE and the RNERE. The LFC is applied to stability analysis of the RNERE model for conditions of initial saturation less than residual.

We study counter-current imbibition, where a strongly wetting phase (water) displaces non-wetting phase spontaneously under the influence of capillary forces such that the non-wetting phase moves in the opposite direction to the water. We use an approximate analytical approach to derive an expression for saturation profile when the viscosity of the non-wetting phase is non-negligible. This makes the approach applicable to water flooding in hydrocarbon reservoirs, or the displacement of non-aqueous phase liquid (NAPL) by water. We find the recovery of non-wetting phase as a function of time for one-dimensional flow. We compare our predictions with experimental results in the literature. Our formulation reproduces experimental data accurately and is superior to previously proposed empirical models.

Reservoirs are often composed of an assortment of rock types giving rise to permeability heterogeneities at a variety of length-scales. To predict fluid flow at the full-field scale, it is necessary to be aware of these different types of heterogeneity, to recognise which are likely to have important effects on fluid flow, and to capture them by upscaling. In fact, we may require a series of stages of upscaling to go from small-scales (mm or cm) to a full-field model. When there are two (or more) phases present, we also need to know how these heterogeneities interact with fluid forces (capillary, viscous and gravity). We discuss how these effects may be taken into account by upscaling. This study focusses on the effects of steady-state upscaling for viscous-dominated floods and tests carried out on a range of 2D models are described. Upscaling errors are shown to be reduced slightly by the increase in numerical dispersion at the coarse scale. We select a combination of three different upscaling methods, and apply this approach to a model of a North Sea oil reservoir in a deep marine environment. Six different genetic units (rock types) were identified, including channel sandstone and inter-bedded sandstone and mudstone. These units were modelled using different approaches, depending on the nature of the heterogeneities. Our results show that the importance of small-scale heterogeneity depends on the large-scale distribution of the rock types. Upscaling may not be worthwhile in sparsely distributed genetic units. However, it is important in the dominant rock type, especially if there is good connectivity through the unit between the injector wells (or aquifer) and the producer wells.

Current theories of multiphase flow rely on capillary pressure and saturation relationships that are commonly measured under static conditions. To incorporate transient behaviour, new multiphase flow theories have been proposed. These include an extended capillary pressure-saturation relationship that is valid under dynamic conditions. In this relationship, the difference between the two fluid pressures is called dynamic capillary pressure, and is assumed to be a function of saturation and its time rate of change. The dependency is through a so-called damping coefficient. In this work, this proportionality between capillary pressure and saturation rate of change is investigated using a pore-scale network model. It consist of a three-dimensional network of tubes (pore throats) connected to each other by pore bodies. The pore bodies are spheres and pore throats are cylinders. Numerical experiments are performed wherein typical experimental procedures for both static and dynamic measurements of capillary pressure-saturation curves are simulated. The value of the damping coefficient is determined for one realisation of our network model. Then, the effect of different averaging domains on capillary pressure-saturation curves is investigated.

Intrinsic heterogeneities influence the multi-phase flow behavior of a dense non-aqueous phase liquids (DNAPL) infiltrating into a natural soil. Typically, we cannot resolve the scale of these heterogeneities so that upscaling techniques are required. The choice of the appropriate upscaling method depends on the averaging scale, since the relative importance of capillary and gravity forces change with scale. We present an easy and quick upscaling approach for cases in which the flow on the length-scale of heterogeneities is dominated by capillary forces.

The approach is based on a percolation model and a single-phase flow-averaging method. We apply the upscaling approach to experimental data of a DNAPL infiltration into a sandbox with artificial sand lenses. The anisotropy of the structure results in anisotropic flow which is amplified by the nonlinear behavior of multi-phase flow. The residual saturation depends on the direction of flow, and the anisotropy ratio of the effective permeability is a function of the DNAPL saturation. Furthermore, it appears necessary to regard the relative permeability-saturation relationship as a tensor property rather than a scalar. The overall flow behavior simulated by the upscaled model agrees well with simulations accounting for the distinct lenses and the experimental data.