Flow-Area Relations in Immiscible Two-Phase Flow in Porous Media (original) (raw)

Relations Between Seepage Velocities in Immiscible, Incompressible Two-Phase Flow in Porous Media

Transport in Porous Media

Based on thermodynamic considerations, we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible two-phase flow in porous media. They necessitate the introduction of a new velocity function, the co-moving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.

Local Statistics of Immiscible and Incompressible Two–Phase Flow in Porous Media

SSRN Electronic Journal

We consider immiscible and incompressible two-phase flow in porous media under steady-state conditions using a dynamic pore network model. We focus on the fluctuations in a Representative Elementary Area (REA), with the aim to demonstrate that the statistical distributions of the volumetric flow rate and the saturation within the REA become independent of the size of the entire model when the model is large enough. This independence is a necessary condition for developing a local statistical theory for the flow, which in turn opens for the possibility to formulate a description at scales large enough for the typical pore size to be negligible using differential equations.

A new set of equations describing immiscible two-phase flow in homogeneous porous media

2016

Based on a simple scaling assumption concerning the total flow rate of immiscible two-phase flow in a homogeneous porous medium under steady-state conditions and a constant pressure drop, we derive two new equations that relate the total flow rate to the flow rates of each immiscible fluid. By integrating these equations, we present two integrals giving the flow rate of each fluid in terms of the the total flow rate. If we in addition assume that the flow obeys the relative permeability (generalized Darcy) equations, we find direct expressions for the two relative permeabilities and the capillary pressure in terms of the total flow rate. Hence, only the total flow rate as a function of saturation at constant pressure drop across the porous medium needs to be measured in order to obtain all three quantities. We test the equations on numerical and experimental systems.

The Co-Moving Velocity in Immiscible Two-Phase Flow in Porous Media

2021

Relative permeability theory for immiscible two-phase flow in porous media assumes a linear dependency of the seepage velocity of each fluid on the pressure gradient. This implies that the average fluid velocity also exhibits such a linear dependence. Recent experimental, computational and theoretical work, however, show that the average flow velocity follows a power law in the the pressure gradient with an exponent in the range larger than one up to two over a wide range of parameters. Such a behavior is incompatible with relative permeability theory. A recent theory based on Euler homogeneity of the volumetric flow rates of the fluids generalizes relative permeability theory in such a way that it is capable of handling this non-linear behavior. A central quantity in this theory is the co-moving velocity which is related to, but not equal to the difference between the seepage velocities of the fluids. In order to close the equation set that ensues from the theory, a constitutive eq...

A statistical mechanics for immiscible and incompressible two-phase flow in porous media

arXiv (Cornell University), 2022

We construct a statistical mechanics for immiscible and incompressible two-phase flow in porous media under local steady-state conditions based on the Jaynes maximum entropy principle. A cluster entropy is assigned to our lack of knowledge of, and control over, the fluid and flow configurations in the pore space. As a consequence, two new variables describing the flow emerge: The agiture, that describes the level of agitation of the two fluids, and the flow derivative which is conjugate to the saturation. Agiture and flow derivative are the analogs of temperature and chemical potential in standard (thermal) statistical mechanics. The associated thermodynamics-like formalism reveals a number of hitherto unknown relations between the variables that describe the flow, including fluctuations. The formalism opens for new approaches to characterize porous media with respect to multi-phase flow for practical applications, replacing the simplistic relative permeability theory while still keeping the number of variables tractable.

A new set of equations describing immiscible two-phase flow in isotropic porous media

arXiv: Fluid Dynamics, 2016

Based on non-equilibrium thermodynamics we derive a set of general equations relating the partial volumetric flow rates to each other and to the total volumetric flow rate in immiscible two-phase flow in porous media. These equations together with the conservation of saturation reduces the immiscible two-phase flow problem to a single-phase flow problem of a complex fluid. We discuss the new equation in terms of the relative permeability equations. We test the equations on model systems, both analytically and numerically.

Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients

Transport in Porous Media, 1990

In order to capture the complexities of two-phase flow in heterogeneous porous media, we have used the method of large-scale averaging and spatially periodic models of the local heterogeneities. The analysis leads to the large-scale form of the momentum equations for the two immiscible fluids, a theoretical representation for the large-scale permeability tensor, and a dynamic, large-scale capillary pressure. The prediction of the permeability tensor and the dynamic capillary pressure requires the solution of a large-scale closure problem. In our initial study (Quintard and Whitaker, 1988), the solution to the closure problem was restricted to the quasi-steady condition and small spatial gradients. In this work, we have relaxed the constraint of small spatial gradients and developed a dynamic solution to the closure problem that takes into account some, but not all, of the transient effects that occur at the closure level. The analysis leads to continuity and momentum equations for the {3-phase that are given by a{E{3}* + v. {(vll)} = 0, at {(VIl)} =-"'!"K$. (v{(p/3)/3}Il-P/3g)-u/3 a{E/3}*-U/3' v ah}*-Because of the highly nonlinear nature of the capillary pressure-saturation relation, small causes can have significant effects, and the treatment of the large-scale capillary pressure is a matter of considerable

A Lattice-Boltzmann study of immiscible two-phase flow in pore networks

± In this paper we study the immiscible two-phase flow in porous media using the Lattice-Boltzmann model proposed by He et al. [Phys. Fluids 11 (1999) 1143-1152] through a series of numerical simulations performed at the Barcelona Supercomputing Center. By considering a set of appropriate boundary conditions for the density distribution function defined in that model, we account for the effect of wettability at solid-fluid interfaces and capillarity in the pores where the fluid-fluid interfaces reside. Different contact angles of the fluid-fluid interface at solid walls can be realized by taking appropriate values for the density distribution function at the solid sites of the porous domain. It is shown that the steady state contact angle is a linear function of the density value assigned to the solid sites. The proposed model is an ideal tool for modeling immiscible two-phase flow in porous media, due both to its ability to incorporate complicated boundary conditions at the pore walls and also capture the physical aspects of the flow in the bulk and the interfaces. Furthermore, the width of the fluid-fluid interfaces is kept less than 3-4 lattice units allowing for simulations in relatively low resolution porous lattices.

Immiscible two-phase flow in porous media: Effective rheology in the continuum limit

arXiv: Fluid Dynamics, 2019

It is becoming increasingly clear that there is a regime in immiscible two-phase flow in porous media where the flow rate depends of the pressure drop as a power law with exponent different than one. This occurs when the capillary forces and viscous forces both influence the flow. At higher flow rates, where the viscous forces dominate, the flow rate depends linearly on the pressure drop. The question we pose here is what happens to the linear regime when the system size is increased. Based on analytical calculations using the capillary fiber bundle model and on numerical simulations using a dynamical network model, we find that the non-linear regime moves towards smaller and smaller pressure gradients as the system size grows.