COMSOL 6.4 - Two-Phase Flow in a Porous Medium: Buckley

Two-Phase Flow in a Porous Medium: Buckley–Leverett Model

This example uses the Multiphase Flow in Porous Media multiphysics interface to model an immiscible displacement process in a porous medium. You can think of the displacement of oil by water in a reservoir. In a one-dimensional setting and under certain assumptions, the equations for the saturations and pressures of the two phases reduce to a single conservation equation for the saturation of one of the phases, the Buckley–Leverett equation. In the model setup described below this equation allows for an analytical solution, and as such, this example also serves as a benchmark model.

Model Definition

A 1D porous medium with a length of 1 meter is considered. It is assumed that both phases present are incompressible, that there are no sources or sinks, and that gravity plays no role. Furthermore, zero capillary pressure and (relative) permeabilities and porosity independent of time and space are assumed.

Initially the porous medium is completely filled with phase 1. At x = 0 phase 2 enters the porous medium with a volumetric flux of 0.001 m/s and displaces phase 1, which is allowed to flow out of the porous domain at x = 1. The volumetric flux of phase 1 is equal to 0 at x = 0. The pressure at x = 1 is set to be equal to 0 Pa. Table 1 collects the relevant material properties. The process in simulated for a time interval of 300 seconds.

From the assumptions and boundary conditions mentioned above it follows that the total volumetric flux u = u1 + u2 is constant in time and space, and that the two-phase flow equations implemented in the Multiphase Flow in Porous Media interface for the saturations and pressures reduce to a single equation for the saturation s1:

(1)

where εp denotes the porosity and λi = κri/μi, with κri and μi the relative permeability and dynamic viscosity of phase i, respectively. This equation allows for an analytical solution (see Ref. 1 for the construction of the analytical solution to this Buckley–Leverett equation). In the Results and Discussion section below, the analytical solution of the Buckley–Leverett equation is compared to the solution obtained with the Multiphase Flow in Porous Media interface.

Table 1: Model Data.
Quantity	Value	Description
ρ1 = ρ2	1000 kg/m3	Density of both phases
μ1 = μ2	0.001 kg/(m·s)	Dynamic viscosity of both phases
εp	0.5	Porosity
κ	10-9 m2	Permeability
κr1	s12	Relative permeability of phase 1
κr2	s22	Relative permeability of phase 2

Results and Discussion

In Figure 1, the profiles of the saturation s2 are shown for different instances in time (20 seconds intervals). Phase 1 is displaced by phase 2, and the solution exhibits a shock traveling through the porous domain. The solution of the two-phase flow equations (solid lines) shows a good agreement with the analytical solution of the Buckley–Leverett equation in Equation 1(dotted lines).

Figure 1: Saturation profiles for phase 2 shown at intervals of 20 seconds, as computed using the Multiphase Flow in Porous Media interface (solid lines), and as obtained analytically as solution of the Buckley–Leverett equation.

Reference

1. R. Helmig, Multiphase Flow and Transport Processes in the Subsurface – A Contribution to the Modeling of Hydrosystems, Springer–Verlag, 1997.

Subsurface_Flow_Module/Verification_Examples/buckley_leverett_model

Modeling Instructions

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New

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Model Wizard

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Geometry 1

Interval 1 (i1)

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In the window for , click

Definitions

Add a piecewise analytic function to visualize the analytic solution of the Buckley–Leverett equation and to compare it to the solution computed using the Multiphase Flow in Porous Media interface.

Piecewise 1 (pw1)

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In the window for , locate the section.

Find the subsection. In the table, enter the following settings:


Start	End	Function
0.7071	1	d(x^2/(x^2+(1-x)^2),x)

Phase Transport in Porous Media (phtr)

Fluid 1

In the window, under > > click .

In the window for , locate the section.

From the ρs1 list, choose . From the μs1 list, choose . Locate the section. From the ρs2 list, choose . From the μs2 list, choose . This sets the density and dynamic viscosity of both phases to the default values ρ = 1000 kg/m3 and μ = 10−3 Pa·s.

Volume Fraction 1

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