Simulating two-phase flow in two-dimensional
porous media
porous media
Introduction
Two-phase displacements in porous media have been much studied over the last two decades. The main reason for this is the great variety of structures observed when changing the physical parameters of the fluids like viscosity contrast, wettability, interfacial tension and displacement rate. Besides being a process of great interest in modern physics, it has a large number of practical applications in many fields of science like oil recovery and hydrology.
We have developed a network simulator modeling immiscible two-phase flow on a two-dimensional lattice of cylindrical tubes. The model is capable of measuring the time dependence of different physical properties and to study the dynamics of the fluid movements. So far we have focused on drainage displacements, i.e. the process where a non-wetting fluid displaces a wetting fluid in a porous medium. Moreover, we have used the model to study the temporal evolution of the pressure due to capillary and viscous forces and the time dependence of the front between the two liquids.
The Porous Medium Model
The porous medium is represented by a square lattice of tubes inclined 45 degrees. The tubes are connected together at nodes, where four tubes meet. There is no volume assigned to the nodes: the tubes represent the volume of both pores and throats. The tubes are cylindrical with equal lengths and their radii are chosen at random in a defined interval. The randomness of the radii represent the disorder of an ordinary porous medium.
The liquids flow from the bottom to the top of the lattice and periodic boundary conditions are applied horizontally. The pressure difference between the first (bottom of system) and the last (top) rows defines the pressure across the lattice. Gravity effects are neglected, and as a consequence we consider a horizontal flow in a two-dimensional network of tubes.
Initially, the model is filled with a defending wetting fluid with a given viscosity. An invading non-wetting fluid with another viscosity is injected from the bottom of the model at a constant injection rate. An interface (a meniscus) is created where the nonwetting and wetting fluid meets in the tubes. Due to this meniscus we apply a capillary pressure (Young-Laplace law), as if the tubes where hour glass shaped. That means, the capillary pressure of the menisci increases at the meniscus moves into the middle and more narrow part of the tube. The flow rate inside each tube is given by Hagen-Poiseulle flow where the pressure across the tube is the pressure difference between the ends of the tubes (the nodes) plus the capillary pressure if a meniscus is present. The pressures at the nodes are calculated by solving the Kirchhoff's equation on the network when conservation of volume flux is assumed. Finally, the flow rate in every tube is calculated the mensici are updated according to a proper time step.
Three Simulation Results
In two-phase fluid displacement there are mainly three types of forces: viscous forces in the invading fluid, viscous forces in the defending fluid and capillary forces due to the menisci between them. The different displacement structures obtained in drainage divide into three major flow regimes: viscous fingering, stable displacement and capillary fingering. Below we present three simulations, one in each of the three above regimes.
Viscous Fingering
(Click on the image to watch a movie of the displacement
simulation)
The top figure shows the result of a simulation in the regime of viscous fingering performed on a lattice of 60x80 nodes. The bottom figure shows the corresponding pressure across the lattice as a function of time. The displacements are done with a high injection rate and the invading fluid is less viscous than the defending wetting fluid.
In viscous fingering the principal force is due to the viscous forces in the defending fluid and the capillary forces at the menisci are less dominant. The displacement pattern shows that the invading fluid creates typical fingers into the defending fluid. Actually, the pattern has a well defined fractal dimesion with D=1.6.
The pressure across the lattice decreases as the less viscous fluid invades the system. Roughly, the pressure appears to decrease linearly as a function of time. However, the slope is non-trivial and results from the fractal development of the fingers.
The small fluctuations in the average decreasing pressure function correspond to the changes in the capillary pressure as a meniscus invades into or retreats from a tube. The fluctuations are small compared to the total pressure, hence, the capillary pressure does not play a significant role in the displacement process.
Stable Displacement
(Click on the image to watch a movie of the displacement
simulation)
The top figure shows the result of a simulation performed in the regime of stable displacement on a lattice of 60x60 nodes. The bottom figure shows the corresponding pressure across the lattice as a function of time. The invading fluid is more viscous than the defending fluid. As in the case of viscous fingering the injection rate is high.
In stable displacement the fluid movements are dominated by the viscous forces in the invading liquid and like viscous fingering the capillary pressure is not importent. The invading fluid generates a compact pattern with an almost flat front between the non-wetting and wetting fluid. The average pressure across the lattice increases accordaing to the amount of the high viscosity invading fluid injected into the system.
The viscous forces dominate the pressure evolution, but fluctuations due to capillary effects are observed. The perturbations have about the same size as for viscous fingering, which is not surprising since the size distribution of the radii of the tubes is the same for the two cases. The threshold pressures setting the strength of the capillary fluctuations, is inversely proportional to the radius of the tubes.
Capillary Fingering
(Click on the image to watch a movie of the displacement
simulation)
The top figure shows the resulting pattern obtained from a simulation in the regime of capillary fingering. The lattice size is 60x60 nodes. In the bottom figure the corresponding pressure across the lattice as a function of time is plotted. The two fluids have equal viscosities and the injection rate is rather low.
In capillary fingering the displacement is so slow that the viscous forces are negligible, with the consequence that the main force is the capillary one between the two fluids. Only the strength of the threshold pressure in the tubes decides if the invading fluid invades that tube or not. Since the radii of the tubes (which determine the threshold pressures) are randomly chosen from a given interval, the non-wetting fluid creats a path of least resistance. The path is fractal with a well defined fractal dimension D=1.82.
The top figure shows a typical rough front between the invading and the defending fluids with trapped clusters of defending fluid left behind the front. As opposed to stable displacement the clusters appear at all sizes between the tube length and the maximum width of the front.
The pressure across the lattice exhibits sudden jumps according to the capillary variation when the non-wetting fluid invades (or retreats) a tube. The fluctuations identify the bursts where the invading fluid proceeds abruptly. The pressure across the lattice increases in stable periods before the threshold pressure in the tube which is going to be invaded is reached. At the threshold pressure the meniscus becomes unstable and the invasion of fluid takes place in a burst accompanied by sudden negative jumps in the pressure.