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Mesh Dependence for Two Phase Drainage Simulation of Water Displacing Oil in a Sphere- Packed Bed
By
Paul Delgado
A Class Project for
MECH5305: Computational Fluid Dynamics by Dr. V. Kumar
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ABSTRACT
INTRODUCTION
PROBLEM DEFINITION
RESULTS & DISCUSSION
CONCLUSIONS AND SUMMARY
REFERENCES
1) ABSTRACT
Two phase flow in porous media is governed by microscale physics which are computationally expensive
to model with high resolution. This study explores the mesh dependence of two phase drainage
simulation of water displacing oil in a sphere-packed bed modeled at the millimeter scale. We observe
the propagation of errors at various timesteps and postulate an optimal mesh size to capture the
interfacial tension effects at this scale.
2) INTRODUCTION
The oil industry is greatly interested in developing high resolution simulations of multiphase flow in a
porous media. To understand the dynamics of this flow, researchers often perform permeability
experiments in packed beds of spherical glass beads using water to displace oil. It is desirable to obtain
macroscopic observations, but the underlying physics governing the flow in oil reservoirs is microscopic
in nature. To reduce the computational cost of a two phase immiscible drainage simulation, we
investigate the effects of both spatial and temporal discretization on the macroscopic patterns observed
in two phase immiscible drainage simulation of water displacing oil in a sphere-packed bed.
3) PROBLEM DEFINITION
Two phase immiscible drainage of water displacing oil in porous media is modeled by the Volume of
Fraction (VOF) method, which calculates the percent xj of each fluid in an element of the mesh. Both
mass and momentum conservation equations are solved using cell averaged density ( and viscosity
(μ). That is,
   xj j
   xj j
Imposing conservation of mass on the cell-averaged system results in the unsteady continuity equation

 j      j u j   0
t
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
where j and u j are the density and velocity of fluid phase j, respectively. Volume fractions are
determined by a finite volume disretization of
1
xj
 

 t x j  j     x j  j u j   0
Neglecting the gravity term to model horizontal displacement only, the momentum equation becomes


 
u
   u u   P   2u  F
t

Where F is the interfacial tension force. Interfacial tension is modeled as a volume force represented
by

 i  j
F   ij 1


2 i   j
where  ij is the interfacial tension coefficient between the two fluids, is the curvature of the interface
(Crandall, 2007)
The goal of this project is to observe the changes in displacement patterns in a two phase drainage
simulation of water displacing oil in small rectangular pore. For simplicity, we assume an oil-wet system
with constant contact angles and surface tension. To reduce computational expense, we only consider
2D horizontal flow only (no gravity) through the flow domain illustrated in figure 1. We imposed wall
boundary conditions at two opposite walls, with wall adhesion and constant contact angle θ = 120o and
constant surface tension 3 dyn/cm1. Pressure outlet and velocity inlet conditions were imposed on the
remaining walls with values 0 Pa and .01m/s, respectively. FLUENT’s default water-liquid (H2O) and gasoil (C16H29) parameters were used as the two fluids.
1
As suggested by Schowalter (1979) to simulate oil-water surface tension for high density crude oil at a reservoir
temperature of 190o F.
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Figure 1: 2D Flow Domain
All simulations were run on FLUENT using non-iterative time stepping scheme in conjunction with a PISO
pressure-velocity coupler, with PRESTO! pressure, QUICK momentum, and Geo-Reconstruction volume
fraction discretizations on triangular meshes. To test the mesh dependence, we held the timestep
constant at 10-6s and varied the interval count as 10, 20, 40, 80, nodes per edge as seen in Figure 2 and
observed the changes in displacement of the interface after 12,000 timesteps. Note that since the inlet
and outlet have smaller width, there is higher resolution near these boundaries than in the interior.
Figure 2: Meshes with 10, 20, 40, and 80 nodes per edge
4) RESULTS & DISCUSSION
Figure 3 illustrates the results of the drainage simulations at t=0.001, 0.005, .01, and .015 seconds. As
seen in the first row of Figure 3, the number of elements separating the oil (red) and water (blue)
saturated regions is initially as minimal as possible. The second row shows that there is a slight increase
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in the number of elements in the interface separating the two saturated regions after the first few time
steps. The third and fourth row indicate that despite this initial increase, the interface remains relatively
stable through the remainder of the timesteps. , all interfaces remained stable until they crossed the
pressure outlet boundary. The 20, 40, and 80 count simulations all preserved relatively constant menisci
and contact angles over all timesteps. The 10 count simulation did not preserve its contact angle or
meniscus.
Figure 3 – Cross comparisons of drainage simulations with a) 10 count, b) 20 count, c) 40 count, and d) 80 count mesh resolutions.
It was also observed that when the interface crosses the pressure outlet boundary at the bottom,
backflow is immediately reported by the FLUENT solver. In literature, this is commonly referred to as
the ‘breakthrough’ time. In reality, we expect that the meniscus should continue to advect through the
domain, preserving the same contact angle. The loss of the completely connected meniscus at the
outlet destabilizes the interface and induces the backflow as seen in Figure 4.
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Figure 4 – Backflow at breakthrough (t=0.0184s) on 80 count simulation.
As expected, the simulation runtime increased greatly with mesh resolution. The average time to finish
the 10, 20, 40, and 80 count simulations with 20,000 timesteps is 8 ,15, 33, and 74 minutes. It is clear
from figure 3 that there is little qualitative difference among the 20, 40, and 80 count simulations. All
preserved a stable interface and contact angle up to breakthrough. Furthermore, the breakthrough
times in all simulations remained the same.
5) CONCLUSIONS AND SUMMARY
In conclusion, all simulations tested in this paper transported a relatively stable an interface throughout
the flow domain until breakthrough. A coarse mesh will preserve a stable interface but may not
preserve a stable contact angle. The relatively indistinguishable displacement patterns in the finer
meshes suggest that we can efficiently perform simulations by refining the mesh such that the meniscus
is distinguishably circular.
Furthermore, simulation results are only valid until the meniscus crosses the outlet boundary. Any
further timesteps immediately result in unphysical backflow. We recommend that any desired flow
domain be augmented by a manifold region to allow the meniscus to advance beyond the desired
‘outlet’ boundary. Careful attention must be taken to prescribe boundary conditions on the manifold
region that preserve the conditions desired at the pressure-outlet region. This will be a topic for future
research.
6) REFERENCES
Crandall, D. (2007). Two Phase Flow in Porous Media and Fractures – Disseration: Clarkson University
Kumar, V. (2012). Computation Fluid Dynamics Lecture Notes – Spring 2012 Course at the University of
Texas El Paso.
Schowalter, T. T. (1979) Mechanics of secondary hydrocarbon migration and entrapment - AAPG Bulletin
v. 63, p. 723-760.