1
B016 SIMULATION OF DISPLACEMENT PROCESSES
BY A LBM VOF-CSF MODEL
JOHANNES STEINER & CHRISTIAN REDL
Christian-Doppler-Laboratory for Applied Computational Thermofluiddynamics,
University of Leoben, Austria
Abstract
The Lattice Boltzmann Method (LBM) is known as a very powerful tool for simulating fluid
flow processes in highly complex structures. The structure of porous media can be resolved in
detail by means of e.g. computer tomography and afterwards used in the LBM. However most
LBM models were restricted to single phase flows or to multiphase flows with only small density
differences between the phases. By means of the LBM VOF-CSF (LBM Volume of FluidContinuum Surface Force) model it is possible to handle multiphase flows with high density
differences. For this purpose a single-phase Lattice Boltzmann model is used in connection with
a volume of fluid method for locating the surface separating the different phases. Surface tension
effects between the phases and wetting characteristics are taken into account by a continuum
surface force model. The single phase model leads to acceptable calculation times. The
implementation of surface forces used in the VOF-CSF model is discussed. Finally several case
studies demonstrate the capabilities of the developed model. This studies cover immiscible
displacement processes in a straight pore channel and in a more complex geometry.
Introduction
Displacement processes are one of the primary targets of reservoir simulation. There exist a
number of analytical or numerical solutions for simple test cases mostly based on the concept of
relative permeabilities1. Despite the fact that the Lattice Boltzmann method (LBM)2,3 is very
well suited for simulating fluid flows in highly complex structures like porous media it has not
gained much interest in the petroleum industry. Probably not competitive for field scale
problems it may suit as a very useful tool for pore scale investigations.
Although there are already some LB models for multi-phase flow available most of them suffer
from severe numerical instabilities especially for density ratios exceeding 2 4. Therefore a
Volume of Fluid(VOF)-Continuum Surface Force(CSF) model has been developed which can be
used also for flows with high density differences5. This model and its integration into the LBM
framework is described in the first part of this paper. Subsequently a few test cases are presented.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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LBGK Method
The Lattice BGK method is a kinetic based numerical technique for solving fluid flow problems
using a finite difference formulation of the discrete Boltzmann equation. The evolution of a
particle distribution function on a lattice with reduced discretisation of the velocity space is
calculated. The use of a strictly regular grid and very simple wall boundary condition makes it
especially suitable for flow simulations within porous materials and highly complex structures.
As a standard incompressible formulation of the Lattice BGK model on a 2D lattice with 9
discrete velocity vectors is used in the current work this model is only described very roughly.
The interested reader is referred to the excellent reviews of Chen & Doolen2 and Luo3.
The non-dimensional form of the evolution equation for the pressure distribution function
governing the fluid flow reads as follows
f i (x + e i , t + 1) − f i (x, t ) =
where:
eq
τ
F
[f
τ
1
eq
i
]
(x, t ) − f i (x, t ) + F
(Eq. 1)
... indicates equilibrium state
... is the relaxation parameter related to the viscosity
... represents a general source term
The evolution equation for the particle distribution function is given by eq. (1). It is solved
explicitly in time and consists of two parts. The first part represented by the left hand side of eq.
(1) describes translation of the particle distribution functions along their corresponding lattice
velocity vectors. The second one is the collision procedure given by the first term on the right
hand side of eq. (1). The collision term which is constructed according to the Bhatnagar-GrossKrook (BGK) model describes the relaxation of the actual state to the equilibrium state which is
given by a kind of Taylor expanded Maxwellian:.
f i eq = t i { p + p 0 (
where:
e i α uα
c s2
+
uα u β eiα e iβ
( 2 − δ αβ ))}
cs
2c s2
t 0 = 4 9 , t1− 4 = 1 9 , t 5−8 = 1 36
(Eq. 2)
... are model dependent weighting factors
t 7 −18 = 136
cs = 1
... is the model speed of sound
3
The lattice vectors are given by:
e0
e1− 4
e5 −8
=
(0,0)
=
(±1,0) (0,±1)
= (±1,±1)
3
The pressure and the macroscopic velocity can be obtained from the moments of the distribution
functions:
p = ∑ fi
p0 u = ∑ f i e i
i
where:
p0
... is
(Eq. 3)
i
a reference pressure.
The relaxation factor in eq. (1) is linked to the fluid viscosity by the following expression:
2τ − 1
ν=
(Eq. 4)
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In addition the evolution of the front is determined by solving a scalar transport equation for the
volume fraction φ within the LBGK framework:
g i (x + e i , t + 1) − g i (x, t ) =
1
τD
[g
eq
i
(x, t ) − g i (x, t )
]
(Eq. 5)
where
g ieq = φt i {1 +
and
ei α u α u α u β e i α e i β
+
( 2 − δ αβ )}
2c s2
c s2
cs
φ = ∑ gi
(Eq. 6)
(Eq. 7)
i
The diffusion coefficient, which is written as
2τ D − 1
,
(Eq. 8)
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should theoretically be equal zero in order to obtain a sharp front, but must be non-zero for
reasons of numerical stability.
D=
Note that the local flow velocity u of the carrier fluid is taken to obtain the equilibrium
distribution for the scalar φ .
It has to be pointed out that except of the translation step all operations are done locally for each
cell. This makes the Lattice BGK method well suited for parallel computing.
A remarkable feature of the Lattice BGK method is that no assumptions about the permeability
are necessary for the simulation as the structure of the porous media is resolved in detail and the
Navier-Stokes equations are solved directly on active fluid cells.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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VOF method
In contrast to surface-tracking (Eulerian-Lagrangian) methods, in volume-tracking (Eulerian)
methods the interface is not explicitly defined or tracked but is reconstructed at every step6. The
VOF method makes use of a fluid fraction variable φ , assigned values of 1 and 0 in the two
phases, which is calculated as a field variable over the domain. The interfacial cells are then
identified as those with fractional values of φ .
In order to evolve the interface, the front is advected passively under the local flow velocity u,
according to:
∂φ
∂φ
+u
=0
∂x
∂t
(Eq. 9)
The most striking disadvantage of the volume tracking schemes is that while very complex front
topologies like mergers and breakups of the interface can be handled, they cannot be treated with
precision. The main difficulty arises in the reconstruction of the interface which involves a
considerable number of logical operations.
CSF model
Recently a technique has been developed to impose surface tension effects in an efficient
manner7, where interfacial surface phenomena are no longer applied as discrete boundary
conditions at a discontinuity (the free surface) but as smoothly varying volume forces acting at
the nodes in the transition region, consisting of a few cells. The basic idea is that one can define
a localized body force Fv, which volume integral in the limit of infinitesimally small transitionregion-thickness h is equal to the surface integral of the tensile force Fs,
lim ∫ Fv ( x)dV = ∫ Fs ( x s )dS
h →0 ∆V
∆S
(Eq. 10)
where xs is a position at the free surface. The surface integral is taken over a part of the free
surface lying within the volume of integration ∆V . In its classical form, surface tension
contributes a surface pressure that is the normal force per unit interfacial area S acting at points
xs on S: Fs (x s ) = ∆p (x s ) . The surface force can be written as
Fs (x s ) = σκ (x s )nˆ (x s )
(Eq. 11)
where κ (x s ) is the curvature at x s and nˆ (x s ) is the unit normal to the surface at the same point.
In order to find the body force Fv(x) that satisfies equation (10), first a step function c(x) is
defined, with the following values
⎧
c1
⎪⎪
c( x) = ⎨
c2
⎪(c1 + c 2 )
⎪⎩
2
in fluid 1
in fluid 2
at the interface
(Eq. 12)
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Hence the interface is an infinitesimally thin line. For creating a transition region between the
two different fluids, the interface is imaginary broadened to a small distance h. For this purpose,
a mollified characteristic function c~ ( x) is introduced, which varies smoothly from c1 over c to
c2 and fulfilles the following relation:
lim c~ ( x) =c( x)
(Eq. 13)
h →0
The volume force Fv is then formulated as
∇c~ ( x)
Fv ( x) = σκ ( x)
,
[c ]
(Eq. 14)
where the normalization factor is given by [c ] = c 2 − c1 .
By setting the color function c~ ( x) in the CSF model equal to the VOF function φ, the body force
can be written as
Fv ( x) = σκ ( x)∇φ ( x).
(Eq. 15)
Note that the normalization factor [φ] is equal to one. Furthermore the relation (10) is also
satisfied if the body force (15) is multiplied by an additional function
g ( x) =
φ ( x)
,
φ
(Eq. 16)
because of the fact, that in the limit of an infinitesimally thin transition region the numerator
equals the denominator.
Thus putting together equations (15) and (16) one obtains the body force
Fv ( x) = σκ ( x) g ( x)∇φ ( x) = σκ ( x)
φ ( x)
∇φ ( x) = 2σκ ( x)φ ( x)∇φ ( x).
φ
(Eq. 17)
Some implementation notes
In order to implement surface tension effects, one has to introduce the lattice version of (17).
This is done by defining a normal vector
n = ∇φ
(Eq. 18)
and the according unit normal
n=
n
n
(Eq. 19)
The free surface curvature κ then follows as
κ = −∇ ⋅ n =
9th
1
n
⎡⎛ n
⎤
⎞
⎢⎜⎜ ⋅ ∇ ⎟⎟ n − (∇ ⋅ n )⎥
⎢⎣⎝ n
⎥⎦
⎠
(Eq. 20)
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Since the volume force is located at cell centers, the curvature κ , the function g(x) and the
normal vector n must also be cell-centered:
Fv ,ij = σκ ij g ij nij
(Eq. 21)
where the indices i and j labels the lattice cell, centered at row i and column j. Since only one
phase is used for the simulations, the VOF function φ, which defines the location of the fluid
interface, also shows the presence or absence of a given fluid (liquid or gas). The above
expressions (21) serve as source term in the LB formulation.
Case studies
Capillary pressure
The first test case should show the ability of the developed method to reproduce realistic
capillary motion. This is done by comparing the capillary pressure jump across the interface
obtained by the simulation with the analytical solution given by
p canalytic =
2σ
cos θ
R
(Eq. 22)
Five different pore diameters ranging from 1 [µm] to 50 [µm] were considered. The physical
properties of a water-air system were used for the simulations. For the static wetting angle an
arbitrary value of θ = 45 [°] was assumed. The surface tension was 0.073 [kg s-2].
Figure 1Capillary pressure vs. pore radius
As it can be seen from fig. 1 the numerical solution corresponds excellent with the analytic one.
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Straight pore channel
The endrainage of a fluid into a straight pore channel initially filled with another fluid is
simulated. The two fluids are immiscible. The diameter of the channel is 100 [µm]. The surface
tension is again 0.073 [kg s-2]. The displacing agent is assumed to be the wetting phase and the
static contact angle is 20 [°]. The pressure is fixed on both sides of the tube.
The temporal evolution of the fluid front is depicted in fig. 2. Three different timesteps are
shown.
Figure 2 Temporal evolution of the displacement front in a straight pore channel
Complex porous structure
Immiscible displacement in a complex geometry is simulated. The physical parameters are the
same as used in the above example. Fig. 3 depicts the temporal evolution of the displacement
process.
Figure 3 Displacement in a complex porous structure
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Conclusion
A new technique for simulating displacement processes based on the Lattice Boltzmann method
was introduced. It was shown that the capillary driven fluid motion in complex geometries can
be simulated. The obtained results are very promising on a qualitative basis.
Nevertheless more simulations using different material properties and parameters are still left to
do. This also means including gravity which can be done by adding additional source terms to
the evolution equation for the particle distribution functions.
The authors are very confident that the LBM with the VOF-CSF model can handle even much
more complex situations.
Bibliography
[1] Heinemann, Z., Flow in Porous Media, Textbook, University of Leoben, 1995.
[2] Chen, S., Doolen, G. D., Annu. Rev. Fluid Mech. 30 (1998), 329.
[3] Luo, L.-S., The Lattice Gas and Lattice Boltzmann Method: Past, Present, Future,
Proceedings Int. Conf. Appl. Comp. Fluid Dyn., Beijing, China, October 17 – 20, 2004.
[4] Succi, S., The Lattice Boltzmann Method for Fluid Dynamics and Beyond, Oxford University
Press (2001).
[5] Steiner, J., Numerical Simulation of Slag Assault on Refractory Material using the Lattice
Boltzmann Method, Thesis, University of Leoben, Austria (2004).
[6] Hirt, C.W., Nichols, B. D., J. Comp. Phys. 39 (1981), 201.
[7] Brackbill, J. U., Kothe, D. B., Zemach, C., J. Comp. Phys. 100 (1992), 335.