Towards Smoothed Particle Hydrodynamics Simulation of Viscous Fingering in Porous Media_Crimson Publishers

Ronchi N and Bertola V*
Laboratory of Technical Physics, University of Liverpool, UK
*Corresponding author: Bertola V, Laboratory of Technical Physics, School of Engineering, University of Liverpool, Brownlow Hill, UK.
Submission: May 05, 2018; Published: May 10, 2018
Towards Smoothed Particle Hydrodynamics
Simulation of Viscous Fingering in Porous Media
Introduction
Viscous fingering, which is observed at the interface between
two immiscible fluids of different viscosities flowing through a
porous medium when the more viscous fluid is displaced by the
less viscous fluid [1,2], is a classical problem of fluid mechanics
with important applications in oil recovery and earth drilling [3-
6] and underpins the study of the wide range of Laplacian growth
phenomena [7,8]. Although this phenomenon had been known
to oil engineers for a long time, the first systematic studies of the
interfacial instability during viscous fluid displacement appeared
in the 1950s, using the Hele-Shaw cell as reference geometry based
on the equivalence between the flow in a porous medium and the
creeping flow between two parallel plates separated by a small gap
[1,2]. Since then, viscous fingering is often referred to as Saffman-
Taylor instability, which occurs when the displacement velocity
exceeds a critical value:
1 2
2 1
2 1
( )
c
g
V
k k
ρ ρ
µ µ
−
=
−
(1)
Where 2 1
µ µ
< . The problem of viscous fingering has been
extensively studied form the experimental [9-11], theoretical [12-
14], and computational point of view [15-17], for both Newtonian
and non-Newtonian [18-20] fluids. Similar to other hydrodynamic
instabilities, simulations of viscous fingering with commercial
CFD codes using Eulerian meshes are difficult because it involves
a moving interface usually characterized by large deformations.
In this paper, Smoothed Particle Hydrodynamics (SPH) is used
to simulate viscous fingering of Newtonian fluids in Hele-Shaw
geometry. SPH is a fully Lagrangian computational technique,
originally developed to simulate non-axisymmetric phenomena
in astrophysics [21-23], that approximates the continuum fluid
medium through the use of virtual particles and does not require a
grid to calculate spatial derivate. In particular, a generic function at
a given position r is represented as [21-23]:
( ) ( ) ( , )
f f W h dV
′ ′
= −
∫
r r r r (2)
Where ( , )
W h
′
−
r r is the smoothing kernel and h is is
the smoothing length that defines its influence region. The
smoothing kernel can be an arbitrary function that (i) satisfies the
normalisation condition, (ii) is identically zero outside the effective
region defined by h, and (iii) tends to the Dirac delta when h→0.
The integral representation of the spatial derivative of an arbitrary
function reduces to the following equation:
. ( ) ( ). ( , )
f f W h dV
′ ′
∇ =
− ∇ −
∫
r r r r (3)
Introducing a smoothing function or smoothing kernel, the
values of functions and spatial derivatives for a specific particle
are approximated considering the interaction of that particle
with a certain amount of neighbouring particles. This means that
the physical quantities of a specific particle can be obtained by
Review Article
114
Copyright © All rights are reserved by Bertola V.
Volume 1 - Issue - 5
Abstract
The mesh-free Lagrangian Smoothed Particle Hydrodynamics (SPH) method is used to simulate the problem of viscous fingering in Hele-Shaw
geometry. Viscous fingering occurs when a lower-viscosity fluid displaces a higher-viscosity fluid in a narrow channel, and is a major concern in the
removal of drilling mud’s from oil well bores and in secondary and tertiary oil recovery. The flow field was modelled using two sets of particles with
different properties, initially placed in the left half and in the right half of the channel, respectively; in particular, the two fluids had the same density and
a viscosity ratio of 1:10. Results show that SPH can reproduce the formation of a low-viscosity finger penetrating into the higher-viscosity fluid during
displacement. Because of its intrinsically Lagrangian, mesh-free nature, SPH is a promising method to simulate viscous fingering in complex geometries;
in addition, it can easily incorporate non-Newtonian constitutive equations to account for shear-thinning and/or viscoplastic behaviours.
Progress in Petrochemical Science
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How to cite this article: Ronchi N, Bertola V. Towards Smoothed Particle Hydrodynamics Simulation of Viscous Fingering in Porous Media. Progress
Petrochem Sci .1(5). PPS.000523.2018. DOI: 10.31031/PPS.2018.01.000523
Copyright © Bertola V
summing the relevant properties of all the particles which lie within
the range of the smoothing function, whose values are smoothed by
the kernel function itself:
1
N
j
i j ij
j j
m
f f W
ρ
=
= ∑ (4)
Through this approximation, the governing equations of fluid
dynamics, i.e. the Navier-Stokes equations, are reduced to a set
of ordinary differential equations with respect to time. Because
of its Lagrangian nature, the SPH method has clear advantages
over traditional grid base Eulerian methods for some fluid flow
calculations, such as complex boundaries flows, multiphase fluids
flows, free surfaces flows and non-Newtonian flows. In fact, since
the particles simply follow their trajectories, fluid advection can
be accomplished without difficulty. In particular, the absence of a
fixed or adaptive mesh makes SPH particularly advantageous to
track free surfaces and interfaces in comparison with, for example,
with the Volume of Fluid method (VOF), where the exact position of
interfacesisnotdeterminedapriori,anddoesnotnecessarilymatch
the grid. However, the SPH method can be more computationally
expensive than alternative techniques for a given application [24].
Historically, SPH took a relatively long time (decades) to become
of widespread use in engineering fluid mechanics applications,
mainlybecauseinitsoriginalformulationsometurbulentinvariants
are not conserved, and because sometimes the implementation
of boundary conditions is not trivial; thus, several years were
necessary to address these issues satisfactorily. Recently, the
SPH method has been applied to simulate flow instabilities at the
interface between two fluids, such as the Rayleigh-Taylor [25,26]
and the Kelvin-Helmholtz instabilities [27,28], as well as the flow
in porous media [29,30].However, the SPH approach has not been
used to simulate the Saffman-Taylor instability so far.
Two-Phase SPH Algorithm
The SPH formulation of fluid dynamic equations is extensively
discussed in the literature [21-23,31,32], and essentially consists
in combining the conservation equations for mass, momentum
and energy for the smoothed particles with a suitable constitutive
equations that relates pressure with density. In most circumstances,
a quasi-incompressible equation of state is used, so that the energy
equation is not necessary. The particle density can be calculated
either using a summation method:
1
N
i j ij
j
m W
ρ
=
= ∑ (5)
Where ( , )
ij i j
W W r r h
= − is the smoothing kernel of particle i
evaluated at particle j (Eq. 10), or alternatively through the mass
continuity equation:
1
.
N
i
j ij i ij
j
D
m v W
Dt
ρ
=
= ∇
∑ (6)
Where ij i j
ν ν ν
= − ; whilst Eq. (6) is more efficient than Eq (5)
from the computational point of view, it does not ensure the mass
conservation exactly [24]. The momentum equation can be written
in the form of Newton’s second law, and expresses the total force
acting on a particle as the sum of a pressure force [23], a viscous
force [24,33], and a body force (e.g., gravity):
2 2 2
1 1
( )
( ) .
N N
j i j i j ij
i
i i j i ij ij i ij i
j j
i j i j ij
p m m v
p
F m m W r W m g
r
µ µ
ρ ρ ρ ρ
= =
+
=
− + ∇ + ∇ +
∑ ∑ (7)
The choice of the constitutive equation and of the smoothing
kernel can be somewhat controversial and depends on the
characteristics of the flow under consideration [23,24,34]. Here, a
quasi-incompressible flow equation of state [24] was chosen:
2
p c ρ
= (8)
Where c2
is an artificial speed of sound, calculated as:
2
2 0 0 0
0
max ; ;
v vv FL
c
L
δ δ δ
 
=  


(9)
In Eq. (9), v0
and L0
are the velocity and the length scales,
respectively, and d is the density variation defined as Δρ⁄ρ.
A quantic spline kernel [31] was chosen as smoothing function:
5 5 5
5 5
5
(3 ) 6(2 ) 15(1 ) 0 1
(3 ) 6(2 ) 1 2
( , )
(3 ) 2 3
0 3
v
R R R R
R R R
W r h
h R R
R
σ
 − − − + − ≤ <

− − − ≤ <

= 
− ≤ <

 ≥

(10)
Where σ is a normalization constant that depends on the
number of dimensions (σ = 7/478 in 2D and σ = 3/359p in 3D,
respectively),  the number of dimensions, R is the ratio between
the modulus of the distance vector, r and the smoothing length h.
Although this choice is computationally more expensive than a
cubic spline kernel, it is more stable for flows with low Reynolds
numbers [24]. Finally, the time step was selected small enough not
only to satisfy the CFL condition, but also to resolve adequately the
particle acceleration and the viscous diffusion:
2
0.5
min 0.25 ;0.25( ) ;0.125
h h h
t
c F v


∆ ≤  
 
(11)
The main idea behind the SPH algorithm is to solve the
Poiseuille problem between two infinite parallel plates where two
different fluids (represented by two distinct sets of particles) are
initially placed side by side, as shown in Figure 1. This means that,
instead of calculating only the interactions between pairs of fluid
particles and those between fluid particles and boundary particles,
the algorithm must evaluate the interactions between pairs of fluid
particles of the same set, the interactions between pairs of fluid
particles belonging to different sets, and the interactions between
particles of each set and boundary particles.
InstandardSPHalgorithms,thepropertiesofboundaryparticles
are calculated based on the nearest fluid particle; however, this can
generate significant errors when boundary particles interact with
fluid particles belonging to different sets. To overcome this issue,
two overlapping sets of boundary particles were used, one for each
set of fluid particles.

116
Progress Petrochem Sci Copyright © Bertola V
Figure 1: Initial particle distribution of the two fluids and boundary particles.
Validation
The code was initially validated against the analytical solutions
of the single-phase, time-dependent Poiseuille flow; the validation
was then repeated placing in the computational domain two fluids
initially separated, identified by different sets of particles but with
the same density and viscosity. A viscous fluid initially at rest starts
to flow between two parallel plates located at y=0 and y=L, driven
by a body force F(i.e. the pressure gradient) parallel to the plates
(x-axis), until it reaches the well-known steady-state solution for
the velocity profile:
1
( ) ( )
2
x
v y Fy y L
v
= − (12)
Where F is the body force per unit mass and  the kinematic
viscosity. The full transient solution is [24]:
2 2 2
3 2
0
4 (2 1)
( , ) ( ) sin (2 1) exp
2 (2 1)
x
n
F FL y v n
v y t y L n t
v v n L L
π π
π
∞
=


 +

= − + + − 


 
+ 
  
∑ (13)
In this simulation, the following parameters were chosen: F
= 2.10-4
m/s2
, L = 1.10-3
m, v = 1.10-6m2
/s so that the Reynolds
number based on the fluid maximum velocity was 2.5•10-2
. The
problem domain was a square of 0.001m x 0.001m. The fluid
was modelled with 399 particles, 19 in the y-direction and 21
in the x- direction, while the two plates were modelled with 105
particles each, 5 in the y-direction and 21 in the x-direction. The
initial smoothing length was taken equal to 1.33 times the initial
particle space, which in turn was calculated as one twentieth of the
distance between the two plates; the time step was set to 0.0001s.
The simulation reached a steady state after around 6000 time
steps however, in order to check the actual effect of the periodic
boundary, a plot of the particle distribution was taken after 50,000
time steps and displayed in Figure 2. The comparison of the results
SPH simulation with the analytical solution (Eq. 13), displayed in
Figure 2, shows a very close agreement, with an average relative
error of 1.4% after 100 time steps and 0.3% after 1000 and 6000
time steps.
Figure 2: Steady-state (t=5s) particle distribution in single-phase Poiseuille flow (A), and comparison of computed transient
velocity profiles with the analytical solution (Eq. 13) (B).
The code was then tested using two fluids characterised by
the same density and viscosity. In this way, the velocity profile
obtained must be identical to the velocity profile of the single-phase
Poiseuille flow. All parameters, including the initial smoothing
length and the time step length, were the same used previously. The
problem domain was a rectangle of dimensions 0.002m x 0.001m,
with the first fluid placed in the half on left side. The latter was
modelled with 380 particles, 19 in the y-direction and 20 in the
x-direction while the second fluid was modelled with 399 particles,
19 in the y-direction and 21 in the x-direction. The first boundary

117
was modelled with 200 particles, 10 in the y-direction and 20 in the
x-direction and the second boundary was modelled 210 particles,
10 in the y-direction and 21 in the x-direction. The same body
force was applied to both fluids particles, and the resulting velocity
profile obtained was exactly the same as the one shown in the
Figure 2b, while the particle distribution after 50000 time steps is
shown in the Figure 3.
Figure 3: Steady-state particle distribution obtained for the Poiseuille flow of two fluids with identical properties.
Viscous Fingering Simulations
SPH simulations of viscous fingering were conducted using
two fictitious fluids characterised by the same value of density and
different values of the kinematic viscosity. The channel containing
the two fluids was a rectangle of length 5x10-3
m and width 10-3
m,
with the lower-viscosity fluid (“fluid_1”) placed in the left half of the
channel and pushing the higher-viscosity fluid (“fluid_2”) initially
at rest in the right half of the channel, as shown in Figure 1. Fluid_1
was modelled with 950 particles, 19 in the y-direction and 50 in the
x-direction, while fluid_2 was modelled with 969 particles, 19 in the
y-direction and 51 in the x-direction. Boundaries were modelled
with 500 particles for fluid_1, 10 in the y-direction and 50 in the
x-direction, and with 510 particles, 10 in the y-direction and 51 in
the x-direction, for fluid_2.
The reference density of the two fluids was set to 1000kg/m3
,
the kinematic viscosity of fluid_1 was kept constant at a value of
10-6
m2
/s while the value of the viscosity of fluid_2 was 10-5
m2
/s.
The Reynolds number was varied in the range between 0.1 and
1; this condition was implemented by calculating the laminar
pressure gradient corresponding to the Reynolds number value,
and applying it as a body force acting on particles corresponding
to both fluids. The upper limit of the Reynolds number magnitude
was determined by the CFL condition (Eq. 11), using a time step of
10-4
s for all simulations. The initial smoothing length was equal to
1.33 times the initial particles spacing, which in turn was calculated
as one twentieth of the channel width. Simulations were run for
10,000 time steps, corresponding to 10s.
Figure 4: Evolution of viscous fingering in fluids with kinematic viscosity ratio 2
/1
=10 and Reynolds number Re=0.1, at t=0.5s
(A), t=5s (B), t=10s (C).
Figure 4-6 display the evolution of particle distribution at three
different Reynolds numbers (Re=0.1, Figure 4; Re=0.5, Figure 5;
Re=1, Figure 6). Unlike in the case of fluids with identical properties,
one can observe an evolution of the interface that indicates the
development of viscous fingering. Remarkably, fingering is obtained
purely as a result of interactions among particles, without explicit
modelling the interfacial tension.

118
Progress Petrochem Sci Copyright © Bertola V
/1
1=10 and Reynolds number Re=0.5, at t=0.5s
(A), t=5s (B), t=10s (C).
/1
1=10 and Reynolds number Re=1, at t=0.5s
(A), t=5s (B), t=10s (C).
Conclusion
A two-phase mesh-free Lagrangian SPH code was developed
and validated to simulate the displacement of fluids in porous media
by means of a fluid having different viscosity. The code features
an original implementation of boundary particles, which create a
virtual channel for each fluid to avoid discontinuities at the contact
point where the interface between fluids meets the channel wall.
Preliminary results show that SPH can reproduce the formation of
a low-viscosity finger penetrating into the higher-viscosity fluid
during displacement. Because of its intrinsically Lagrangian, mesh-
free nature, SPH is a promising method to simulate viscous fingering
in complex geometries; in addition, it can easily incorporate non-
Newtonian constitutive equations to account for shear-thinning
and/or viscoplastic behaviours.
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