8/8/2019 2003 - CChE - Jain et al
1/16
Monte Carlo simulation of flow of fluids through porous media
Semant Jain, Madhav Acharya 1, Sandeep Gupta, Ashok N. Bhaskarwar*
Department of Chemical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
Received 30 July 1999; received in revised form 26 August 2002; accepted 17 September 2002
Abstract
This simulation employs Monte Carlo technique for studying fluid flow through a porous medium in the capillary regime. Themedium has been modelled as a 2 or 3-dimensional network of elements, some of which are randomly closed to the fluid flow.
Dijkstras algorithm has been employed to identify the least-resistance pathway, which is instrumental in determining the minimum
pressure required to achieve break-through across the network. At higher pressures, network resistance has been calculated by
determining the manner in which the cluster forms and by accounting for the nature of flowpaths. The simulation yields a linear
relationship between the pressure applied across the network and flowrate showing similarity to Darcys law. Polynominal fitting of
the data on the fraction of openable pores open as dependent on pressure applied across the network has been carried out and the
coefficients determined.
# 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Monte Carlo; Simulation; Porous media; Fluid flow; Darcys law; Scaling
1. Introduction
Monte Carlo technique is a method of computer
simulation of a system with many degrees of freedom. It
makes use of random numbers to numerically generate
probability distributions, which might otherwise not be
explicitly known since the considered systems are so
complex (Binder, 1979). Monte Carlo simulation pro-
vides a good comparison between data from experi-
ments on real systems to those from the model. It is used
in areas like simulation of thermodynamic properties of
fluids, crystal growth, combustion of coal particles etc.
Its classical application includes evaluation of multiple
integrals in statistical mechanics.
The term porous media encompasses a wide variety
of contacting devices such as packed towers, sand beds
and substances like limestone rock, filter paper and
catalytic particles. It is desirable to classify the porous
media according to the types of pore spaces they
contain. A proposed classification was by dividing the
pore spaces into voids, capillaries and force spaces
(Manegold, 1937). Void spaces are characterized by
the fact that walls have little or no effect on hydro-
dynamic properties in the interior; in capillaries, the
walls do affect the hydrodynamics but do not bring the
molecular structure of the fluid into evidence; and in
force spaces, the molecular structure of the fluid is of
considerable importance. This work concentrates on
pores of the size of capillaries.
Flow through a porous medium requires a description
of both the medium and the flow. A porous medium can
be represented as an extremely complicated network of
channels, including those containing obstructions and
dead ends too (Bernsdorf, Brenner & Durst, 2000). The
distribution of channels is obtained from assumed
statistical descriptions. The pores in the network can
be interconnected or non-connected, depending on
whether they are a part of a continuous network of
pores that exists within the medium or not. Put another
Abbreviations: APO, actual number of pores that have opened up;
FOPO, fraction of openable pores open.
* Corresponding author. Tel.: '/91-11-6591028; fax: '/91-11-
6581120.
E-mail addresses: [email protected] (S. Jain),
[email protected] (M. Acharya),
[email protected] (A.N. Bhaskarwar).1 Current address: Catalyst Technology Laboratory, ExxonMobil
Refining and Supply Company, Process Research Laboratories, 1545
Route 22 East, Annadale, NJ 08801, USA.
Computers and Chemical Engineering 27 (2003) 385/400
www.elsevier.com/locate/compchemeng
0098-1354/03/$ - see front matter# 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 8 - 1 3 5 4 ( 0 2 ) 0 0 2 1 1 - 9
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]8/8/2019 2003 - CChE - Jain et al
2/16
way, any channel classified as interconnected will
ultimately be filled by fluid flowing through the
medium, while non-interconnected elements will re-
main devoid of any fluid flow. The fraction of inter-
connected channels gives an indication of the accessible
porosity of the medium.
To describe the flow through a porous medium, we
also need to specify two parameters*/applied pressure
across the network and the flowrate (i.e. the net amount
of fluid passing through the network per unit time).We have divided the paper in six main sections. The
theory section has an overview of percolation theory,
capillary pressure and pore structure models. The
section following that describes the approach we used
to develop the simulation code. It covers the algorithm
used to identify the least resistant pathway, how the
status of a pore changes from closed initially to open
to finally part of a flowpath, the manner in which
subsequent flowpaths are identified and finally the
equations used to compute the flowrate through the
medium. After briefly mentioning our constraints, we
describe our results and finally present the conclusions.
2. Theoretical background
2.1. Percolation theory
There are many physical phenomena in which a fluid
spreads randomly through a medium, e.g. it may be a
solute diffusing through a solvent, molecules penetrating
a porous solid, or electrons migrating over an atomic
lattice. Besides the random mechanism, external forces
may also govern the process, as in the case of water
flowing through limestone under gravity.
According to the nature of the problem, the random
mechanism can be attributed to the fluid or the medium.
The former falls under the category of diffusion
processes. A typical example is the motion of one
molecule in a gas as it undergoes collisions with other
molecules. In case of a dilute gas, each collision event is
totally random as it is not influenced by other collisions
that have occurred in the past. In other words, the
medium has no memory of its past history. Also, themedium (which is essentially the molecules), is continu-
ously varying after each collision and so is not in variant
in time.
The other, relatively less common, is known as
percolation. In percolation processes (such as a fluid
soaking into a porous medium), there is a distinction
between the fluid particles and the scattering medium.
This medium, although it varies in random fashion from
point to point, is invariant in time. Thus memory
effects cannot be neglected as in diffusion, and the
random scattering of the particles of the fluid must be
treated as being an inherent property of the medium.This difference between percolation and diffusion can
be mathematically understood through the 1-dimen-
sional Polya walk. The medium is described as a set of
points placed at equal intervals along a straight line and
the particles of fluid can move in steps of unit length in
either direction with equal probability. In the case of
diffusion, the points constitute the fluid as well as the
medium and so they can move in random fashion
without any constraints. In the corresponding percola-
tion process, the points of the medium are assigned a
direction to the left or right with equal probability. A
particle entering the medium moves in accordance with
Nomenclature
Small letters
fb fraction of pores that have been blocked
n number of flowpaths present
r capillary radius
sg size of the grid along one dimension or sideCapital letters
Pa applied pressure (on inlet face of network)
Pc capillary pressure
Pcr break-through pressure
Pmax maximum pressure
Q total flowrate
Qi flowrate of ith flowpath
Qmax maximum flowrate through the network
Rc radius of curvature of meniscus
Ri resistance of ith flowpath
Greek symbols
s surface tension
f contact angle
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400386
8/8/2019 2003 - CChE - Jain et al
3/16
the arrows at each point and so the medium plays the
active role. As can be seen from Fig. 1, a particle can get
trapped and be forced to oscillate indefinitely between
two points (whereas this does not occur in the case of
diffusion). In such a case, no percolation path can be
struck between two ends of a line.
This simple 1-dimensional illustration can be ex-
tended further to 2 and 3-dimensional networks of
elements, where some elements may also be blocked
off to fluid flow. Consider a 2-dimensional matrix
through which a path has to be found (Fig. 2). If all
elements are open to fluid flow, then they form part of a
single percolation path for the entire medium. As
elements are blocked off, the size of the percolation
path is reduced, but it still connects both ends of the
network. At a certain critical fraction of closed elements,
the percolation path ceases to exist. It has been observed
that this fraction is 0.41 for 2-dimensional networks and
0.69 for 3-dimensional ones (Efros, 1982) (Fig. 3).
2.2. Theory of capillary pressure
Consider the hydrostatics of two immiscible fluids orphases that exist simultaneously in a porous medium
(Greenkorn, 1983; De Weist, 1969). In general, one
phase will wet the solid. The entrance of one fluid into a
small pore against the other fluid is opposed by surface
tension forces between the wetting fluid and the pore
walls (Scheidegger, 1963; Muskat, 1982). The result is
that a certain pressure differential in the displacing
phase versus the displaced phase will have to be
produced to maintain equilibrium. This pressure is
called the capillary pressure. In a single capillary, the
curvature Rc of the interface gives rise to the pressure
differential equal to
Pc02s=Rc (1)
The radius of curvature of the meniscus is equal to
Rc0r=cos f (2)
So that for a single circular capillary
Pc02s cos f=r (3)
As can be seen from the above expression, the surface
tension force is inversely related to capillary radius.
Hence, capillary pressure can be regarded as the
resistance offered by a capillary to the flow of fluid
through it*/the larger the capillary radius, the lower the
Fig. 1. Particles a and b are trapped in the medium due to
orientation of arrows.
Fig. 2. Percolation pathway found in the network. Pores part of
pathway have been shown connected with a dashed line. Black filled
circles are inter-connected pores. Open circles are blocked pores.
Arrows indicate direction of motion of a particle at a point in the
medium.
Fig. 3. Intermediate stage of percolation in 2-dimensional network.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400 387
8/8/2019 2003 - CChE - Jain et al
4/16
resistance. In this simulation, the resistance of an
element is dimensionless and so surface tension values
are irrelevant, i.e. the analysis is system non-specific.
2.3. Pore structure models
This simulation uses a simplified model of porousmedium. The model consists of a network of elements
that represent cylindrical capillary tubes of different
diameters and equal lengths. A single pore, then, is a
series of elements placed one after another and so
incorporates the effect of varying diameter along its
length. Our model has all pores of same length. The
effect of arrows (as in Polya walk) is obtained by the
random generation and assignment of resistances (i.e.
through assigning radii values) to elements of the
medium. The concept of least resistance is used to
determine the percolation path of the fluid.
If pressure is applied to a fluid-filled porous medium,or to the fluid at the entrance to a capillary system, the
fluid will penetrate those capillaries whose capillary
pressure is lower than the applied pressure. In other
words, the largest-diameter capillaries would be filled
first and at increasing pressures, the smaller capillaries
would get filled. This is referred to as the concept of
least resistance.
In actual porous media, the pores can be fully filled,
partially filled or be completely empty. Although the
simulation assumes a pore under consideration as being
either fully filled or completely empty, it is possible to
model a partially filled pore as a combination of two
adjacent pores*/one being fully-filled and the othercompletely empty.
In this simulation, the pressure applied across the
network is incremented by a very small amount at every
iteration. This approach is validated by experimental
observations where the effect of hysteresis was dimin-
ished or eliminated by carrying out the experiment
sufficiently slowly (Dullien, 1992). Varying pore geome-
try in a porous medium essentially implies a pore of a
greater or smaller capillary pressure than a correspond-
ing pore of uniform diameter. This implies that this pore
requires a greater or smaller pressure for break-through.
Such effects have been accounted for by selecting theminimum and maximum pore radii at the beginning of
the simulation. The random selection of a radius value
between the above limits for a pore incorporates all
media effects that can affect the resistance of a pore
(Dullien).
Thus, though this simplified model does not account
for the presence of films, permanent hysteresis (Dul-
lien, 1992) or different connectivities of the pore
elements with their neighbours (Cordero, Rojas &
Riccardo, 2001) in the system, it incorporates most
features of porous media that would have a strong
bearing on the flowrate.
3. Simulation
3.1. Initialization
The simulation requires a random generation of
resistances of the elements that mimic the porous
medium. The parameters of relevance are the size ofthe network or lattice and the desired phase fraction of
blocked elements (Monteagudo, Rajagopal & Lage,
2002). The resistances are generated over a range of
values as would exist in a porous medium, but the actual
distributions as reported in literature have not been
used.
At a given instant, each pore is either closed, open
or part of a flowpath. To start with, all nodes are
marked closed. It should be noted that there might be
pore(s) having finite resistance but are surrounded
completely by blocked-off pores. As these pores cannot
be reached from the entry face, their status wouldremain unchanged from closed throughout the dura-
tion of the simulation. As the pressure is increased and
the fluid begins to percolate in the medium, pores that
are filled are marked open. For a pores status to
become part of a flowpath, it must become a part of
either a dependent or an independent flowpath. It is
possible for pores to remain open and yet not part of a
flowpath because these pores could be dead ends or a
sequence of pores that have as yet not succeeded in
forming a flowpath.
3.2. Least resistant pathway
The simulation employs Dijkstras algorithm to
determine the least resistant pathway. Since both
pressure and resistance are dimensionless, the resistance
value of the least resistant pathway can be equated to
give the break-through or the minimum pressure
required to cause the first flowpath to appear.
The simulation has an entry face for the fluid to enter
into the network which can be regarded as a single-
source for the algorithm. By definition, the resistances
of all pores are positive implying positive edge weights
for the graph. The concept of edge weight is equivalent
to the resistance of a pore and is stored in the pore itself.The simulation maintains a priority queue that
contains all the pores whose source distance is yet to
be finalized. All pores present in the queue are sorted in
ascending order of their source distance. To begin with,
all pores are assigned a source distance value as infinite.
Then the source distance of the pores on the entry face is
equated to the resistance of the pore. This is followed by
repeatedly selecting a pore with the least source distance
in the priority queue and relaxing all pores that are
connected to it. The process of relaxation of the source
distance of a pore involves updating the current value of
the source distance with the sum of the resistance of the
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400388
8/8/2019 2003 - CChE - Jain et al
5/16
pore and the source distance of the neighbouring pore
(that has just been popped from the queue), if the value
of the latter is smaller (Cormen, Leiserson & Rivest,
2000).
After all the pores have been assigned the minimum
possible source distance value, the source distance of all
pores on the exit face is compared and the one havingthe least value is selected. This is the value of resistance
of the least resistant pathway or the break-through
pressure required across the network.
For a pathway to exist between the entry and exit face
of the network, the value of break-through pressure
must be finite. If a pathway has been found, the
elements are stored in order of their appearance in the
percolation path and consequently, the order in which
they will be filled up by the fluid.
3.3. Opening of a node
The next step is to carry out a depth-first search
across all pores having finite resistance which have
opened up. A pore is deemed to have opened-up when
its capillary pressure is smaller than the effective
pressure available at that pore. Effective pressure at a
pore is the difference of the cumulative resistance of all
pores that precede the current pore from the applied
pressure across the network. The recursion for a pore
ends when the capillary resistance is greater than the
effective pressure or all adjacent pores having finite
resistance have been explored.
The pressure is incremented in steps until all the pores
that have finite resistance and are reachable haveopened-up. Although the pressure is incremented
slowly, at a given increment there can be multiple pores
that open-up. This is similar to the morphological
approach used to study fractal dimensions (Hilpert &
Miller, 2001).
3.4. Additional flowpath determination
The process of repeatedly increasing applied pressure
would eventually lead to other flowpaths opening up. A
new flowpath comes into existence when an independent
or dependent flowpath comes into existence. An in-dependent flowpath consists of a series of pores that
have opened-up from the entry to the exit face, while a
dependent flowpath is a sequence of pores from the
entry face to a pore already part of an existing flowpath.
The resistance offered by an independent flowpath is
the sum of the resistances of all pores that form the
flowpath from the entry to the exit face.
For a dependent flowpath, the simplest case would
involve just one sequence of pores from the entry face to
a pore already part of an existing flowpath. Here, the
resistance of the newly opened flowpath equals the
cumulative resistance of all pores that have just recently
opened-up and the flow resistance of the pore already
a part of some existing flowpath. Flow resistance of a
pore is defined as the cumulative resistance of all the
pores starting from the pore under consideration to a
pore on the exit face. Other than the above mentioned
case, there is a distinct possibility of the existence of
multiple series of pores that have opened up from a poreinside the network and joined existing flowpaths. In
order to take into account the branching of flow from
such pores, the resistance of each branch is computed
separately by linear addition of the resistances of the
pores that have opened-up (starting at the pore from
which the branching begins till a pore which is an
immediate neighbour of a pore whose status is part of a
flowpath) with the flow-resistance of the pore which is
a part of a flowpath. Thus, the flow resistance of the
pore from which the branching begins is the parallel
summation of the resistances of all the branches from
that pore onwards. The effective resistance of thedependent flowpath is thus the linear addition of the
resistances of all pores from the entry face to the pore
from which branching begins and the flow resistance of
that pore.
3.5. Flowrate
In the case of an actual porous medium, the physical
quantity flowrate is defined only when the fluid
actually exits from the pores at the end opposite to the
one at which it entered (assuming 1-dimensional perco-
lation). The simulated network, in reality, has severalexit points, each of which has its own flowrate. The
flowrate through the network is defined as the cumula-
tive flowrate from all the pores on the exit face.
In this simulation, mass and volume conservation
have been assumed to hold. The pores do not rupture in
the pressure range being studied. Since fluid enters from
one face and leaves from the opposite face, the
cumulative flowrate could be alternatively defined as
the sum of the flowrates entering the network through
the pores on the entry face.
When the first flowpath is obtained (using Dijkstras
algorithm), the net flowrate through it (and in this case,
through the network) is zero. As the inlet pressure isincreased, a new sequence of pores starts getting filled
by the fluid, which may result in another flowpath
joining one of the existing flowpath or flowpaths
emerging from the exit face. All flowpaths have the
same inlet and outlet pressures at any given time and
vary only in their individual resistances. Thus, the net
pressure driving force across a particular path, rather
than the inlet pressure is taken for calculation of the
flowrate. The flowrate through a path is then computed
by dividing the driving force by the path resistance.
Pi0Pa(Ri (4)
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400 389
8/8/2019 2003 - CChE - Jain et al
6/16
Qi0Pi=Ri (5)
The total flowrate is then the sum of individual
flowrates and can be expressed as
Q0Xn
1
Qi (6)
The flowrate is calculated each time a pressure
increment is made and also a new pore opens up.
The pressure after which no more pores open up is
called maximum pressure. Once maximum pressure is
attained, the matrix is regenerated and the process
repeated a few dozen times over.
3.6. Constraints
The random nature of the trials causes variation in the
values of break-through and maximum pressures.
Both these pressures vary with matrix size and fraction
of pores blocked-off.
The choice of matrix size and blocked fraction is
dictated by two factors*/computer memory size and
critical phase fraction values. The memory factor limits
Fig. 4. Fraction of openable pores open vs. applied pressure for a 150)/150 grid at 20 % blockage. Continuous line represents the least-squaresaveraged trend line.
Fig. 5. Flowrate vs. applied pressure for a 150)/150 grid at 20% blockage. Continuous line represents the least squares averaged trend line.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400390
8/8/2019 2003 - CChE - Jain et al
7/16
the maximum network size that can be simulated but
there is also a certain minimum value below which the
simulation is unable to generate statistically significant
data to mimic the flow properly. At these finite sizes,
the randomness associated with Monte Carlo simulation
cannot be used effectively and errors result.
4. Results
In this work, we have a pore throat diameter
distribution (pore size distribution) which is generated
by a pseudorandom number generator according to a
uniform distribution between the minimum and max-
Fig. 6. Fraction of openable pores openvs. Applied pressure for 80)/80 grid at 30% blockage. Continuous line represents the least-square averaged
trend line.
Fig. 7. Flowrate vs. applied pressure for 80)/80 grid at 30 % blockage. Continuous line represents the least-squares averaged trend line.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400 391
8/8/2019 2003 - CChE - Jain et al
8/16
imum values. These values were assumed to be 1 and 10,
respectively. The pseudo-random numbers are used for
assigning values to the required throat diameters, so that
those are uncorrelated, i.e. the size of one throat is
independent of the size of any other throat. Porous
media may have polymodal and/or spatially correlated
pore size distributions, that can also be handled by the
model, but these are not considered in this work.
The flowrate vs. applied pressure relationships ob-
tained from the model for different sets of parameters
show a linear dependence similar to Darcys law (Figs. 5,
7, 9 and 11). In these plots, the values of the y -intercept
must be negative. It is indicative of the fact that only at a
certain finite positive pressure does the flow through the
porous medium begin to take place. The values of
parameters that have been estimated using linear least-
squares technique, show a consistency in the predicted
values regardless of the matrix size for a specified
blockage fraction (Tables 1/5). Due to simulation
time constraints we could not carry out runs at grid
sizes greater than 200 for 2-dimensional networks.
However looking at the graphs of the dominant para-
Fig. 8. Fraction of openable pores open vs. applied pressure for 70)/70 grid at 40% blockage. Continuous line represents the least-squares averaged
trend line.
Fig. 9. Flowrate vs. applied pressure for a 70)/70 grid at 40% blockage. Continuous line represents the least-squares averaged trend line.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400392
8/8/2019 2003 - CChE - Jain et al
9/16
meter vs. grid size (Fig. 12) and cumulative flowrate vs.
grid size at a specific applied pressure (Fig. 13), we
believe that an infinite medium can be simulated
satisfactorily in the 200/300 grid size range for 2-
dimensional networks.
The flowrate is dependent on the number of flowpaths
and that in turn on the accessible porosity of the
medium. In our view, fraction of openable pores open
(FOPO) is a good measure of the accessible porosity. To
calculate FOPO, we noted the actual number of pores
that have opened-up (APO) while incrementing the
pressure and then divided it by the number of pores that
are not blocked-off.
For 2-dimensional systems;
Openable Pores0 s2g(1(fb)(7)
For 3-dimensional systems;
Openable Pores0 s3g(1(fb)(8)
FOPO0APO=Openable Pores (9)
On plotting FOPO vs. applied pressure for different
configurations, i.e. grid sizes and blockage fractions, we
note an initial linear rising trend, attaining an asympto-
tic value subsequently (Figs. 4, 6, 8 and 10). This trend is
Fig. 10. Fraction of openable pores open vs. applied pressure for 18)/18)/18 grid at 40% blockage. Continuous line represents the least-squaresaveraged trend line.
Fig. 11. Flowrate vs. applied pressure for a 18)/18)/18 grid at 40% blockage. Continuous line represents the least-squares averaged trend line.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400 393
8/8/2019 2003 - CChE - Jain et al
10/16
8/8/2019 2003 - CChE - Jain et al
11/16
8/8/2019 2003 - CChE - Jain et al
12/16
8/8/2019 2003 - CChE - Jain et al
13/16
8/8/2019 2003 - CChE - Jain et al
14/16
8/8/2019 2003 - CChE - Jain et al
15/16
expected and holds irrespective of the grid size or the
fraction of pores blocked. This is similar to the observed
trend when calculated cumulative mercury intrusion
volume was plotted against applied pressure (Bryntes-
son, 2002). When the least-squares analysis is conducted
and seven parameters of the polynomial estimated, it is
observed that the parameter values gradually attain
asymptotic values on increasing the grid size (Tables 1/
5). This observation is in line with the fact that an ideal
simulation would have all grid dimensions as infinite.
At low blocked fractions, there are few chances of
pores with finite resistance being surrounded by
blocked-off pores. Thus all pores having finite resis-
tance can be considered openable. The asymptotic
value of FOPO attained in all such cases is near 1 (Figs.
4 and 6). As the blockage fraction increases, the
Fig. 12. Slope of flowrate vs. grid size at different blockage fractions. Simulations with grid size 40 and above rate for 2-dimensional grids, while
stimulations smaller than this size were for 3-dimensional grids.
Fig. 13. Flowrate vs. grid size at different blockage fractions at an applied pressure of 1000 units. Simulations with grid size 40 and above were for 2-
dimensional grids while simulations smaller than this size were for 3-dimensional grids.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400 399
8/8/2019 2003 - CChE - Jain et al
16/16
probability of finite- resistance pores being surrounded
by blocked-off pores is higher and thus, a greater
variation is seen in the asymptotic values of FOPO. This
observation is most stark at blockage values within 5%
of the critical value (Fig. 8). This observation can be
explained by the fact that near critical values there is a
sharp decrease in the probability of finding a flowpath,the implication being that there is a greater variation in
the probability of finding a flowpath for a given
configuration. For this to be true, it would be natural
to expect a greater variation in the asymptotic value of
the openable pores. These observations have been
validated by the regression analysis while computing
the parameter values. The R2 trend values indicate that
there is greater deviation from 1 at blockage fractions
near critical values, particularly at low grid sizes (Tables
1/5). As grid size increases the R2 values improve
indicating that the simulation is closer to that of an
infinite medium.
5. Conclusions
This simulation is a simplified but innovative way of
dealing with flow through porous media. Using the
Monte Carlo technique and concepts of percolation
theory, simulations have been carried out for 2 and 3-
dimensional models of porous media in capillary flow
regime. The flowrate vs. applied pressure results have
been found to be in agreement with Darcys law.
Parameter estimation of fraction of openable poresthat have opened up vs. applied pressure has also
been shown to be consistent. More accurate results
could be obtained by using actual resistance distribu-
tions of porous media. It might then be possible to
predict flowrate vs. applied pressure behaviour for
systems with different physical properties (e.g. surface
tension and contact angle).
Actual experimental data for flow in the capillary
regime have not been reported in literature, where the
major focus has been on statistical description of porous
media. Also, experiments that reproduce the exact
conditions of the simulation would be difficult to
perform. Therefore, the results mentioned herein should
be treated as being of a qualitative nature.
The parameter values have been computed after
considering several thousands of points for each config-
uration (matrix size and blocked fraction). Runs were
concluded once the curve predicting FOPO vs. applied
pressure exhibited an asymptotic behaviour. It has been
our observation that until this asymptotic behaviour is
observed the runs should not be concluded, otherwise
the parameter values would be misleading. This is also in
accordance with the random nature of the simulation
because theoretically an infinite number of runs are
required before presenting any trend.
Further work in this field would include extension of
the simulation to the region of Hagen /Poiseulle flow,
where the resistance is inversely related to the fourth
power of the radius of the element. The resistance
beyond the laminar flow region decreases slightly andthen reaches a constant value at very high flowrates in
the turbulent zone. This can be modelled as two separate
flow regions, with elements having different (but con-
stant) resistances in each region.
Acknowledgements
We are grateful to Dr M.N. Gupta, Senior Manager,
Computer Services Centre, IIT, Delhi, for the generous
allocation of computational time. We would also like to
thank all staff members of CSC, IIT, Delhi, for theircooperation throughout this project. In particular, we
would like to mention Mr. Gopal Krishen, the system
programmer, and Mr. Gulshan Naveriya, the hardware
engineer, for their invaluable assistance.
References
Bernsdorf, J., Brenner, G., & Durst, F. (2000). Numerical analysis of
the pressure drop in porous media flow with lattice Boltzmann
(BGK) automata. Computer Physics Communications 129 , 247/
255.
Binder, K. (Ed.), Monte Carlo methods in statistical physics (1979).
Springer-Verlag.
Bryntesson, L. M. (2002). Pore network modelling of the behaviour of
a solute in chromatography media: transient and steady-state
diffusion properties. Journal of Chromatography A 945 , 103/115.
Cordero, S., Rojas, F., & Riccardo, J. L. (2001). Simulation of three-
dimensional porous networks. Colloids and Surfaces A: Physico-
chemical and Engineering Aspects 187/188 , 425/438.
Cormen, T. H., Leiserson, C. E., & Rivest, R. L. (2000). Introduction to
algorithms. EEE: Prentice Hall of India.
De Weist, R. J. M. (Ed.), Flow through porous media (1969). Academic
Press.
Dullien, F. A. L. (1992). Porous media*/ fluid transport and pore
structure. Academic Press.
Efros, A. L. (1982). Physics and geometry of disorder*/percolation
theory . Mir Publishers.
Greenkorn, R. A. (1983). Flow phenomenon in porous media . Marcel
Dekker.
Hilpert, M., & Miller, C. T. (2001). Pore-morphology-based simula-
tion of drainage in totally wetting porous media. Advances in Water
Resources 24 , 243/255.
Manegold, V. E. (1937). Uber Kapillarsysteme. Kolloid-Z 80 , 253.
Monteagudo, J. E. P., Rajagopal, K., & Lage, P. L. C. (2002).
Simulating oil flow in porous media under asphaltene deposition.
Chemical Engineering Science 57, 323/337.
Muskat, M. (1982). Flow of homogenous fluids through porous media .
IHRDC.
Scheidegger, A. E. (1963). The physics of flow through porous media .
University of Toronto Press.
S. Jain et al. / Computers and Chemical Engineering 27 (2003) 385/400400