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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.

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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

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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.

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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


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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)


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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.


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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.


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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.


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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.


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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.


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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.

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