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APPLICATION OF THE PRECONDITIONED CONJUGATEGRADIENTS METHOD IN THE SIMULATION OF RELATIVEPERMEABILITY PROPERTIES OF POROUS MEDIAAPOSTOLOS KANTZAS a & IOANNIS CHATZIS aa Department of Chemical Engineering , University of Waterloo , Waterloo, Ontario, N2L 3G1,CanadaPublished online: 25 Apr 2007.

To cite this article: APOSTOLOS KANTZAS & IOANNIS CHATZIS (1988) APPLICATION OF THE PRECONDITIONED CONJUGATEGRADIENTS METHOD IN THE SIMULATION OF RELATIVE PERMEABILITY PROPERTIES OF POROUS MEDIA, Chemical EngineeringCommunications, 69:1, 169-189, DOI: 10.1080/00986448808940611

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Chem. Eng. Comm. 1988, Vol. 69. pp. 169-189Reprints available directly from the publisher.Photocopying permitted by license only.© 1988 Gordon and Breach Science Publishers S.A.Printed in the United States of America

APPLICATION OF THE PRECONDITIONEDCONJUGATE GRADIENTS METHODIN THE SIMULATION OF RELATIVE

PERMEABILITY PROPERTIESOF POROUS MEDIA

APOSTOLOS KANTZAS and JOANNIS CHATZISt

Department of Chemical EngineeringUniversity of Waterloo

Waterloo, Ontario, Canada N2L 3Gl

(Received January 9,1987; infina/form November /7,1987)

A model that is after the determination of the relative permeability characteristics of porous media ispresented. It is part of a general model that deals with the simulation of capillary phenomena andimmiscible fluid flow behaviour in porous media. The relative permeability characteristics in a waterwet porous medium are simulated with the use of three-dimensional (3-D) network models of porestructure with pore body and pore throat size distributions. The major assumption involved is that acubic network of pore bodies connected by pore throats with pore body and pore throat sizedistribution respectively, is a realistic representation of the pore structure of a porous medium. Thephysical laws that apply in real media are simulated in the network analysis, and the computed resultsare compared with experimental findings. A new algorithm was developed that leads to the solution ofa large set of linear equations, with a sparse and positive definite coefficient matrix. Results obtainedwith the application of the Preconditioned Conjugate Gradients method and numerical aspects of thesimulation are presented and discussed. Comparison of this method with other available numericalmethods is also made. It is concluded that the Preconditioned Conjugate Gradients method isadvantageous for large networks with regards to time of solution. convergence, and accuracy. Thevalidity of the algorithm is tested against other methods in the literature.KEYWORDS Relative permeability Numerical simulation Conjugate gradients method

Porous media

INTRODUCTION

The most important property of any porous medium is its permeability. It isdefined as the ability of the medium to transmit fluids and it is describedmathematically by Darcy's law. For a uni-directional flow in a core sample,Darcy's law is well known by the equation:

Q/A = - (K//l)(.6.P/L) (1)

where Q is the volumetric flow rate, A is the cross-sectional area of the core, /l isthe viscosity of the fluid, k is the permeability of the medium and t1P is the

t To whom correspondence should be addressed.

169

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170 A. KANTZAS AND 1. CHATZ1S

pressure drop across the length L of the sample. The permeability to a fluid at100% saturation is called specific or absolute permeability. Effective permeabilityis the measured permeability to a fluid in the presence of a second or third phasein the medium. The ratio of effective to absolute permeability is called relativepermeability. Knowledge of relative permeabilities is an integral part of thesimulation of the time history of oil recovery. There is need of a model that willgive an accurate prediction of relative permeability as a function of phasesaturation and saturation history.

During the past thirty years many investigators tried to simulate capillary andtransport phenomena in porous media and an extensive list of past work has beengiven elsewhere (Dullien, 1979; Kantzas, 1985). The most appropriate models arethe network models of pore structure (Chatzis and Dullien, 1977; Dullien, 1979;Chatzis, 1980; Chatzis and Dullien, 1985; Diaz, 1985; Kantzas, 1985). In thesemodels the porous medium is represented as a random network of pores ofvarious sizes. The physical aspects of such a network must follow some theoreticalconsiderations that deal with the porosity of the medium, the pore to porecoordination number, pore shapes and sizes, etc., which are characteristicproperties of a porous medium (Kantzas and Chatzis, 1988). Rock fluid andfluid-fluid interaction are also important in the case of multiphase flow. Networkformulations are based on percolation theory concepts and its applications toproblems in porous media research (Shante and Kirkpatrick, 1971; Kirkpatrick,1973; Chatzis and Dullien, 1977, 1985; Chatzis, 1980; Larson et al., 1981). Amodel that incorporates the above considerations has been developed by Chatzis(1980), Chatzis and Dullien (1977, 1985), Diaz (1985) and Kantzas (1985).According to the aforementioned workers, a three dimensional random networkmodel generated using the cubic lattice is a topological representation of athree-dimensional irregular network with pore coordination number equal to 6.The network is generated using the bond correlated site percolation scheme(Chatzis, 1980; Chatzis and Dullien, 1985) which involves the notion that the sizeof a pore throat is correlated to the size of the smallest of the pore bodies itconnects. Generalized number based pore accessibility functions have beenobtained for this type of network models (Chatzis, 1980; Chatzis and Dullien,1985) and applied to model capillary pressure saturation relationships formercury-air systems (Chatzis and Dullien, 1985), and water-oil systems withentrapment (Diaz, 1985). Having specified the number based pore body sizedistribution, the pore throat size distribution and the pore geometry, there arealgorithms available to convert the number based pore accessibility data tovolume based capillary pressure curve data (Diaz, 1985; Chatzis and Dullien,1985). Moreover, the model has been applied successfully to predict the dendriticto flow non-wetting phase saturation during primary drainage in mercury-airexperiments (Chartzis and Dullien, 1985) and the relative permeability charac­teristics in mercury permeametry experiments (Kantzas, 1985), respectively.Recently the capability of the model was expanded in modeling fluid distributionsof the wetting phase and the non-wetting phase, respectively, as a function ofsaturation and saturation history (Diaz, 1985). The pore accessibility history datawere used to simulate the saturation history in the form of capillary pressure

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PERMEABILITY OF POROUS MEDIA 171

curves. The number based information of fluid distributions in the networksimulation of quasi-static immiscible oil-water displacements, is used in this workfor the simulation of the relative permeability characteristics in porous media(Kantzas, 1985).

The Hagen-Poiseuille equation is assumed to be the governing equation offlow in a single pore. The pores are assumed to have lamelar shapes (e.g. prismswith rectangular cross-sections). In this case the H.P. equation is written asfollows:

Q/A' =Q/BW = (1/12)(B 2//lo)(l:1P/L') (2)

where A' is the cross section, B is the height, W is the width and L' is the lengthof any capillary slit. The extension of the H.P. equation to pore level dimensionsis justifiable based on the numerical solution of the Navier-Stokes equation inperiodically constricted tubes (Azzan and Dullien, 1977; Dullien, 1979). FromEqs. (1) and (2) the terms

(3)and

(4)

represent the conductivity to flow of the medium and of one pore respectively.Equation (3) defines the macroscopic conductivity of the medium, while Eq. (4)defines the microscopic conductivity of a pore. The procedure for the computa­tion of the permeability, k, from Eq. (3) involves principles for the computationof the overall conductivity of a network of resistors with individual conductivitesgiven by Eq. (4).

The electrical analog to Darcy's and H.P.'s laws, is Ohm's law which states:

I=VG (5)

where I is the current passing through a resistor with conductivity G, and V is thevoltage drop across the resistor. It can be seen from Eqs. (1), (2) and (5), that Qis analog to I, and l:1P to V, while the analog of G is the qo and qi respectively.As a result, one can use the similarity between flow of electricity and fluid flow tocalculate the overall flow conductivity of a porous medium with the same strategyas that used for a passive resistor network. The overall conductivity can be solvedwith the application of Kirchhoff's laws in a network of electrical resistors.Equation (4) is used for each individual pore element as the analog to Eq. (5),and Eq. (3) is used for the calculation of the permeability value. Equation (5) ismodified in Kirchhoff's laws as:

(6)

for every node (i) in the network, connected to m (m = 1, 2, ... , 6) neighbouringnodes. Gim is the conductivity of the resistor connecting node (i) with node (m)and \lim is the voltage difference (\Ii - \l;n)' Equation (6) leads to a linear set ofequations, that can be represented in matrix notation as

111= IVI . IGI (7)

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172 A. KANTZAS AND I. CHATZIS

if the conductivities are used, or

IVI = IRI· III (8)

if the resistances are used. In the above two equations III is the current vetor, IVIis the voltage vector, while IRI and IGI are the resistance and conductivitymatrices respectively.

For optimum numerical solution, Eq. (7) is preferred because of less memoryspace and less execution time requirements. For a network that consists of Nnodes, Eq. (7) leads to a set of N linear equations. In this set, all the elements ofthe current vector have the value of zero. All the elements of the voltage vectorare unknown. The elements of IGI are directly related to the conductivities of theindividual elements of the network. Therefore, the conductivity matrix IGIcontains the coefficients of the unknowns in the linear equations and it is calledthe coefficient matrix. However, the above matrix is not positive definite and it isnot easy to obtain a solution. Therefore, it is necessary to have an additionalindependent equation that relates the overall conductivity of the network, Go, theoverall voltage drop, Vo , across the network and the overall current, 10 , Thisbecomes available if the network is connected to external nodes as in the schemeof Figure 1, where 10' Vo and Go are related through the equation:

(9)

If the value of the overall current 10 is fixed, then a linear and positive definite setof equations is obtained and can be solved for the voltage vector. If 10 isarbitrarily set equal to 1, then the overall conductivity of the network can beautomatically calculated, since

(10)

Therefore, according to the above scheme, if the conductivity of each element(resistor or pore) is known, the overall conductivity of the network can becalculated by use of Eq. (10). The questions that arise now are: 1) What is theexact structure of the conductivity matrix? and 2) Which is the most efficient wayof solving this linear set of equations? The answer to these questions is given inthe present work. It should be noted that the electrical analog of fluid flowproblems has been used in the past (Dullien, 1979). However, a comprehensiveexamination of the capabilities of this analog has not been shown before.Realistic physical assumptions have not been implemented in previous publica­tions and the speed and accuracy of the proposed methods were not discussed. Inthis work emphasis is given in the numerical aspects of the simulation. Thephysical aspects of modeling two phase immiscible flow are discussed in detail in asubsequent publication (Kantzas and Chatzis, 1988).

THE NEW ALGORITHM

The conductivities of the network elements in our algorithm are first stored in theproper positions of the coefficient matrix IG I and then a numerical method is

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PERMEABILITY OF POROUS MEDIA

-::J00DOC~OOC)OC~

(a) A 3xS pore body pore network

-173

'--------1 f-:--.------.-J'--SOURCE

(b) Symbolic representation as an etectrtcat network of resistors.

FIGURE 1 A pore network and its symbolic representation as an electrical network of resistors.

called upon for the solution of the linear set of equations. A description of thealgorithm is presented below:

1 Algorithm for the Creation of the IGI Matrix

The IGI matrix is a two dimensional (2-D) representation of the conductingelements of the 3-D pore network. It also provides a picture of the interconnec­tedness of these elements. The elements of the IGI matrix, a(i, j), are made upwith the conductivity values of the bonds of the pore network (i.e. the porethroats of the network) in such a way that Kirchhoff's laws are satisfied. In whatfollows, we explain in detail how the elements of the coefficient matrix IGI areset-up.

(a) Calculation of the diagonal elements of the coefficient matrix Let us considera cubic network with dimensions X nodes long, Y nodes wide and Z nodes high.Consider a node of the network NOD (J, J, K). The sum of the conductivities ofall the bonds that are connected to NOD (J, J, K) should be in the position ail'

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174 A. KANTZAS AND I. CHATZ1S

i = 1 + J + K, or the coefficient matrix. This arrangement satisfies Eq. (6) for eachspecific node.

(b) Calculation of the off-diagonal elements of the coefficient matrix Eachoff-diagonal array of the coefficient matrix is filled with the individual conduc­tivities of the bonds that connect two nodes. The number of the off-diagonalarrays is equal to all the possible ways that two adjacent nodes can be connected.In the cubic network there are six bonds that connect each node to all itsneighbours and, as a result, six off-diagonal arrays have to be filled, three on eachside of the diagonal. In general, for a coordination number z = n, n off-diagonalarrays are to be filled. The rest of the arrays are going to have elements with zerovalues. Extra care is sometimes required for the boundaries of the network. Forthe specific case of the cubic network, the procedure of allocating the conductivityvalues in the off-diagonal arrays of the coefficient matrix is as follows:

The first off-diagonal array is occupied by the conductivities of the bonds thatconnect the nodes at positions (1, J, K) and (1, J + 1, K). Those elements holdthe (i, i + 1) and (i + 1, i) positions in the coefficient matrix. The secondoff-diagonal array is occupied by the conductivities of the bonds that connect thenodes at positions (1, J, K) and (1 + 1, J, K). If the network has dimensions X byY by Z, the elements of the matrix that represent the bonds mentioned above willhold the (i, i + X) and (i + X, i) positions in the coefficient matrix. The thirdoff-diagonal array is occupied by the conductivities of the bonds that connect the(1, J, K) and (1, J, K + 1) nodes. These elements will hold the (i, i + X· Y) and(i + X . Y, i) positions of the coefficient matrix. Finally, the conductivities of thebonds that connect the (1, J, Z) nodes to the external node, occupy the(i, X· Y· Z + 1) and (X· y·z + 1, i) positions in the matrix. All of the aboverefer to the case where the sides of the network that are parallel to themacroscopic flow direction are considered to be impervious. If other boundaryconditions are taken into account, care must be taken when inputing theconductivities of the bonds that connect the (1, 1, K) and (1, Y, K) nodes, and the(1, J, K) and (y. (X - 1), J, K) nodes, respectively.

After the values of the elements of the coefficient matrix and their (i), (j)positions have been determined, they are stored in three vectors. For example,the element aU, j) =m will be stored as A(l) =m, X(/) = i and Y(/) =i. where1= 1, 2, 3, ... , I', and I' is the total number of non zero elements. The value ofI' is calculated through the algorithm by incrementing. With this arrangement,only the non-zero elements of the coefficient matrix IGI are stored in computermemory, without having to allocate memory space for the whole coefficientmatrix. As a result the memory requirements for the solution of the linear set isdrastically reduced.

Figures 2a and 2b show how the nonzero elements of the coefficient matrix arelocated for the case of impervious boundaries and cyclic boundary conditionsrespectively. These two figures were made by applying the construction rules for a3 x 3 x 3 cubic network. Both matrices are symmetric and highly structured.Errors that can arise through improper positioning of the elements of the matrixor erroneous circuit arrangement in space, can be detected easily if the matrix is

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PERMEABILITY OF POROUS MEDIA

: zero elements ali

.. nonzero elements

...... .......

•

•·····................(a) Impervious lateral boundaries

in a 3x3x3 cubic network

...... ... ...... ... ..

·...... ... .... .· .· ............(b) Case of cyclic boundary conditions

in a 3x3x3 cubic network

175

FIGURE 2 Symbolic representation of the coefficient matrix in a 3 x 3 x 3 cubic network: a) case ofimpervious boundaries; b) case of cyclic boundaries.

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176 A. KANTZAS AND I. CHATZIS

not properly constructed. The error can be detected either through inspection,since all the elements of the network are assigned unique positions in thecoefficient matrix, or through the solving procedure since only the properpositioning gives a positive definite matrix.

(c) Memory requirements for storage The memory requirement for storage ofthe elements of the coefficient martix is directly related to the network size understudy. It has been proved that the smallest network size which can give realisticpredictions of the porous media behaviour, is a cubic network that consists of15 x 15 x 15 = 3,375 nodes (Chatzis, 1980). When a N by N network is used forrelative permeability simulations, it will give a coefficient matrix with (N 3 + I?elements. The memory required for the storage of all these elements is enormous,especially when double precision arithmetic has to be used. In addition, most ofthese elements have the value of zero and are of no use for the solution of thelinear set. Fortunately, the problem is greatly simplified by the use of specialalgorithms and subroutines that can treat the so-called sparse matrices. Suchsoftware can be found in many of the commercially available libraries.

Sparse matrices are matrices that contain many zero elements. The subroutinesthat deal with them usually need only the nonzero elements of the matrix and, asa result, the amount of storage required is reduced dramatically. For example, anetwork of 10 x 10 x 10 nodes will be transformed to a coefficient matrix that has1,002,001 elements. The nonzero elements are only 3,801 or 4,001, depending onthe boundary conditions (i.e. impervious or cyclic) applied to the network. Inorder to minimize storage requirements and speed of solution it was decided torun the simulations with a 10 x 10 x 10 network. This network is the central partof a larger network (e.g. 20 x 20 x 20, or 30 x 30 x 30) which was used tosimulate pore accessibilities (Diaz, 1985).

2 Algorithms for Solving Sparse Matrices

In the process of investigating the appropriate solving techniques, many differentmethods (direct and iterative) were tested. The method used for this work had tosatisfy the following requirements:

- minimum storage,- high precision, and- maximum speed of solution.

An iterative method is preferable compared to a direct one because theround-off error in the final solution is much smaller (Nash, 1979). In addition, thecomputer memory and execution time are less for large sparse systems. Thecoefficient matrix in this specific problem is symmetric and positive definite. Onecan take advantage of these properties to obtain optimum solutions. Commer­cially available packages include:

(i) IMSL (International Mathematical and Statistical Libraries) routines forvarious cases of linear sets.

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PERMEABILITY OF POROUS MEDIA 177

(11)

(ii) Y12M is a set of routines available as a part of the NAG (NumericalAnalysis Group, Oxford University) libraries.

(iii) The Waterloo Sparse Linear Equations Package (SPARSPAK), is anothercollection of methods for solving sparse systems of linear equations (Georgeand Liu, 1981).

(iv) The HARWELL subroutine library (1981).

After careful consideration of the capabilities of the various methods, as it willbe shown later, we decided that the best method for the solution of the linear setwas the Preconditioned Conjugate Gradients method and best subroutine wasHARWELL MA31AD/BD.

The Conjugate Gradient method had been a controversial method for manyresearchers (Westlake, 1968; Nash, 1979). For a case of a positive definite andsymmetric matrix A, and a linear set Ax = b, Hestenes and Stiefel (Westlake,1968) developed the first iterative procedure based on the Conjugate Gradients.The optimized method was the result of research on the minimization ofquadratic functions. Today the algorithm has become very popular (Nash, 1979).It requires five working vectors to store the problem, the intermediate results andthe final solution. This is exclusive of any space needed to store or generate thecoefficient matrix. Moreover, the HARWELL version of the algorithm, needssome additional parameters to be introduced by the user. The parameters ofspecial interest are the parameters C, D, and EPS.

Let us provide a background of the aforementioned parameters: C is a doubleprecision variable whose value is specified by the user. A new entry, aij. createdduring factorization is ignored if its numerical value is less than ICI Vaijajj. If Chas a non-negative input value, it may be changed during the decomposition withthe aim of using the available space to obtain an accurate factorization. D is alsoa double precision variable. The active matrix is considered as a full matrix whenit has density D, and it can be set by the user. EPS is an array where EPS (1)must be set by the user to be equal to the desired accuracy of solution. For thepurposes of this work, the desired accuracy was 1 x 10-10

• Another subroutineparameter of interest if IFLAG. This is an integer variable set by the subroutineto designate an error code. It is set to IFLAG = 0 on a successful return,otherwise it will have a non-zero value. Another acceptable value is the value ofIFLAG = 2, indicating that it has been necessary for the subroutine to modify thediagonal element d;j to remain a positive value. The new entry of djj is set as

[ (") Idj j = max la;i I • k > j + 1

Many simulations that gave error return IFLAG =2 were repeated withdifferent C and D values in order to get a solution with IFLAG = O. Theconductivity value obtained with IFLAG = 0 was not different from that withIFLAG = 2 in the specified level of accuracy (e.g. 10- 10

) . Therefore the resultswith IFLAG = 2 were considered acceptable.

Our algorithm was first tried out with the WATFIV compiler along with thestandard set of 1,001 equations. A group of 40 to 50 runs were made forgenerating data of a complete relative permeability curve. Therefore, for the

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178 A. KANTZAS AND I. CHATZIS

purposes of this simulation, several hundred groups of 1,001 equations weresolved. All of them had coefficient matrices with the same construction, but thevalues of non-zero elements were changing as the saturation history of thenetwork was changed.

The input information for the coefficient matrix obeyed the following generalrule. The non-zero elements of the matrix were given certain values that weregenerated using Eq. (4) for each bond of the network. The overall conductivitywas calculated next. After this was done, the procedure was repeated at differentnetwork conditions for which not all of the bonds in the complete network wereconducting. The non-conducting bonds were assigned very small conductivities,(e.g. G =10-4

) in order to avoid computational problems. In addition to thecomputed value of the overall conductivity, the algorithm provided solutions forthe voltage (or pressure) at each node of the network and statistical informationabout the execution time, the number of iterations, messages about thefactorization, etc. After the debugging of the algorithm was completed in theWATFIV compiler, the FORTVS compiler was used. The FORTVS compilerwas found to be approximately ten times faster than WATFIV. The reader shouldtake this into account in the interpretations of the results.

RESULTS AND DISCUSSION

The results presented in this paper address two key areas, namely: a) numericalaspects of the new algorithm and b) physical aspects of the algorithm for relativepermeability calculations under various conditions.

For the investigation of numerical aspects of the new algorithm, we did thefollowing computer experiments in order to determine the speed and the accuracyof various subroutines:

1. statistical analysis of the time to solve and the time to factorize;2. investigation of the optimum values of the subroutine parameters C and D;

and3. comparisons among various subroutines.

In the section devoted to the investigation of physical aspects of the newalgorithm, we present and discuss the relative permeability simulation results.The detailed study of relative permeability predictions under various immiscibletwo-phase fluid flow conditions is given elsewhere (Kantzas, 1985; Kantzas andChatzis, 1988).

(a) Investigation of the Numerical Aspects of the New Algorithm

(1) Statistical analysis A random sample of 66 runs was used in order to obtainsome descriptive statistics on the behaviour of the subroutine. The sampleconsisted of seven complete runs of network emptying and filling respectively. Wedefine as "emptying" the procedure that begins with the conductivity calculationson a fully connected cubic network (i.e. conducting) and proceeds to conductivity

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PERMEABILITY OF POROUS MEDIA 179

20,------------------, 20,-------------------,

roo: LINEAR EOUATIONSFORTVS COMPILER

CI I FLAG = 0

eIFLAG=2

•... z

af­<[N

a:af­uit.a:au, f;

1001 LINEAR EQUATIONSFORTVS COMPILER

o I FLAG =0.IFLAG=2

JiIllij> 'l,

~ .....D•.. ~ '-

CI CI •••DC,", • •• •

NUMBER OF ITERATIONS ITERATIONS

70

FIGURE 3 Primary drainage simulation runs with different values of the parameters C and D.

calculations on networks that are created with the removal of nodes and bonds ofthe network until the condition of a fully disconnected network is reached."Filling" is qualitatively the opposite procedure. Filling and emptying are basedon aspects of the bond correlated site problem of percolation theory (Chatzis andDullien, 1985; Diaz, 1985). Using porous media terminology, the emptyingprocedure corresponds to the partial displacement of one phase in the porousmedium by another phase under quasi-static capillary equilibrium conditions. Thenowetting phase invades the medium which is fully saturated with the wettingphase, simulating the primary drainage type of displacement. Then the wettingphase displaces the nonwetting phase and the secondary imbibition is simulated.Finally the nonwetting phase invades again and secondary drainage is simulated.The details of this simulation are discussed in detail elsewhere (Kantzas, 1985;Diaz et al., 1987; Kantzas and Chatzis, 1988).

After performing hundreds of runs and taking the aforementioned randomsample, we found that 90% of the runs needed less than 60 iterations in order toconverge to a 10-10 accuracy (Figure 3). Although there is a linear relationshipbetween the number of iterations and solving time, there is no obviouscorrelation between solving time and time for factorization. Factorized matricesrequiring a large number of iterations in order to converge were those withIFLAG = 2. It was expected that the plot of time of factorization versus time ofsolving should be a vertical bar. The observed scattering is most probably aneffect of the different values of the alj elements of the IG I matrix used as inputdata. Therefore, a user cannot make accurate predictions about the expectedbehaviour of the subroutines.

Next we consider another statistical test where a sample of 70 linear sets of 9complete runs with IFLAG return equal to zero, were examined in the same way

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180 A. KANTZAS AND 1. CHATZIS

as before. It was found that 90% of the runs converge at 10 iterations maximum.This indicates that the routine performs with ease where non-disturbed factoriza­tion is performed (that is the case where there is no need of applying Eq. (11».The degree of network filling is considered to be the main parameter whichaffects the speed of solution.

The conclusion drawn from these results if that no prediction can be made withregards to the time for factorization. Non-disturbed factorization (IFLAG = 0)reduces significantly the number of iterations. The user has to apply a trial anderror technique to find optimal solution pathways. These optimum routes areinvestigated by using the two input parameters C and D.

(2) Investigation of the parameters C and D Many different runs for a fullyconnected cubic network and for a disconnected network were performed usingdifferent values of C and D. The results plotted as solving time and factorizationtime versus D for different C are given in Figures 4a and 4b. For the completenetwork, there is a region of C and D values for which IFLAG = O. This region,seen in Figure 4a, can be combined with the region having minimum solving time,

m,,------------, '0,------------,

9f-zo •;:« ,N

~ 0...~ 5

0: 'o"- 3

w

" a;:

C.Oool Qe...« ,N

~ 6...u ,~

0: '

f2 3

w

" e>=

F"QfITV5 CO",PIl.EAMOSTLY ["'''TY N[T\IoQAK

o I Fl,'£•• 2

°00 01 02 OJ 04 0.5 06 07 08 ~;;-0-;0';;'----;;0.';-'-;0-;;3----;;0.4;-;0;':'-0~0"---;;0';-'-;!0'o VALUE 0 VALUE

(a) Effect of parameters C and 0 on factorlutlon time

'00,------------, '00,------------,rOllTVS COloOPIL[fl

90 rULLY rONO/EcTED '1(fWOR~ ~rOll'vS (0"'''11..(11

MOSTLY ["'PTY NETWORK

a I FLAC, • 2:

0-- --e ~C'Qoo:'Q

~

c• 01

90

~,:::o-'0':"-. -:::0':-'-'0"'3---70,0-:0:':-,---7:-::':---:"o VALUE

"' '0

~70«~GO...- 50

13'0

0:w"'30

"~ 20

'0

C'OO!

: "Q ''''':c,(5001,OJ 04 05 06 07 09

o VALUE

• I rL .. t • 0o I rL.ol,G • 2

00 01 DC!

~60!;i 700:W 00!::"- '0o0: '0Wca 3D

"i 20

'0 ia;~;~;;~=~1o '

(b) Effect of parameters C and 0 on number of iterations

FIGURE 4 Effect of C and D values on a) the time for factorization and b) the numer of iterations.

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PERMEABILITY OF POROUS MEDIA 181

seen in Figure 4b, for the numerical solution that will give an optimum solutiondefined as the overall minimum time for factorization plus solving. Thedisconnected network is not giving any point with IFLAG = 0 for the range ofvalues examined. The choice of the operating C and D values was based ontroubleshooting tests that gave good performance of the algorithm with regards toexecution time. However, the overall conductivity results, Go' obtained fordifferent values of C and D, were exactly the same in all cases. Therefore there isno issue of increased or decreased accuracy to talk about, but only the issue offaster or slower convergence. The aforementioned results indicate that the pointswith faster convergence can be identified using such a test for any different matrixconfiguration. Nonetheless, the effort of finding an optimum solution can belengthy and expensive (e.g. in the case of the disconnected network). It isrecommended, therefore, to use small and positive values of C and D (i.e D == 0.5and C = 10- 2

) . With this in mind, the subroutine will definitely converge,although it may be slower than optimum for a few cases.

(3) Comparisons of various subroutines The subroutines chosen for the com­parison tests were the HARWELL MA16, MA31 and MA28 routines andSPARSP AK. The main criterion used was that the algorithm had to run on amachine with 1024 K maximum memory space capacity. MA16 and MA28 gavefast and good results for small sets of equations. Both routines failed to convergein reasonable amounts of time for the set of 1,001 equations.

SPARSPAK routines gave very good results. Three out of the five availablemethods needed memory space less than that available by the machine andSPARSPAK routines gave very accurate results, compared to MA31. If weconsider the fact that the routines are in single precision, then the SPARSPAKpackage seems to be even more attractive. However, the execution timerequirement of the methods was considerably higher than that of MA31.

The MA31 subroutine was undoubtedly the best one. Approximately 11.5 CPUseconds were sufficient for the IBM-4381 mainframe computer to create thenetwork, calculate the conductivity values, construct the coefficient matrix, anddecompose and solve a linear set of 1,001 equations. Often, the total timerequired was down to 4 CPU seconds if proper C and D values were chosen.

Tables I and II give a picture of the results of the solution of 1,001 equations.These results are obtained by use of the subroutines: HARWELL MA31 (indouble precision), SPARSPAK B5 (Quotient Minimum Degree Ordering andGeneral Sparse Solve), SPARSPAK A5 (Nested Dissection Ordering andGeneral Sparse Solve) and SPARSPAK B3 (Retired Quotient Free Ordering andImplicit Block Solve). All four methods had memory requirements less than theimposed constraint. These results were obtained by calculating first the conduc­tivity of a fully connected network which, in subsequent stages, is emptiedthrough a series of eight steps. As can be seen from Table I, the computed valuesof the voltage drop by all methods are in very good agreement for networksaturation conditions up to Ps = 0.68 which corresponds to the percolationthreshold of the wetting phase in a two phase simulation (Chatzis and Dullien,1985; Diaz et al., 1987). At Ps values greater than 0.68, the wetting phase looses

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182 A. KANTZAS AND I. CHATZIS

TABLE I

Comparison of numerical results for 1001 equations, given by various subroutines

Overall voltage drop across the network of wetting phase (volts)Network

Saturation Subroutiney, p, HARWELL MA31 SPARSPAK 5 SPARSPAK B5 SPARSPAK B3

0.000 0.00 0.0000397651 0.0000397085 0.0000397332 0.00003972890.354 0.40 0.0000577268 0.0000576765 0.0000576908 0.00005769580.371 0.42 0.0001161882 0.0001160934 0.0001161123 0.00011613960.422 0.45 0.0002125266 0.0002122263 0.0002123363 0.00021242000.620 0.65 0.0010278311 0.0010264690 0.0010266760 0.00102766400.652 0.68 0.0037961813 0.0037853478 0.0037872121 0.00379598070.663 0.70 1.9024424059 1.7352218600 1.4100742300 1.87533760000.668 0.75 1.3657640127 1.3581485700 1.3419389700 1.3462886800

TABLE II

Time and storage requirements for the above routines. The results are for the eight saturation pointsin Table I, using the FORTVS compiler

Method

Total area available (bytes)

Array area assigned (bytes)

Total exec. time CPU

B5

609232

492058

264.6

B3

609232

492058

622.6

A5

609232

492058

382.3

MA3J

609232

492058

97.9

hydraulic continuity across the network. As a result the voltage drop increasessignificantly, as computed, and there is higher degree of disagreement among thecomputed result of overall voltage drop by the various methods. The subroutineMA31 should be the more reliable one, since it is the only double precisionmethod. Table II demonstrates the speed of solution for each of the methods andjustifies the choice of subroutine MA31.

(b) Investigation of Physical Aspects of the New Algorithm

Simulation of relative permeabilities The first set of computer experimentsperformed, were based on a bond correlated site percolation problem (Chatzis,1980; Kantzas, 1985). The 10 x 10 x 10 cubic network was generated and nodes ofindex j (j = 1, 2, ... , k, ... , 100) larger than a certain index k were removedfrom the network. The procedure of removal consisted of assigning a very smallconductivity value to the elements that were considered to be non-conductive.Thus, the non-conducting elements, e.g. bonds of index j greater than k, could beassumed to have negligible conductivity, and therefore it was as if they wereactually removed. The conductivity values were assumed to be 100 for the

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PERMEABILITY OF POROUS MEDIA 183

1.0,...------------.-----....,

(,. - ?rase"\. >;0'-"o - -:c"'~;;<;;vcc:"

OO!--~_c':~,L....,..,___-_"..:__-...,...,...--o 20 40 50 80 100PERCENT OF CONDUCTING NODES PRESENT

01

0.2

09

08

07>-I-s>=06<J:>Q

~05<Jw204

S::t 03

FIGURE 5 Comparison of the relative conductivity characteristics of a bond correlated cubicnetwork with Kirkpatrick's results when the resistors present in the subset network have the sameconductivity value.

conductive bonds, and 0,01 for the non-conductive ones, The overall conductivityof the network was calculated for each step, and then all the calculatedconductivities were normalized to the value that corresponds to a fully conductingnetwork. The relative conductivity curve that was obtained in our work wascompared to that reported by Kirkpatrick (1973). Figure 5 shows the results ofthis computer experiment. It can be seen that the agreement between the twomethods is very good, especially if we consider that Kirkpatrick had used a largernetwork, one of 15 x 15 x 15 nodes, and a different mathematical approach.

With regards to the problem of two phase flow in porous media, the firstcomputer experiment represents the relative conductivity characteristics of thenonwetting phase in a fluid pair system such as mercury and air where no trappingmechanisms exist. The nonwetting phase (e.g. mercury) will occupy the relativelylarge pores of the medium which in the simulation are represented by nodes inthe network assigned small indices (e.g. indices 1,2,3, represent the largest, thesecond largest and the third largest group of pores in the medium, respectively).This is shown by the results shown by curve (a) in Figure 6. After this firstsuccessful test of the relative conductivity problem, another set of computerexperiments was performed. In this case, we started with a fully connected cubicnetwork of indices j = 1, 2, ... , 100. Bonds of index j less than or equal to k wereremoved and bonds of index j greater than k were allowed in the network. For

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184 A. KANTZAS AND I. CHATZIS

I.0r--.;:------------"j

09

0.8

~ 0.7...s~ 06::>Q

~05<.>w~ 04:5wII: OJ

02

0.1

6 - 5ubseL ao - Subs e t b() - Ki.r-kpot.r-Lck

j

00'-__,------"c--,---~-__:_,___-"o-__,_:I

o 20 40 60 ao 100PERCENT OF CONDUCTING NODES PRESENT

FIGURE 6 Relative conductivity characteristics of subset networks of a bond correlated cubicnetwork as a function of the percent of nodes allowed for accessibility. (Note: subset network gcharacterized by the fraction p, of nodes present in the lattice; subset network 11 is the complementaryof subset g network; all resistors arc of the same size.)

different simulation steps k, k = 10, 20, ... , 90, representing different saturationconditions, the overall conductivity of the remaining network with bonds of indexj greater than k was calculated for each step k; normalized relative conductivitywas calculated in the same manner as before. The results of the second set ofcomputer experiments are shown in curve (b) of Figure 6. These results could notbe compared to Kirkpatrick's results because the displacement mechanism isdifferent. However, these two sets of experiments are useful, not only for resistoror semiconductor networks, but for flow in porous media simulations as well. Therelative conductivity characteristics of the second set of computer experimentssimulate qualitively the conductivity behaviour of the wetting phase within aporous medium when there is no trapping of the wetting phase. The wettingphase occupies the relatively smaller pores of the medium. In this networkanalysis, the small pores are represented by the nodes and bonds in the networkwhich have been assigned relatively large indices (e.g. indices 68,69,70, ... 100).An example of this is the case of mercury (n.w.p.)-air (w.p.) system of fluid pairsat saturation conditions where all of the removed bonds and nodes of index j lessthan or equal to k are part of the largest cluster involving bonds and nodes of indexj = 1, 2, ... , k. Comparison of the relative conductivity curves in Figure 6 showsthat the conductivity characteristics are not symmetric.

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PERMEABILITY OF POROUS MEDIA 185

LEGEND--With Size

Distribution_._.- All Throat

Sizes Equal

>­I-

>t=u::JClZoU

W>

jwa:

II

II

II

I/

//

a .06----"-~----'-""--.J'--'---·--'---.:::.,...____¢.____'l~

a 20 40 60 80 100PERCENT OF CONDUCTIVE NODES PRESENT

FIGURE 7 Effect of pore throat size distribution on relative conductivity of subset networks ofnonwetting phase (a) and wetting phase (b).

Pore throat size distribution functions and therefore different conductivityfunctions were used in a new set of computer experiments to investigate theeffects of throat size distribution (Kantzas, 1985; Kantzas and Chatzis, 1988). Theresults are shown in Figure 7. It can be seen that the pore throat conductivitydistribution has a very strong effect on the relative conductivity characteristics ofthe network compared to the case where all pore throats had the sameconductivity value (c.f. Figure 6). For a given fraction of nodes present (Chatzisand Dullien, 1978, 1985), p" the relative conductivity (GIGo) of the n.w. phase inthe case of throat size distribution is greater than the corresponding GIGo valuewhen all pore throats are of the same value. Note that the symbol Go is beingused to characterize the conductivity of the complete network. The relativeconductivity characteristics with size distribution shown in Figure 7 can be used tosimulate relative permeability to mercury curves. In order to verify this, wecompare the results of curve (a) in Figure 7 to the simulation results of Chatzisand Dullien (1985), and to other experimentally determined mercury relativepermeability data (Dullien, 1979; Chatzis and Dullien, 1985). Very goodagreement between the simulation scheme of this work and that of Chatzis andDullien (1985) was found, as can be seen from Figure 8. The interestingconclusion of this work is that the mathematical transformation of Chatzis andDullien (1985) gives the true prediction of the relative conductivity of thenon-wetting phase, such as mercury in primary drainage displacement conditions.

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186 A. KANTZAS AND I. CHATZIS

20 40 60 80MERCURY SATURATION ('I,P.VJ

II - BERER BE-!.·51 MEINRAD 112(I-NOXle 1290- CHAlZIS AND QUI-LIEN• - PRESENT ..OR",

0.0L-...-<'''----:------:c=-----:-:c-~o 100

02

0.1

1·0.-------------.7""1

0·8

0.9

>- 0.7~

::;

~ 0.6

"':;ffi 0.5CL

"'>;: 04:3"'II: OJ

FIGURE 8 Comparison of simulations to the experimental results of relative permeability curves tomercury of Berea, 51. Meinrad and Noxie sandstone samples (Chatzis and Dullien, 1985).

We demonstrated by the aforementioned computer experiments that bond­correlated network models of pore structure can accurately simulate the relativepermeability data to mercury for primary drainage conditions. This complementspast work by Chatzis and Dullien (1985) in the sense that with a physically soundnetwork model and by the incorporation of realistic pore body and pore throatsize distributions (Diaz et al., 1987), one can simulate more than one property ofthe medium (e.g. Capillary pressure curves, relative permeability curves, den­dritic to flow saturation, etc.). This approach, however, is not followedconsistently by other researches in this field (Larson et al., 1981; Heiba et al.,1982; Koplik, 1982; Mohanty and Salter, 1982). One example of this inconsis­tency is the simulation of oil-water or gas-oil relative permeability curves usingthe relative conductivites of random network models after the bond percolationproblem without using trapping mechansims for the wetting and the nonwettingphase. For the simulation of relative permeability curves and capillary pressurecurves, they used pore throat conductivities proportional to D~ (or D?) and thevolume of a pore proportional to D" where D, is the pore throat diameter.Although the pore conductivity relationship is physically sound, the pore volumerelationship to pore throat size is not. This is a major drawback of the networkmodels of pore structure built after the bond percolating problem because theyrequire to have unrealistic pore throat volume to pore throat size relationships forthe simulation of capillary pressure-saturation relationship (Chatzis and Dullien,

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PERMEABILITY OF POROUS MEDIA 187

1977). It is believed by the authors that successful simulation of relativepermeability curves for water-oil or gas-oil fluid pairs in porous media will beachieved by doing successfully the simulation of the trapping behaviour duringprimary drainage, secondary imbibition and secondary drainage types ofdisplacement.

CONCLUSIONS

The reported results illustrate that the Precondioned Conjugate GradientsMethod is a powerful tool for numerical simulation of problems that involvelinear equations with sparse and positive definite matrices. It is the fastest of allthe different methods that have been tested in this work, and requires the leastmemory space. Its accuracy is demonstrated by the good reproducibility of resultsobtained by others who used different numerical methods. Efficient use of theworking parameters of the available subroutine can result in optimum use ofcomputer time and space. The algorithm was successfully applied to simulatedrainage relative permeability curves of sandstones to mercury.

ACKNOWLEDGEMENT

The financial support of this work by the Natural Science and EngineeringResearch Council of Canada is geatly appreciated by the authors.

NOMENCLATURE

A

A'

A(J, J)

B

b

C

D

D(J)

djj

G

GoG(J, J)

I

cross sectional area of porous medium

cross sectional area of a single pore

square matrix

matrix element

slit height

right hand side vector in a linear set of equations

Harwell MA31 working parameter

Harwell MA31 working parameter

main diagonal of the coefficient matrix A(J, J)

diagonal entry of matrix A(J, I)

electrical conductivity

overall conductivity in a complete network

conductivity coefficient matrix

current

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188

IFLAG

kL

L'

L

L(T)N

NOD(I, J, K)

Q

Ps

R

V

W

X

Y

z

Greek

REFERENCES

A. KANTZAS AND I. CHATZIS

Harwell MA31 error code return

permeability

porous medium length

length of a pore

lower triangular matrix of A(I, I)

transpose of lower triangular matrix

number of nodes in the network

space representation of the nodes of a network

volumetric flow rate

overall conductivity of the medium (Eq. (3»

conductivity of a pore (Eq. (4»piezometric pressure

fraction of nodes in the complete network permitted foraccessibility to the n.w. phase

electrical resistance

voltage

slit width

network length (no. of nodes)

network width (no. of nodes)

number based network saturation of the non-wetting phase(fraction of the total number of nodes of the completenetwork that make the subset network where nodes areoccupied by the non-wetting phase; 1 - V. is the networksaturation of nodes by the wetting phase).

coordination number of the network

viscosity

difference

I. Azzarn, M.tS., and Dullien, F.A.L., "Flow in Tubes with Periodic Step Changes in Diameter: ANumerical Solution", Chern. Engn. Sci., 32, 1445 (1977).

2. Chandler, R.. Koplik, J., Lerman, K., and Willemsen, J.F .. "Capillary Displacement andPercolation in Porous Media", J. Fluid Mec, 119,249 (1982). '

3. Chatzis, I., "A Network Approach to Analyze and Model Capillary and Transport Phenomena inPorous Media", Ph.D. Thesis, University of Waterloo, Waterloo, Ontario, Canada (1980).

4. Chatzis, I. and Dullien, F.A.L., "Modelling Pore Structure by 2-0 and 3-D Networks withApplication to Sandstones", J. of Can. Pel. T., 16,97 (1977).

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PERMEABILITY OF POROUS MEDIA 189

5. Chatzis, I., and Dullien, F.A.L., "A Network Approach to Analyze and Model Capillary andTransport Phenomena in Porous Media", Proceedings of the IAHR Symposim "Scale Effects inPorous Media" held in Thessaloniki, Greece, Aug. 29-Sept. I (1978).

6. Chatzis, I., and Dullien, F.A.L., "The Modelling of Mercury Porosimetry and the RelativePermeability of Mercury in Sandstones Using Percolation Theory", I.Ch.E., 25, 1,47 (1985).

7. Diaz, CE., "Simulation of Fluid Distributions and Oil-Water Capillary Pressure Curves in WaterWet Porous Media Using Network Modelling Techniques", M.A.Sc. Thesis, University ofWaterloo, Waterloo, Ontario, Canada (1985).

8. Diaz, C.E., Chatzis, I., and Dullien, F.A.L., "Simulation of Capillary Pressure Curves UsingBond Correlated Site Percolation on a Simple Cubic Network", Transport in Porous Media, 2,215 (1987).

9. Dullien, F.A.L., Porous Media Fluid Transport and Pore Structure, Academic Press, (1979).10. George, A., and Liu, J., W.-H., Computer Solution of Large Sparse Positive Definite Systems,

Prentice Hall, (1981).11. HARWELL Subroutine Library, Computer Science and Systems Division, Atomic Energy

Research of England, Harwell, England, (1981).12. Heiba, A.A., Sahimi, M., Scriven, L.E., Davis, H.T., "Percolation Theory of Two Phase

Relative Permeability". SPE paper No. 11015, presented at the 57th Annual Fall TechnicalConference of SPE-AIME, New Orleans, (Sept. 26·-29, 1982).

13. Kantzas, A., "Computer Simulation of Relative Permeability Properties of Porous Media",M.A.Sc. Thesis, University of Waterloo, Waterloo, Ontario, Canada (1985).

14. Kantzas, A., and Chatzis, I., "Network Simulation of Relative Permeability Curves Using a BondCorrelated-Site Percolation Model of Pore Structure", Chem. Eng. Comm. 69, 191 (1988).

15. Kirkpatrick, S., "Percolation and Conduction", Rev. M. Phys., 45(4), 574 (1973).16. Koplik, J., "Creeping Flow in Two Dimensional Networks", J. Fluid Mec., 119,219, (1982).17, Larson, R.G., Scriven, L.E., and Davis, H.T., "Percolation Theory of Two Phase Flow in Porous

Media", Chem. Engng. Sci., 36, 57 (1981).18. Larson, R.G., Scriven, L.E., and Davis, H.T., "Displacement of Residual Non-Wetting Fluid

from Porous Media", Chem. Engng. Sci., 36,75, (1981).19. Mohanty, K.K., and Salter, S.J., "Multiphase Flow in Porous Media II. Pore Level Modeling".

SPE paper No. 11018, presented at the 57th Annual Fall Technical Conference and Exhibition ofSPE of AIME held in New Orleans in LA (Sept. 26-29, 1982).

20. Nash, J.C, Compact Numerical Methods for Computers: Linear Algebra and FunctionMinimization, Adam Higler Ltd., (1979).

21. Shante , V.K.S., and Kirkpatrick, S., "An Introduction to Percolation Theory", Adv. Phys., 42,385 (1971).

22. Westlake, J.R., A Handbook of Numerical Matrix Inversion and Solution of Linear Equations,John Wiley and Sons, (1968).

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