Application of the Lanczos algorithm to the simulation of
groundwater ¯ow in dual-porosity media
K. Zhang
a,*, A.D. Woodbury
a, W.S. Dunbar
ba
Department of Civil and Geological Engineering, University of Manitoba, Winnipeg, Canada
b
Department of Mining and Mineral Process Engineering, University of British Columbia, Vancouver, Canada Received 5 May 1999; received in revised form 4 October 1999; accepted 20 October 1999
Abstract
Groundwater ¯ow in fractured porous media can be realistically described using a dual-porosity approach. A popular numerical approach for simulation of groundwater ¯ow in dual-porosity media is the use of spatial discretization procedures based upon the ®nite element techniques. The computational eort for this technique strongly depends on both the number of unknowns and the number of time steps required to obtain an accurate and stable solution. In this paper we develop a modal decomposition technique based on the Lanczos algorithm to solve the equations of transient groundwater ¯ow in fractured dual-porosity media. The Lanczos algorithm uses orthogonal matrix transformations to reduce the ®nite element equations to a much smaller tridiagonal system of ®rst-order dierential equations. By using this method, problems with large node number can be reduced into equivalent systems of much smaller size. Consequently, large savings in computer time can be realized, especially for the problems requiring many time steps. The eciency is further achieved by using a recursion method to compute the ¯uid exchange between matrix blocks and fractures. In addition, this paper shows how time-dependent boundary conditions or multiple sources or sinks can be realized for the Lanczos method. In order to verify the proposed numerical technique and show its eciency, two examples are presented: one is for a homogeneous aquifer and the results are compared to the analytical solutions and the other shows a multiple well system of dierent time histories in a synthetic dual-porosity aquifer. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords:Lanczos reduction method; Dual-porosity; Numerical modelling; Groundwater ¯ow; Fractured media
1. Introduction
A number of mathematical models describing groundwater ¯ow in fractured porous media have been developed over the past three decades. The distinction between these models arises from dierences in the conceptual model upon which they are based and the methods used to solve the governing equations. One of the most popular and appealing conceptual models is that of a dual-porosity media. The dual-porosity ap-proach was ®rst introduced by Barenblatt et al. [1]. Such an approach assumes two overlapping continua; two porosities and permeabilities are associated with the fractured medium. The primary porosity and perme-ability are those of the porous, but low-permeperme-ability blocks that are separated by fractures. The secondary porosity and permeability are associated with fractures,
where the permeability is generally high and the porosity is low. In order to use the dual-porosity approach, two sets of properties must be known, one for the fractures and another for the blocks. The two systems are linked through a leakage term representing the ¯uid or solute mass exchange between them. It is also assumed that the porous matrix blocks act as sources (or sinks) that feed (or drain) the fractures with ¯uid or solute mass. Examples of using dual-porosity approach to simulate ¯ow in fractured media include Huyakorn et al. [9], Douglas and Arbogast [2] and Fillion and Noyer [6].
In this paper, we extend the Lanczos modal reduc-tion method to solve the equareduc-tions of transient groundwater ¯ow in dual-porosity media. This method uses an orthogonal matrix transformation to reduce a set of ®nite element equations to a much smaller tridi-agonal system of ®rst-order dierential equations. The new system can be solved by a standard tridiagonal algorithm with little computational eort. Similar approaches have been used by Nour-Omid [13] for heat conduction analysis, Dunbar and Woodbury [4] for
*
Corresponding author.
E-mail address:[email protected] (K. Zhang).
solving the equations of groundwater ¯ow in porous media and Li et al. [12] for solving the equations of decay chain transport. The implementation of multi-group time dependent boundary conditions or source/ sink terms has been an impediment for the Lanczos method (see discussion by Gambolati [7], and Dunbar et al. [3]). In this paper a detailed formulation of the Lanczos method for multi-group time-dependent boundary conditions is presented. The implementation method is eective and converges quickly. In our approach, an unsteady state treatment of the facture± matrix leakage term is adopted. Both parallel and blocky fracture conceptual approaches are used and a very ecient scheme for calculating the leakage term using a recursion technique is developed.
The main focus in this paper is to demonstrate the high eciency of the algorithm in solving the ground-water ¯ow equation for the dual-porosity media. Two examples are presented. The ®rst example is for a ho-mogeneous aquifer and the results are compared to the analytical solutions. The second example shows a mul-tiple well system of dierent time histories in a synthetic dual-porosity aquifer with heterogeneities and aniso-tropic properties.
At this point we wish to impress upon the reader that the Lanczos modal reduction method described in this work should not be confused with iterative solvers such as conjugate gradient methods. The dierence between these will be made obvious later in this paper.
2. Theoretical development
The partial dierential equations describing ground-water ¯ow in dual-porosity fractured media are ob-tained by considering a representative elementary volume (REV) consisting of a suciently large number of matrix blocks and fractures. The dimensions of each of these blocks are assumed to be small relative to the scale of the problem. The geometry of the matrix blocks is taken here to consist of either parallel slabs or spherical ``blocks'', as shown in Fig. 1. Because the matrix block dimensions are assumed to be small
rela-tive to the scale of the problem, the hydraulic heads on the surface of each matrix block will be approximately uniform and ¯ow in the block will be essentially one-dimensional. At each point in the domain, there are two overlapping continua, one for the matrix blocks and the other for the fractures. Detailed description of this ap-proach can be found in the work of Barenblattet al. [1]. The partial dierential equation of ¯ow in fracture continuum takes the form [9]
o
wherehis the hydraulic head in the fractures, Tij andS
are the fracture transmissivity and storage coecient of the formation, C is the volumetric rate of ¯uid trans-ferring from porous matrix blocks to fractures per unit volume, and qw is the volumetric rate of ¯uid ¯ow via sinks or sources.
The term C represents the ¯uid±¯ux interaction be-tween the porous matrix and the fracture. In general, it is a function of both time and space. There are three-popular alternative mathematical models for describing
C. Detailed descriptions of these models can be found in the works of Huyakorn et al. [9]. In this paper, we use both the unsteady parallel fracture and blocky fracture model, although the approach we discuss in this paper can be extended to any fracture conceptual models. The equations developed for the parallel frac-ture model are presented below. The corresponding formulae for the blocky fracture model can be found in Appendix B
The ¯uid interaction at the interface of the matrix block and fracture can be obtained by solving a one-dimensional governing equation with appropriate initial condition and boundary conditions at the rock matrix± fracture interface. Ccan be expressed as
C
where a, b are the half thickness of matrix block and fracture, H is the aquifer thickness, K0 is the hydraulic conductivity of the matrix, and an is a constant which
can be determined by
anp2 2n1 2
K0= 4S0sa2; 3
whereS0
sis the speci®c storage of the rock matrix.
The numerical solutions of the dual-porosity ¯ow governing equations are obtained by applying the standard Galerkin ®nite element method for spatial discretization. Consider now the following ¯ow equa-tion obtained by substituting (2) to (1).
o Fig. 1. Dual-porosity conceptual models: (a) parallel fracture model;
where
andrfor a parallel fracture model is
r 2K
0H
a ab: 6
If we adopt a ®nite dierent approach to solve (4) and if the time step size is assumed to be a constant Dt, at time step k+1, the time integration in (5) can be ap-proximated by note the groundwater heads at dierent time steps. The assumption of a constantDtis not necessary. However,
without this assumption, Eq. (7) cannot be written in a recursive form, and we would have to use a less ecient scheme to compute the leakage terms than what is suggested herein.
From (7), it is easy to show that the relation between
Ik
õ andIk 1
õ can be expressed as
Iõk1exp ÿaõDtIõk
1ÿexp ÿaõDt
aõDt
hk1ÿhk:
8
Note that Eq. (8) is in a form of recursion and can be rewritten as a function of groundwater hydraulic head history.
In order to use the Lanczos reduction method, it is as-sumed here that the storage coecient S is a constant for all the fractures. After substituting (9) into (4), dis-cretization of (4) by Galerkin ®nite element method leads to the following system of ®rst-order dierential equations
wherehis a vector of hydraulic heads at nodes of a ®nite element mesh, K and M are conductivity matrix and capacity matrix respectively. Both of these matrices are symmetric and positive de®nite. The right-hand side vector f includes the eects of source terms as well as boundary conditions. In the next section, we will show how (10) can be reduced in size by way of the Lanczos approach.
3. Lanczos reduction method
Numerical solutions of the governing equations for groundwater ¯ow can be complicated due to strong contrasts in the material properties that are likely to exist between the fractures and the porous matrix. In addition, a detailed multi-dimensional numerical grid can easy involve many tens to hundreds of thousands of unknowns. Therefore, an ecient and robust numerical technique is necessary to solve the governing equations. Dunbar et al. [3] have shown that the Lanczos algorithm is well suited for solving large groundwater ¯ow prob-lems, particular when the time duration is long. The Lanczos algorithm uses orthogonal matrix transforma-tions to reduce the ®nite element equatransforma-tions to a much smaller tridiagonal system of ®rst-order dierential equations. A standard tridiagonal solution algorithm can solve this new system with little computational ef-fort. The original solutions at any desired time steps can be obtained from the new system solutions through a matrix-vector multiplication .
Background of the method: The Lanczos method uses the transformationhQw, whereQ is anM orthogo-nal matrix (i.e., QtMQI, the identity matrix). The matrixQconsists ofmLanczos vectors with dimensions
nm. These vectors are an orthogonal set of vectors known as Lanczos vectors [13], which are constructed during a Lanczos recursion process. The Lanczos re-duction process may be started with a vector r0Kÿ1f
properties of the Lanczos vectors, the ®nal reduced equation can be obtained. The Lanczos algorithm and the reduction process are presented in Appendix A. Detailed discussions of the Lanczos reduction method can be found in the following Refs. [10,14,15].
The Lanczos method does not require an actual in-version of the stiness matrix K[3], even though there are many references to Kÿ1 in the Lanczos algorithm. The Kÿ1f are actually computed by solving the system
Kxf. This can be accomplished by a variety of linear algebraic methods.
In this paper we use a Cholesky decomposition and a series of back-solves to carry out the Lanczos reduction. This is, of course, a direct as opposed to iterative scheme. In addition, we typically use time-marching by the Crank±Nicolson method and direct solvers for the solution of (10) as a basis for overall timing compari-sons. It has been pointed out that perhaps iterative solvers such as conjugate gradients should be used in-stead of direct solvers for the solution of (10). This topic has been discussed thoroughly by Dunbar and Wood-bury [4] and Farrell et al. [5]. We recommend that iter-ative solvers be used to carry out the Lanczos reduction in place of direct solvers if the Lanczos method is ap-plied to three-dimensional problems, or where an arbi-trary sparse matrix structure is encountered. It should be noted that whatever method is used to compute the solutions toKxf, the overhead cost will be the same for both the direct integration of (10) and the Lanczos method. The total computation eort of a Lanczos process is approximately equal to solving equation
Kxf for m time steps, no matter what solution method is used. We have developed an iterative imple-mentation of the Lanczos algorithm and applied it to the discrete fracture ¯ow problem, which will be reported seperately.
During the Lanczos decomposition, each Lanczos vector generated is subject to loss of M orthogonality with respect to earlier vectors due to computer run-o error and cancelation. This means that QtMQ6I. Therefore, the orthogonality must be monitored. When loss of orthogonality has occurred, reorthogonalization must be implemented.
Another important issue for the Lanczos method is the stopping criterion or terminating the recursion when a sucient number of Lanczos vectors has been computed. In this work we use an approach described by Dunbar and Woodbury [4]. Numerical experiments have shown the number of Lanczos vectors needed is dependent on the inhomogeneity of aquifer hydraulic properties and element grid size. Typically for a problem of 5000 nodes, less than 100 vectors are needed.
Following the Lanczos reduction method, a reduced system of equations is created by lettinghQwin (10) which results in
wherewis the solution vector with a length ofmin re-duced space, HQtMKÿ1MQ, an mm tridiagonal matrix, and Q is amn matrix. Note that mis much less than total number of original equations, n. This is due to the fact that the recursion for determining the Lanczos vectors is terminated after mn steps and these basis states capture the essence of the solution.gis the reduced right-hand side vector of the equation sys-tem which equals to QtMKÿ1f. The total computation work for formation ofQandHis approximately equal to the work of solving equationKxfformtime steps. The procedure for forming the matrix Qand His pre-sented in Appendix A.
The reduced system equations can be solved by any time integration technique. Note the solution at each time level does not need to be transformed back to the original unknowns. Only the original solutions h at the desired time steps or locations are computed by the matrix-vector multiplication hQw. The Lanczos re-duction method therefore yields a large saving in com-puter memory storage and solving of the reduced system is more ecient than time-marching of the original equation system (10). Another advantage of the reduc-tion method is the great saving in disk space. For a medium-size of real ®eld problem with several thousands to several ten thousands solving time steps, its solution data for all time steps can easy involve hundreds of megabytes to even several gigabytes. However, the total reduced solution data (w) are much less than total orig-inal solution data (h). Typically, the total reduced solu-tion data may be only 1.0% or less of the total original solution data. The original solution for any desired time step or location can be retrieved from the reduced solu-tion data in future with very little computasolu-tion eort.
Initial and boundary conditions: Proper implementa-tion of initial condiimplementa-tions and boundary condiimplementa-tions is key to eciently using the Lanczos reduction method. When the boundary conditions are complicated, the reduction method may lose its advantage by having to re-evaluate the vector g. This subject is discussed in more detailed below. If all the boundary conditions can be written in terms of one time varying function, the reduction method will be most ecient. This situation may include a single well system, or multi-well system with the same linearly dependent pumping rates. For dierent time-dependent boundary conditions, these can be divided into several groups and each group can be implemented separately.
Initial conditions must also be treated properly. They can be eliminated from the solution by writingh as the sum of the initial conditionsh0, plus a transient partv t
Substituting the above equations into (10), one can obtain a dierential equation for the transientv
Mv_KvÿrM
Eq. (13) has the same form of left-hand side as the original equation (10) forh. The only dierence between (10) and (13) is that the new equation has a right-hand side offÿKh0. This is becauseMh_00, sinceh0is not a function of time. In this way the initial vectorw0can be
set equal to the zero vector,0.
The right-hand vector f is time-dependent if the boundary conditions are time-dependent or wells have a non-constant pumping history. Therefore the right-hand side vector of the reduced equation g is also time-de-pendent, which is de®ned as QtMKÿ1 fÿKh0. Unfor-tunately, the vectorgwould have to be evaluated at each time step during the solution of the small tridiagonal system. This would be very time consuming and com-pletely negates any bene®ts in eciency aorded by the small system. The Lanczos decomposition is spatial in nature. Therefore, to retain the time-dependent eects which arise in the vector g, the vector f must be de-composed into spatial and temporal components, i.e.,
fbp t. Herep trepresents the temporal behavior of the boundary conditions or sink/sources. The vector
Kÿ1b then becomes the starting vector which is trans-formed to the constant part of vectorg.
For multiple time-history boundary conditions, Dunbar and Woodbury [4] proposed a scheme that groups the boundary conditions and wells into Nparts with each part having same individual time history pattern
fX
N
j1
fjpj t f0ÿKh0: 14
Its corresponding reduced right-hand side vector has the form
gX
N
j1
gjpj t g0: 15
At each time step, g is updated individually for every group boundary conditions. Our investigation indicates that this scheme is appropriate for one dimensional problems; however, it may not converge (or converge slowly) for multi-dimensional problems.
In this paper we develop a new scheme for the mul-tiple-group time-history boundary condition problems based on the principle of superposition. Similar ideas have been proposed by Li [11]. According to Dunbar's scheme, we start by writing the right-hand side of (13) in the form of (14).
Due to homogeneous initial conditions for (13) (the in¯uence of initial conditions has been eliminated from the left-hand side of the equation), (13) can be split into several systems of equations equivalent to
Mv0_ Kv0ÿrM
S u0f0Kh
0; 16
Mvi_ KviÿrM
S uipi tbi i1;. . .;N; 17
whereuiis in compact notation for the leakage terms,N
the total dierent time history group number, and
vPN
i0vi. The right-hand side of Eq. (16) is not a
time dependent function. It denotes the in¯uences of the steady state boundary conditions and the initial condition.
The modi®ed reduction method generates Lanczos vectors by using their corresponding starting vectors
Kÿ1 f0ÿKh0 and Kÿ1bi for each equation system. Qi
andHiare produced for all the equation systems. After the Lanczos reduction process, Eqs. (16) and (17) will be reduced to:
H0w0_ w0ÿrH0
S z0g0; 18
Hiwi_ wiÿrHi
S zipi tgi i1;. . .;N; 19
wherezidenotes a reduced vector of the leakage terms in
compact notation, andviQiwi.
In this approach, N Lanczos processes are needed, but only one decomposition of matrix K needs to be computed. For a limited number of boundary condition time histories, this approach is expected to be very e-cient. In most cases boundary conditions can be grouped into similar time history patterns. Therefore the Lanczos method is still ecient. The computation eort for each Lanczos process is about the same as solving the original equation system for m time steps. When more Lanczos processes are required or when
N1 m>tn(tn is the total required solution time steps), the Lanczos method would not be ecient.
Computer implementation: The Lanczos method for solving the dual-porosity ¯ow problem is set up in the following stages:
1. The coecientsdõandõ for updating of the leakage
terms are computed.
2. The matrices K, M are formed. The initial and boundary conditions are applied to the discretized equation system to form the right-hand side vectorf. 3. The matrixKis factored by the Cholesky
decomposi-tion method.
4. The Lanczos decomposition is performed. The Lanc-zos vectors Q, right-hand side g, and matrix H are formed.Qis then stored on a disk ®le for future ref-erences.
5. The small mm system of ®rst-order dierential equations is decomposed using a standard tridiagonal solver.
6. The leakage terms are evaluated and a back solve is performed at each time step.
7. If there are more than one boundary condition time history patterns, repeat step 4 to 6 for the next group boundary conditions.
8. A matrix-vector multiplication computes the solution to the original problem for all the desired time or a dot product of twomentry vectors to obtain the orig-inal solution for any location. Computation for the original solutions can be performed during solving of the reduced system or any time after solving of the system.
9. If there are more than one group time-dependent boundary conditions, add the solutions of dierent groups together.
Updating the leakage terms of the reduced equation system is needed at each time step, and they are related to the solution history. By using the following equation, it is possible to avoid the storage of all the hydraulic head time history data.
Evaluation of the leakage terms ofIõinvolves an in®nite exponential series summation. Therefore, updating the right-hand side requires the storage of three extra arrays forIõ,dõandõwith a length of the number of truncated
terms for the in®nite exponential series. Huyakorn et al. [9] showed that the rate of convergence of the series depends directly on the dimensionless parameter t
K0t= S0
sa
2. It is suciently accurate to determinea0such
that t0:5 and a06a. They indicated that the
com-putational eciency can be greatly improved by re-placing the actual semi-thickness of the matrix block,a, with an eective thickness,a0. Only within this eective
thickness the hydraulic head h0 in the matrix block is
aected by the change of head at the block-fracture interface. Using a0 instead of a, very small truncation errors are introduced by retaining only the ®rst several terms of the series. Our experiences indicate that with less than 20 truncated terms a very high accuracy can be achieved. Updating the leakage terms in reduced space is based on the reduced leakage terms and solutions of the last time step. If 20 truncated terms are used, at each time step only about 22m m multiplications are required to update the leakage terms. It is very economic compared to the computation for original equation system which needs 22n n multiplications by using Eq. (21) to update the leakage terms. This recur-sive scheme is also more ecient compared to the iter-ative scheme proposed by Huyakorn et al. [9].
4. Examples
To investigate the robustness and eciency of the proposed approach the Lanczos technique is applied to two examples. The ®rst example examines the numeri-cal solution algorithm by simulating a problem involving transient ¯ow to a pumped well fully pene-trating a fractured, con®ned aquifer. Both parallel and blocky fracture unsteady fracture±matrix interaction models are applied to the example. The numerical so-lutions are compared to the available analytical solu-tions. The second problem investigates accuracy and eciency of the Lanczos method for a synthetic aquifer system with multiple wells of dierent time history. The computed hydraulic head ®eld is compared to the results computed from the direct solution method for example two.
Example 1. The ®rst example is a homogeneous, iso-tropic and in®nite aquifer system with a well at the center. The problem is identical to Theis problem, ex-cept the aquifer consists of a dual-porosity media. The dual-porosity aquifer is considered to be single hori-zontal fracture which is analogous to the aquifer±aqui-tard system. The problem is geometrically symmetric, so only one quarter of the ¯ow ®eld is simulated with 1/4 of the original pumping rate. The dimension of the model is set large enough so that the in¯uence of boundary cannot be reached in a short time and can be considered of in®nite extent. Values of various parameters em-ployed in the simulation are given in Table 1. Fig. 2 shows the grid used. The cross-section for this example is shown in Fig. 3(a).
el-ements and 550 nodes. The grid has a size about 0.1 m near the well and 300 m at the outside boundary. Fig. 3 shows the computed dimensionless time versus dimen-sionless drawdown at r48:3 and 266.22 m and the analytical solutions. The analytical solutions are ob-tained from the solutions of classical problem of well ¯ow in a con®ned aquifer±aquitard system [8]. Note that the parallel fracture dual-porosity model is analogous to the short-term response of the con®ned aquifer±aquitard system which can be written as
s Q=4pTH u;b;
wheresis drawdown andHis the leaky well-function.
By selecting an appropriate b, the analytical solutions can be read from a column of the table of function
H u;b. From Fig. 4 it can be seen that overall, the
results of Lanczos reduction method agree very well with the analytical curves. For this extreme example, which has a ratio of 3000 between maximum and min-imum grid size, it is expected that more Lanczos vectors may be needed. However only 30 vectors are required. For this small problem with 600 solving time steps, the reduction method can still achieve about 75% reduction in time (1:4 in time) compared to the direct solution method.
The same hydraulic parameters are applied to blocky fracture model with exception that the prismatic matrix blocks are now replaced by spherical blocks of 5-m radius (see Fig. 3(b)). Computed drawdowns atr48:3 and 266.22 m were plotted in Fig. 5 against time in di-mensionless values. The leakage parameterb is used to identify the type curves. In view of the unavailability of an analytical solution for the problem of well ¯ow in a spherical porous matrix fractured system, a comparison
Table 1
Values of various parameters for Example 1
Parameter Value
Number of nodes 550
Number of elements 980 Dimension of aquiferr 2000 m Fracture transmissivityTx,Ty 18.2 m2/d Pumping rate,Q 250 m3/d
Fracture storage coecient,S 0.002 Hydraulic conductivity of matrix,K0 0.0005 m/d
Speci®c storage of matrix,S0
s 0.005 mÿ
1
Thickness of matrix block, 2a 10 m Fracture aperture, 2b 0.01 m
Fig. 2. Finite element mesh for Example 1. Only one quarter of the total ®eld is shown.
Fig. 3. Example 1: two dimensional ¯ow in dual porosity media.
Fig. 4. Dimensionless fracture drawdown versus dimensionless time, showing the comparison of numerical and analytical solutions for the parallel fracture model.
of this solution with the solution of traditional ®nite element method with Crank±Nicolson scheme, which had been veri®ed by Huyakorn et al.[9], is performed. The comparison indicates that both methods agree with each other very well. Compared to the drawdown of parallel fracture model for the same parameters, the blocky model has a less drawdown. This is due to the fact that the parallel fracture model has less fracture space than the blocky model in the same domain. The fracture space determines the conductivity of the frac-tured aquifer system and the higher conductivity causes less drawdown.
Example 2. In order to investigate the ecency and robustness of the Lanczos method for multiple well system with dierent time history, a relatively large con®ned dual-porosity aquifer problem has been solved. Fig. 6 shows the synthetic aquifer system. The selected domain is a dual-porosity version of the example given in [16]. The example is chosen to include several of the commonly occurring complexities in groundwater analysis. It includes Dirichlet boundary conditions and two groups of time-dependent sources with dierent pumping histories. The fractured rock aquifer system is
heterogeneous and anisotropic with respect to the frac-ture transmissivity. The model parameters are given in Table 2.
To obtain the numerical solution for the problem, the ¯ow domain was discretized into 10 085 triangular ele-ments and 5120 nodes. The eleele-ments have a size about 10 m near the pumping wells and 100 m in the boundary area. The initial hydraulic head in the fractures is 60 m. A fully penetrating river ¯ows through the area and it is considered to be a ®rst type boundary with a constant head of 60 m. Three pumping wells with dierent pumping rate are distributed in the area (see Fig. 6). Well 2 and 3 have same time-dependent pumping his-tories, so the three wells and constant heads can be grouped into two groups of time-dependent boundary conditions for the Lanczos method.
Table 3 shows the comparison of performance be-havior of the Lanczos and direct solution methods. The maximum error, which is de®ned as the maximum dif-ference between the two solution methods at each time level, is expected to occur at one of the pumping wells. The root-mean-square (RMS) in the table is computed based on the drawdown data at the three wells for all
Fig. 6. Synthetic dual-porosity aquifer system.
Table 2
Synthetic aquifer model parameters
Parameter Value
Dimension of aquifer 2000 m Fracture TransmissivityTx,Ty
Area 1 10, 10 m2/d
Area 2 100, 30 m2/d
Pumping rate,
q1: (06t<1 d) 150 m3/d
(16t<2 d) 100 m3/d
(tP2 d) 0
q2: (06t<1 d) 100 m3/d
q3: (06t<1 d) 250 m3/d
q2;q3: (tP1 d) 0
Fracture storage coecient,S 0.002 Hydraulic conductivity of matrix,K0 0.0005 m/d
Speci®c storage of matrix,S0s 0.005 m
ÿ1
Thickness of matrix block, 2a 5.0 m Fracture aperture, 2b 0.01 m Total thickness of the aquifer 20.04 m
Table 3
Comparison between the Lanczos and direct solution method, a Pen-tium II 333 MHz CPU with 128M memory personal computer was used for the simulation
Items Direct solution Lanczos Time step size 0.001 d 0.001 d Time step number 3000 3000 Equation matrix size n;m n5120 m80 Total solution time 92 min 34 sec 5 min 12 sec Maximum drawdown dierence
between the two methods 0.039 m (1.9 %) (in percentage)
3000 time steps. The maximum computed drawdown dierence between the two methods is 0.039 m and oc-curs at well 3 at the ®rst time step. The results of the comparison are very encouraging. For the case of 3000 time steps and 80 Lanczos vectors used, the Lanczos method is approximately 18 times faster than direct solution method and the accuracy is almost the same. Even higher eciency can be achieved if the problem requires more time steps. In addition, if the wells have same time history of pumping rate, only one Lanczos process is needed. The solution time for this problem is still reduced by about 45% compared to the solution time for the original problem which requires two Lanczos processes. The Lanczos method is also a great saving in computer memory. Through the operating system, we found that the peak computer memory requirement of Lanczos method for this example is about 60% of the memory requirement for solving the original system.
The number of Lanczos vectors determines the accuracy of the method. The maximum error always occurs in the ®rst several time steps or the moment of pumping rate changed at a particular well. Fig. 7 shows maximum errors in the ®rst 10 time steps compared to the number of Lanczos vectors for both homogeneous and heterogeneous situations. For a problem of homo-geneous and isotropic hydraulic properties and simple boundary conditions, less Lanczos vectors may be re-quired. Fig. 7 indicates that if the domain is homoge-neous, we need less Lanczos vectors to achieve a similar accuracy. In addition, if the heads of ®rst several time steps are not important for a problem, one can use less Lanczos vectors. For the example, less than 1% draw-down dierence (or 0.018 m) will occur by using only 60 vectors after 10 time steps, or 7% for 40 vectors. The time needed to solve the problem is 3 min, 32 s using 40 vectors and 4 min, 20 s using 60 vectors.
In order to show the in¯uence of domain discretiza-tion on the number of Lanczos vectors, we use a
dif-ferent discretization scheme. The example domain was discretized into 28 514 triangular elements and 14 456 nodes without a re®nement near the pumping wells. The new equation system has about three times the nodes as the ®rst scheme. However, it requires even less Lanczos vectors to achieve the similar accuracy as the ®rst scheme (see Fig. 8). By using 60 vectors, a maximum error of 0.074 m and RMS of 2:1610ÿ3m is reached.
The total solution time for 3000 time steps is about 6.7 min. This result indicates that the discretization scheme is one of the most important factors in determining the number of Lanczos vectors required.
5. Conclusions
The Lanczos reduction method has been successfully developed for the solution of problems involving groundwater ¯ow in heterogeneous and anisotropic dual-porosity media. The main advantages of the method are twofold: (1) the equation system is solved in a reduced space, which is much smaller than the original system and (2) the ¯uid leakage terms are calculated by a recursion scheme in the reduced space. The solution time for the Crank±Nicolson scheme with a direct so-lution method is proportional to nnb, where n is the
total unknowns of the original equation system andnbis the half bandwidth of the matrices Kand M, however solution of the reduced tridiagonal system is only pro-portional to m, which is the unknown number of the
reduced equation system. In addition, the Lanczos e-ciency is such that a dual-porosity solution presents a negligible additional computational burden compared to that for a single-continuum simulation. Consequent-ly, the decrease in solution eort is pronounced, par-ticularly on large problems or problems requiring more time steps.
By examining Appendix A, the reader will note that embedded within the Lanczos algorithm is a modi®ed
Fig. 7. The maximum drawdown dierences between Lanczos and direct solution method for homogenous and heterogeneous cases using dierent Lanczos vectors up to 10 time steps.
Gram±Schmidt process. This process, as mentioned, requires (as a principal computation overhead) m
for-ward solutions ofKxf. Again, we wish to emphasize that we did not use (perhaps) more ecient solvers such as the preconditioned conjugated gradient method or the multigrid method to carry out the solution to
Kxf. The reader may ask, what about comparing the Lanczos method to the iterative solution of the system of equations that would arise from Crank±Nicolson time-marching of (10)? We use a direct solution method to solve systems of the formKxfand it is reasonable to compare the Lanczos approach to the classic time marching of (10) using the same solvers. Nevertheless, if one alternatively uses an iterative method in the Gram± Schmidt process to solve Kxf and we compare this new approach to the same iterative solution to the time-marching of (10) then exactly the same time compari-sons as described herein would apply. It is apparent that the Lanczos approach could achieve high eciency no matter what solution scheme is implemented.
Comparisons with exact analytical solutions have indicated that the Lanczos method is capable of yielding a highly accurate solution even when a relatively less Lanczos vectors are employed. The example simulation for the case of multiple wells with dierent pumping time history indicates that the reduction method is very robust. Both examples have shown that only a small number of Lanczos vectors were required to accurately match the drawdown computed by direct solution method. For the homogeneous problems, experiments have shown the ratio of n to mcan be as high as 200.
However,mis not a function ofn. For a problem with a
large n, the ratio is expected to be larger. Thus the
method aords an ecient means of solving large problems.
Multiple well systems with dierent pumping time histories or time-dependent boundary conditions have been implemented in the reduction method using a su-perposition principle. The results of the second example have veri®ed that the implementation method is correct and ecient. We note that the reduction method could be applied to any realistic single and dual-porosity groundwater ¯ow model. The Lanczos algorithm per-forms the reduction process based on the matrixKand
M and the starting vector. No matter what dimensio-nality of the problem, the properties of these matrices and vectors are the same. Therefore, the theory devel-oped in this paper is readily applicable for a fully three-dimensional groundwater ¯ow problem.
Acknowledgements
The authors would like to acknowledge Dr. Henian Li for providing technical assistance. This research was funded by a grant from the Natural Sciences and
Engineering Research Council of Canada for Dr. Allan Woodbury.
Appendix A. The Lanczos algorithm
The reader is referred to [4,13,14] for more details. Start from an initial vector r0Kÿ1b and q00, then nal matrixHmare formed
Hm
From the above algorithm, the following relationships hold:
where Im is themmidentity matrix and the m-vector
em is themth column of matrixIm.
The two equations in (24) refer to the M-orthogo-nality property. The Rayleigh±Ritz reduction process is then used to reduce the size of Eq. (13). Multiplying equation (13) byQtmMand substituting the approximate transformation
hQmw 25
into the original equation, and applying (23) and (24), the reduced system (11) is obtained, wheregis given by
gb1e1p t 26
Appendix B. Formulae for the conceptual model of spherical matrix block
Cfor the spherical matrix is given by
where
anp2n2K0= Ss0a
2 28
andais the radius of spherical `block'.
Corresponding to Eq. (6),rfor spherical block model is given by
r 6K
0H
a ab: 29
References
[1] Barenblatt GI, Zheltov IP, Kochina IN. Basic concepts in the theory of seepage of homogeneous liquids in ®ssured rocks strata. J Appl Math Mech Engl Transl 1960;24(5):1286±303.
[2] Douglas J, Arbogast T. Dual-porosity model for ¯ow in naturally fractured reservoirs. In: Dynamics of ¯uids in hierar-chical porous media. San Diego, CA: Academic Press, 1990. p. 177±221.
[3] Dunbar WS, Woodbury AD, Nour-Omid B. Comment on `On time integration of groundwater ¯ow equations by spectral methods' by G. Gambolati. Water Resour Res 1994;30(3):2347-2352.
[4] Dunbar WS, Woodbury AD. Application of the Lanczos algorithm to the solution of the groundwater ¯ow equation. Water Resour Res 1989;25(3):551±8.
[5] Farrell DA, Woodbury AD, Sudicky EA. Numerical modelling of mass transport in hydrogeologic environments:performance
com-parison of the Laplace Transform Galerkin and Arnoldi modal reduction schemes. Adv. Water Resour 1998;21:217±35. [6] Fillion E, Noyer ML. Flow modeling in a dual porosity domain
with automatic mesh generation and parameter calibration: application to the Aspo site. J. Hydrol. 1996;180(1-4):1±19. [7] G. Gambolati. On time integration of groundwater ¯ow equations
by spectral methods. Water Resour Res 1993;29:1257±1267. [8] Hantush, MS. Hydraulics of wells. Adv. Hydrosci. 1964;1:281±
432.
[9] Huyakorn PS, Lester BH, Faust CR. Finite element techniques for modeling groundwater ¯ow in fractured aquifers. Water Resour Res 1983;19(4):1019±35.
[10] Lanczos C. An iteration method for the solution of the eigenvalue problem of linear dierential and integral operators. J Res Nat Bur Standards 1950;45:255±82.
[11] Li H. Development and application of the unsymmetric reduction Lanczos method. Ph.D. Dissertation, The University of Manito-ba, 1996.
[12] Li H, Woodbury A, Aitchison P. Application of the unsymmetric Lanczos method to radionuclide decay chain transport in porous media. Int J Numer Meth Eng 1999;44:355±72.
[13] Nour-Omid B. Lanczos method for heat conduction analysis. Int J Numer Methods Eng 1987;24:251±62.
[14] Nour-Omid B. Application of the Lanczos method. Comput Phys Commun 1989;53:157±68.
[15] Simon, Horst D. Analysis of the symmetric Lanczos algorithm with reorthogonalization methods. Linear Algebra Appl 1984;61:101±131.
