Finite element solutions of the non-self-adjoint convective-dispersion equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 11, 269-287 (1977)

FINITE ELEMENT SOLUTIONS OF THE

EQUATION NON-SELF-ADJOINT CONVECTIVE-DISPERSION

ANAND PRAKASH

Senior Water Resources Engineer, Sargent di Lundy Engineers, Chicago, Illinois, U.S. A.

SUMMARY

Finite element formulations based on the Galerkin and variational principles have been developed for the self-adjoint and non-self-adjoint problems represented respectively by the flow and convective-dispersion equations in the cylindrical polar system of co-ordinates. The formulation based on the variational principle is shown to be restricted to dispersion-dominant transports only. The Galerkin method is demonstrated to be more versatile and free from convergence and stability problems. The computational scheme based on the Galerkin principle is shown to be equally valid for both convection and dispersion dominant transports. The numerical results obtained are verified with known analytical solutions. It is concluded that the suggested scheme can be used in solving a variety of field problems involving groundwater dispersion.

INTRODUCTION

The objective of this paper is to discuss alternative finite element formulations for the solution of the non-self-adjoint convective-dispersion equation. Mathematical development of the classical finite element formulation based on the variational principle is presented and the insurmountable numerical difficulties experienced in pursuing this method on a digital computer are discussed. An alternative solution technique is developed which is based on the Galerkin method and is found to yield satisfactory results. Although the solution technique is applicable to the general system of orthogonal co-ordinates, the discussion here is limited to the cylindrical polar (axisymmetric) forms of the governing equations.

In cylindrical co-ordinates with axial symmetry, the convective-dispersion equation describ- ing the transport of a decaying adsorbent substance through a saturated isotropic elastic porous medium can be written as follows :

1 1-4 - ( rD , , C, + rD , 2Cz}r + { D, 2Cz + D, , C,}, - uC, - uC, - IC r

where, 4 is the medium porosity, I is the decay constant, C is the concentration of the substance per unit volume of the liquid phase, a,C is the mass of the substance adsorbed per unit volume of the solid phase, rand z are the radial and vertical co-ordinates and u, u are the seepage velocities in the r and z directions respectively. The dispersion coefficients D, ,, D,, , D,, , and D z l are

Received I August 1975 Revised 15 January 1976

8 1977 by John Wiley & Sons, Ltd.

269

270 ANAND PRAKASH

defined by the following expressions :

where, p is the mass density of the fluid in which the decaying substance is dissolved, z is the tortuosity of the porous medium, D d is the coefficient of molecular diffusion, a, and a,, are the longitudinal and lateral dispersivities of the porous medium and V is the resultant of the velocity components u and u. u is a constant of proportionality quantifying the variation of the mass density of the fluid with the concentration of the decaying substance in the linear relationship,

P = POL1 + B P - Po11 + U(C - CO) (3) where, po, Po and C, are the reference fluid density, fluid pressure and concentration, /? is the fluid compressibility and p, P and C are the new level density, pressure and concentration respectively.

Any solution of equation (1) involves a priori description of the flow field defined by the flow equation,'

where, $o is the reference porosity, C, is the aquifer compressibility, k is the intrinsic permeability of the porous medium, p is the dynamic viscosity of the fluid and g is the acceleration due to gravity.

Non-dimensionalization of equations (1) and (4) yields,

k [ F . SRi],+[SR2] x r and,

c, = ;{F(b:,cF+ b:2c,)}F+ {B,*2e,+b;,eF}P- ii*ci- fi*c,- xc

S = - pk; R = P + p z ; Y = $,(~+C,)P+a$,C

where $o has been assumed to be invariant and,

-

B The scaling variables used for the above non-dimensionalization are,

P O L O . Po = pogL,; t o = - , vo=- Po&, P O

lo = l t , ; a:, - --

- v

Rd ij* = - and R d =

For convenience, the asterisks and bars are dropped from hereon.

THE NON-SELF-ADJOINT CONVECTIVE-DISPERSION EQUATION 27 1

VARIATIONAL FUNCTIONALS

The eigen value problem defined by the linear homogeneous differential operators M,, and N,,,

is said to be self-adjoint if the following conditions hold for any two arbitrarily chosen functions uo and uo

Bi(uo) = 0; Bi(uo) = 0, i = 1,2,3,. . . , m 1

According to the above criteria, equation (5 ) is self-adjoint but equation (6) is not. In general, non-self-adjoint problems are not directly derivable from a variational functional. However, linear and homogeneous non-self-adjoint systems of the type,

a+rr + 2Wrz + c+zz + d+r + + f + + g = 0 (9) where the coefficients a, b, c, d, e,fand g are functions of r and z, are reducible to a self-adjoint form, if a reducing factor A(r, z), satisfying the following partial differential equations can be found,,

aA, + bA, = (d - a, - b,)A

bAr + cA, = (e - b, - c,)A

4 , z ) = r exp (8)

(10)

(11)

The reducing factor corresponding to equation (6) is

where, p-arc ut+uz p = --

p ' a,V+ D,,T

Thus, the variational functionals for equations (5 ) and (6) are,

and,

where q is the flow rate per unit area of the porous matrix. Finite element formulation of equation (12) is straightforward. For equation (13), it is expedient to introduce a new variable + such that,

+ = Cexp(8/2) (14)

272 ANAND PRAKASH

With this transformation, the variational functional is,

Thc corresponding convective-dispersion equation is

where

Transformed to the local co-ordinate system, the variational functional for the convective- dispersion equation is,

where, t; and q are the local co-ordinates referred to the centroid of an element and F" is the global r-co-ordinate of the centroid.

BOUNDARY CONDITIONS

The flow domain is illustrated in Figure 1. The boundary conditions are designed to simulate salt water intrusion into a fresh water aquifer underlain by a saline reservoir of large depth induced by continuous or intermittent pumping by a partially penetrating well. Thus,

(i) The surfaces c1 and c2 represent the Dirichlet boundary conditions where the nodal point values of H, C and $ remain invariant with time. c1 is a horizontal section of the aquifer below which the influence of the well is not felt. c2 is the outer boundary of the cone of influence

H = Hi(r , z); C = Ci(r,z); $ = $i(r, Z) V t 2 0 i = 1,2, 3 , . . . (19)

(ii) The surfaces c3 and c5 represent the Neumann boundary conditions. c3 is the bottom of the upper confining layer and c5 is a vertical stream surface below the well across which there is no solute transport due to radial symmetry.

v* S H , = 0; v = 0, D2,C,+D2,C,= 0 and D21t,br+D22$z-- = 0 2

along c3 V t > 0 u* S H , = 0; u = 0, D 1 2 C , + D , , C , = 0 and D 1 2 $ 2 + D 1 1 $ r - - = 0 2

along c5 V t > 0

THE NON-SELF-ADJOINT CONVECTIVE-DISPERSION EQUATION 273

r = o r = ro

Figure 1. Schematic of flow boundaries

(iii) The su.rface cq represents a flow boundary s c h that,

FINITE ELEMENT FORMULATION (BASED ON THE VARIATIONAL PRINCIPLE)

Following the usual minimization process for equation (1 8), the following expression is obtained for the non-self-adjoint convective-dispersion equation pertaining to the mth element,

+ [SB] + [SC]+ [SD] + [SE] + 0-5 x [SH] + 0.5 x [SI] + [SJ] + [SKI

+ [ A+-- ' iac. qgT2LL1] . {[SAA] + [SALPHA] + [SBETA] + [SGAMA]

+ [STHETA]} {$}"'+ { [SAA] + [SALPHA] + [SBETA] + [SGAMA]

(22)

1 + [STHETA]} { $,}"' + { Q}"' = 0

274

where,

ANAND PRAKASH

D, = all/

All other symbols are defined in Appendix I. In a compact symbolic notation equation (22) may be written as,

[SLI {$}" + [PLI i+r}"' + [Ql" = 0 (23)

Similar equations are written for all the elements in the domain of transport. Assemblage of such equations yields,

[SKI {$} + [PKI i + r } + [QKI = 0 (24)

Note that both the matrices [SKI and [PK] are banded and symmetric and therefore permit considerable saving in computation time and storage requirements.

LIMTTATlONS OF THE VARIATIONAL APPROACH For any practical prob!cm of saltwater movement towards a partially penetrating ix!!,

p-aC

P x 1 ; D, x 0, u =flu, fl << 1 and zo = f2r0, f2 -= 1 ;

(25)

Usual values of ro and a, are in the range of 500 ft (1 52.4 m) to 2,000 ft (609.6 m) and 0401 ft (0-030 cm) to 0.1 ft (3.05 cm) respectively. Therefore, for the Dirichlet boundary conditions on the surface cl, the $ values corresponding to the same concentrations C at nodes (rw, zo) and (ro, zo) of Figure 1, would be in a ratio range of exp (2,500) to exp (1,000,OOO). Such largely varying values of * undergo different orders of truncation errors in the process of matrix operations and vitiate the results beyond tolerable limits. For UNIVAC 1106 computers the limiting permissible argument of exponential functions is about & 88 for single precision and about k710 for double precision. For CDC 6400 computers the limiting value is k741.67 (single precision). Therefore, in most practical situations, the exponential arguments go out of range and the solution does not proceed beyond the computation of $ values for different nodal points


in the domain of transport unless the value of uI is hypothetically high. This means the method is applicable to highly dispersion-dominant transports only. This lacuna greatly restricts the use of the method. The general criterion for comparing the contributions of convection and dispersion is the magnitude of the variable E, where,

It would be seen that for most problems of convective-dispersion through porous media, the values of E are too small to qualify the transport as dispersion-dominant. Note that these limita- tions are not circumvented even if the transformation to the variable $ is not attempted and the solution is continued directly from equation (13) in terms of the variable C itselEg Even in this case, the reducing factor e8, with associated problems occurs in all the elements of the matrices [SKI and [PK]. The reverse step of transforming back from the predicted values of $ to C is, however, obviated. Therefore, for tractable dispersion dominant transports, this device does reduce the truncation errors somewhat; nevertheless, it does not expand the scope of applicability of the method.

FINITE ELEMENT FORMULATION (BASED ON THE METHOD O F

An alternative solution of the non-self-adjoint convective-dispersion equation is obtained by the Galerkin method.6 The following equation is obtained by minimizing the integrated value of the weighted residual of equation (6).

INDEPENDENT PARAMETERS-THE GALERKIN APPROACH)

-uC,-vC,-IC-C, rdrdedz = 0 1 Using divergence theorem and collecting the volume and surface integrals separately,'

Using appropriate boundary conditions, introducing a shape function defined by C = [A] {C}'" and transforming to the local coordinate system,

where [Wl has been replaced by the shape function [A] and taking advantage of radial symmetry, the integral with respect to 8 has been-omitted.

276 ANAND PRAKASH

Evaluating the integrals in equation (29), the following system of equations is obtained,

[[SA] + [SB] + [SC] + [SF] + [SG] + [SH] + [SI] + A{[SAA] + [SALPHA] + [SBETA]

+ [SGAMA] + [STHETA]}] {C}" + [[SAA] + [SALPHA] + [SBETA] + [SGAMA]

+ [STHETA]] {C,}m = - [Q] + [R] (30) The various symbols used in equation (30) are defined in Appendix I and [R] refers to the

unknown values on the right-hand side of equations (28) and (29) pertaining to the Dirichlet boundary conditions. Note that [R] is a null matrix everywhere except along the geometric boundaries c1 and c 2 .

Assembling similar equations for all the elements in the domain of transport gives

[[SKI + AFKl1 {C} + F K I {CJ + [QKI = [RI (31)

[SKI and [PK] in this case are banded and non-symmetric matrices. For the time-domain solution, following the Galerkin residual process yields,

1 1 [$SKl+A[PKl) + ,,[pKl {C}, = [Rl-[QKl+ z[pKl-3{[SKl+XPKl) {C}, (32)

I 1 " where the subscripts 0 and 1 refer to two consecutive time steps At apart.

GALERKIN FORMULATION FOR THE FLOW EQUATION

Following the Galerkin residual process6 in the space domain, equation (5) yields,

JJJv [SH,( W,), + SH,( W,), + W,l3 dr do dz = Js WpSH,nl dS + Is W,SH,n3 dS (33)

Using appropriate boundary conditions and introducing a local co-ordinate system and a shape function as before, r r r

J J u [ A I ~ A , ) ~ + s[AI,(A,),I {HY + A,[AI { XYX? + 5) d t dv = J A , w , dv + [RI (34) Rm =?

Evaluating the integrals, it may be seen that equation (34) is identical to the equation obtained by minimizing equation (12) except the unknown terms [R]. These terms are non-zero only for the geometric boundaries and so do not affect the solution at all. This leads to the conclusion that for self-adjoint problems both the Galerkin and variational approaches result in the same system of simultaneous equations. The resulting system may be written as,

[FSL] {H}" + [FPLEI { x}" = [QI" + [Rl"

[SKI {HI + F K I {Yt> = [QKI + [RI

I; 2: E . H , where E = $,(P+C,) (37)

(35)

(36)

Assemblage of the elemental matrices yields,

[SKI and [PK] here are banded and symmetric matrices : another characteristic of self-adjoint problems. Assuming,

and introducing the time-domain formulation as before,

[ [PKI . z + p l {HI, = [QKl+[Rl+ P K l . t - p I {HIo E 2 1 1 (38)


NUMERICAL SOLUTION

The algorithm used for the solution of the non-self-adjoint convective-dispersion equation in the cylindrical polar system is an iterative scheme commonly known as the leap-frog technique." The various steps in the scheme are:

1. Generate a mesh layout for the region of transport. 2. Initialize the nodal point values of the variables p, 4, H and C ; elemental values of the

3. Solve the flow equation (equation (38)) by Gauss-elimination for banded and symmetric

4. From the nodal point values of the variable [HI1, obtain the corresponding values of the

variable k and spatially invariant values of the variables p, a,, a,,, p, C,, Dd and z.

matrices and obtain the nodal point values of the variable [HI, at the new time step t + At.

variable [PI from the equation,

H = P + p z

5. Update the nodal point values of the variable 4 from the relation

4 = 4011 + CA~-Pol 6. Obtain the elemental seepage velocities for the new time step using Darcq law as

where 6 are the elemental values of the porosity. 7. Compute the elemental values of the dispersion coefficients D I I , Ol2, DZl andDzz from

equation (2) . 8. Solve the non-self-adjoint convective-dispersion equation (equation (32)) by Gauss-

elimination for banded and non-symmetric matrices and obtain nodal point values of the variable. [C], at the new time step.

9. Update the nodal point values of the variable p using equation (3). 10. Substitute the updated values of p into equation (38) and repeat the process over and over

again until the required time level is reached. For axi-symmetric problems with a linear source or sink in the field, the mesh has to be finer

in the vicinity of the source or sink and coarser in the remote regions of low potential gradients. A special algorithm is used for such a mesh layout. The entire flow field is divided into a number of loops. Each loop is subdivided into a number of well-conditioned quadrilaterals depending upon the requirements of the element sizes in different portions of the flow field. The quadri- lateral elements are subdivided into triangles by joining the shorter diagonals of each. The co- ordinates of each nodal point are computed and sequential numbers are assigned to all the elements and nodal points in the mesh. This, however, results in a very large bandwidth. In order to reduce bandwidth, two subroutines' are introduced in the mesh generation program. The first subroutine simply arranges the node numbering in a form suitable for the use of the second subroutine. The second subroutine generates trial node numbering schemes starting with a different node each time and checks the resulting bandwidth until the smallest bandwidth is obtained. The final element and node numbering scheme is printed/punched as output of this program. This program was used to generate the element layout for the flow field around a partially penetrating well comprised of 320 elements and 183 nodal points (Figure 2). The band-

278 ANAND PRAKASH

width with the initial node numbering scheme was 133. The reduced bandwidth turned out to be 16 only.

Test runs with program FLOW (finite element formulation for the flow equation) indicated that space discretization in the form of triangular elements, results in predicting unequal potentials along verticals even in the case of pure radial flow induced by a fully penetrating well. To minimize this discretization error, the [SKI and [PK] matrices for the flow and convective- dispersion equations are generated twice. The first set corresponds to the elements formed by joining shorter diagonals of the quadrilaterals and the second to the elements formed by joining the larger diagonals. The final [SKI and [PK] matrices are taken to be the arithmetic averages of the above two sets. This scheme predicted equal potentials along the verticals for the case of pure radial flow.

TESTS OF NUMERICAL SCHEME

In order to test the validity of the numerical scheme, results of program FLOW for a fully penetrating well were compared with the analytical solution given by."

s = -W(u) 47c T

t SrZ 4Tt

u = -

~ ( u ) = sum f du J In equation (411 s = drawdown at time t and distance r from the well, S = storage coefficient, T = aquifer transmissivity and Q is the well discharge. A comparison of the numerical and analytical solutions of the flow equation for the simplified case of pure radial flow towards a fully penetrating well is illustrated in Figure 3. Note that the two solutions are in close agreement except at the well face. The difference at the well face can be minimized by refining the mesh and choosing a larger radius of influence. The element layout for this case comprised of 380 elements and 288 nodal points with a bandwidth of 5. The compilation time on a CDC 6400 machine was 8.622 seconds and the total C.P. time for 16 iterations was 150.106 seconds.

Two test cases were run to demonstrate the convergence of the solution technique with mesh refinement. Representative values for these cases are shown in Table I. The computed values of H at points away from the well face are in close agreement with the analytical solution for both the cases and are not appreciably affected by mesh refinement. On the other hand, the computed values of H at the well face (r = 0-5 ft) exhibit a marked convergence to the analytical solution as the mesh is refined from 240 elements to 380 elements.

The numerical scheme is found to be sensitive to the choice of the initial time step. This sensi- tivity is larger for the flow equation than the convective-dispersion equation. It has been found that the instability caused by a wrong choice of the initial time step size can be avoided by adopt- ing a very small initial time step and increasing it gradually after every 10 consecutive iterations. The improvement obtained by this device is demonstrated by the results shown in Table 11.

Basically, the programs for both the flow and convective-dispersion equations are similar. The algorithms for the assemblage, multiplication and Gauss-elimination of the symmetric and non-symmetric matrices are, however, different. For the symmetric matrices only the upper


Figure 2. Typical element layout

diagonal elements within the bandwidth are stored. For the non-symmetric mat 'ces, both the

of the original (NUMNP x N U M N P ) matrix becomes the element [ i , j - 1 +(ZBAND+ 1)/2] of the contracted ( N U M N P X I B A N D ) matrix. The variables N U M N P and ZBAND denote the

upper and lower diagonal elements within the bandwidth are stored such that th !! element (i, j ]

0.m 2.005 3.005 4.005 s.0os 6.005 7.005 8.005 9.005 0.005

r B -

Figure 3. Analytical and numerical solutions of flow equation

280 ANAND PRAKASH

Table I. Values of H after 55.5 seconds of pumping

Numerical solution

r (ft.) Analytical solution 240 Element mesh 380 Element mesh

0.5 50.5

100.5 200.5 300.5 400.5 500.5 950.5

7558.60 7795.97 7641.49 7951.43 7946.93 7945.48 801 2- 1 8 8004.24 8004.06 807060 8059.99 805960 808246 8088.12 8087.86 8 108.07 8 104.29 8104.11 8 124.88 8 1 13.76 8113.61 8125.00 8 124.59 8125.00

Table 11. H values at well face with different time steps

DT* (sec.) Time (sec.) H DTt (sec.) Time (sec.) H

5.0 5.0 7505.77 0.1 1 .o 7639.59 10.0 79484 1 1 .o 10.1 7434.39 15.0 750750 10.0 110.1 7227.94 20.0 7889.13 100.0 1110.1 7 168.44 25.0 7508.71

Time step was maintained constant at 5.0 seconds. t Time step was gradually increased from 0.1 to 100.0 seconds.

number of nodal points and the bandwidth respectively. All the operations are performed on the contracted matrix. The validity of the formulation for the non-symmetric matrices included in program DISPER (finite element formulation for the convective-dispersion equation) was tested by modifyindthe input so that the convective-dispersion equation becomes identical to the flow equation treating the variable C as synonymous with H. The two programs FLOW and DISPER were then run on identical data. Comparison of the output at the 7th iteration at t = 2.5 seconds is shown in Table 111.

The formulation of program DISPER was also tested by comparison with a known approxi- mate analyticsll solution for radial dispersion4 in a situation where a,, = 0, D, = 0, u = 0 and p = constant. The anaiytical solution is given by,

Table 111. Comparison of program FLOW and DISPER

Values of the variable H or C

r (ft.) FLOW DISPER Radial distance

0.5 7935960144 7935.960001 50.1 8075276853 8075.276814

100.5 8 1 13.059404 8113.059394 200.5 8124-933966 8124.933966 300.5 8 124.999562 8 124.999562

THE NON-SELF-ADJOINT CONVECTIVE-DISPERSION EQUATION 28 1

where,

1 = depth of well penetration and Q = well discharge. It may be readily seen that equation (42) satisfies the boundary conditions,

C(r = 0, t > 0) = C, and C(r = co, t 2 0) = 0 but does not satisfy the initial condition

C(r, t = 0) = 0 Nevertheless, for points away from the source, the solution has been found to yield satisfactory

results? Comparison of the numerical results with those given by equation (42) is presented in Figure 4. To illustrate the effect of varying dispersivities, three different values a, = 0.1 ft (3.05

t (SECONDS)

Figure 4. Analytical and numerical solutions of convective-dispersion equation

cm), l-Oft (30.5cm) and 10.0ft (304-8cm) were used It can be seen that the agreements are equally satisfactory for all the three cases. The element layout for the cases analysed in Figure 4 comprised of 160 elements and 105 nodal points with a bandwidth of 13. The compilation time on a CDC 6400 machine was 10-844 seconds and the total C.P. time for 38 iterations was 110.088 seconds. The same problem took 1-66 minutes of C.P.U. time for 40 iterations on a UNIVAC 1106 computer.

In order to solve the combined flow and convective-dispersion problem, program FLOW and DISPER are run in sequence following an iterative scheme. For operational simplicity, both these programs are reconstituted into one program FFLOW. In the transient stage, fluid transport is a process faster than the tracer transport. Therefore, the time-step for the time-domain solution of the flow equation is kept as one-tenth of the time step for the convective-dispersion

282 ANAND PRAKASH

equation. The fluid transport attains steady state earlier than the tracer transport. At this stage, the time-step for the flow equation is made equal to that for the convective-dispersion equation. Choice of appropriate time step is a matter of trial and error. For the examples attempted, an initial time step of 0.1 seconds gave sufficiently accurate and stable results for the flow equation. The time step was successively increased to ten times its previous value after every ten iterations without exhibiting any marked oscillations in the results.

ILLUSTRATIVE FIELD PROBLEMS

Two example problems were solved with program FFLOW with different boundary conditions. The first pertains to the fluid and solute transport around a fully penetrating well and the second around a partially penetrating well. The data used for each case are shown in Table IV. The resulting propagation of concentration for these two cases is illustrated in Figures 5 and 6.

0.5 0.6 0.7 0.6 0.9 1.0 1.1 I .2 1.3 1.4 1.5

t ( 10' SECONDS 1

Figure 5. Propagation of concentration for a fully penetrating well

The values 780.5 ft (237.9 m) and 510-5 f t (155.6 m) used for the radius of influence of the well in these cases are smaller than those expected for a field situation. This was done to save computer time and storage. Even with these expedients, the concentration propagated only 160 ft (48.8 m) over a period of 2.29 x 108 seconds (7-26 years) for the fully penetrating well. This took 300 iterations and 2,000 seconds of C.P. time on a CDC 6400 computer. The compilation time was only 14.685 seconds. For these iterations, the time step was successively increased from 0.1 second at the first iteration to 1,000,OOO seconds at the 80th iteration; thereafter it was maintained at this value through the 300th iteration. For the partially penetrating well, it took 2.2 x lo7 seconds (255 days) for the concentration to propagate 90 ft (27-4 m) involving 320 iterations and 3,600 seconds of C.P. time. The compilation time was only 14.714 seconds. As in

2 rn

Tabl

e IV

. D

ata

for

illus

trat

ive p

robl

ems

Z

Q

zmax

Pe

netr

atio

n a,

a,

, D*

H

Pene

trat

ion

NU

MN

P N

UM

EL

(c

fs)

(ft)

(ft)

(ft)

(ft)

(ft

2/se

c)

(ft Ib

) CI

CO

r 2 780.5

1.0, r 2 780.5, t 2 0

7 E! 8125.0,

Full

200

312

2.0

100.0

100.0

0.1

0.0

0.0

H U z is

r 2 510.5

54000.07

t 2 0

1.0, 2

Q 0, t

0

9 L:

r.

Part

ial

198

340

4.0

300.0

1 20.0

1.0

0.1

O*oOOl

54000*0, z Q

0, t

2 0

?

54000.0* 2

> 0, t

= 0

i?! 8

0.0,

2 >

0, t

= 0

r <

510.5

Pene

trat

ion =

Dep

th o

f wel

l pen

etra

tion.

z

max

= M

axim

um d

epth

to w

hich

the i

nflu

ence

of t

he w

ell e

xten

ds.

NU

ME

L =

Num

ber

of e

lem

ents

rv

= W

ell r

adiu

s = 0

.5 ft

.

284 ANAND PRAKASH

C c o -

t ( loc SECONDS 1

Figure 6. Propagation of concentration for a partially penetrating well

the previous case, the time step was successively increased from 0.1 seconds at the first iteration to 100,OOO seconds at the 110th iteration; thereafter it was maintained at that value through the 320th iteration.

The qualitative trends depicted by these runs indicate that for a normal size well, the computer time required for the concentration to appear in the well discharge might be of the order of 10,000 seconds or so and the actual time of travel might be in the range of a few years depending upon the aquifer properties.

CONCLUSION

Galerkin and Rayleigh-Ritz (based on the variational principle) finite element solutions of the non-self-adjoint convective-dispersion equation are developed. The variational approach is found to be feasible only for highly dispersion-dominant transports. For convection-dominant transports the Galerkin residual process is found to be more appropriate. The former has a computational advantage that it produces a system of banded and symmetric matrices. The Galerkin approach yields a system of banded but non-symmetric matrices for non-self-adjoint problems. This requires a little less than twice the storage for one and the same problem. Even after using the bandwidth reduction scheme, storage requirements for some field problems may become prohibitive for moderate size machines. However, this difficulty can be overcome by taking recourse to auxiliary tape storage. An algorithm is developed to iterate the solutions for the fluid and tracer transports alternately. A mesh generation program is developed to produce a suitable element layout for the problem with option for bandwidth reduction. Validity of the numerical simulator is tested by comparison with known solutions and the versatility is demonstrated by solving example field problems.


It is shown that the solution technique is capable of handling low, moderate and high values of the geometric dispersivities of the porous medium. The formulations in the space and time domains for both the flow and convective-dispersion equation are inherently implicit and exhibit no instability provided the initial time step is kept sufficiently small. Gradual increase in the time step size in successive iterations is found to economize computer time without introducing instability. The solution converges to the true values with greater and greater refinement of the finite element mesh.

For the illustrative field problems, the flow equation attained steady state after 1.2 x lo8 seconds (3.8 years) of pumping by the fully penetrating well. The dispersion process remained transient even after 2.29 x lo8 seconds (7.26 years) of pumping (2,000 seconds of C.P. time). For the partially penetrating well, the flow equation attained steady state after 7.2 x lo6 seconds (83.3 days) of pumping but the dispersion process remained unsteady even after 2.2 x lo7 seconds (254-6 days) of pumping (3,600 seconds of C.P. time). It is concluded that the Galerkin formulation developed here can be used successfully for the solution of both convection and dispersion dominant field problems.

ACKNOWLEDGEMENTS

This study was supported by Colorado State University Experiment Station Project 110 and the unsponsored research funds of the Civil Engineering Department, Colorado State Univer- sity. Thanks are due to R. A. Longenbaugh of Colorado State University and R. M. Sio, S. Bhamidipaty and V. S. S. Annambhotla of Sargent & Lundy for their co-operation and encouragement.

APPENDIX I

C = [A]{C}" where,

and values of aij, aZj , a,j, a l k , a2k and a3k follow in cyclic sequence.

2A"= 1 ti qj i'

286 ANAND PRAKASH

i and j refer respectively to the lower and upper nodes of element m lying along the well face.

q: = rwq - -1 -+- (q j -q i )

qi* = rwq --1 -+- ( q j - q i )

(; )(; 2) (; )(: :)

A = Area of the element

A,, = aZmaZn n = i, j , k ; m = i,j, k

B,, = a3,aZn+ aZma3, n = i, j , k ; m = i, j , k

I m n = a lma3n

a


P m n = a2ma3n+ a3ma2n

E X X E T = JJRm 12q d l dg

EXETT [STHETA] = 4 A m A m

REFERENCES

1. Anand Prakash, ‘Galerkin simulation of hydrodynamic dispersion’, Ph.D. dissertation, Coloradc Sta:c University,

2. F. B. Hildebrand, Methods of Applied Mathematics, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1965, 362 p. 3. J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, EnviroiliiX!!!al Science Series, i 972, 764 p. 4. J. A. Hoopes and D. R. F. Harleman, ‘Dispersion in radial flow from a recharge well’, Jour. ofGPoph. Research,

5. G . Arfken, Mathematica1Me:hcds for Physicists, Academic Press, New York, 1970,815 p. 6. 0. C. Zienkiewicz, The Finite-Element Method in Engineering Science, McGraw-Hill, London, 1971, 521 p. 7. R. J. Collins. ‘Bandwidth reduction by automatic renumbering’, Int. J. num. Meth. Engng, 6, No. 3,345-356, 1973. 8. S . H. Crandall, Engineering Analysis, McGraw-Hill, New York; 1956,417 p. 9. L. R. Gary Guymon, ‘Note on the hiteelement solution of the diffusion-convection equation’, Water Resources

10. D. W. Peaceman and H. H. Rachford, Jr, ‘Numerical calculation of multidimensional miscible displacement’,

11. W. C. Walton, Ground Water Resources Evaluation, McGraw-Hill, New York, 1970,664 p.

Fort Collins, 1974,182 p-

-72, No. 14,3595-3607, Juiy 1967.

Research, 8, No. 5,1357-1360, October 1972.

Society of Petroleum Engineers Journal, 2, No. 4. 327-339, Dec. 1962.

Finite element solutions of the non-self-adjoint convective-dispersion equation

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