Calculation of Resistance and 

Error in an Electric Analog 

of Steady Flow Through 

Nonhomogeneous Aquifers

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1544-G

Property of

US. GEBLOGML SURVEY

WATER RESOURCES DIVISION 

GROUND WATER BRANCH
Calculation of Resistance and 

Error in an Electric Analog 

of Steady Flow Through 

Nonhomogeneous Aquifers

By ROBERT W. STALLMAN

GENERAL GROUND-WATER TECHNIQUES

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1544-G

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.G.
CONTENTS

Page

_____________.._______._.________________________________ Gl

Introduction _______________________________________________________ 1

Equations of flow__-.____-__________________-_____-________________- 3

Resistance values.-_________________________________________________ 5

One-dimensional flow solutions_____-_-_________-_________--_-_-__--- 8

Design of model of two-dimensional flow_____________________________ 12

Calculating state of the model from measurements of potential. _________ 15

Computation of errors in potentials observed on the model____________ 16

Summary_______________________________________________________ 19

References...--_-____.___-__________________________-_-___--___--- 20

ILLUSTKATIONS

Page 

FIGURE 1. Model relations.________-____________--__-_____---__--__ G4

2. Grid reference, T= 3*_______________________________ 5

3. Flow in rectangular region___________-_-______-_--_-__--_- 13

4. Localerrors____-___-___________-______________---__---_- 18

TABLES

Page

TABLE 1. Head distribution for one-dimensional flow through an aquifer 

with T=lX104 (l+a;), as determined from theory and model 

observations. _ ____________________-_--_____-_--___-_-_- G10

2. R/C values calculated from equations 6a and 6b for the space 

between nodes 2 and 3 with T=3Z and the node configuration 

of figure 2, keeping node 2 at x=2 ____ ____________________ 11

3. Computed heads and heads observed on electric models for

one-dimensional flow with T=3*-.-----~------- __________ 12

4. Selected measurements of potential from a model of a rectangular

flow field with T= ______________ _ ____________

100 (1-fz)

5. Aquifer characteristics calculated from selected potential 

measurements, assuming T=ax-\-by. _____________________

15

16

in

GENERAL GROUND-WATER TECHNIQUES

CALCULATION OF RESISTANCE AND ERROR IN AN 

ELECTRIC ANALOG OF STEADY FLOW THROUGH 

NONHOMOGENEOUS AQUIFERS

By ROBERT W. STALLMAN

ABSTRACT

Several types of nonhomogeneous aquifers were modeled using resistance 

elements to represent finite sections of the aquifer. Various design equations 

were used to calculate model resistance in selected problems. Flow solutions 

obtained from the models qualitatively demonstrate the desirability of carefully 

selecting the design equation so as to minimize the model size for attaining a 

given accuracy of solution. Criteria are established which show the types of 

variations of transmissibility that can be modeled by making resistance inversely 

proportional to transmissibility without regard for spacing of the model grid. 

Areal trends of transmissibility are computed from the model data to illustrate 

numerical techniques. Head values computed by numerical techniques are 

compared with those observed on the analog to determine the accuracy of the 

analog solution.

INTRODUCTION

One type of electric model of ground-water flow is constructed as 

an assemblage of fixed resistance elements, each of which represents a 

large block of aquifer material. Each element in the model represents 

an analogous region in the aquifer system. The accuracy of the 

ground-water flow data derived from such a model is highly dependent 

on the accuracy with which the aquifer is represented by the model. 

The electric analog model, constructed as an assemblage of discrete 

resistance elements, is essentially equivalent to a set of finite-difference 

equations. Thus, errors in flow-problem solutions obtained from 

models of this type will be at least as great as the errors in a mathematical solution of the finite-difference equations representing the 

continuous aquifer.

Several sources of error must be considered in evaluating the 

accuracy of analog or finite-difference solutions. Those frequently 

cited are: (a) truncation or roundoff errors, (b) observational errors 

made in measurement of the model, (c) errors due to instability of 

model or analog control equipment, (d) errors due to representing the

Gl
G2 GENERAL GROUND-WATER TECHNIQUES

continuous aquifer as a group of finite elements, and (e) inaccurate 

proportioning between the aquifer properties and model components. 

Mathematical investigations (Blanch, 1953; Douglas, 1956; Garza, 

1956; Greenspan, 1957; Landau, 1956; Lawson and McGuire, 1953; 

Milne, 1949; Paschkis and Heisler, 1946; Walsh and Young, 1953; 

and Wasow, 1952) have defined errors due to the above sources (a) 

and (d) for certain finite-difference solutions. By proper design of 

the model, matrix error of the types (a) and (d) may be reduced to 

insignificance with certainty only if those factors producing error can 

be evaluated completely before the model is built. Unfortunately, 

this evaluation can be made in only a few problems in which errors 

have been or can be defined by mathematical studies. No general 

criteria as yet exist for exactly predicting errors of these types. This 

is because the errors are dependent on the fineness of the model grid 

representing the aquifer, on the manner in which the characteristics 

of the aquifer change in space, and on the curvature of the potential 

distribution being investigated. In nearly all flow-problem solutions 

performed as engineering studies, one or more of these factors is 

unknown. Thus, the model design initially can only be guided by 

intuitive reasoning from general knowledge of design versus error 

characteristics.

Although definition of the error inherent in an electric model 

solution is important, tolerance for error is ordinarily quite high in 

hydrologic studies. This is because the aquifer characteristics are 

seldom known with great accuracy, and therefore the model matrix 

itself can never be an exact replica of the aquifer. Because the hydraulic characteristics of aquifers are not likely to be known within 

± 10 percent, it seems rather ludicrous to strive for analog accuracies 

on the order of 1 percent in general-purpose investigations of groundwater flow. Wide tolerances notwithstanding, no analog solution 

obtained may be considered a sufficiently accurate forecast or description of flow unless an error evaluation shows the solution accuracy 

to be within the tolerance demanded by final application of the solution. Error analysis has generally been made from a viewpoint 

that looks toward an unknown solution. For most ground-water 

flow problems this position is untenable. The viewpoint adopted here 

is that a solution should be obtained through model design guided 

by experienced intuition, and error evaluations should be made from 

the completed solution.

On the following pages, criteria are discussed for designing the model 

resistance elements which represent given blocks of the aquifer 

prototype. The interrelationship between the aquifer segment and 

the electric model component is discussed on the basis of a similitude 

between the electrical flow in the model and of ground-water motion
ELECTRIC ANALOG, NONHOMOGENEOUS AQUIFERS G3

in the aquifer. A few features of model design, as related to aquifer 

characteristics, are studied in detail to indicate type problems for 

which caution in selecting the grid subdivision must be exercised. 

Finally, the use of finite-difference methods for evaluating the total 

error in the analog solution is proposed and discussed.

EQUATIONS OF FLOW

At any given point in a nonhomogeneous aquifer, two-dimensional 

steady flow may be defined by the following differential equation:

-- .- 

2 VJ ^ dzdy by

where T is the aquifer transmissibility, h is the height of the water 

level above an arbitrary horizontal reference plane, and x and y are 

the coordinates of the point at which h is defined. The differential 

terms of equation 1 may be written in finite-difference form (Southwell, 1946), whence

The subscript notation of equation 2 is identified on figure IA, which 

shows a small segment of the aquifer subdivided by a rectilinear grid, 

with spacing Aa? and A?/.

The model resistors connecting points of the analogous grid intersections, or nodes, of figure IA are shown in figure IB. To afford a 

correct analogic relation between the systems of I A and IB, the values 

of resistance of the elements in figure 1 B must be compatible with the 

transmissibility distribution about the nodes of figure IA. The equation of steady electrical flow to the junction (p, ri) in figure IB may 

be expressed in simplest form by the following:

,n i p+l p, n \ n l P, n <n+l P,n _ r> I 5 I 5 I 5 u

in which e is the voltage at the junction indicated by the subscript, 

and R is the resistance of the element between junction (p, n) and the 

junction indicated by the subscript. Equation 3 may be rewritten 

in the following form:

g n-1 en+l r_l_ 1

? n ^PP » I 15i" Z? 7? ' I? -CVn-1 -ttn

To permit a more easily visualized comparison of the equations of
G4 GENERAL GROUND-WATER TECHNIQUES

t

Ay

I

p -1

<- Ax -»

n+1

P, n

n -1

p+1

n-l

8

FIGURE 1. Model relations at a point in a two-dimensional field 

of flow. A, grid reference in the aquifer; B, analogous resistance 

junctions of an electric model,

flow of ground water and electricity, let Ax Ay and rewrite equation 

2 as

Let

n-^)+hn+l(TP. n+Tn+l 4 Tn-l^-4hp, nTp. re=0 (5) 

TP. n -Tp+l~ Tp - 1=j^- (6a)
ELECTRIC ANALOG, NONHOMOGENEOUS AQUIFERS G5

_ TI n 1 L - 1 °

Tv, n ^ n+l~ n - 1 ^- (6d) 

4 # +!

in which (7 is an arbitrarily selected constant relating model resistance 

to aquifer transmissibility. From equations 6a-d,

4-tp, «=C/ p +^p +15 +15 (6e)

\_Rv -\ -iip+i -ttn-l -tt«+lj

On the basis of the analogy between Ohm's and Darcy's laws, e may 

be taken to be proportional to h. Substitution of the latter proportionality and the relations 6a-e into equation 4 transforms equation 

4 into equation 5. Thus, it is evident that equations 6 analogically 

relate the finite-difference equations of ground-water flow to the equations of electrical current flow for a junction like (p, n) in figure IB.

RESISTANCE VALUES

Equations 6 may be used for calculating the resistance values 

around each junction of the model if the transmissibility distribution 

in the aquifer system is known. However, unless the transmissibility configuration and the node spacing are such that the value of 

a given resistance element, say, Rp ^i of figure IB, is the same regardless of whether it is calculated using node p,n or node p l as the 

reference for equations 6, the model itself cannot be an exact finitedifference replica of the aquifer. The transmissibility distribution 

over the interval 0<x<3, shown on figure 2, may be used to illustrate

x= o i 2 3 4 5

7= 1 3 9 27 81 243 

FIGURE 2. Grid reference and corresponding values of T along #=constant for T=SX.

this point. There, it has been assumed that T=3X and Ax=l. 

Thus, T is assumed constant along lines parallel to the y axis. Taking 

x=2 in figure 2 for the p, ^-reference node of equations 6, the resistance 

of the element to be placed between x= 1 and x 2 may be computed 

from equation 6a as follows:

660452 6
G6 GENERAL GROUND-WATER TECHNIQUES

But the value of this same resistor is also defined by equation 6b if 

the #,n-reference node for equations 6 is taken at x=l. Thus, from 

equation 6b,

-7^=3+ =5

a result markedly different from that found with the #,n-reference for 

equations 6 at x=2.

Rather than calculating R values by the finite-difference techniques, 

Karplus (1958, p. 181) has suggested making the values of the model 

resistors proportional to the integral of resistivity between nodes. 

Although the latter approach is justifiable as a much better approximation of the continuous aquifer media than equations like 6, error 

dependent on the flow regime will still arise. The latter error will be 

dependent mainly on the relative magnitude of the higher differentials 

of head and probably will be insignificant in many analog solutions.

Analysis of the form of T in space and integration of that form 

between all pairs of adjacent nodes according to Karplus' method for 

resistor design may be a laborious process. Study of equations 6 

reveals what may be a less arduous means for calculating resistance

m __rp

values. Note that terms like p+1 are simply J of the AT

between two successive grid intervals. Thus, if T changes linearly, 

or almost linearly, over a span of three successive grid intervals, the 

resistance values between nodes, as given by equations 6, are simply 

inversely proportional to the arithmetic average of T between nodes. 

For all linear forms of T, this is true irrespective of which node is used 

as the p, n-reference in equations 6. It is also true for a host of other 

functional relations describing variations of T in space. The general 

nature of such functions can be discerned by defining T in the form of 

Taylor's series (see, for example, Scarborough, 1958, p. 338).

If equations 6 are to afford a single value of resistance between 

model nodes, regardless of which node is used for reference, then, 

from equations 6a and 6b for the nodes of figure 2,

 0-2-3

or

From Taylor's series, it can be shown that

-12~-* 3 b^ ^~W 2^3T~ b? 2^5! 

and
ELECTRIC ANALOG, NONHOMOGENEOUS AQUIFERS G7

&T kx-* .,.

+ 81 + ' ' ' (8b)

From a substitution of equations 8a and 8b in equation 7,

&T b5T ($T 0.1458 ^3 Aar3+0.02070 ^ Aar5+0.001128 |^- AaT7+ ... =0 (9)

without a loss of generality. An equation identical to the form of 9 

may also be derived for the y axis. Equation 9 will always be satisfied if the differentials included are zero everywhere in the flow field. 

Thus if variations of T in space are defined by either linear or quadratic forms, equations 6 will always yield a single value of R for the 

space between a given pair of nodes irrespective of the reference 

position adopted for computing and irrespective of the magnitude of 

Ace. For other functional relations defining T in space, the series 

represented by equation 9 must be made insignificant by adopting 

an appropriately small value of Ax if equations 6 are used for computing 

values of resistance.

Equation 9 is helpful to the model designer only in that it specifies the type of variations of T for which resistance values can be 

computed as simple averages without concern for the magnitude of 

node spacing. However, for those situations in which the criterion 

of equation 9 is not satisfied along both the x and y axes, a model 

design based on equations 6 might be much less efficient than one 

based on Karplus' integration of hydraulic resistivity from one node 

to another. Under some circumstances, the latter approach may 

lead to a considerable amount of computation to define the resistor 

grid, but it may also lead to a greatly improved model efficiency by 

reducing the number of resistance elements required to obtain a given 

accuracy of the solution.

Consider, for one example, the case in which

T=a+bx (10)

To represent T in the model we may use the fundamental model 

relation

R= (11)

in which R0 is the resistivity of the model medium at the point where 

T=T0 . Following the form used by Karplus (1958, p. 181) with

equations 10 and 11, dR~ , °

/»Z-(P+!)AJ; ftx=(p+l)Ax Jv

= dR=-T0R0b ^2* (12)

Jx=pAx Jz=pbx \CL-\-VX)
G8 GENERAL GROUND- WATER TECHNIQUES

in which R& is the resistor representing the aquifer between nodes 

p and p+1. From equation 12

Rfx=TeBiAx I", , , . , ,* , , A x 1 (13) w o- L(«-f6^Ax+6Aa;)(a+6^Aa;) J v '

or

In a relation like T=a-\-bx, the finite-grid model cannot be designed 

by lumping resistivity characteristics between grid intersections of 

the prototype. This is evident from equations 13 and 14. Letting 

Ax approach zero results in the resistance element value being inversely 

proportional to the square of transmissibility, at variance with the 

basic model relation given by equation 11. Equations 13 and 14 

illustrate that some model elements must be designed only as a direct 

function of the finite-difference equivalents of the lumped transmissibility relations between two specified nodes. The latter approach 

may be taken by utilizing a familiar modified form of Darcy's law. 

For T=a-\-bx, one-dimensional flow may be expressed as

v=-(a+bx)^ (15)

where v is the gross ground-water velocity. Integration of equation 

15 between two points p and p-\-l along the x direction leads to

, , a-\-bxp+i_hp 

Comparing equation 16 with Ohm's law, the left side of equation 16 

is directly analogous to the resistance between points p and p-\-l if 

voltage is taken proportional to k, and if electrical current flow is 

proportional to v. Thus, for T a-^bx, the model resistance between 

nodes p and p-\-l could be made directly proportional to the value of 

the left side of equation 16.

ONE-DIMENSIONAL FLOW SOLUTIONS

The solutions of a particular problem of one-dimensional flow 

obtained are compared here by using the various procedures already 

discussed for computing resistance values. From equations 6, the 

resistance values are independent of Ax for the relation T=a-\-bx, 

because equation 9 will always be satisfied. Thus, one set of values 

of model resistance might be computed simply by making them 

inversely proportional to the average T between two nodes. Another
ELECTRIC ANALOG, NONHOMOGENEOUS AQUIFEKS G9

set may be constructed from equation 16. Still another, but erroneous, 

set might be developed from equation 13. Further discussion of the 

latter set is included only to illustrate the error committed. 

Analog solutions of the head distribution were obtained for

T=lX104 (l+z) (17)

with a grid spacing of Ax 1.0 assumed for the interval 0<x<10.0, 

with h Q at x=0, and A. 10.00 at x W.O. An analytical solution 

for the distribution of head is easily obtained as follows: v, the 

velocity at any point x, is obtained from Darcy's law and equation 17 

as

0=_lX104 (l+aO^ (17a)

For steady flow v is constant with x. Equation 17a can be integrated 

between xa and xp, from which

10* (17b)

1 -j-J>a

Substituting the boundary conditions xa=0, £0=10, ha Q, and hp 

= 10.00 in equation 17b yields

(170 2.3979 

Substituting equation 17c in equation 17b and letting xa =0 shows

(17d)

Equation 17d is the analytical solution of equation 17 for the boundary 

conditions stated. Head values calculated from equation 17d are 

given in the second column of table 1 .

Equations 6, 13, and 16 were used individually to design three 

electric models of the aquifer defined by equation 17 over the interval 

0<x<10. Each model consisted of a series of variable resistors set to 

within 0.5 percent of the resistance indicated by the design equation. 

Voltage control and model readout were accurate to 0.1 percent or 

better. The theoretically correct head at any point x for each of the 

model designs is given by and was computed from the relation

X-AX

X)

x=0
G10 GENERAL GKOUND-WATER TECHNIQUES

TABLE 1. Head distribution for one-dimensional flow through an aquifer with 

T=1~X.10* (l-\-x), as determined from theory and model observations

X

0

1

2

3

4

5

6

7

8

9

10

Analytical

solution

0

2.891

4.582

5.781

6.712

7.472

8. 115

8.672

9. 163

9.604

10. 000

Equation 6

Obs.

0

2.81

4.50

5.70

6.64

7.41

8.06

8.63

9. 12

9. 59

10.00

Theor.

0

2.817

4.507

5.714

6.652

7.421

8.071

8.634

9. 131

9.598

10. 000

Equation 16

Obs.

0

2.89

4.57

5.77

6.70

7.46

8. 10

8.66

9. 15

9.60

10.00

Theor.

0

2.891

4.582

5.781

6.712

7.472

8. 115

8.672

9. 163

9.604

10. 000

Equation 13

Obs.

0

5.51

7.34

8.26

8.82

9. 19

9.46

9. 66

9.79

9.92

10.00

Theor.

0

5.500

7.333

8.250

8.800

9. 167

9.429

9.625

9.778

9.900

10. 000

Heads observed on the three models and the theoretically correct 

heads calculated by means of equation 18 are given in table 1 for 

comparison with the analytical solution. All the model observations 

agreed with the corresponding theoretical model analysis to within 

±0.5 percent. The electric model based on equation 16, as expected, 

produced the most accurate solution at a cost of greater time for 

computing resistance values. For this particular flow problem, the 

maximum error arising from the model based on equations 6 is only 

about 3.5 percent. This is not a significant error for most field 

applications.

Modeling T by the most fundamental finite-difference approach, as 

stated by equations 6, can sometimes lead to an unnecessarily large 

model to gain a given accuracy of the solution. The functional relation T=3X was used earlier as an example in which resistance values 

for model design cannot be computed by equations 6 unless Ax is 

comparatively small. This is because the higher differentials of 

T=3X are all larger than the first and second differentials, and therefore equation 9 cannot be satisfied, even approximately, unless Ax is 

very small. The effects of Ax on the values of R, calculated by means 

of equations 6, are illustrated in table 2. The resistance value was 

determined from equations 6 for the space between nodes 2 and 3 on 

figure 2, using both nodes alternately as reference, and letting Ax take 

several different values as node 2 was held at x 2. Where equation 

6 is applicable, all the calculated resistance values in any one column 

of table 2 should be alike. Thus, for this particular functional relation, a net spacing of Ax 1 is much too large to permit the simplified 

calculation of R by averaging T values between adjacent nodes. 

With Ax=0.10, R calculated by equations 6b and 6c differs from the 

R calculated by averaging T values by only about 0.3 percent. The
ELE'CTRIC ANALOG, NONHOMOGENEOTJS AQUIFERS Gil

TABLE 2. R/C values calculated from equations 6a and 6b for the space between 

nodes 2 and 8 with T=S* and the node configuration of figure 2, keeping node 2 

at x = 2

Ar =

Gale, for 

Calc. for

Gale. J?2

node 2 

node 3

3/C 

3/ T3

2

Calculated K/C values for indicated units of A a;

1.0

0. 06667 

0. 1111

0. 05556

0.5

0. 08622 

0. 09019

0. 08134

0.10

0. 10531 

0. 10532

0. 10501

0.01

0. 110505 

0. 110505

0. 110501

error introduced in a solution will be dependent on these deviations to 

some unknown extent. If R is to be computed for T=3Z on the basis 

of average T, it would be judicial to select a Ax of less than 0.1 so as to 

gain a solution accuracy probably within one percent.

Further study of modeling relationships reveals a more simplified 

means for modeling T=3* and increasing the solution accuracy 

obtained for a given number of resistance elements. Following much 

the same approach used in deriving equation 12 for calculating R&,

( 

=

J v

( (P

J p&x

for T=2>x . From equation 19

For Ax = 1.0 equation 20 is

(19)

(20)

(21)

Equation 21 is a relatively simple form for calculating R; it eliminates 

the need for averaging T values at adjacent nodes, and is a more 

exact replica of conditions between nodes because of the integration 

performed. Contrary to the results from equations 13 and 14, 

equation 21 shows resistance to be inversely proportional to T. 

Therefore, it is in agreement with the basic analogy given by equation 

11 and can be used directly for model design.

If resistors are designed using equations 6 and 21, an electric analog 

solution for one-dimensional flow over the interval 0<£<i4, with 

h=Q at x=0 and h=W at x=4, will produce the values of head 

given in table 3. The greatest error for the study reported in column 2 

was about 2 percent for x smaller than 1.5. For x greater than 1.5, 

the observed values of head compared more favorably with the 

analytical solution. For example, five times the number of resistors 

were required for the solution given in column 2, where Az=0.2, than
G12 GENERAL GROUND-WATER TECHNIQUES

TABLE 3. Computed heads and heads observed on electric models for one-dimensional

flow with T=3X

(1)

X

0

1

2

3

4

(2)

Model by eq. 6 

Ar=0.2

0

6. 62

9.00

9. 74

10. 00

(3)

Model by eq. 20 Ar=l

0

6.75

9.00

9.76

10.00

(4)

Analytical 

solution

0

6. 75

9.00

9. 75

10.00

were employed for the solution given in column 3. Yet, the solution 

accuracy appears to be considerably better in the data of column 3. 

Thus, for the function T=3*, equation 21 yields a design approach 

much superior to the finite-difference design embodied in equations 6.

DESIGN OF MODEL OF TWO-DIMENSIONAL FLOW

A mathematically rigorous description of model resistance for twodimensional flow would require detailed knowledge of the flow characteristics about each point in the flow field. However, such knowledge 

is not ordinarily available and approximations to the more rigorous 

model design must be applied. One such approximation can be 

formed by considering the velocity components along the x and y 

directions separately for the purpose of calculating model resistance.

d% d2/t This approach is equivalent to assuming that ^ and ^ ^ are very

small compared with ^ and ^ at all points in the flow system, and

dT that the higher spatial derivatives of T are much smaller than z or

dy These conditions are satisfied in most aquifers. Thus, model

resistance along the x and y axes can be computed independently 

of each other by integrating T along each axis in turn, as was done 

along the x axis in the derivation of equation 16.

Take as an example the aquifer wherein T is constant with y, and 

along the x axis

T= 1 (22) 100(l+z)

over the rectangular region 0<jc<^9 and 0<j/<O6 shown in 

figure 3. Applying the same principle used in the development of
4 « 

II 3

M e

Q>

-^

X

V

r ^

/

<

«!

51

r / f/

^b

/-

i

</ A

S

1

A

/

f

/A/ i

^^&

 /

/

-Outer boundary of flow
G14 GENERAL GROUND-WATER TECHNIQUES

equation 16, the resistance values along the x direction should be

1 -x|) (23)

Similarly, the model resistors along the y axis have the following 

values:

^=100(l+x)(2/re+1 -2/re ) (24)

in which R$y represents the resistance between adjacent nodes n-f-1 

and n.

Kesistors between nodes along the y direction represent an aquifer 

width of Ax between p % and p-\-%. They must represent the hydraulic resistivity of the space in the aquifer between p % and p-^-Yz] 

and according to equation 22, T is nonlinear over this interval. 

Equation 24 is a first approximation in which nonlinearities in T 

have been ignored. Nonlinearities in T may be accounted for partly 

by defining the average velocity along the y axis as

Vy= \ fl  

f 

* LAccJpdh

in which the terms in brackets represent the effective T along the 

y axis for flow along that axis between (p, n-f-1) and (p, n) or between 

(p, ft,) and (p, ft 1); and Tx is the functional relation between ^and 

x only. From equation 25, following the technique used for obtaining 

equation 16, it can be shown that

S (26) 

$

for Ax=&y=l. Although equation 26 is a better approximation for 

R^v than equation 24, it results in resistance values only 10 percent 

different from those specified by equation 24 along x=l. At larger 

values of x, the difference between equations 24 and 26 is negligible. 

Thus, equation 24, which is fundamentally equivalent to equations 

6, was accepted for calculating R%v in the model of figure 3. The 

technique for calculating R$x and R%v leading to equations 23 and 

24 may be used for any functional variation of T in space. Resistance 

values computed by means of equations 23 and 24 are indicated on 

figure 3. These values were used in the construction of a model on 

which potentials, resulting from input and output through opposite 

corners of the system, were observed at each node. Only the equipotential lines developed from these observations are shown on figure 3.
ELECTRIC ANALOG, NONHJOMOGENEOUS AQUIFERS G15

CALCULATING STATE OF THE MODEL FROM MEASUREMENTS OF POTENTIAL

Provided the model design and measurements were all correct, 

analysis of the observed potential distribution summarized on figure 

3 should yield the original description of the T distribution, namely, 

equation 22. Such an analysis would provide a qualitative test of 

the accuracy of the model solution. To effect this test, a computing 

procedure previously outlined was followed (Stallman, 1956). Rather 

than use equation 22 for computing the spatial rate of change of T, 

it was assumed that the relation T=ax-\-by would approximate 

satisfactorily transmissibility over a small area of the flow field of 

figure 3. The latter approximation contains two unknowns, a and b. 

By algebraic manipulation (StaUman, 1956), the same form can be

stated in terms of two other unknowns, a and -jp ^ > which perhaps

have a more easily visualized physical significance, a is the angle

Aa; 5!T between the x axis and the gradient of T, and - ^ is a nondimensional

number whose magnitude is highly related to flow-field distortion 

caused by nonhomogeneity. A minimum of two equations is required

to solve directly for the two unknowns, a and -m ^r 

Potentials observed on the model and used in the computations are 

given in table 4. The two nodes marked with code 1 in table 4 were 

used as centers for constructing two finite-difference expressions

TABLE 4. Selected measurements of potential from a model of a rectangular flow

field

13

12

11

10

9

5

30.48

6

36.70

*3, *5 

33.23

30.00

7

46. 16

*1, *5 

40.82

*1, *2, *3, 

*4,*5 

36. 12

*2, *5 

32.05

28.56

8

44.84

*4, *5 

38.72

33.74

9

40.44

"Code numbers identify setsjrf nodes used for finite-difference computations. See table 5.
G16 GENERAL GROUND-WATER TECHNIQUES

TABLE 5. Aquifer characteristics calculated -from selected potential measurements)

assuming T=ax+by

Computation 

code (see 

table 4)

1

2

3

4

5

Centers of 

node sets

x= 7 

2/=12. 5

x- 7 

y=11.5

x = 6. 5

y=12

x= 7. 5 

2/=12

x= 7 

2/=12

AZ&T

Tdx 

from eq. 22

-0. 125

-0. 125

-0. 133

-0. 105

-0. 125

AZ&T

Tdx 

calculated from 

observed 

potentials

-0. 110

-0. 142

-0. 391

-0. 162

-0. 150

a

calculated from 

observed 

potentials

+ 4°-45'

-4°-27'

-22°-27'

-8°-5'

OO Acyt

O TTirf

Number 

of equations in 

set solved

2

2

2

2

5

Ax 5T needed for calculating a and - - ^ Under code 5, five finitedifference equations were constructed and normalized to solve for 

the two unknowns, the latter process being used to minimize the 

effects of random errors in observed potentials. Calculated results 

given in table 5 are keyed to the data of table 4 by code numbers 1 

through 5. According to equation 22, all calculated values of a 

should be zero. With the exception of code 3, the trend of T computed 

from the potential measurements agrees reasonably well with the 

trend computed from equation 22. The results given in table 5 

indicate that under some conditions reasonably good estimates of 

T variations in aquifers could be obtained from water-level altitudes 

without other detailed knowledge of the aquifer characteristics.

A# dT Values of a and -^ ^ calculated from the observed potentials are 1 ox

highly sensitive to errors in the potential distribution. A comparison 

of such calculated values with the correct values given by the equations 

defining the aquifer is useful for indicating areas of the model where 

even slight local errors have been made. However, calculated values

of a. and -jf Y~ cannot be used for directly obtaining a quantitative 

evaluation of total error in the observed potential.

COMPUTATION OF ERRORS IN POTENTIALS OBSERVED

ON THE MODEL

In passing from equation 1 to equation 2, simple finite-difference 

approximations of the differentials in equation 1 were accepted as
ELECTRIC ANALOG, NONBOMOGENEOUS AQUIFERS G17

adequate without qualification. Then the approximations inherent in 

equation 2 were applied as a basis for model design. Thus, errors 

due to the finite-difference approximations occur in the potential 

distribution observed on the model. A more accurate statement 

relating finite-differences to the differentials than was used in developing equation 2 may be obtained from Taylor's formula (Scarborough, 

1958, p. 338) after letting Ax=^y X, whence

. , 

4! dyV 6! +d^/V ' ' ' l '

i h-! x2 dx 5% , -,, ' ' ' ^ '

2x 3! cte3 5! dx5 ' ' '

- 23 4 , ,

' ' ( } dy 2x 3! cty3 5! d ' ' 

The infinite series of differential terms on the right side of each of 

equations 27a-c has not been accounted for in the model design 

based on equation 2. Error due to this omission is called truncation 

error, which will be represented in the following by e z . An expression 

for e t at any point (p, ri) can be obtained by substituting in equation 

1 the equations 27a-c, and subtracting the following modified 

form of equation 2 from the result:

n -i <>T

X2 2x dx 2x

(2a) 

Putting all the e t into the potential at (p, n), it can be seen that

!^_n^ /^_I_<^_L^ ^_L X2 ~L4! W1"^"1 " 6! Vdx^d

rx2

+L3!

As fixed by the assumptions leading to equation 28, e t is the error 

in potential at (p, ri) due to inadequate theoretical representation of 

potentials in the model region bounded by p l,p-}-l, n l, andr^+1. 

It may be evaluated approximately by obtaining finite-difference 

estimates (see Southwell, 1946, p. 229-237) of all the derivatives 

of significance in equation 28 from the problem solution and values
G18 GENERAL GROUND-WATER TECHNIQUES

>* rr\ ^k frt

of T, j > and ^ from the model design criteria. Obviously from

equation 28, the truncation error is dependent on both the degree 

of curvature of the potential surface and on the degree of nonhomogeneity of the aquifer represented in the model.

In addition to the truncation errors, the observed potentials contain 

errors due to other sources such as faulty or inaccurate readout equipment, inaccurate resistance settings or selections, inadequate representation of the aquifer transmissibility by the model resistance, 

and others. The magnitude of local error from all these sources 

combined can be found by solving for hPtn in equation 2a, after 

substituting in it the potentials observed on the model at points

p+1, p 1, n-{-l, and n 1, and the values of T, 3 ? and 3 from

ox oy

the model design criteria. The difference between the value of hp>n 

thus obtained, hp>n (ea. 2a> and the value of hp, n observed on the 

model, hpin (0bs.), equals the sum of all local errors other than those 

arising from truncation of the finite-difference expressions for potential. 

The total local error, AehPi , at point (p, n) is comprised of only two 

parts, a component due to truncation and another component due to 

other omissions or inaccuracies, and can be expressed as

q. 2a) ">p, »(obs.) T" ft (29)

The terms in equation] 29 are [illustrated schematically in profile 

through points (p 1, ri), (p, ri), and (p+1, n) on figure 4.

The total error at any point in the model is dependent on the 

distribution and magnitude of AehPtn as just defined, on the spatial 

variations of T, and on the boundary conditions imposed on the 

region of flow. Thus, it is evident that the total error of the solution 

at any point will be a complicated function of all the variables of

h=0

p 1, n p, n p + 1, n 

FIGURE 4. Profile of potential distribution defining local errors.
ELECTRIC ANALOG, NONHOMOGENEOUS AQUIFERS G19

flow and cannot be found from any simple relationship. However, 

equation 2 can be corrected to account approximately for local error 

as follows:

As2

(Tp+i -t p-i) . (hn+i hn -i) (Tn+i Tn -i) __ ""

2Az 2Az 2Ay 2A?/

The addition of the term kjiv, n in equation 2, as shown in equation 

2b, could be simulated on the model of equation 2 by adding 

current from an external source at points (p, ri) in proportion to 

Tpi n(AfhPin}/Ax~2. The resulting potential distribution in the model 

would be a solution of equation 2b, and it would be a better approximation than that obtained from the model based on equation 2. 

However, it would not be an exact solution because a corrected solution 

based on equation 2b would alter all the potential values, which in 

turn would alter the values of AfhPtTl . If the solution based on 

equation 2 is found to be relatively accurate, however, it is likely 

that only one correction will ordinarily be needed for an assessment 

of total error at any point. The total error may be obtained by 

subtracting the potentials observed on the model of equation 2 

at all (p, n) from the potential observed on the model of equation 2b. 

Such a procedure seems inordinately complicated and time consuming. However, for most analog solutions, brief studies of error, 

as defined by equations 28 and 29, will obviate the need for a 

complete and detailed error analysis. Using the data from table 4 

for the point x=7, y ll of figure 3 as an example, the value of 

^(7,ii) (eq. 2a) is found to be 36.12 from equation 2a. This equals 

exactly the value of A?, n ( 0bS .)> and therefore, from equation 29, all 

the detectable local error arises from truncation. From an analysis 

for truncation error by equation 28, using the basic data of the 

solution presented in figure 3, it was found that e, is of the order of 

 0.001 at z=7, y=ll. Unless e z is larger than this over a significant 

area of the field of flow in that problem, it appears doubtful that 

the total error in potential at any point on the solution would be of 

importance in hydrologic studies.

SUMMARY

Resistance elements of an electric model of two-dimensional steady 

ground-water flow through nonhomogeneous aquifers may be assigned 

values inversely proportional to the aquifer transmissibility if variations of transmissibility in space can be represented accurately by 

linear or quadratic equations. The solution error is independent of 

the analogic distance between resistor units, except for errors due to 

truncation of the finite-difference approximations of head, in models
G20 GENERAL GROUND-WATER TECHNIQUES

of aquifers with, variations of transmissibility of first or second degree. 

Where transmissibility changes in space are of third or higher degree, 

model size for a given solution accuracy can be made small by selecting resistance values inversely proportional to an integral form of 

transmissibility. For some types of aquifers, the latter design procedure may be preferred so as to yield solutions of adequate accuracy 

from a comparatively small number of components.

A procedure for computing local error at each point in the analog 

solution has been proposed, using finite-difference techniques. An 

approximation of the total error in the analog solution can be obtained by injecting currents proportional to the calculated local error 

at each point on the model.

REFERENCES

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Douglas, J., Jr., 1956, On the errors in analogue solutions of heat conduction

problems: Quat. Appl. Math. Quart., v. 14, p. 33-335. 

Garza, A. de la, 1956, Error bounds for a numerical solution of a recurring linear

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Greenspan, D., 1957, On a "Best" 9-point difference equation analogue of Laplace's equation: Franklin Inst. Jour., v. 263, p. 425-430. 

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434 p. 

Landau, H. G., 1956, A simple procedure for improved accuracy in the resistor

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Lawson, D. I., and McGuire, J. H., 1953, The solution of transient heat flow

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Milne, W. E., 1949, The remainder in linear methods of approximation: Natl.

Bur. Standards, Jour. Research, v. 43, n. 5, p. 501-511. 

Nelson, R. William, 1960, In-place measurement of permeability in heterogeneous

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Paschkis, R., and Heisler, M. P., 1946, The accuracy of lumping in an electrical

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Scarborough, J. B., 1958, Numerical mathematical analysis: (4th Ed.), Baltimore,

Johns Hopkins Press, 576 p. 

Southwell, R. V., 1946, Relaxation methods in theoretical physics: London,

Oxford University Press, 522 p. 

Stallman, Robert W., 1956, Numerical analysis of regional water levels to define

aquifer hydrology: Am. Geophys. Union Trans., v. 37, p. 451-460. 

Walsh, J. L., and Young, David, 1953, On the accuracy of the numerical solution

of the Dirichlet problem by finite differences: Natl. Bur. Standards Jour.

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Wasow, Wolfgang, 1952, On the truncation error in the solution of Laplace's

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O