Calculation of resistance and error in an electric analog of steady flow through nonhomogeneous aquifers

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 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