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An analytical method for computing return flow or drainage from irrigation applications is presented and discussed. Computations consider aquifer properties, drain spacing, and deep percolation. A graph for determining the fractional part of percolation remaining in transient storage is included. By using the graph and simple digital procedures, realistic estimates of irrigation return flow can be determined. The method was used to determine monthly quantities for 13 yr of return flows in the Mesilla Valley, New Mexico-Texas. This study is reported.

A model basin for the study of sandy aquifer properties under controlled conditions is described. Effective porosity determinations show that the time of the first arrival of radioactive tracers injected in an observation well to a pumped well should be considered in porosity calculations. The experiments show the increase of the dispersion of tracers with increasing the distance of their path and lowering the discharge rate. (Activity, time) curves obtained experimentally are not in agreement with theories dealing with the problem.

Radioactivities in ground water collected at Nevada Test Site since the underground nuclear explosions in Operations Plumbbob and Hardtack Phase II have not been significantly above background, due to the formation of silica glass by the explosion, and the ion-exchange properties of fission products in normal earth minerals and water. The glass formed by the high shock pressures releases less than 10 per cent of its radioactivity when leached with water, and the rock minerals crushed by the explosive-forces adsorb several tens of milliequivalents of fission products per 100 g. The distribution coefficients for adsorption range from 50 to 500,000, depending on the particular fission product and mineral. A theory pertaining to the flow of activity in an ion-exchange medium is developed, and equations for application of laboratory data to field experiments are included. Some of the available distribution coefficients are tabulated from the literature on waste disposal. It is concluded from existing information that hazards arising from underground explosions in other media will not give rise to serious contamination problems. Until more experience is gained, measurements of aquifer properties should be made prior to planned explosions to determine that safe conditions exist in each particular case.

Electric-analog or digital-computer models are used to compute the effect of ground-water withdrawal or recharge on streamflow. The results can be generalized on a map showing lines of equal elapsed time. The lines indicate the time of recharging or discharging that is needed to affect the streamflow by a given fraction of the amount pumped or injected. The generalization is based on the similarity in shape of the relations between pumping time and stream depletion for (1) semi-infinite homogeneous aquifers drained by a straight, fully penetrating stream, and (2) complex heterogeneous aquifers. Response curves from a model reflect the combined effect of stream sinuosity, irregular impermeable boundaries, areal variation in aquifer properties, and imperfect hydraulic connection between the stream and aquifer. The elapsed-cime lines are identified by sdf (stream depletion factor) values. These values can be calculated from observations made on an electric-analog model and then may be used in a digital-computer program for determining the effects of ground-water pumping or recharge on streamflow.

Abstract The objective of this work was to use computational techniques advantageously in planning aquifer tests, and evaluating test data for the quantitative description of geologic and hydrologic parameters affecting well and aquifer behavior. A digital computer program was developed to compare the actual pressure performance of aquifer systems to the calculated pressure behavior based on analytical analogs, and to select the analog which best fits the test data; and predict the pressure response of an aquifer discharging through one or more wells to help design proper lest procedures and to monitor test activities in the field. Computer processed and evaluated test data from two aquifers are presented to demonstrate the usefulness of well tests and to show that automatic data interpretation is an excellent time- and money-saving tool in the exhaustive search for the best available mathematical models for the aquifer systems. Introduction This paper develops a digital computer program, based mostly on the concepts of analytical mathematics as used in describing the phenomenon of fluid flow in porous media to facilitate aquifer test planning and test data evaluation. The basic objective of aquifer tests is to evaluate the occurrence, availability and quality of groundwater. When analyzed with appropriate equations or computing techniques, aquifer tests also yield information relative to the geologic and hydrologic characteristics of the aquifer systems. There are many different types of aquifer tests. The constant terminal rate discharge tests followed by pressure build-up tests are of particular interest to engineers who develop natural gas storage pools in aquifers or those who work with producing gas and oil fields subject to water drive or encroachment. These tests are performed by pumping water from one or more wells completed in the aquifer and noting the unsteady-state pressure response of the discharging well and a number of observation wells that are strategically located at various distances from the discharging well. Due to the low compressibility of an aquifer system, pressure transients created by groundwater flow toward the discharging well can be transmitted considerable distances during relatively short periods of time. Thorough knowledge of the behavior of aquifer-aquiclude systems under static and dynamic conditions is indispensable when such systems, associated with structural and/or stratigraphic traps, are considered for the storage of natural gas. Generally, gas injection tests are used as a pilot project to study the confining ability of the aquiclude (confining caprock). The object of the pump tests is to ascertain whether there is leakage through the aquiclude, and to investigate the effectiveness of the porous and permeable strata in receiving, retaining and releasing the gas. However, the element of risk is reduced considerably if the gross characteristics of the aquifer-aquiclude system are dynamically tested by pump test data is much more to gas injection. Also pump test data is much more susceptible to rigorous analysis than gas injection test data. Classical methods of analytical mathematics can often be used to analyze pump test data, and to determine in situ parameters and geometry of the aquifer-aquiclude systems. Such analytical methods apply to relatively uniform systems with simple geometry. A large number of calculations based upon the available structural and stratigraphic data are made for careful planning of pump tests to prevent misuse of test methods and equipment. Analysis of pump test data also requires many computations before a quantitative description of the hydrologic characteristics of an aquifer-aquiclude system is possible. Digital computer methods of planning aquifer tests, interpreting test data and constructing ideal aquifer models of the real aquifers are excellent time- and money-saving tools at the disposal of engineers in their study of aquifer gas storage prospects and producing hydrocarbon fields subject to water drive or encroachment. Typical costs of running computer analysis for a set of well test data are given in the program application section. JPT P. 827ˆ

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