Groundwater Hydraulics Research Articles

A Method for Determining the Exponent Value in a Forchheimer-Type Flow Equation

This article presents a direct method for determining the exponent in a Forchheimer-type flow equation. Forchheimer proposed the following expression as the "law of flow" in porous media: P n = au + bu, .................................(1) L where p/ L = pressure gradient u = volumetric velocity (flow rate per unit area) a, b = constants for a particular porous media n = velocity exponent It is well established in oil and ground-water flow that velocities are generally sufficiently low such that Darcy's law is usually valid as the law of flow. That is, P = cu,........................................(2) L where c = a constant. In those cases where Darcy's law is not valid, Muskat states that the empirical flow data can be expressed in all cases by an equation of the form of Eq. 1, with n 2 when flow velocity is sufficiently high to be in the turbulent flow region. Eq. 1 is also applicable when flow is laminar, but kinetic energy losses are large relative to viscous forces. Although it is not the purpose of this article to discuss the range of validity of either Eqs. 1 or 2, it should be noted that the application of Eq. 1 is appropriate more often than is commonly recognized. Many high-rate Middle Eastern wells, particularly those in fractured reservoirs, are described by Eq. 1, as first reported by Baker. Using Eq. 2 in place of Eq. 1 can result in errors on the order of 50 percent or more in predicting well rates. It is well documented in the literature that a rate equation based on Eq. 1 also applies to many gas wells. Nevertheless, the use of the empirical Rawlins-Schellhardt backpressure flow equation remains common. Cornelson has recently presented a discussion of the potential error in predicting gas rates and reserves when the Rawlins-Schellhardt equation is used in place of Eq. 1 for the flow equation. In addition to reservoir flow considerations, well screens, perforations, and other near-wellbore effects generally cause the pressure loss to be proportional to n th power of the flow rate. Investigators in the field of ground-water hydraulics note that n may deviate significantly from 2 and should be determined separately for each well. In such cases, n is usually determined by trial-and-error curve fitting of well-test data. A direct solution for n, as well as the associated flow coefficients, is possible using the following procedure. Run the well-flow test such that at least three pressure drawdown values are obtained for three rates at equal flow times. P. 883

Erratum for “Ground-Water Hydraulics in Aquifer Management”

Erratum for “Ground-Water Hydraulics in Aquifer Management”

Groundwater hydraulics of extensive aquifers : Edelman, JH Textbook. Figs. Bull. 13. Int. Inst. Land Reclam. and Improvement, Netherlands, 1972, 216P

Groundwater hydraulics of extensive aquifers : Edelman, JH Textbook. Figs. Bull. 13. Int. Inst. Land Reclam. and Improvement, Netherlands, 1972, 216P

Ground-Water Hydraulics in Aquifer Management

The method first replaces the differential equations of ground-water flow by finite-difference approximations that include unknown sink/source terms. The resulting system of algebraic linear equations has a rectangular matrix of coefficients. This system, together with linear inequalities relating sink/source terms, heads or both, and together with an objective function, forms a linear programming (LP) model. The method is applied to small-scale models of confined and unconfined saturated flow for steady-state and transient cases. The steady-state LP models are solved using available computer codes. For the transient confined model, the Crank-Nicolson scheme is used, and a single LP problem is solved covering all of the time steps. For the transient unconfined model, a predictor technique is used, and a LP problem is solved at each corrector step. The optimal solutions are consistent with the results of traditional analyses.

Application of groundwater hydraulics to a basaltic water-table aquifer

Analysis of pump-test data of open wells in a basaltic water-table aquifer is made with methods based on both equilibrium and non-equilibrium formulae. The study has indicated that the Thiem's equilibrium formula and the ratio method of Narasimhan appear to be more suitable as compared with Jacob's and Chow's methods. Of the two, Narasimhan's ratio method, which does not require the graphical procedure, is more reliable than the Thiem's method since the latter involves an assumption of the value for the radius of influence. From the value of T (coefficient of transmissibility) obtained by Narasimhan's method it is also possible to calculate the value of the radius of influence by Thiem's method, which can be used in determining the required spacing between wells in the basaltic terrain in order to avoid mutual interference.

THE RELATION BETWEEN THE LOWERING OF THE PIEZOMETRIC SURFACE AND THE RATE AND DURATION OF DISCHARGE OF A WELL USING GROUND‐WATER STORAGE

When a well is pumped or otherwise discharged, water-levels in its neighborhood are lowered. Unless this lowering occurs instantaneously it represents a loss of storage, either by the un-watering of a portion of the previously saturated sediments if the aquifer is nonartesian or by release of stored water by the compaction of the aquifer due to the lowered pressure if the aquifer is artesian. The mathematical theory of ground-water hydraulics has been based, apparently entirely, on a postulate that equilibrium has been attained and therefore that water-levels are no longer falling. In a great number of hydrologic problems, involving a well or pumping district near or in which water-levels are falling, the current theory is therefore not strictly applicable. This paper investigates in part the nature and consequences of a mathematical theory that considers the motion of ground-water before equilibrium is reached and, as a consequence, involves time as a variable.

Erratum for “Unsaturated Flow in Ground-Water Hydraulics”

Erratum for “Unsaturated Flow in Ground-Water Hydraulics”

History of the Dupuit-Forchheimer Assumptions on Groundwater Hydraulics

History of the Dupuit-Forchheimer Assumptions on Groundwater Hydraulics

Application of the Dupuit-Forchheimer Assumptions in Groundwater Hydraulics

Application of the Dupuit-Forchheimer Assumptions in Groundwater Hydraulics

Unsaturated Flow in Ground-Water Hydraulics

Water tables as zero-pressure surfaces do not separate saturated hydraulic conductivity from impermeability in systems of flow of water through soil. Hydraulic conductivity relationships on the negative pressure side of the water table are a property of the soil and must be experimentally determined. The various methods for obtaining these relations are examined, and curves relating hydraulic conductivity to negative soil-water pressure are shown for 28 soils. These curves, which are generally sigmoid, can be simplified to step functions for including flow at negative soil-water pressure in the analysis of steady flow systems. Such systems are then solved with a certain negative pressure (called the critical pressure head), rather than with zero pressure, at boundaries where the system is in contact with atmospheric air in the soil. Critical pressure heads calculated for the 28 soils show that this factor may vary from -15 cm of water or more for sands to -100 cm water or less for structureless loams and clays. The procedure is demonstrated for systems with mainly horizontal flow, downward flow, and for flow through restricting layers.

Discussion of paper by J. N. Luthin and J. W. Holmes, “An analysis of the flow of water in a shallow, linear aquifer, and of the approach to a new equilibrium after intake”

This paper [Luthin and Holmes, 1960] represents another significant contribution to the rapidly growing literature in the field of transient ground-water hydraulics. It is worthy of note, however, that the complement of the error function sufficiently describes water-table levels in the aquifer from time zero until the water begins to flow from the outlet end of the aquifer for the conditions set forth by the authors.

Present status of our knowledge regarding the hydraulics of ground water

Darcy's law, that the flow of water through porous materials increases directly as the hydraulic gradient, has been given much investigation. Investigations made in the Hydrologic Laboratory of the United States Geological Survey show that Darcy's law holds down to a gradient of two or three inches to the mile, and there are indications that it holds to less than one inch to the mile. On the basis of the work of Tolman and Poland it is evident that Darcy's law applies to the movement of ground-water under high gradients and high velocities except under very unusual conditions. It is now clear that Darcy's law applies to laminar flow, in which essentially no energy is lost in producing eddies.