By Milton E. The movement of groundwater is a basic part of soil mechanics. It is an important part of almost every area of civil engineering, agronomy, geology, irrigation, and reclamation. Moreover, the logical structure of its theory appeals to engineering scientists and applied mathematicians. This book aims primarily at providing the engineer with an organized and analytical approach to the solutions of seepage problems and an understanding of the design and analysis of earth structures that impound water. It can be used for advanced courses in civil, hydraulic, agricultural, and foundation engineering, and will prove useful to consulting engineers — or any public or private agency responsible for building or maintaining water storage or control systems.

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By Milton E. The movement of groundwater is a basic part of soil mechanics. It is an important part of almost every area of civil engineering, agronomy, geology, irrigation, and reclamation. Moreover, the logical structure of its theory appeals to engineering scientists and applied mathematicians.

This book aims primarily at providing the engineer with an organized and analytical approach to the solutions of seepage problems and an understanding of the design and analysis of earth structures that impound water.

It can be used for advanced courses in civil, hydraulic, agricultural, and foundation engineering, and will prove useful to consulting engineers — or any public or private agency responsible for building or maintaining water storage or control systems. Among the special features of this book are its coverage of previously unavailable Russian work in the field, an extensive appendix of concepts in advanced engineering mathematics needed to deal with physical flow systems, and numerous completely worked-out and solved examples coupled with over problems of varying difficulty.

The aim of this work is primarily to present to the civil engineer the means of predicting the exigencies arising from the flow of groundwater. The specific problems which are to be dealt with can be divided into three parts:. When writing on a subject as broad as groundwater and seepage, it is necessary to presume a minimum level of attainment on the part of the reader. Hence it will be assumed that the reader has a working knowledge of both the calculus and the rudiments of soil mechanics.

For example, the problem of the stability of an earthen slope subject to seepage forces will be considered as solved once the upper flow line has been located and the pore pressures can be determined at all points within the flow domain. The actual mechanics of estimating the factor of safety of the slope will be left to the reader.

Several texts on the subject of soil mechanics can be found in the references [, ]. Although the fundamentals of groundwater flow were established more than a century ago, it is only within recent years that the subject has met with scientific treatment.

As a result of the trial-and-error history of groundwater-flow theory, its literature is replete with empirical relationships for which exact solutions can be and have been obtained. Advocates of the empirical approach have long reasoned that the heterogeneous nature of soils is such that rigorous analyses are not practical.

This is not so; as will be seen, much of the subject lends itself readily to theoretical analysis. Recent developments in the science of soil mechanics coupled with more precise methods of subsurface soil explorations have provided engineers with greater insight into the behavior of earth structures subject to groundwater flow.

The mathematics of the theory of functions of complex variables, once the arid theory of imaginary numbers, now allows the engineer to solve problems of otherwise overpowering complexity.

The engineer can now formulate working solutions which not only reflect the interaction of the various flow factors but also allow him to obtain a measure of the uncertainties of his design. In groundwater problems the soil body is considered to be a continuous medium of many interconnected openings which serve as the fluid carrier. The nature of the pore system within the soil can best be visualized by inference from the impermeable boundaries composing the pore skeleton.

For simplicity it will be assumed that all soils can be divided into two fractions which will be referred to respectively as sand and clay. In general, sands are composed of macroscopic particles that are rounded bulky or angular in shape. They drain readily, do not swell, possess insignificant capillary potential see Sec.

Clays, on the other hand, are composed of microscopic particles of platelike shape. They are highly impervious, exhibit considerable swelling, possess a high capillary potential, and demonstrate considerable volume reductions upon drying. Sands approach more nearly the ideal porous medium and are representative of the soils primarily to be dealt with in this book.

This may appear to introduce serious restrictions; however, in most engineering problems the low permeability of the clays renders them relatively impervious in comparison to the coarser-grained soils.

Extensive studies have been undertaken by many investigators [13, 45, 75, 84, ] to calculate the permeability and porosity of natural soils based on their sieve analyses and various packings of uniform spheres. While it is not possible to derive significant permeability estimates from porosity measurements alone [86—88, 99], the pore characteristics of these ideal packings do present some of the salient features of natural soils.

Let us assume that the soil particles are all of uniform spherical shape. Calling the total volume V and the volume of voids Vv , we have for the porosity. For a cubical array of spheres Fig. For a rhombohedral packing Fig. Figures a and b show the pore volume available for flow for the cubic and rhombohedral array, respectively [45]. It should be noted from these figures that even in the ideal porous medium the pore space is not regular but consists of cavernous cells interconnected by narrower channels.

Natural soils contain particles that can deviate considerably from the idealized spherical shape as in the case of clay and, in addition, are far from uniform in size. The true nature of the pore channels in a soil mass defies rational description. Fortunately, in groundwater problems we need not concern ourselves with the flow through individual channels.

We are primarily interested in macroscopic flow wherein the flow across a section of many pore channels may be considered uniform as contrasted to the near-parabolic distribution of the flow through a single pore. The discharge velocity is defined as the quantity of fluid that percolates through a unit of total area of the porous medium in a unit time. As flow can occur only through the interconnected pores of saturated soils Fig.

The quantity of flow also called the quantity of seepage, discharge quantity , or discharge is called the seepage velocity. Let us investigate the nature of m. Designating Ap z as the area of pores at any elevation z Fig. Hence the average value of m is the volume porosity n and will be so designated.

In our work we shall deal mainly with discharge velocities, i. The ratio. In most groundwater problems the velocity heads kinetic energy are so small that they can be neglected. Hence, Eq. Prior to , the formidable nature of groundwater flow defied rational analysis. In that year, Henry Darcy [25] published a simple relation based on his experiments on les fontaines publiques de la ville de Dijon, namely,. In soil mechanics, the coefficient of proportionality k is called the coefficient of permeability and, as shown in Eq.

It is precisely this equivalency that permits the subsequent development of groundwater flow within the theoretical framework of potential flow Sec. Visual observations of dyes injected into liquids led Reynolds [], in , to conclude that the orderliness of the flow was dependent on its velocity.

At small velocities the flow appeared orderly, in layers, that is, laminar. With increasing velocities, Reynolds observed a mixing between the dye and water; the pattern of flow became irregular, or turbulent. With the advent of turbulence, the hydraulic gradient approached the square of the velocity. These observations suggest a representation of the hydraulic gradient as. From studies of the flow of water through columns of shot of uniform size, Lindquist [99] reports n to be exactly 2.

There remains now the question of the determination of the laminar range of flow and the extent to which actual flow systems through soils are included.

Such a criterion is furnished by Reynolds number R a pure number relating inertial to viscous force , defined as. The critical value of Reynolds number at which the flow in soils changes from laminar to turbulent flow has been found by various investigators [99] to range between 1 and It is interesting to note that the laminar character of flow encountered in soils represents one of few valid examples of such flow in all hydraulic engineering.

An excellent discussion of the law of flow and a summary of investigations are to be found in Muskat [99]. However, because of the influence that the density and viscosity of the pore fluid may exert on the resulting velocity, it is of some value to isolate that part of k which is dependent on these properties. To do this, we introduce the physical permeability k 0 square centimeters , which is a constant typifying the structural characteristics of the medium and is independent of the properties of the fluid.

The relationship between the permeability and the coefficient of permeability as given by Muskat [99] is. Equation 3 may be used when dealing with more than one fluid or with temperature variations.

Laboratory and field determinations of k have received such excellent coverage in the soil-mechanics literature [81, 99, , ] that duplication in this book does not appear to be warranted. Some typical values of the coefficient of permeability are given in Table Although the physical cause of capillarity is subject to controversy, the surface-tension concept of capillarity renders it completely amenable to rational analysis.

If a tube filled with dry sand has its lower end immersed below the level of free water Fig. Consideration of the equilibrium of a column of water in a capillary tube of radius r Fig. It can be thought of as being analogous to the tension in a membrane acting at the air-water interface meniscus and supporting a column of water of height hc.

If atmospheric pressure is designated as pa , then the pressure in the water immediately under the meniscus will be. As the capillary water rises, air becomes entrapped in the larger pores and hence variations are induced in both hc and k. On the basis of Eqs. Some typical values for the height of capillary rise are presented in Table In Fig.

Thus they represent functions of the independent variables x , y , z , and t. For a particular value of t , they specify the motion at all points occupied by the fluid; and for any point within the fluid, they are functions of time, giving a history of the variations of velocity at that point.

Let the pressure of the center point A x,y,z of Fig. In groundwater flow the common body force is that due to the force of gravity. With a pressure p at point A x,y,z , the force on the yz face of the element nearest the origin is. When the state of flow is independent of the time steady state , the left sides of Eqs. For example, Eq. Hence, substituting for p from Eq. Equations 6 indicate that for steady-state and laminar flow the body forces are linear functions of the velocity.

Assuming that the coefficient of permeability k is independent of the state of flow and substituting Eqs. Although the steady-state assumption is generally made to set aside the inertia terms in Eqs.

To demonstrate this, we note that each of Eqs. Thus, the remaining equation is. Physically, this implies that the speed of flow through natural soils for laminar flow is so slow that changes in momentum are negligible in comparison with the viscous resistance to flow. If one recalls Sec. Equation 8 a contains the four unknowns u , v , w , and h.

Hence one more equation must be added to make the system complete.

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## Groundwater and Seepage

Essentials of Engineering Hydraulics pp Cite as. Underground is the most important reservoir for some of the most valued fluids of the earth — water, crude oil, natural gas and others. In this chapter we discuss the occurrence of water underground and the basic principles governing its movement with a view to understanding the problem of extraction and recharge of groundwater and seepage through embankments. The discussions will be limited to the zone in which the water occupies all the voids within a geologic stratum. As this is a basic undergraduate introductory course theoretical considerations are also limited mainly to steady flow situations. Unable to display preview.

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