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Dynamics of Fluids in Porous Media

Dynamics of Fluids in Porous Media

by Jacob Bear, Engineering

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This classic work by one of the world's foremost hydrologists presents a topic encountered in the many fields of science and engineering where flow through porous media plays a fundamental role. It is the standard work in the field, designed primarily for advanced undergraduate and graduate students of ground water hydrology, soil mechanics, soil physics, drainage


This classic work by one of the world's foremost hydrologists presents a topic encountered in the many fields of science and engineering where flow through porous media plays a fundamental role. It is the standard work in the field, designed primarily for advanced undergraduate and graduate students of ground water hydrology, soil mechanics, soil physics, drainage and irrigation engineering, and petroleum and chemical engineering. It is highly recommended as well for scientists and engineers already working in these fields.
Throughout this generously illustrated, richly detailed study, which includes a valuable section of exercises and answers, the emphasis is on understanding the phenomena occurring in porous media and on their macroscopic description. The book's chapter titles reveal its comprehensive coverage: Introduction, Fluids and Porous Matrix Properties, Pressures and Piezometric Head, The Fundamental Fluid Transport Equations in Porous Media, The Equation of Motion of a Homogeneous Fluid, Continuity and Conservation Equations for a Homogeneous Fluid, Solving Boundary and Initial Value Problems, Unconfined Flow and the Dupuit Approximation, Flow of Immiscible Fluids, Hydrodynamic Dispersion, and Models and Analogs.
"Systematic and comprehensive . . . a book that satisfies the highest standards of excellence. . . . Will undoubtedly become the standard reference in this field." — R. Allen Freeze, IBM Thomas J. Watson Research Center, Water Resources Research.

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Dover Publications
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Dover Civil and Mechanical Engineering Series
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Dynamics of Fluids in Porous Media

By Jacob Bear

Dover Publications, Inc.

Copyright © 1972 American Elsevier Publishing Company, Inc.
All rights reserved.
ISBN: 978-0-486-13180-1



1.1 Aquifers, Ground Water and Oil Reservoirs

Flow through porous media is a topic encountered in many branches of engineering and science, e.g., ground water hydrology, reservoir engineering, soil science, soil mechanics and chemical engineering. Although our objective in this book was to present only the fundamental aspects of this topic, common to all these scientific and applied fields, we thought it appropriate to begin by presenting examples of porous media, and fluids in them, as encountered in practice. The aquifer, which is the porous medium domain treated by the ground water hydrologist, and the oil reservoir, which is the porous medium domain treated by the reservoir engineer, will serve as typical examples for this purpose. Following is a brief description of these domains and the fluids present in them.

1.1.1 Definitions

An aquifer (a ground water basin) is a geologic formation, or a stratum, that (a) contains water, and (b) permits significant amounts of water to move through it under ordinary field conditions. Todd (1959) traces the term aquifer to its Latin origin: aqua, meaning water, and –fer from ferre, meaning to bear.

In contradistinction, an aquiclude is a formation that may contain water (even in appreciable amounts), but is incapable of transmitting significant quantities under ordinary field conditions. A clay layer is an example. For all practical purposes, an aquiclude is considered an impervious formation.

An aquitard is a semipervious geologic formation transmitting water at a very slow rate as compared to the aquifer. However, over a large (horizontal) area it may permit the passage of large amounts of water between adjacent aquifers, which it separates from each other. It is often referred to as a leaky formation. An aquifuge is an impervious formation that neither contains nor transmits water.

Ground water is a term used to denote all waters found beneath the ground surface. However, the ground water hydrologist, who is primarily concerned with the water contained in the zone of saturation (par. 1.1.2), uses the term ground water to denote water in this zone. In drainage of agricultural lands, or agronomy, the term ground water is used also to denote the water in the partially saturated layers above the water table. In this book we shall use the term ground water mainly in the sense employed by the ground water hydrologist.

That portion of rock not occupied by solid matter is the void space (also pore space, pores, interstices and fissures). This space contains water and/or air. Only connected interstices can act as elementary conduits within the formation. Figure 1.1.1 (after Meinzer 1942) shows several types of rock interstices. Interstices may range in size from huge limestone caverns to minute subcapillary openings in which water is held primarily by adhesive force. The interstices of a rock can be grouped in two classes: original interstices, mainly in sedimentary and igneous rocks, created by geologic processes at the time the rock was formed, and secondary interstices, mainly in the form of fissures, joints and solution passages, developed after the rock was formed.

1.1.2 The Moisture Distribution in a Vertical Profile

Subsurface water may be divided vertically into zones depending on the relative proportion of the pore space occupied by water: a zone of saturation, in which all pores are completely filled with water, and an overlying zone of aeration, in which the pores contain both gases (mainly air and water vapor) and water.

Figure 1.1.2 shows a schematic distribution of subsurface water. Water (e.g., from precipitation and/or irrigation) infiltrates the ground surface, moves downward, primarily under the influence of gravity, and accumulates, filling all the interconnected interstices of the rock formation above some impervious stratum. A zone of saturation is thus formed above this impervious bedrock. The saturated zone (fig. 1.1.2) is bounded above by a water table, or phreatic surface. This is a surface on which the pressure is atmospheric. It is revealed by the water level in a well penetrating the aquifer, in which the flow is essentially horizontal (sec. 6.6). Actually, saturation extends a certain distance above the water table, depending on the type of soil (par. 9.4.2). Wells, springs and streams are fed by water from the zone of saturation.

The zone of aeration extends from the water table to the ground surface. It usually consists of three subzones: the soil water zone (or belt of soil water), the intermediate zone (or vadose water zone) and the capillary zone (or capillary fringe).

The soil water zone is adjacent to the ground surface and extends downward through the root zone. The moisture distribution in this zone is affected by conditions at the ground surface: seasonal and diurnal fluctuations of precipitation, irrigation, air temperature and air humidity, and by the presence of a shallow water table. Water in this zone moves downward during infiltration (e.g., from precipitation, flooding of the ground surface or irrigation; sec. 9.4), and upward by evaporation and plant transpiration. Temporarily, during a short period of excessive infiltration, the soil in this zone may be almost completely saturated (gravitational water).

After an extended period of drainage without a supply of water at the soil surface, the amount of moisture remaining in the soil is called field capacity (par. 9.4.3). Below field capacity, the soil contains capillary water, in the form of continuous films around the soil particles, held by surface tension. This water is moved by capillary action and is available to plants. Below a moisture content—called the hygroscopic coefficient (maximum moisture that an initially dry soil will adsorb when brought in contact with an atmosphere of 50% relative humidity at 20°C)—the water in the soil is called hygroscopic water. Water is then unavailable to plants as it forms thin films of moisture adhering strongly to the surface of soil particles (fig. 9.4.2).

The intermediate zone extends from the lower edge of the soil water zone to the upper limit of the capillary zone. It does not exist when the water table is too high, in which case the capillary fringe may extend into the soil water zone, or even to the ground surface. Nonmoving, or pellicular, water in the intermediate zone is held in place by hygroscopic and capillary forces. Temporarily, water moves downward through this zone as gravitational water.

The capillary fringe extends upward from the water table. Its thickness depends on the soil properties and on the uniformity of pore sizes. The capillary rise ranges from practically nothing in coarse materials to as much as 2 ÷ 3 m and more in fine materials (e.g., clay). A detailed discussion on the capillary rise is given in paragraph 9.4.2. Within the capillary zone there is usually a gradual decrease in moisture content with height above the water table. Just above the water table, the pores are practically saturated. Moving higher, only the smaller connected pores contain water. Still higher, only the smallest connected pores are still filled with water. Hence, the upper limit of the capillary fringe has an irregular shape. For practical purposes some average, smooth surface is taken as the upper limit of the capillary fringe (par. 9.4.2), such that below it the soil is practically saturated (say, > 75%).

In the capillary fringe, the pressure is less than atmospheric and vertical as well as horizontal flow of water may take place. When the saturated zone below the water table is much thicker than the capillary fringe, the flow in the latter is often neglected. However, in most drainage problems the flow in the unsaturated zones may be of primary importance.

Obviously, numerous complications are introduced into the schematic moisture distribution described here by the great variability in pore sizes, the presence of permeability layers and by the temporary movement of infiltrating water.

1.1.3 Classification of Aquifers

The following brief review of some geological formations that serve as aquifers is based on a work by Thomas (1952).

Most aquifers consist of unconsolidated or partly consolidated gravel and sand. They are located in abandoned or buried valleys, in plains and in intermontane valleys. Some are of a limited area; others may extend over large areas. Their thickness may also vary from several meters to several hundred meters.

Sandstone and conglomerate are the consolidated equivalent of sand and gravel. In these rocks the individual particles have been cemented together, thus reducing permeability.

Limestone formations, varying widely in thickness, density, porosity and permeability, serve as important aquifers in many parts of the world, especially when sizeable proportions of the original rock have been dissolved and removed. Openings in limestone may range from microscopic original pores to large fractures and caverns forming subterranean channels. By dissolving the rock along fractures and fissures the water tends to enlarge them, thus increasing permeability with time. Ultimately, a limestone terrane develops into a karst region. Its macroscopic behavior (i.e., on a large scale) is probably similar to that of a sand and gravel aquifer; on a smaller scale, this similarity is questionable (sec. 1.2). Figure 1.1.1 e and f show examples of rocks rendered porous by solution and fracturing.

Volcanic rocks may form permeable aquifers. Basalt flows are very permeable. The pore space of a basalt aquifer may not be as large as that of loose sands and gravels, but the permeability, owing to the cavernous character of the openings, may be many times greater. Most shallow intrusive rocks in the form of sills, dikes or plugs are low in permeability, and many of them are impervious enough to serve as barriers to ground water flow.

Crystalline and metamorphic rocks are relatively impervious and constitute poor aquifers. When such rocks occur near the ground surface, some permeability may develop by weathering and fracturing.

Clay and coarser materials mixed with clay, although in general having a high porosity, are relatively impervious owing to the small size of their pores.

Aquifers may be regarded as underground storage reservoirs that are replenished naturally by precipitation and influent streams, or through wells and other artificial recharge methods. Water leaves the aquifer naturally through springs or effluent streams and artificially through pumping wells.

The thickness and other vertical dimensions of an aquifer are usually much smaller than the horizontal lengths involved. Therefore, throughout this book, all drawings describing flow in aquifers are highly distorted. The reader should not be misled by the distorted scales of such figures.

Aquifers may be classed as unconfined or confined depending upon the presence or absence of a water table.

A confined aquifer (fig. 1.1.3), also known as a pressure aquifer, is one bounded above and below by impervious formations. In a well penetrating such an aquifer, the water level will rise above the base of the confining formation; it may or may not reach the ground surface. A properly constructed observation well (or a piezometer) has a relatively short screened section (yet not too short with respect to the size of the openings; see sec. 1.3) such that it indicates the piezometric head (sec. 3.3) at a specific point (say, the center of the screen). The water levels in a number of observation wells tapping a certain aquifer define an imaginary surface called the piezometric surface (or isopiestic surface). When the flow in the aquifer is essentially horizontal, such that equipotential surfaces are vertical, the depth of the piezometer opening is immaterial; otherwise, a different piezometric surface is obtained for piezometers that have openings at different elevations. Fortunately, except in the neighborhood of outlets such as partially penetrating wells or springs, the flow in aquifers is essentially horizontal.

An artesian aquifer is a confined aquifer (or a portion of it) where the elevations of the piezometric surface (say, corresponding to the base of the confining layer) are above ground surface. A well in such an aquifer will flow freely without pumping (artesian or ITL{flowing well}ITL). Sometimes the term artesian is used to denote a confined aquifer.

Water enters a confined aquifer through an area between confining strata that rise to the ground surface, or where an impervious stratum ends underground, rendering the aquifer unconfined. The region supplying water to a confined aquifer is called a recharge area.

A phreatic aquifer (also called unconfined aquifer or water table aquifer) is one with a water table (phreatic surface) serving as its upper boundary. Actually, above the phreatic surface is a capillary fringe, often neglected in ground water studies. A phreatic aquifer is recharged from the ground surface above it, except where impervious layers of limited horizontal area exist between the phreatic surface and the ground surface.

Aquifers, whether confined or unconfined, that can lose or gain water through either or both of the formations bounding them above and below, are called leaky aquifers. Although these bounding formations may have a relatively high resistance to the flow of water through them, over the large (horizontal) areas of contact involved significant quantities of water may leak through them into or out of a particular aquifer. The amount and direction of leakage is governed in each case by the difference in piezometric head that exists across the semipervious formation. Obviously, the decision in each particular case whether a certain stratum overlying an aquifer is an impervious formation, a semipervious one, or simply another pervious formation having a permeability that differs from that of the aquifer considered, is not a clear-cut one. Often, a layer that is considered semipervious (or leaky) is thin relative to the thickness of the main aquifer.

A phreatic aquifer (or part of it) that rests on a semipervious layer is a leaky phreatic aquifer. A confined aquifer (or part of it) that has at least one semipervious confining stratum is called a leaky confined aquifer.

Figure 1.1.3 shows several aquifers and observation wells. The upper phreatic aquifer is underlain by two confined ones. In the recharge area, aquifer B becomes phreatic. Portions of aquifers A, B and C are leaky, with the direction and rate of leakage determined by the elevation of the piezometric surfaces of each of these aquifers. The boundaries between the various confined and unconfined portions may vary with time as a result of changes in water table and piezometric head elevations. A special case of a phreatic aquifer is the perched aquifer (fig. 1.1.3) that occurs wherever an impervious (or relatively impervious) layer of limited horizontal area is located between the water table of a phreatic aquifer and the ground surface. Another ground water body is then built above this impervious layer. Clay or loam lenses in sedimentary deposits have shallow perched aquifers above them. Sometimes these aquifers exist only a relatively short time as they drain to the underlying phreatic aquifer.

1.1.4 Properties of Aquifers

The general properties of an aquifer to transmit, store and yield water are further defined numerically through a number of aquifer parameters. A detailed analysis of these parameters is given throughout this book. Here we shall present a brief general description of some of them in order to supplement the definition of aquifers given above.

The hydraulic conductivity indicates the ability of the aquifer material to conduct water through it under hydraulic gradients. It is a combined property of the porous medium and the fluid flowing through it (sec. 5.5). When the flow in the aquifer is essentially horizontal, the aquifer transmissivity indicates the ability of the aquifer to transmit water through its entire thickness. It is the product of the hydraulic conductivity and the thickness of the aquifer (sec. 6.4).

The storativity of an aquifer (sometimes called the coefficient of storage) indicates the relationship between the changes in the quantity of water stored in an aquifer and the corresponding changes in the elevations of the piezometric surface (or the water table in an unconfined aquifer).


Excerpted from Dynamics of Fluids in Porous Media by Jacob Bear. Copyright © 1972 American Elsevier Publishing Company, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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