Updated: May 20, 2001.
Copyright © 2000 by Jacob Bear, Haifa, Israel.
All Rights Reserved.

This is an incomplete version for demo purposes only

Computer-Mediated Distance Learning

Course on

MODELING GROUNDWATER FLOW AND CONTAMINANT TRANSPORT

INSTRUCTOR:

JACOB BEAR

Professor Emeritus

Faculty of Civil Engineering

Technion-Israel Institute of Technology

Haifa 32000, Israel

TOPIC B:
GROUNDWATER AND AQUIFERS (Contd.)

In the first two lectures of this topic, we have discussed:

Moisture Distribution in the Subsurface.

Classification of Aquifers.

Earlier, we have stated that our role, as geohydrologists, is to predict the behavior of an aquifer system, in response to any natural (e.g., natural recharge from precipitation), or man-made (e.g., pumping) excitations. Essentially, the relationship between excitation and response takes the form of a water balance. A water balance is a very simple and obvious concept: given any spatial domain, the difference between all inflows and outflows during any specified period of time -- the balance period -- is stored in that domain. In hydrology of groundwater, we consider two possible domains:

A finite spatial domain, e.g., an aquifer, or part of an aquifer.

A very small domain (infinitesimally small) surrounding a given point within an aquifer domain. We really think of a "point in space".

Similarly, we may consider two possible balance periods:

A finite period of time, say a year, or a season.

An infinitesimally small time interval. In the latter case, we really think of a "point in time".

Later in the course, we shall give an interpretation of what we mean by "a water balance for a point in a spatial domain and at a point in time".

Since for any selected domain and time interval, a "water balance" must always be maintained, any change in one of the balance components, whether natural or man-made, must inevitably affect all other components. This is the basis for our statement that the water balance can be used as a tool to predict the response of the system (say, an aquifer domain) to any excitation.

As we continue, we shall return to the details of what we have just outlined. Our objective in what follow is to discuss the components of a water balance; as well as the terms "inflows" and "outflows" used above. How do we calculate, or estimate, them? How is water stored in an aquifer?

As a case of particular interest, we shall consider aquifer domains, basing our discussion on the concept of essentially horizontal flow, to be presented in detail in TOPIC C: MOTION EQUATIONS, and LECTURE 3: Aquifer Transmissivity.

The groundwater part of the hydrological cycle was presented in Topic A, Lecture 1. The figure there schematically describes the entire hydrological cycle.

In the management of groundwater resources, man intervenes in the hydrological cycle in order to achieve beneficial goals. This intervention takes the form of modifications imposed on the various (inflow, outflow, and storage) components of the water balance.

Water and pollutants carried with it may enter an aquifer, or a considered portion of one, in the following ways:

Groundwater inflow through aquifer boundaries and leakage from overlying or underlying aquifers.

Spring discharge.

Evapotranspiration.

The difference between total inflow and total outflow of water and of pollutants during any selected period is stored in (or removed from) the aquifer, causing a rise (or drop) in water levels and in the concentration of pollutants, respectively.

Let us review these components. We have to understand them, as they will later appear in the water balance equations that constitute the core of the models that we intend to construct.

The boundaries of a considered aquifer domain may be

the natural boundaries of a groundwater basin (e.g., an impervious boundary, a water divide, or a river fully penetrating an aquifer), or

any closed boundary drawn on a map (say, for administrative reasons).

In the present lecture, we shall use some of the aquifer concepts and definitions that we intend to discuss in detail in later lectures, assuming that you are familiar with them, at least in a general way, from previous studies, reading, or work.

The water balance, or budget, discussed below is only for groundwater in the saturated zone. It is, obviously, possible to discuss a water balance for an entire subsurface domain, including water in the unsaturated zone, or even a regional water balance, which will also include surface water.

LECTURE 3:

REGIONAL GROUNDWATER BALANCE

We consider an aquifer, or part of it, and we wish to establish a groundwater balance for it. There is no need for much discussion in order to reach the conclusion that such a balance may take the form:

All terms in this equation are expressed as volume of water per the balance period, where the balance period may be a year, a season, 3 months, etc.

Most of these balance components are obvious and need no further discussion. Let us focus only on some of them.

A. INFLOW AND OUTFLOW THROUGH AQUIFER BOUNDARIES

Let us consider an aquifer domain, which may encompass an entire aquifer, or part of it. When the boundary of such a domain is pervious, groundwater may enter the considered domain through it from the outside: from another aquifer, or from the remaining portion of the same aquifer. The rate of flow is governed by the gradient of the water table, or the piezometric head, and the hydraulic conductivity along the considered boundary.

Water may also enter an aquifer from a river or a lake that serves as its boundary. We shall discuss this case in a later lecture, where we shall discuss boundary conditions.

The figure below shows a contour map and a portion of an aquifer, ABCD, for which a water balance is being established. (green) Arrows indicate inflow and outflow. Groundwater flows into the aquifer through the boundary segments DA and AB, and leaves it through the segments BC and CD. Recall that we are assuming here two-dimensional flow in the horizontal plane.

Inflow and outflow through aquifer boundaries

QUESTION 1: Study the figure. Do you agree with the statement "Groundwater flows into the aquifer through the boundary segments DA and AB, and leaves it through the segments BC and CD"?

How do we calculate net inflow into a specified domain?

Because the hydraulic gradient varies in magnitude and direction along the boundary, and because, in general, streamlines are not perpendicular to the boundaries, we divide the entire boundary of the considered domain (here: the rectangle ABCD) into segments (like MN) of length W_j each (the figure is shown again below). Segments are chosen such that along each of them, the component of the hydraulic gradient normal to W_j can be, satisfactorily, represented by an average value, J_nj .

Similarly, the transmissivity along the j-th segment is represented by an average value, T_j, over that segment. The inflow through W_j is then given by the product

W_j¥ T_j¥ J_nj

We may then sum such expressions for the entire boundary, with a positive value of J_nj for inflow and a negative one for outflow. We obtain

This expression provides the instantaneous net inflow through the aquifer boundaries.

In general, the instantaneous net inflow may vary with time. Denoting the average net inflow during

Groundwater entering a balance area carries with it solutes (which may be polluting elements) present in the formation (or in the adjacent portion of the aquifer) outside the boundary. Groundwater leaving a balance area carries with it solutes that are present in the aquifer on the inner side of the boundary. Often, especially if the balance area is not too large, it is assumed that complete mixing of groundwater takes place in the balance area, so that the water leaving it carries with it the average concentration in the balance area.

We shall discuss solute balances in later lectures.

B. LEAKAGE INTO AND OUT OF AN AQUIFER

As in the discussion on the previous balance component, do not worry if AT THIS TIME you do not understand the mathematical expression that I shall present for the leakage term. We shall come to this topic at a later lecture, and then I'll call your attention to the material presented here.

We are discussing the case in which the considered portion of an aquifer is overlain, and/or underlain by an aquitard. We have already introduced the term aquitard to denote a layer that overlies or underlies an aquifer, provided it is characterized by:

It is much thinner than the considered (underlying or overlying) aquifer.

Its permeability is much lower than that of the considered (underlying or overlying) aquifer.

The leakage, q_v(= volume of water that leaks per unit horizontal area and per unit time) through a semi-permeable layer (or an aquitard) from an overlying (or an underlying) aquifer, in which the piezometric head is h_ext into the aquifer, where the piezometric head is h, is given by:

In Eq. (4), K' is the hydraulic conductivity of the aquitard, and B' is its thickness.

The figure below shows the general case of leakage from one aquifer to an overlying or underlying aquifer. The leakage is driven by the piezometric head difference h₂-h₁, across the resistance B'/K',

where K' and B' are the hydraulic conductivity and the thickness, respectively, of the semipervious layer. If q_v as calculated by Eqn. (4) is negative, we have leakage out of the aquifer.

As leakage may vary from point to point, we may divide the (horizontal) area of the aquitard into elementary areas, DA_i, through each of which we calculate an average leakage, <q_vi> during a time interval, Dt. Then, the total net inflow (= leakage) into the aquifer during Dt is given by

Dt S_(i)(DA_i <q_vi>).

As we lower the piezometric head in a pumped aquifer, the leakage may reverse its direction from outflow to inflow.

The remarks made above with respect to inflow and outflow of pollutants, which now are carried with the leaking water, are valid also here.

C. NATURAL REPLENISHMENT FROM PRECIPITATION

Natural replenishment, or recharge, is a term used to denote the rate at which an aquifer is replenished directly (from that portion of the precipitation that infiltrates into the subsurface and percolates downward) or indirectly, via another aquifer as part of the hydrological cycle. We may speak of replenishment as volume of water in terms of millions cubic meters per year, or as depth (in meters) per year, assuming a uniform distribution over the horizontal area of the replenished aquifer. Later we'll discuss artificial recharge.

Phreatic aquifers can be replenished, or recharged, from above by precipitation falling directly over the ground surface overlying the aquifer, provided the ground surface is sufficiently pervious. Part of the area may be almost completely impervious (e.g., covered by houses, streets, highways, or an impervious rock or topsoil which is practically impervious) and does not contribute to the natural replenishment of the aquifer beneath it.

It is well known that with the expansion of urbanization, highways, parking lots, etc., the area available for infiltration is reduced. A larger fraction of the precipitation is diverted to the rainfall drainage system. It may still be used downstream as surface water (e.g., in rivers and lakes), but the amount of natural replenishment reaching an underlying aquifer is certainly reduced.

Confined aquifers are replenished by groundwater inflow from an adjacent phreatic aquifer which, in turn, is replenished from precipitation (e.g.. aquifers B and C in the cross-section shown in the figure in Lecture 2 of Topic B.

The relationship between natural replenishment and total precipitation is governed, among others, by the following factors:

Type of precipitation.

Climatic conditions.

Soil moisture prior to rain event.

Rain event characteristics (duration, intensity, peak intensity).

Topography of ground surface.

Perviousness of ground surface.

Vegetation cover of ground surface.

Moreover, in general,

for the purpose of management of a groundwater system, and

in view of the buffer effect of the large volume of water in storage in the aquifer at any time,

we are not interested in the variability in infiltration during any individual rain event (= storm), and not even that resulting from all the storms during a year, taking each storm as an instantaneous pulse. For most regional management purposes, we are interested only in annual or seasonal replenishment. Here, "season", which could mean 2-6 months, is used as a time unit in the management model.

Within the framework of management models, we often assume that the natural replenishment is uniformly distributed throughout the year, or throughout the rainy season. In certain cases, where more details are required, monthly averages are used.

Several methods are available for estimating natural replenishment from annual or seasonal precipitation data. Note the use of the term "estimating", rather than "determining", to emphasize that, in view of the complexity of the subject, one can expect at most to derive an estimate of the replenishment, rather than any precise value. This fact will become clearer as we learn more about the complexity and uncertainty associated with any investigation related to groundwater.

We shall mention two methods for estimating aquifer replenishment:

Relating replenishment to precipitation by means of certain parameters. These parameters depend on local conditions. We estimate values of these parameters by an appropriate parameter estimation, or inverse, or model calibration, method.

Using some kind of a watershed model.

Relationship between replenishment and precipitation

This is an obvious, almost "natural", approach, as precipitation is the source of replenishment. Also, in general, much more data is available on precipitation, than, say, on water levels -- input required in any application of an inverse method. The idea is based on the observation that except for precipitation that varies from one year to the next, all other factors which affect replenishment (see list presented above) are constant in time (except antecedent conditions when considering infiltration from an individual rain event), or vary only gradually (e.g., due to changes in land use). Hence, rather than refer to annual replenishment as the unknown parameter, the natural replenishment is often first related to precipitation, for which a sufficient amount of data is usually available. One way is to assume that annual or seasonal infiltration, or replenishment, is a certain percentage of the annual, or seasonal precipitation. It is simple, but not a too valid assumption. Another possible such relationship is

where N is the annual natural replenishment, a is a coefficient, P is annual precipitation, and P₀ is threshold precipitation. For example, for a = 0.9, P = 650 mm/year, we obtain N = 405 mm/year, for P₀ = 200 mm/year. In this way, the number of unknown parameters defining natural replenishment is reduced to only two: a and P_0.. These are then regarded as parameters of the aquifer model, to be estimated within the framework of model calibration. The number of parameters to be determined becomes larger if we assume that a and P₀ vary throughout a considered aquifer.

QUESTION 2: What may cause replenishment to vary from one part of an aquifer to another, for

Recall that the alternative is to estimate the natural replenishment for EVERY year, as variations in precipitation also induce variations in natural replenishment.

Watershed Models

Another method often used for estimating natural replenishment, when detailed data on precipitation are available, is the use of any of the so called Watershed Models. The Stanford Warershed Models developed by Crawford and Linsley (1966) may serve as an example. The Hydrocomp Simulation Program (Hydrocomp, 1968) is a more sophisticated version of this model. Many watershed models have been developed and published since the pioneering work of Linsley and Crawford (see a survey of such models given by Fleming, 1975, p. 190).

Like the Stanford model, which since its first publication has gone through a large number of development phases, most models of this kind simulate the hydrologic cycle, using a moisture accounting procedure of one form or another. A system of equations describes the interrelationships among the various elements of the model. During the simulation, a running record is maintained of all moisture entering the basin (or the considered part of it), stored in it, and leaving it as evapotranspiration, surface runoff, and groundwater. The latter is the natural replenishment considered here.

D. CHANGED STORAGE WITHIN THE AQUIFER

If total inflow (from all sources) exceeds total outflow, the difference is stored within the considered aquifer domain. At this point we have to make a distinction between a phreatic aquifer and a confined one.

[Comment: The subject of water storage in an aquifer will be discussed in detail in later lectures. Here, we shall briefly introduce the subject for the purpose of completeness, without too many details.]

Storage in a phreatic aquifer:

As water accumulates in the considered portion of an aquifer, the water table rises (similar to what happens in any vessel).

We may ask: what is the volume of water that has to be added to storage in a phreatic aquifer, per unit (horizontal) area, in order to cause a water table rise by one unit (e.g., one meter). Similarly, what is the volume that has to be withdrawn from a unit area of aquifer in order to produce a unit drop in the water table.

Water is stored in a phreatic aquifer column of unit cross-sectional area mainly due to the fact that water fills up the void space as the water table rises.

The above figure shows a column of (horizontal cross-sectional) area A, and water levels at times t and t + Dt, during which the water table rises a distance Dh.

So what is the volume that can be stored per unit horizontal area, when the water table rises by one unit? Write your answer, and check with my answer.

The storativity, S, of a phreatic aquifer is defined as

the volume of water, DU_w , released from storage (or added to it) per unit horizontal area, A, of an aquifer and per unit decline (or rise) in the water table, Dh.

Because water is already present in the void space as the water table rises, and water remains in the void space as the water table drops, the specific yield is less than the porosity, by an amount referred to as "irreducible moisture content, q_wr , such that S_y= f - q_wr.

Storage in a Confined aquifer:

We have seen above that the change in water storage in a phreatic aquifer is associated with filling and emptying of the void space. What about a confined aquifer? How can water be stored in the aquifer, if it is completely full with water and remains so?

The ability of a confined aquifer to store additional water stems from the fact that water is slightly compressible and so is the aquifer material. Hence, water can be added into an aquifer that is already fully saturated, causing a rise in pressure (which means also a rise in the piezometric head).

We define storativity of a confined aquifer, S, as

the volume of water, DU_w , released from storage (or added to it) per unit horizontal area, A, of an aquifer and per unit decline (or rise) in the piezometric head, Dh.

This relationship is shown in the figure below.

Note that S is dimensionless.

Typical values of storativity of a confined aquifer are of the order of 10^-4 - 10^-6, roughly 40% of elementary areas, A_i, within each of which we have the local values of Dh_i and S_yi, and obtain the total change in the stored volume of water in the form:

U_w = S_(i)A_i Dh_i S_yi.

You may find more material on the components of a regional water balance in: Bear, (1979).

You are now ready to continue to the LECTURE on
TOPIC C, LECTURE 1: Darcy's Law,
where we focus our attention on the motion of water in the saturated zone.
As always, you can go back to the table of and choose a LECTURE.

You may e-mail me questions and comments.
Jacob Bear
E-mail address: cvrbear@tx.technion.ac.il
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ANSWER TO QUESTION 2:

Such variations are due to the fact that the various factors that affect the relationship between precipitation and infiltration (and also surface runoff), e.g., slope, vegetation, soil type, etc., also vary spatially.

INSIGHT:

The role of the hydraulic gradient is associated with the basic idea expressed by Darcy's law, namely, that flow takes place from high to low piezometric head. Consider the case shown in the following figure:

Look at Figure (a). You see two contours: h' and h'' (< h'). Consider point A. What is the flow, in direction and magnitude, at this point? In two-dimensional flow, it is Q' = T ¥ J, i.e. the product of the transmissivity and the hydraulic gradient. What is J here? It is the value of the difference between the piezometric heads h' and h'', divided by the distance between the two contours. This distance is normal to the two (straight-line) contours. It is the shortest distance between the two (parallel) contours. Note that AB is shorter than AC, AD, or AE. This gives the direction and magnitude of the hydraulic gradient. It is the direction and magnitude of the steepest descent in piezometric head at point A.

In Figure B, in which we see curved equipotentials (i.e., curves of contours of equal piezometric head), we implement the same idea. We look for the steepest descent at A. Actually, in both cases, we have to take the average hydraulic gradient at a point which is mid-way between the two contours (i.e., between A and B). In an isotropic porous medium (i.e., one that has the same transmissivity in all directions), the direction of the hydraulic gradient is also the direction of the streamline at that point.

In other words, to obtain flow normal to a line segment, we need the component of the hydraulic gradient normal to that segment!

ANSWER:

I am not sure, but I have a feeling that your answer is "porosity, f". I am saying so, because in my years of teaching, this is the first answer that I usually get. It is not so bad, as it assumes that as the water table rises , the volume of water stored is equal to the void space. The ratio between volume of void space and volume of porous medium is indeed the porosity. However, this answer does not take into account the fact that the void space in the unsaturated zone is not completely free of liquid water. There is always some moisture (in the form of liquid water) in the void space, as the water table starts to rise, so that the volume required to saturate the unit volume of soil is less than the porosity. Similarly, as water is drained out, accompanied by a decline in the water table, some water remains in the void space as a result of capillary forces. We shall go into details of these phenomena at a later stage in this course. Right now, it is suffiecient that you recognize the fact stated above.

The conclusion is, therefore, that the volume of water stored in the void space per unit rise of the water table is somewhat less than the porosity.

ANSWER TO QUESTION 1:

Although we shall learn about Darcy's law at a later stage, it seems reasonable to assume that flow takes place form high water levels to lower ones. Also, you may probably recall from your course in fluid mechanics, or hydraulics, that streamlines are perpendicular to equipotentials, a role played here by the contours.

The contours show lines of equal piezometric head. The piezometric head is defined as the energy of the water (due to elevation and pressure), per unit weight of water. Flow takes place from high energy to low energy, i.e., from a high to a low piezometric head. You'll learn all this later.

Examining the figure, we note that the statement is indeed correct.

Compare with what you thought.

You may e-mail me questions and comments.
Jacob Bear
E-mail address: cvrbear@tx.technion.ac.il
Copyright Notice