15 Groundwater Hydrology-II

Brijesh Kumar Yadav

epgp books

 

 

 

Objectives

 

In this module the students will able to learn about:

 

1.      Fundamentals of groundwater flow

2.      Aquifer storage and compressibility

3.      Groundwater interaction with streams and lakes

 

15.1. Fundamentals of Groundwater Flow

 

Ground water moves from one region to another to eliminate energy differentials. The flow of ground water is controlled by the law of physics and thermodynamics. The motion of water requires energy. This energy can be expressed as a head above a datum. The elevation of this datum is arbitrary. This is because the difference in energy or the difference in head is the concern. It is therefore important that the energies be measured with respect to the same datum. In groundwater engineering the mean sea level (MSL) is usually taken as the datum. The hydraulic head is defined as the energy per unit weight measured relative to the datum.Water can possess several forms of energy. Perhaps the most obvious is the energy that water possesses by virtue of its elevation above the datum. This is the potential energy. A mass m of water at an elevation z above the datum has a potential energy mgz, where g is the acceleration due to gravity. This is the work necessary to move the mass m from the datum to the elevation z. If ρ is the density of the water, a unit volume of water has a mass ρ and a weight ρg and a potential energy ρgz. The potential energy per unit weight, that is the elevation head, is thus ρgz/ρg= z. Note that the head has the unit of length.

 

The energy that water possesses by virtue of its motion is the kinetic energy. A mass m of water that moves with a velocity v has a kinetic energy ½ mv2. Thus the kinetic energy per unit volume is ½ ρv2 and the kinetic energy per unit weight or velocity head is ½ ρv2/ρg= v2/2g. The velocity head has the dimension of length. When groundwater is flowing through the pores of the rock or soil formation, the velocity is very small, perhaps of the order of centimeters per year, and the velocity head is usually negligible with respect to the other forms of energy. One exception is near wells where the velocity increases significantly. Another exception is in certain karst conduits where groundwater can flow fast enough that the velocity head is important. The energy that water possesses by virtue of its pressure is the pressure energy. The pressure intensity of the fluid, p, acting on an area dA produces a force p dA. If the area is displaced by a distance ds, in the flow direction, then the force produces an amount of work p dAds known as flow work. The volume dAds has a weight ρgdAds and the flow work per unit weight is p dAds/ρgdAds = p/ρg known as the pressure head. The sum of the elevation head and the pressure head is known as the piezometric headh = z + p/ρg.

 

15.2.  Streamlines and Flow Nets

 

A flow line is an imaginary line that traces the path that a particle of ground water would follow as it flows through an aquifer. Groundwater is seen to have the same direction as –Ψ gradas long as K is constant. Thus, the streamlines, which are lines everywhere tangent to the velocity vector, are perpendicular to lines of Ψ= constant or equipotential lines. The streamlines and the equipotential lines are orthogonal. A network of streamlines and equipotential lines form the flow net, which is a useful tool in the analysis of 2D flows. Thus, it is a network of streamlines and equipotential lines that intersect at right angles.

 

Physically, all flow systems extend in three dimensions. However, in many problems the features of the motion are essentially planar, with the flow pattern being substantially the same in parallel planes. For these problems, for steady-state incompressible, isotropic flow in the xy plane, it can be shown (Harr,1962) that the governing differential equation is:

Here the function h(x, y) is the distribution of the total head (of energy available to do work) within and on the boundaries of a flow region, and kx and ky are the coefficients of permeability in the x– and y-directions, respectively. If the flow system is isotropic, kx= ky, and Equation 2.1 reduces to-

Equation 2.2 called Laplace’s equation, is the governing relationship for steady-state, laminar-flow conditions (Darcy’s law is valid). The general body of knowledge relating to Laplace’s equation is called potential theory. Correspondingly, incompressible steady-state fluid flow is often called potential flow. The correspondence is more evident upon the introduction of the velocity potential φ, defined as-

 

where h is the total head, p/γw is the pressure head, z is the elevation head, and C is an arbitrary constant. It should be apparent that, for isotropic conditions,

and, Equation 2.2 will produce

The particular solutions of Equation 2.2 or Equation 2.5 which yield the locus of points within a porous medium of equal potential, curves along with h(x, y) or φ(x, y) are equal to a series of constants, are called equipotential lines.

 

In analyses of groundwater flow, the family of flow paths is given by the function ψ(x, y), called the stream function, defined in two dimensions as (Harr, 1962).

where vx and vy are the components of the velocity in the x– and y-directions, respectively. Equating the respective potential and stream functions of vx and vy produces-

 

Differentiating the first of these equations with respect to y and the second with respect to x and subtracting, we obtain Laplace’s equation

 

Consider AB of Figure 1 as the path of a particle of water passing through point P with a tangential velocity v. We see from the figure that:

 

Hence, the quantity of flow (also called the quantity of discharge and discharge quantity) between any pair of streamlines is a constant whose value is numerically equal to the difference in their respective ψ values. Thus, once a sequence of streamlines of flow has been obtained, with neighboring ψ values differing by a constant amount, their plot will not only show the expected direction of flow but the relative magnitudes of the velocity along the flow channels; that is, the velocity at any point in the flow channel varies inversely with the streamline spacing in the vicinity of that point.

 

15.3. Unconfined Flow

 

The analysis of unsteady groundwater flow requires the introduction of the concept of compressibility, a property which describes the changes in volume (or strain) induced in a material under an applied stress.

 

Case of Unconfined flow

 

Flow in aquifers is often modeled as two-dimensional in the horizontal plane. This can be done because most aquifers have an aspect ratio like a thin pancake, with horizontal dimensions that are hundreds or thousands of times greater than their vertical thickness. In most aquifers, the bulk of the resistance encountered along a typical flow path is resistance to horizontal flow. When this is the case, the real three-dimensional flow system can be modeled in a reasonable way using a two-dimensional analysis. This is accomplished by assuming that h varies with x and y, but not with z, reducing the spatial dimensions of the mathematical problem to a horizontal plane. This simplifying assumption for modeling aquifer flow as horizontal two- dimensional flow is called the Dupuit–Forchheimer approximation, named after the French and German hydrologists who proposed and embellished the theory (Dupuit, 1863; Forchheimer, 1930). Dupuit and Forchheimer proposed the approximation for flow in unconfined aquifers, but the concept is equally applicable to confined aquifers with small amounts of vertical flow. They understood their approximation to mean that vertical flow was ignored. Fetter (1994) clarified the concept, pointing out that there may be vertical flow in Dupuit–Forchheimer models, but that resistance to vertical flow is neglected. To picture what a Dupuit–Forchheimer model represents in a physical sense, imagine an aquifer perforated by numerous tiny vertical lines that possess infinite hydraulic conductivity. The vertical lines eliminate the resistance to vertical flow, but the resistance to horizontal flow remains the same. In models using this approximation, the head distribution on any vertical line is hydrostatic (∂h/∂z = 0). Figure 4 illustrates the differences between actual three-dimensional flow and flow modeled with the Dupuit–Forchheimer approximation.

 

The general equations for two-dimensional aquifer flow will be derived in a manner similar to that used in the previous section on the three-dimensional flow equations. First, equations will be derived for one-dimensional aquifer flow in the x direction, then they will be extended to two-dimensional flow in the x, y plane. In this derivation, we perform a volume balance instead of a mass balance, which is equivalent to assuming that following equation holds.

 

Consider an elementary volume that is a vertical prism of cross-section x × y, extending the full saturated thickness of the aquifer (b), as sketched in Figure 5. First consider the discharge (volume/time) flowing through the face that is normal to the x axis at the left side of the prism. Using Darcy’s law the flow (volume/time) into the prism at coordinate x is:

where Kx(x) is the x-direction hydraulic conductivity at coordinate x, b(x) is the saturated thickness at x, and ∂h/∂x(x) is the x-direction component of the hydraulic gradient at x. For a uniform, single-layer aquifer, transmissivity is defined as T = Kb, so the above expression can be simplified to-

 

Where Tx(x) is the x-direction transmissivity. Equation 3.3 applies regardless of whether the aquifer consists of a single layer as in Eq. 3.2 or has some more complicated distribution of transmissivity such as multiple layers with varying Kx. The flow out of the right side of the prism at coordinate x+Δx is similarly defined as-

 

The net volume flux (volume/time) into the element through the top and bottom of the prism is given as-

 

 

Where N is the net specific discharge coming in the top and bottom. The dimensions of N are volume/time/area [L/T].The time rate of change in the volume of water stored in the element (volume/time) is

Balancing the volume fluxes given by the previous four expressions results in

Equation 3.10 is the general equation for two-dimensional aquifer flow, allowing for anisotropy and spatial variations in T.

 

15.4. Transient Flow

 

In case of unsteady flow, in confined aquifers, some fluid is released to or goes into, storage with time.

 

 

 

Equations (4.6) and (4.7) are partial differential equation governing unsteady flow of water in a confined aquifer of thickness b. (The corresponding equations for unconfined aquifers are nonlinear in form). However, the above equations can still be used provided the drawdown in small in relation to the aquifer thickness.

 

15.5. Flow in Fracture Rock

 

Flow in fractured rock is difficult to analyze for several reasons. For one, flow occurs along discrete fractures, the distribution and properties of which are mostly unknown. It is generally not possible to map the location and orientation of the important water-bearing fractures in the subsurface, or to know their aperture (width) and roughness. Flow in some larger fractures is turbulent as opposed to laminar, so Darcy’s law should not be applied to these. Two approaches to analysing flow in fractured rock are (1) analysis of flow in discrete fracture(s), and (2) treating the network of fractures as a continuum. The following are some common techniques for both methods, which assume laminar flow in the fractures. The laminar flow in a single smooth-walled planar fracture of uniform aperture b, and length w normal to flow was presented by Romm (1966) as:

where ρw is the density of water, g is gravity acceleration, μ is the dynamic viscosity of water, and ∂h/∂x is the hydraulic gradient in the direction of flow (Figure 6). This equation is called the cubic law, since Qx is a function of b3. Rock fractures are not perfectly smooth, and various studies have been performed to incorporate roughness into equations like Eq. 4.1. In general, Qx decreases as roughness increases.

 

This discrete fracture approach can be used in cases where the scale of the problem is not much bigger than the scale of fracture spacing. It is necessary to characterize the distribution, orientation, and aperture of fractures in the problem area, which is not an easy task. This approach is used, among other things, to analyze geotechnical problems of rock slope stability, seepage into tunnels, and seepage under dams. With the continuum approach, the location of particular fractures is not accounted for, and the rock mass is assumed to be equivalent to a porous medium with homogeneous conductivities. To use this approach, the scale of the problem analyzed must be macroscopic (larger than the representative elementary volume). According to Snow (1969) effect of parallel sets of fractures can be incorporated by assigning anisotropic conductivity to the continuum. Snow (1969) derived an equation for estimating the macroscopic hydraulic conductivity Kx for a set of uniform fractures oriented parallel to the x direction, using the cubic law of Eq 4.1.

where N is the number of fractures per unit width normal to the fracture planes, and b is the aperture of each fracture. Real fractures do not occur in perfectly planar and uniform sets, but this equation can give rough estimates where hydraulic testing is lacking. Often the representative elementary volume in rock is large and difficult to define, making results using the continuum approach quite uncertain. To further complicate matters, the aperture of a fracture fluctuates in response to changes in the water pressure in the fracture. When heads decline, water pressure declines and aperture decreases, and vice versa. Therefore, the conductivities of individual fractures and of the rock mass as a whole are dependent on head to some degree.

 

Example 1: Estimate the equivalent continuum Kx for a granite that has, on average, one fracture parallel to the x direction per 2 m distance normal to the fractures. The average aperture of each fracture is 0.3 mm.

 

Using Eq. 4.2 in this case gives-

Since b is raised to the third power in these equations, aperture has tremendous impact on the result. Uncertainty in the value of average b magnifies into large uncertainty in Kx.

 

15.6. Aquifer Storage and Compressibility

 

The compressibility of water-bearing rock and soil at some internal point is affected by both external and internal stresses and pressure of entrained water within the pores. The stress-balance equation is given as-

 

where σt is the total vertical stress acting downward on the point of interest and includes the pressure of overlying soil or rock and its contained water as well as that from buildings, trees, and the like on the surface. The effective stress or resisting stress from the skeleton of the solid grains, that is, the matrix, is σe, and Pp is the pore pressure of the water in the pores.

 

Different geologic materials compress different amounts under similar changes in vertical effective stress. The compressibility α is a measure of one-dimensional (vertical) matrix stiffness; the smaller α is, the stiffer the medium is. Compressibility is defined as:

 

 

where b0 is the initial vertical thickness, db is the change in vertical thickness, and dσve is the change in vertical effective stress. Compressibility here is defined in terms of vertical strain db/b0, which for one-dimensional compression is equal to volume strain dVt/Vt0, where Vt0 is initial total volume.

 

In addition to water compression and matrix compression, there is a third way to change the amount of water stored within. This third way involves raising or lowering the boundary between the saturated and unsaturated zones within the volume. When heads decline, the upper part of the saturated zone drains and the water table and saturated/unsaturated boundary shift downwards. When heads rise, the lower part of the unsaturated zone is flooded and the water table and saturated/unsaturated boundary shift upwards. This type of storage is called water table storage or phreatic storage. The specific storage Ss is the basic storage property of saturated materials. In words, Ss is the amount of water expelled from a unit volume of saturated material when the pore water is subject to a unit decline in head.

 

The mathematical equivalent of this word definition is:

 

where dVw is the volume of water expelled from aquifer volume Vt when the head changes by dh. The negative sign is there because Ss is a positive constant and when head declines, dh is negative and dVw is positive. For a unit volume (Vt= 1) and a unit decline in head (dh = −1), Ss= dVw, as the word definition above states.

 

15.7. Groundwater Interaction with Streams and Lakes

 

Streams also gain and lose water in the same manner as lakes and ponds (Figure 7a and b). Perennial streams, that is, streams that flow year-round, are usually gaining streams. They gain water from base flow through the sides of the streams as it flows through from the groundwater system. This kind of stream is situated such that the water table is always above the stream surface.

Gaining streams in the cross section (Figure 7a) will usually show a stagnation point in the subsurface similar to that of lakes. However, in three dimensions, it is in reality a stagnation line that runs under the stream along its length. Most streams are gaining streams; losing streams (Figure 7b) are generally those intermittent streams which flow only after significant precipitation and during runoff periods. They lose water to the subsurface because the water table is below the stream surface. Therefore, losing streams are generally found in mountainous areas on alluvial fans debouching onto pediments, on sand and gravel surfaces where the water table is low, or on steep slopes. In nearly all cases, the precipitation rate is insufficient and/or the subsurface is too permeable to maintain a water table high enough to support gaining streams. Streams in certain geological settings may contain stretches which are gaining for some distance where the water table is above the stream surface, and then become losing stretches when they flow over a low water-table area; or vice-versa. These streams are common in karst terrains where a stream may be losing in its upstream reaches, and may even disappear beneath the ground surface into a cave; downstream, it may later emerge to flow on the surface as a gaining stream. In glaciated areas, streams may flow over till with high water tables as gaining streams, but after crossing a contact between till and outwash sand and gravel, the streams may become losing streams as their water infiltrates downward into the permeable outwash. In areas where there are heavy demands for water for irrigation, industry, and large cities, damming of streams and pumping of aquifers has dramatically altered natural stream/groundwater interactions. Damming often raises the water table upstream of the dam, causing some losing streams to become gaining streams; and lowers the water table downstream, causing gaining streams to become losing. Heavy pumping has caused some small gaining streams to become losing because of drastic lowering of the water table. When a gaining stream does become losing, it can introduce surface contaminants into the groundwater system.

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References and Suggested Books

 

  • Bear, J. Hydraulics of groundwater. Courier Corporation, 2012.
  • Brutsaert, W. Hydrology: an introduction. Cambridge University Press, 2005.
  • Dupuit, J. Études Théoriques et Pratiques sur le Mouvement des Eauxdans les Canaux Découverts et à Travers les Terrains Perméables. Dunod, Paris. 1863.
  • Fetter, C. W. Applied Hydrogeology. 3rd Edition, Prentice Hall, Englewood Cliffs, NJ, New York. 1994.
  • Fetter, C. W. Applied hydrogeology. 4th Edition, Prentice Hall, Upper Saddle River, N.J., London. 2001.
  • Forchheimer, P. Hydraulik. Teubner Verlagsgesellschaft, Stuttgart. 1930.
  • Freeze, R.A. and Cherry, J., Groundwater, Prentice Hall Inc., 1979.
  • Harr, M. E. Ground Water and Seepage. McGraw-Hill Inc., New York. 1962.
  • Jacques W. Delleur. The Handbook of Groundwater Engineering, CRC Press LLC. 1999.
  • Karanth, K.R., Groundwater, Assessment, Development and Management, MC Graw Hill Publishing Company, 1987.
  • Kruseman, G. P. and Deridder, N.A., Analysis and Evaluation of Pumping Test Data, ILRI Publication No. 47, 1991.
  • Raghunath, H. M. Hydrology: principles, analysis and design. New Age International, 2006.
  • Rushton, K. R. Groundwater hydrology: conceptual and computational models. John Wiley & Sons, 2004.
  • Schwartz, F. W. and Zhang, H., Fundamentals of Groundwater, John Wiley & Sons, 2003. Snow, D. T. Anisotropie permeability of fractured media. Water Resources Research 5(6):1273- 1969.
  • Todd, D. K. Ground water hydrology. John Wiley and Sons, Inc, New York, 1959.
  • Todd. D. K. and Mays, L.W., Groundwater Hydrology, John Wiley & Sons, 2005.