16 Groundwater Hydrology-III

Brijesh Kumar Yadav

Objectives:

The students will able to understand:

1.      Well hydraulics and well flow equations

2.      Types of wells

3.      Aquifers testing

4.      Geophysical methods in ground water exploration

1. Introduction and Aquifer Types:

Estimating the physical properties of water-bearing layers is an essential part of groundwater studies. One of the most effective ways of determining these properties is to conduct and analyse aquifer tests. The principle of an aquifer test is simple: water is pumped from a well tapping the aquifer, and the discharge of the well and the changes in water levels in the well and in piezometers at known distances from the well are measured. The change in water level induced by the pumping is known as the drawdown. In the literature, aquifer tests based on the analysis of drawdowns during pumping are commonly referred to as pumping tests. In the analysis and evaluation of aquifer test data, three main types of aquifer are distinguished: the confined, unconfined, and leaky aquifer (Figure 1a, b, c and d).

The confined aquifer: A confined aquifer is a completely saturated aquifer having upper and lower boundaries are aquicludes. In confined aquifers the pressure of the water is usually higher than that of the atmosphere, and the water level in wells tapping such aquifers stands above the top of the aquifer (see Figure 1a). The piezometric surface is the imaginary surface to which the water will rise in wells penetrating the aquifer. When the water level in wells tapping such aquifers stands above the ground surface, they are called free- flowing wells or artesian wells.

The unconfined aquifer: Anunconfined aquiferis a partly saturated aquifer bounded below by an aquiclude and above by the free water table or phreatic surface (see Figure 1b). At the free water table, the pressure of the groundwater equals that of the atmosphere. The water level in a well penetrating an unconfined aquifer does not, in general, rise above the water table, except when there is vertical flow.

The leaky aquifer: A leaky aquifer, also known as a semiconfined aquifer, is a completely saturated aquifer that is bounded below by an aquiclude and above by an aquitard. The overlying aquitard may be partly saturated when it extends to the land surface (Figure 1c, d) or is fully saturated when it is overlain by an unconfined aquifer that is bounded above by the water table. The piezometric level in a well tapping a leaky aquifer may coincide with the water table if there is a hydrologic equilibrium between the aquifer’s recharge and discharge; it may rise or fall below the water table in areas with upward or downward flow, in other words, in discharge or recharge areas. A multi-layered aquifer is a special case of the leaky aquifer.

2. Physical Properties:

Properties of materials that are responsible for the resistance to flow usually show variations through space, i.e. from point to point. In addition, they may also show variations with the direction of measurement at any given point in a geologic formation. The first property of spatial variation is called Heterogeneity, whereas second property is termed as Anisotropy.

2.1 Homogeneity and Heterogeneity:

If the hydraulic conductivity K is independent of position within a geologic formation, the formation is Homogeneous. If the hydraulic conductivity K is dependent on position within a geologic formation, the formation is Heterogeneous. If we set up an x-y, z coordinate system in a homogeneous formation, K(x,y,z)=C. C being a constt; whereas in a heterogeneous system, K(x,y,z)≠C.

2.2 Isotropy and Anisotropy:

If the hydraulic conductivity K is independent of the direction of measurement at a point in a geological formation, the formation is Isotropy at that point. If K is dependent (or varies) with the direction of measurement at a point in a geological formation, the formation is Anisotropic at that point.

3. Groundwater Flow Equations:

There are two types of flow equations: the equation describing steady state flow and those describing unsteady state flow.

3.1 Steady state flow:

Steady, or equilibrium, flow occurs when there is equilibrium between the discharge of the pumped well and the recharge to the aquifer by an outside source (outside source may be water table aquifer contributing water through a semi pervious layer, or exposure of the confined aquifer on surface at considerable distance). Steady state is reached if in the piezometer, the changes in drawdown with time have become negligible, or that the hydraulic gradient has become constant. In steady state flow, at any point in a flow field, the magnitude and direction of the flow velocity are constant with time.

3.2 Unsteady state flow:

(or Non Equilibrium flow) occurs from the moment pumping starts till the steady state is reached. Consequently, in an infinite, horizontal, completely confined aquifer (of constant thickness), which is pumped at a constant rate, there will always be an unsteady state. In practice, well flow is considered to be in unsteady state as long as the changes of the water level in the piezometers with time, due to pumping alone are measurable or as long as the hydraulic gradient changes in a measurable way.

Steady State Saturated Flow

Consider a unit volume of porous media (fig 2). Such an element is usually called an elemental control volume or representative elemental volume (REV). The law of conservation of mass for steady state flow through a saturated porous medium requires that the rate of fluid mass flow into any elemental control volume (through the three pairs of faces) be equal to the rate of fluid mass flow out of the REV.

The equation of continuity that translates this law into mathematical form can be written from fig. 2 as under: to give the net inward flux:

The solution of the above equation is a function h(x, y, z) that describes the value of the hydraulic head h at any point in a three dimensional flow field. A solution to equation (3.4) allows us to produce a contoured equipotential map of h, and with the addition of flow lines, a flow net. For steady state, saturated flow in a 2 dimensional field of flow, say in the x-z plane, the central term of equation (3.4) would drop out & the solution would be a function h(x, z).

TRANSIENT GROUNDWATER 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.

Unsteady flow: Case of Confined flow

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

We know that, Fluid compressibility

4.      TYPES OF WELL

4.1 Dug Well: Hacking at the ground with a pick and shovel is one way to dig a well. If the ground is soft and the water table is shallow, then dug wells can work. Historically, dug wells were excavated by hand shovel to below the water table until incoming water exceeded the digger’s bailing rate. The well was lined with stones, brick, tile, or other material to prevent collapse, and was covered with a cap of wood, stone, or concrete. They cannot be dug much deeper than the water table.

4.2 Driven Well: Driven wells are still common today. They are built by driving a small-diameter pipe into soft earth, such as sand or gravel. A screen is usually attached to the bottom of the pipe to filter out sand and other particles.

4.3 Drilled Well: Most modern wells are drilled, which requires a fairly complicated and expensive drill rig. Drill rigs are often mounted on big trucks. They use rotary drill bits that chew away at the rock, percussion bits that smash the rock, or, if the ground is soft, large auger bits. Drilled wells can be drilled more than 1,000 feet deep. Often a pump is placed in the well at some depth to push the water up to the surface.

5. WELL FLOW EQUATIONS:

5.1 Confined Aquifers:

When a fully penetrating well pumps an extensive confine aquifer at a constant rate, the influence of the discharge extends outward with time. The rate of decline of head, multiplied by the storativity and summed over the area of influence, equals the discharge. The Theis equation, which was derived from the analogy between the flow of groundwater and the conduction of heat, is written as:

where s(r,t) is the drawdown in m measured in a well, Q is the constant well discharge in m3/day, T is the transmissivity of the aquifer in m2/day, W(u) is the dimensionless Theis well function, r is the distance in m from the pumped well, S is the dimensionless storativity of the aquifer, and t is the time in days since pumping started. In Figure 2, the Theis well function W(u) is plotted versus 1/u on semi-log paper. This figure shows that, for large values of 1/u, the Theis well function exhibits a straight-line segment. The Jacob method is based on this phenomenon. Cooper and Jacob (1946) showed that, for the straight-line segment, Equation (5.1) can be approximated by:

In most handbooks on this subject, the condition to use Equation (5.3) is taken as 1/u > 100. From personal experience, this limiting condition can often be relaxed to 1/u > 10.

When a confined aquifer is pumped, the cone of depression will continuously deepen and expand. Even at late pumping times, the water levels in the piezometers will never stabilize to a real steady state. Although the water levels continue to drop, the cone of depression will eventually deepen uniformly over the area influenced by the pumping. At that stage, the hydraulic gradient has become constant; this phenomenon is called pseudo-steady state. For this situation, Thiem (1906), using two or more piezometers, developed an equation, the so called Thiem–Dupuit equation, which can be written as:

Equation (5.4) can also be derived by applying Equation (5.3) to two piezometers at distances r1 and r2 at large times.

5.2 Unconfined Aquifers:

Figure 4 shows a pumped unconfined aquifer underlain by an aquiclude. There are the following basic differences between unconfined and confined aquifers when they are pumped:

(1)    an unconfined aquifer is partly dewatered during pumping, resulting in a decreasing saturated aquifer thickness, whereas a confined aquifer remains fully saturated, (2) the water produced by a well in an unconfined aquifer comes from the physical dewatering of the aquifer, whereas in a confined aquifer it comes from the expansion of the water in the aquifer due to a reduction of the water pressure, and from the compaction of the aquifer due to increased effective stresses, and (3) the flow toward a well in an unconfined aquifer has clear vertical components near the pumped well, whereas there are no such vertical flow components in a confined aquifer, provided, of course, that the well is fully penetrating.

In an unconfined aquifer, the water levels in piezometers near the well often tend to decline at a slower rate than that described by the Theis equation. Time-drawdown curves on semi-log paper therefore usually show a typical S shape: a relatively steep early-time segment, a flat intermediate segment, and a relatively steep segment again at later times, as depicted in Figure 5.     Nowadays, the widely used explanation of this S-shaped time-drawdown curve is based on the concept of delayed yield. It is caused by a time lag between the early elastic response of the aquifer and the subsequent downward movement of the water table due to gravity drainage.

During the early stage of an aquifer test a stage that may last for only a few minutes-the discharge of the pumped well is derived uniquely from the elastic storage within the aquifer. Hence, the reaction of the unconfined aquifer immediately after the start of pumping is similar to the reaction of a confined aquifer as described by the flow equation of Theis.

Only after some time does the water table start to fall and the effect of the delayed yield becomes apparent. The influence of the delayed yield is comparable to that of leakage: the average drawdown slows down with time and no longer conforms to the Theis curve. After a few minutes to a few hours of pumping, the time-drawdown curve approaches a horizontal position. The late-time segment of the time-drawdown curve may start from several minutes to several days after the start of pumping. The declining water table can now keep pace with the increase in the average drawdown. The flow in the aquifer is essentially horizontal again and as in the early pumping time the time-drawdown curve approaches the Theis curve. Jacob (1950) showed that if the drawdowns in an unconfined aquifer are small compared to the initial saturated thickness of the aquifer, the condition of horizontal flow toward the well is approximately satisfied, so that Equations (5.1) to (5.3) can also be applied to determine the physical properties. The only changes required are that the storativity S be replaced by the specific yield Sy of the unconfined aquifer, and that the transmissivity T be defined as the transmissivity of the initial saturated thickness of the aquifer. When the drawdowns in an unconfined aquifer are large compared with the aquifer’s original saturated thickness, the observed drawdowns need to be corrected before this equation can be used. Jacob (1944) proposed the following correction:

Wheresc(r,t) is the corrected drawdown in m, s(r,t) is the observed drawdown in m, and D is the saturated aquifer thickness in m prior to pumping. This correction is only needed when the maximum drawdown at the end of the test is larger than 5% of the original saturated aquifer thickness. As with confined aquifers, the cone of depression will continuously deepen and expand in unconfined aquifers. For pseudo-steady-state conditions, Equation (5.4) can also be used to describe the drawdown behaviour in two piezometers in an unconfined aquifer, provided that the observed drawdowns are corrected according to Equation (5.5).

5.3 Leaky Aquifer:

When a leaky aquifer is pumped (Figure 6), the piezometric level of the aquifer in the well is lowered. This lowering spreads radially outward as pumping continues, creating a difference in hydraulic head between the aquifer and the aquitard. Consequently, the groundwater in the aquitard will start moving vertically downward to join the water in the aquifer. The aquifer is thus partially recharged by downward percolation from the aquitard. As pumping continues, the percentage of the total discharge derived from this percolation increases. After a certain period of pumping, equilibrium will be established between the discharge rate of the pump and the recharge rate by vertical flow through the aquitard. This steady state will be maintained as long as the water table in the aquitard is kept constant.

According to Hantush and Jacob (1955), the drawdown due to pumping a leaky aquifer can be described by the following equation:

where W(u,r/L) is the dimensionless Hantush well function, L is the leakage factor or characteristic length in m, c(= D′/K′) is the hydraulic resistance of the aquitard in days, D′ is the saturated thickness of the aquitard in m, K′is the vertical hydraulic conductivity of the aquitard in m/day, and the other symbols as defined earlier.

Equation (5.6) has the same form as the Theis equation (Equation 5.1), but there are two parameters in the integral: uand r/L. Equation (5.6) approaches the Theis equation for large values of L, when the exponential term r2/(4L2y) approaches zero. In Figure 7, the Hantush well function W(u,r/L) is plotted versus 1/u on semi-log paper for an arbitrary value of r/L. This figure shows that the Hantush well function exhibits an S shape and, for large values of 1/u, a horizontal straight-line segment indicating steady-state flow. For the steady-state drawdown in a leaky aquifer, De Glee (1930, 1951) derived the following equation:

Where sm(r) is the steady-state, stabilized drawdown in m and Ko(r/L) is the dimensionless modified Bessel function of the second kind and of zero order (Hankel function). Hantush (1956,1964) noted that if r/L is small (r/L<0.05) and L >3D, Equation (5.7) can, for all practical purposes, be approximated by:

It is important to note that the flow system in a pumped leaky aquifer consists of a vertical component in the overlying aquitard and a horizontal component in the aquifer. In reality, the flow lines in the aquifer are not horizontal but curved (i.e., there are both vertical and horizontal flow components in the aquifer). The above equations can therefore only be used when the vertical-flow component in the aquifer is so small compared to the horizontal-flow component that it can be neglected. In practice, this condition is fulfilled when L > 3D.

5.4 Recovery Well-Flow Equations:

The well-flow equations describing the drawdown behaviour during the recovery period are based on the principle of superposition. Applying this principle, it is assumed that, after the pump has been shut down, the well continues to be pumped at the same discharge as before, and that an imaginary recharge, equal to the discharge, is injected into the well. The recharge and discharge thus cancel each other, resulting in an idle well as is required for the recovery period. For any of the well-flow equations presented in the previous sections, a corresponding recovery equation can be formulated.

On the basis of Equation (5.1), Theis (1935) developed his recovery method for confined aquifers. After a constant-rate pumping test, the residual drawdown during the recovery period is given by:

Where s (r,t) is the residual drawdown in m, u =r²S /(4Tt’ ), S is the dimensionless storativity of the aquifer during recovery, and t is the time in days since pumping stopped. In Figure 8, the Theis recovery well function W(u)–W(u ) is plotted versus u /u on semi-log paper. This figure shows that, for small values of u /u, the expression exhibits a straight-line segment. For this segment, Equation (5.9) can be approximated by:

Neuman (1975) showed that Equation (5.10) can also be used to describe the residual drawdown behavior in unconfined aquifers, because its delayed water table response to pumping is fully reversible (no hysteresis effects). The only changes required are that the storativity S be replaced by the specific yield Sy of the unconfined aquifer, and that the transmissivity T be defined as the transmissivity of the initial saturated thickness of the aquifer.

For leaky aquifers, the principle of superposition can also be applied. This results in the following equation:

Equation (5.11) can only be solved by numerical simulation. Hantush (1964) showed that, if pumping and recovery times are short, i.e., if t+t <(L²S)/20T or t+t <cS/20, Equation (5.9) can also be used.

6.      Geophysical Methods in Groundwater Exploration

Geophysical methods continue to show great promise for use in groundwater. Geophysics can be defined several ways. In the broadest sense, geophysics is the application of physical principles to studies of the Earth. This general definition of geophysics encompasses a wide range of disciplines, such as hydrology, meteorology, physical oceanography, seismology, tectono physics, etc. The geophysical methods employed to obtain subsurface information from surface-based measurements include resistivity, electromagnetic induction, ground-penetrating radar, magnetometer, self-potential, seismic, gravity, radioactivity, nuclear magnetic resonance, induced polarization, etc.

Geophysical methods can be classified as passive or active. There is no artificial application of energy with passive geophysical methods. On the other hand, active geophysical methods do require the artificial application of some form of energy. The three geophysical methods predominantly used for agricultural purposes are resistivity, electromagnetic induction, and ground-penetrating radar. Resistivity methods measure the electrical resistivity, or its inverse, electrical conductivity, for a bulk volume of soil directly beneath the surface. Resistivity methods basically gather data on the subsurface electric field produced by the artificial application of electric current into the ground. With the conventional resistivity method, an electrical current is supplied between two metal electrode stakes partially inserted at the ground surface, while voltage is concurrently measured between a separate pair of metal electrode stakes also inserted at the surface. The current, voltage, electrode spacing, and electrode configuration are then used to calculate a bulk soil electrical resistivity (or conductivity) value.Self-potential methods collect information on a naturally occurring electric field associated with non-artificial electric currents moving through the ground. Unlike resistivity methods, no electric power source is required. Naturally occurring electric potential gradients can arise a number of different ways, including the subsurface flow of water containing dissolved ions, spatial concentration differences of dissolved ions present in subsurface waters, and electrochemical interactions between mineral ore bodies and dissolved ions in subsurface waters. Self-potential methods are fairly simple operationally. All that is required to obtain information on a natural electric field below ground is the voltage measurement between two non-polarizing electrodes placed or inserted at the ground surface.

6.1 SP METHOD

At constant potential, the current is inversely proportional to the solution’s resistance. The measured conductance is a consequence of the solution’s salt concentration and the electrode geometry whose  effects  are  embodied  in  a  cell  constant.  The  electrical  conductance  is  a reciprocal of the resistance as shown in Equation (6.1):

ECT = k/RT………………………………. (6.1)

Where ECT is the electrical conductivity of the solution in dS m−1 at temperature T (°C), k is the cell constant, and RT is the measured resistance at temperature T. Electrolytic conductivity increases at a rate of approximately 1.9 percent per degree centigrade increase in temperature. Customarily, EC is expressed at a reference temperature of 25°C for purposes of comparison. The EC measured at a particular temperature T (°C), ECT, can be adjusted to a reference EC at 25°C, EC25, using the below equations:

EC25 = fT × ECT………………… (6.2)

where fT is a temperature conversion factor. Approximations for the temperature conversion factor are available in polynomial form.

The SP method measures natural potential found within the earth. Measurements, usually in milli-volts, are obtained from the recoding potentiometer connected to two like electrodes. One electrode is lowered in an uncased well and the other is connected to the ground surface as in Figure 9. The potential are primarily produces by the electrochemical cell formed by the electrical conductivities differences of the drilling mud and groundwater where the boundaries of permeable zones intersect a borehole. In some instance electro kinetic effects of the fluids moving through permeable formation are also responsible for SP.The potential depends on the ration of the salinities of the drilling mud to the formation water. SP resulting from electrochemical potential can be expressed by:

SP= -(64.3+0.239T) logρf/ρw…………… (6.3)

Where the ρf is the drilling fluid resistivity in ohm-m, ρw is the groundwater resistivity.

It should be noted that the resistivity of the groundwater is the reciprocal of its specific conductivities. The relation is the form:

ρw= 104/EC……………… (6.4)

Where ρw is the the groundwater resistivity in ohm-m and the EC is specific conductance in μS/cm. The TDS (total dissolved solids) is calculated by the following relation:

TDS= 0.627 * EC……………………(6.5)

Where the TDS is ppm and mg/l. and EC is μS/cm.The resistivity of the groundwater depends on ionic concentration and ionic mobility of the salt solution. The ion mobility of a sodium chloride solution is several times that of a comparable calcium carbonate solution.

A contaminant plume in the ground is typically a thermodynamic system out of equilibrium. There are several sources of self-potential signals. They are (1) diffusion potentials associated with concentration gradients, (2) redox potentials associated with a gradient in the electro activity of the electrons, and (3) streaming potential associated with ground water flow

2.2 RESISTIVITY METHOD

Electrical resistivity methods involve the measurement of the resistance to current flow across four electrodes inserted in a line on the soil surface at a specified distance between the electrodes (Figure 10). The resistance to current flow is measured between a pair of inner electrodes while electrical current is caused to flow through the soil between a pair of outer electrodes. Although two electrodes (i.e., a single current electrode and a single potential electrode) can also be used this configuration is highly unstable, and the introduction of four electrodes helped to stabilize the resistance measurement. According to Ohm’s Law, the measured resistance is directly proportional to the voltage (V) and inversely proportional to the electrical current (i):

R=V/I  ………………………………. (6.6)

Where resistance (R) is defined as one ohm (ω) of resistance that allows a current of one ampere to flow when one volt of electromotive force is applied.

The conductance (C, ω−1 or Siemens, S) is the inverse of resistance, and the ECa (dS m−1) is the inverse of the resistivity:

EC=1/ρ=1/R(1/a) =C(1/a)……………….(6.7)

When the four electrodes are equidistantly spaced in a straight line at the soil surface, the electrode configuration is referred to as the Wenner array (Figure 11). The resistivity measured with the Wenner array is shown in Equation 6.8:

ρ = 2πa∆V/i=2πaR……………………..(6.8)

And the measured EC is as shown in Equation (6.9):

EC= 1/2πaR ………………………(6.9)

Where a is the inter electrode spacing (m). There are a variety of electrode configurations, called electrode arrays, most of which are linear, with the Wenner, Schlumberger, and dipole-dipole arrays the ones employed the majority of the time. The traditional Schlumberger array is symmetric, and as shown in Figure 11, typically has the current electrodes on the outside of the array and the potential electrodes placed within the arrays interior. The spacing between the current electrodes is by a large factor greater than the spacing between the potential electrodes. Being able in practice to move the outer current electrodes further apart, while potential electrode positions are kept constant, makes the Schlumberger array one of the best available for determining variations of resistivity with depth.

Where a is a distance value, and n is a factor by which that distance is multiplied.

The dipole-dipole array (Figure 12) is normally configured to have a relatively large separation between the pair of current electrodes on one side of the array and the pair of potential electrodes on the other side of the array. The dipole-dipole array is employed both for mapping lateral changes in ρa and for assessing the variation of resistivity with depth. When using this array for areal mapping of ρa, the spacing between electrodes remains constant as the array is moved from one location to another, but for measuring resistivity changes with depth, the array midpoint stays at the same location and the current electrode pair and the potential electrode pair are moved further apart. The apparent resistivity for the dipole-dipole array (Figure 12) is expressed as follows:

ρ= ∆V/I π(n)(n+1)(n+2)a……………….(6.11)

Organic and inorganic contaminants can be the origin of groundwater pollution within the surrounding area of waste deposits and abandoned industrial sites. Geophysical mapping strategies of contaminant plumes depend on the contaminant-groundwater interaction: inorganic contaminants dissolve, whereas organic contaminants are in general not dissolvable and so do not affect the physical properties of the groundwater. Abandoned industrial sites are mainly characterized by remnants of production facilities (pipelines, tanks, cables, building foundations) and contaminated soil. If geoelectrical or electromagnetic measurements are possible without perturbation by power lines or metallic objects, resistivity mapping can show an overview of the previous installations of that place. Soil contamination, e.g. through mineral oil or slags, is common on previous industrial sites. Considered here are only the contaminations in the vadose zone above the groundwater table, whereas the groundwater contaminations will be treated in the next paragraph. Hydrocarbons such as diesel fuel normally have low conductivities. However, long term hydrocarbon pollution in the vadose zone can undergo biodegradation processes under aerobic conditions resulting in high conductivities or low resistivity’s due to an increase of cation exchange capacity.

Summary:

After going through this module the students learn about:

Well hydraulics and well flow equations

Types of wells

Aquifers testing

Geophysical methods in ground water exploration

References

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