Estimation of Hydraulic Properties of Aquifers

  • B. B. S. Singhal
  • R. P. Gupta
Chapter

Abstract

Hydraulic properties of rock materials can be estimated by several techniques in the laboratory and in the field. The values obtained in the laboratory are not truly representative of the formation. However, the advantage of laboratory methods is that they are much less expensive and less time consuming.

Keywords

Hydraulic Conductivity Observation Well Type Curve Unconfined Aquifer Slug Test 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hydraulic properties of rock materials can be estimated by several techniques in the laboratory and in the field. The values obtained in the laboratory are not truly representative of the formation. However, the advantage of laboratory methods is that they are much less expensive and less time consuming.

9.1 Laboratory Methods

The laboratory techniques are based on (a) indirect, and (b) direct methods.

9.1.1 Indirect Methods

In unconsolidated material, hydraulic conductivity can be determined from grain size analysis. The hydraulic conductivity of unconsolidated material is found to be related empirically to grain-size distribution by a number of investigators (e.g. Hazen 1893; Krumbein and Monk 1942; and Uma et al. 1989, among others). Hazen, as far back as 1893 developed the empirical relation (Eq. 9.1) between hydraulic conductivity (K) and effective diameter (de)

$$ K = Cd_e^2 $$
(9.1)

where C is a coefficient based on degree of sorting (uniformity coefficient) and packing. If K is in centimetre per second and de is in cm, the value of C in Eq. 9.1 ranges between 45 in very fine poorly sorted sand to 150 in coarse well sorted sand; a value of C = 100 is used as an average.

Effective diameter, (de), is the diameter of the sand grain (d10) such that 10% of the material is of smaller size and 90% is of larger size. It can be estimated by plotting the grain size distribution curve. Uniformity coefficient being the ratio of d60 to d10 is a measure of the degree of sorting (Fig. 9.1).
Fig. 9.1

Grain size distribution curve of unconsolidated material based on sieve analysis data

On comparison of values of K obtained from pumping tests and Hazen’s formula, Uma et al. (1989) showed that Hazen’s formula gave consistently higher values of K which is perhaps due to the reason that degree of compactness is not considered in the Hazen’s formula. They suggested that for unconsolidated and poorly cemented sandy material, value of C in Hazen’s formula is 6.0, for moderately cemented sandstones 3.8, and for well compacted and cemented sandstones its value is 2.0.

One of the well known equations for determining hydraulic conductivity by indirect method is Kozeny–Carman equation (Lee and Farmer 1993), which has the form

$$ K = \frac{1}{{{C_T}{C_0}S_{SP}^2}}\frac{{{e^3}}}{{1 + e}}\frac{\gamma }{\mu } $$
(9.2)

where CT is the pore tortuosity factor, and C0 is a pore shape factor; Ssp, specific surface area, and e, void ratio, were defined earlier in Chap. 8.

Values of hydraulic conductivity of unconsolidated sands obtained from grain size analysis, slug test, pumping test and numerical modelling show variations of one to two orders of magnitude (Cheong et al. 2008).

9.1.2 Direct Method

In the laboratory, hydraulic conductivity can be measured directly in three ways: (a) steady or quasi-steady flow techniques, (b) hydraulic transient flow tests, and (c) mechanical transient flow tests. The last two methods which analyse the time dependent behaviour can be used for the estimation of both hydraulic conductivity and specific storage. Steady state flow techniques are applicable in rocks of high conductivity, viz. sand, sandstone etc. while transient flow tests are recommended in low permeability tight formations like crystalline rocks, carbonate rocks and shales. Mechanical flow tests are generally used in compressible media such as clays.

Steady or Quasi-steady Flow Techniques

These involve use of various types of permeameters—constant and variable head types. In these methods the rate of fluid flow through the specimen and the hydraulic gradient across the specimen are measured. Darcy’s Law can be used to calculate hydraulic conductivity. The description of these methods is given in many texts on groundwater hydrology, e.g. Todd (1980), and Fetter (1988).

When the conductivities are very small (of the order of 10 10m s 1), steady state flow can be achieved by taking small lengths of samples in the flow direction and using large hydraulic gradients (Neuzil 1986). Falling (variable) head permeameters are also used for the estimation of hydraulic conductivity of both coarse grained and tight formations. Some researchers have used a closed reservoir instead of an open stand pipe which is advantageous for clayey soil (Neuzil 1986).

In low permeability formations, e.g. hard rocks, use of gas permeants can be advantageous. Gas permeameters are more commonly used in oil industry. Compressed air is a satisfactory fluid in most cases. Other gases such as nitrogen, oxygen, hydrogen, helium and carbon dioxide have also been used in special cases. The advantage of using gas is that they have low viscosity and they do not react significantly with the rock material in the dry state. Therefore, problems of swelling of clay minerals, bacterial growth and other chemical changes which can considerably affect permeability, are avoided.

Hydraulic Transient Flow Test

The transient groundwater flow equation in one dimension can be written as

$$ K = \frac{{{\partial ^2}h}}{{\partial {z^2}}} = {S_S}\frac{{\partial h}}{{\partial t}} $$
(9.3)

Transient tests are used for estimating very low permeabilities (10 10 to 10 17 m s 1). Neuzil (1986) performed these experiments on shale specimens for estimating permeability, k and specific storage, Ss. In most of the cases a constant lateral and longitudinal load on the specimen was applied. In order to study the effect of rock deformation on permeability and specific storage, experiments based on transient pressure pulse method were designed by Read et al. (1989). Trimmer et al. (1980) used transient technique for determining permeabilities of intact and fractured granites and gabbros in the laboratory under high confining and pore-water pressures to have an understanding of fluid flow behaviour in igneous rocks at large depths from the point of view of their suitability as host rock for radioactive waste disposal.

Mechanical Transient Flow Test

Permeability and specific storage can also be determined from transient mechanical behaviour of the specimen due to drainage of pore water as a result of loading. In soil mechanics, such a type of test (consolidation test) is used for low permeability compressible media. The values of permeability obtained by consolidation test and hydraulic test vary considerably in the case of highly deformable media such as clays.

Scale Effects

The values of permeability determined in the laboratory from core samples especially of fractured rocks, are usually several orders of magnitude lower than those existing under natural conditions due to the smaller size of the sample and in-situ formation heterogeneity (Clauser 1992; Rovey and Cherkauer 1995, Sanchez-Vila et al. 1996, Ilman and Neuman 2003; Shapiro 2003). In this context a reference may be made to Fig. 13.12. The main factors which influence permeability values determined in the laboratory on core samples are:
  1. 1.

    Core (specimen) length: If the fracture spacing is more than the core length, the measured permeability will be representative of the matrix only.

     
  2. 2.

    Fracture orientation and connectivity: The permeability estimates are considerably influenced by the orientation of fractures in relation to flow direction. The radial flow to a well during field test will be quite different from the linear flow through the sample examined in the laboratory.

     
  3. 3.

    Aperture size: As permeability is dependent on aperture size, an estimate from core samples obtained from deeply buried rocks will not be representative of in-situ condition.

     
  4. 4.

    Duration of testing: The values of permeability are found to decrease with time during extended testing in the laboratory. This is attributed to clogging of pore spaces by finer particles, swelling of clay minerals and other chemical reactions between permeant and pore fluids.

     

Therefore, as compared to laboratory methods, the field or in-situ methods provide a better estimate of hydraulic characteristics of rock formations as a larger volume of the material is tested.

In a recent study, Shapiro (2003) has shown that although the hydraulic conductivity measured from borehole tests in individual fractures varies over more than six orders of magnitude (10 10–10 4 m s 1), the magnitude of the bulk hydraulic conductivity of the rock mass was the same from aquifer tests over 10s of meters and kilometrer-scale estimates inferred from groundwater modelling. In contrast, the magnitude of the formation properties controlling chemical migration viz., dispersivity and matrix diffusion increases from laboratory size tests to field tests on kilometre scale. A reference may be made to the concept of REV (representative elementary volume) which has been discussed in Sect. 8.4.

9.2 Field Methods

Table 9.1 gives a list of commonly used field methods. The choice of a particular method depends on the purpose of study and scale of investigations. For small scale problems, as in geotechnical investigations, seepage of water to mines and contaminant transport problems, especially in fractured rocks, packer tests, slug tests, cross-hole tests and tracer tests are preferable. In case of groundwater development and management on a regional scale, pumping test methods should be preferred. The choice is also governed by practical limitations or expediencies. The applicability of these methods in geothermal reservoirs has been discussed in Chap. 18.
Table 9.1

Field rest methods for the estimation of hydraulic characteristics of aquifers. (Modified after UNESCO 1984a)

Purpose of investigation

Size of area under investigation

Distribution of fractures

Test method

Geotechnical investigations, mine drainage, waste disposal, etc.

A few square kilometres

Random

Systematic fractures of 1, 2 or 3 sets

Packer (Lugeon) test; slug test, tracer injection test

Modified packer test; crosshole hydraulic test; tracer injection test

Groundwater development; water resources investigation

>100 km2

Random and closely interconnected

Pumping test

Geothermal and petroleum reservoirs

A few square kilometres

Random

Well interference test; tracer injection test

9.2.1 Packer Tests

A packer test, also known as injection test, is used in uncased borehole to determine the hydraulic conductivity of individual horizon by isolating it with the help of pakcers (Fig. 9.2). This method is widely used for estimating the hydraulic characteristics of fractured rocks in various geotechnical and waste disposal investigations (Louis 1974; Black 1987; Levens et al. 1994, Kresic 2007; Novakowski et al. 2007, among others). In a packer test water is injected under pressure at a constant hydraulic head in an isolated portion of the borehole measuring the flow rate at a steady-state condition. Generally a two-packer system having a single isolated zone is adequate for testing moderately fractured rocks (Fig. 9.3). The borehole should be flushed before hand to remove coatings from the wall of the borehole in order to get reliable results. The following three types of packer test could be used in fractured rocks, depending on details of the information required:
Fig. 9.2

Standard Lugeon test with single packer

Fig. 9.3

Influence of direction of fractures on the flow during Lugeon tests a fractures parallel to the borehole, b fractures not parallel to the borehole

  1. 1.

    Standard Lugeon test, which gives average hydraulic conductivity.

     
  2. 2.

    Modified Lugeon test, which gives directional hydraulic conductivity on the basis of relative orientation of the test hole to the system of fractures, and

     
  3. 3.

    Cross-hole hydraulic test, described under Sect. 9.2.5.

     

Standard Lugeon Test

The Lugeon method of testing was introduced by Maurice Lugeon, a French engineer, mainly for rock grouting in geotechnical works. It is relatively a low cost method especially for determining variations in hydraulic conductivity with depth and also in different strata. The test is made either in a completed borehole, or as the hole advances during drilling.

Lugeon test can be carried out by using either one or two packers (Figs. 9.2, 9.3). In the single packer method, the packer is placed at some selected distance above the bottom of the hole. After the test is over, drilling can further be resumed and the test can be repeated in deeper horizons. The single packer method is recommended when the rock mass is weak and intensely jointed and there are chances of the hole to collapse. In a completed borehole, two packers can be used to isolate the required section (3–6 m long) of the hole from the rest of it (Fig. 9.3) Tests can be carried out to depths up to 300 m.

Water is injected under pressure into the test section with increasing pressure from 0 to 10 bars (0–1 MPa) and then it is decreased from 1 to 0 MPa, in fixed steps at prescribed time intervals. However, testing at pressures as high as 1 MPa is questionable in estimating permeability, as such high pressures are likely to increase the permeability locally by inducing new fractures and widening the existing ones. It also enhances the possibilities of turbulence. The time interval is commonly 15 min. The flow rate of water in the borehole is measured under a range of constant pressures.

The flow pattern around the test zone depends on the orientation of the fractures in relation to the axis of the borehole (Fig. 9.3). The flow rates will be very high when the intercepted fracture is parallel to the borehole.

In a test where the flow is cylindrical and an observation well (piezometer) is used (Fig. 9.4), hydraulic conductivity (K) can be determined from Eq. 9.4
Fig. 9.4

Standard Lugeon test—experimental setup and definitions necessary to interpret test results

$$ K = \frac{{{Q/L}}}{{2\pi ({{h_0} - h})}}\,\,ln \frac{r}{{{r_w}}} $$
(9.4)

where, K is hydraulic conductivity perpendicular to the axis of the bohrehole, in (m s 1), Q is rate of inflow, in (m3 s 1), L is thickness of the test zone, in (m), h0 and h are the piezometric heads in (m) measured in the test well and at distance (r) in the observation well, and rw is radius of the test borehole, in (m).

Specific permeability (q) which is also a measure of rock permeability can be obtained from Lugeon test. It is defined as the volume of water injected into the borehole per 1 m length during 1 min under a pressure of 1 m of water (0.1 atm)

$$q= \frac{Q}{{H l}}\; \left( {L\;\!mi{n^{ - 1}}\;\!{m^{ -1}}\; at\; 0.1\; atm} \right)$$

where Q is the injection rate, H is the effective injection pressure, and l is the length of the test interval.

In order to reduce the cost of the test, very often piezometers are not installed. For such conditions, Eq. 9.4 becomes

$$ K = \frac{{{Q/L}}}{{2\pi \Delta h}}\,ln \frac{R}{{{r_w}}} $$
(9.5)

where, Dh is the hydraulic head, in (m) in the test well causing flow and R is the radius of influence i.e. distance (m) at which initial water-level conditions do not change due to injection. Errors in the evaluation of R do not affect test results very much as R >> rw. Therefore,

$$ \frac{{ln ( {{R/{{r_w}}}})}}{{2\pi }} = constant $$
(9.6)

As, lnR/rw does not vary much, it can be assumed that (UNESCO 1984b)

$$ ln ( {{R/{{r_w}}}})\, \cong \,7.$$
(9.7)

Therefore, for numerical computations, Eq. 9.5 can be rewritten as

$$ K = 1.12 \times \frac{{{Q/L}}}{{\Delta h}} $$
(9.8)

where units are the same as in Eq. 9.4.

If it is assumed that only one fracture intersects the test section, an equivalent single fracture aperture, aeq, can be determined from the test results using Cubic law (Novakowski et al. 2007).

$$ {a_{eq}} = {\left[ {T.\frac{{12\mu }}{{\rho g}}} \right]^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} $$
(9.9)

The vertical distribution of T and aeq can be determined by using packers at the desired levels.

Some of the important rules which should be followed in conducting a packer test are:
  1. 1.

    The test should be carried out in a saturated zone.

     
  2. 2.

    Each test at each pressure is continued until steady state conditions are reached.

     
  3. 3.

    The test may be repeated with two or three increasing steps of pressure but care should be taken that it may not cause fracture-dilation or hydro-fracturing.

     
  4. 4.

    Pressure is measured in piezometres around the test hole.

     

Hydraulic conductivity, can be expressed in Lugeon unit (Lu). Lugeon is defined as the rate of flow of water per minute under a pressure of water injection of 1 MPa m 1 length of the tested material, i.e.

$$ 1\;\!{\rm{L}}_u = 1\,{\rm{l}}\;\!{\rm{min}}^{-1}\;\!{\rm{m}}^{-1}\quad {\rm{under}}\quad1\;\!{\rm{MP}}_{\rm{a}} $$

or

$$ 1\;\!{{\rm{L}}_{\rm{u}}} = 1 \times {10^{ - 7}}{\rm{m}}\;\!{{\rm{s}}^{ - 1}}\,({\rm{approx}}.). $$

It is generally assumed that the test section intercepts a number of fractures. Therefore, the estimated value of K represents an average value of hydraulic conductivity of the rock mass in the plane perpendicular to the borehole. The conductivity of individual fracture, Kf can be expressed by

$$ {K_f} = \frac{K}{N} $$
(9.10)

where N is the total number of fractures in the test section as determined from borehole core logs. It is assumed that fracture apertures are constant.

A comparison of transmissivity values determined from packer test and those obtained from pumping test indicates that although the two values are correlatable, the values obtained from pumping test tend to be higher (Fig. 9.5). This appears to be due to the reason that in a packer test only the properties in the immediate vicinity of the borehole are reflected but pumping test covers a larger volume of rock mass around the borehole. Therefore, pumping tests are preferred to packer tests in oder to obtain representative values of hydraulic parameters of rocks for the purpose of groundwater development.
Fig. 9.5

Comparison of transmissivity values obtained from Lugeon tests and pumping tests. (After Karundu 1993)

In a standard packer test, the flow rate (Q/L) is plotted against applied pressure (P) as a flow–pressure curve. The flow–pressure curve will show different characteristics, depending on the permeability of the formation and changes brought about by water injection during the test (Fig. 9.6). A typical flow–pressure curve shows a lower flow rate during the increase in pressure as compared with values during the fall in pressure (Fig. 9.6a). A similar pattern (Fig. 9.6b) is also expected from the cleaning of the existing fractures or development of new fractures due to hydraulic fracturing during the test. A reverse situation might also be observed due to the clogging of the fractures during the test (Fig. 9.6c).
Fig. 9.6

Characteristic flow–pressure curves for packer (Lugeon) test: a normal type with reversible cycle, b cleaning of existing fractures or development of new fractures, c reverse type due to the clogging of fractures

Modified Lugeon Test

In the standard Lugeon test, the borehole is vertical irrespective of the position and orientation of fractures in the rock mass. This is the only alternative when fractures are randomly disttributed and are very large. Under such conditions the rock mass can be considered continuous. When there are well defined fracture sets as in Fig. 9.7 (F1, F2 and F3), it is necessary to determine the hydraulic conductivities of each fracture set separately, i.e. K1, K2 and K3 especially in geotechnical and contaminant transport problems. This can be achieved by having separate test holes for separate fracture sets; keeping the orientation of borehole perpendicular to the considered fracture set.
Fig. 9.7

a Modified Lugeon test in rock mass with three sets of fractures. Borehole (1) is parallel to F 2 and F 3 for testing the fracture set F 1

A triple hydraulic probe to measure the directional permeabilities was proposed by Louis (1974) In this method records are taken in three appropriate directions and measurements are made at different pressures. In order to measure the hydraulic heads during the test in the vicinity of the testing area, piezometers are installed. Four piezometric measurements enable to determine the three dimensional distribution of the anisotropic permeabilities.

Some of the problems in conventional packer test are (Louis 1974).
  1. 1.

    Turbulence effect

     
  2. 2.

    Deformation of the medium due to high injection pressure

     
  3. 3.

    Influence of other fracture sets, i.e. effect of F2 and F3 when test hole is perpendicular to F1

     
  4. 4.

    Entrance losses.

     

A Multi-function Bedrock Aquifer Transportable Testing Tool (BAT3) has been developed at the USGS for carrying out a single-hole hydraulic test in fractured rocks by withdrawing/injecting water. It is also used to identify hydraulic head as a function of depth and to collect discrete-interval groundwater samples for chemical analysis. The water should be withdrawn in case of highly transmissive fractures and injected when fractures are less transmissive. BAT3 can estimate transmissivity ranging over approximately eight order of magnitude (Kresic 2007).

9.2.2 Slug Test

A slug test involves sudden injection of a known volume (or slug) of water into a well and measurement of fall of water-level with time. Alternatively a known volume of water can be withdrawn and rise of water-level is noted.

The advantage of slug test is that it is cheap, as it requires less equipment and manpower and the duration of the test is also relatively short. Observation wells are also not required. It can be carried out even if nearby wells are being used. Contaminated aquifers can also be tested as there will be no problem of extraction and disposal of contaminated water.

Slug tests can also provide reliable values of T in fractured rocks in spite of rock heterogeneties. Discrete fractures, if suitably located by geophysical or other methods can be tested by using packers (Butler 1998). Slug tests have been also performed for site selection of nuclear waste repositories suspected to behave as a double-porosity medium (Butler 1998). Values of hydraulic conductivity determined from slug tests, fluid injection tests and pumping tests in fractured crystalline rocks are quite comparable (Shapiro and Hsieh 1998; Nativ et al. 2003). However, as slug tests stress only a small volume of the aquifer, this method may not be used to interpret formation heterogeneity or large-scale formation properties. Although slug tests provide reliable values of T but estimates of S may have dubious values (Beckie and Harvey 2002). Therefore, slug test cannot replace the conventional pumping test as the latter have several additional advantages.

The duration of the slug test depends on the permeability of rocks—it has to be larger in low permeability media. The well radius also influences the duration of the slug test. In low permeability formations, it is necessary to have boreholes or standpipes of small diameter in order to reduce the duration of the test.

The major factors affecting test duration for a partially penetrating well is given as (Butler 1998)

$$ \beta= \frac{{{K_f}bt}}{{r_c^2}} $$
(9.11)

where b is dimensionless time factor, Kf is radial component of the hydraulic conductivity, b is effective screen length, t is the total time since the start of the test, and rc is effective casing radius. Equation 9.11 indicates that the test duration (t) is inversely proportional to the effective screen length (b) and directly proportional to the square of the effective casing radius, rc.

Since the hydraulic conductivity calculated from a slug test is an estimate for a small part of the aquifer around the well, skin effect (especially skin with low permeability) will greatly influence the computed hydraulic conductivity. Both field slug test results and numerical modelling demonstrate that well bore skin significantly affects water-level recovery and drawdown in low permeability geologic formations. Therefore, only properly developed wells will provide reliable estimates of aquifer characteristics from slug tests. Wellbore skin effect can be minimized by using the proper segment of the time-drawdown curve. However, there is no general agreement on which segment will have no influence from the wellbore skin effect. Values of permeability estimated from slug tests are generally low as compared with those obtained from pumping tests which is attributed to incomplete well development and skin effect (Butler and Healey 1998).

The commonly used methods of analysing slug test data for confined aquifer is given by Cooper et al. (1967) described below. Bredehoeft and Papadopulos method which is especially suited to low permeability rocks is also given in this text. Readers may refer to Kruseman and de Ridder (1990) and Butler (1998) for other methods.

Cooper’s Method

Cooper et al. (1967) developed a curve-matching method for slug test in fully penetrating wells in confined aquifers. This method does not require any pumping or observation well. The sudden injection of a known volume of water, V in a well of diameter 2rw will cause an immediate rise of water-level, h0 in the well (Fig. 9.8) which can be expressed as
Fig. 9.8

Slug test in a fully penetrating well of finite diameter in a confined aquifer

$$ {h_0} = \frac{V}{{\pi r_c^2}} $$
(9.12)

The change in water-level with time is given by Eq. 9.13.

$$ \frac{{{h_t}}}{{{h_0}}} = F\!\left( {\alpha ,\beta } \right) $$
(9.13)

where,

$$ \alpha = \frac{{r_w^2\,S}}{{r_c^2}} $$
(9.14)
$$ \beta = \frac{{Tt}}{{r_c^2}} $$
(9.15)
h0, ht, rc and rw are as shown in Fig. 9.8.
Fig. 9.9

Family of Cooper’s type curves. (After Papadopulos et al. 1973)

Values of function F(a, b) against a for different values of a and b are plotted on semi-log paper as family of type curves (Fig. 9.9). The value of h0 is calculated from Eq. 9.12 based on the volume of water injected into or withdrawn from the well. Ratio of ht/h0 is calculated for different values of t.

A field data curve (log ht/h0 against corresponding values of t) is superimposed over the family of Cooper’s type curves (Fig. 9.9) to find the best match of the field data plot with one of the type curves. The value of b for the corresponding type curve is noted. The value of t for corresponding value of b = 1 is read from the data curve. T is calculated by substituting values of t, rc and b = 1 in Eq. 9.15. The value of S can be determined from the value of a of the type curve with which data curve is matched using Eq. 9.14.
Fig. 9.10

Comparison of cones of depression formed due to pumping in a confined aquifer of a high transmissivity, and b low transmissivity

The type curves especially for small values of a are quite similar in shape and therefore it is difficult to select the type curve for a unique match. This may result in large errors in calculating S. However, T can be determined with greater accuracy but it will be representative of the formation in the immediate vicinity of the well.

Bredehoeft and Papadopulos Method

A modified slug test, i.e. pressure pulse method was suggested by Bredehoeft and Papadopulos (1980) in which the test well is filled with water to the surface, and suddenly pressurized with an additional amount of water. The water is then shut-in and the fall in pressure is noted with time. Packers are used to isolate the intake part of the open hole. This method is advantageous in reducing the time of test which is the main problem in tight formations. Bredehoeft and Papadopulos (1980) showed that a formation of T = 10 11 m2s 1 and S = 4 ´ 10 4, can be tested adequately in a few hours using the pressure pulse method, whereas years may be required by the conventional slug—testing methodology. Therefore Bredehoeft and Papadopulos method is feasible for in-situ estimation of hydraulic properties of tight formations, viz. shales and massive crystalline rocks having low hydraulic conductivities of the order of 10 13–10 16 m s 1. Bredehoeft and Papadopulos also argued that for a > 0.1, type curve of F(a, b) vs. the product, ab are more suitable for analysing test data than the Cooper’s type curves (Fig. 9.9) where F(a, b) is plotted against b only. The same procedure of matching the field data curve (ht/h0 vs. t) plotted on semi-logarithmic paper with the type curve F(a, b) against ab can be adopted for estimating either T and S separately or only the product TS.

This method is based on the assumptions, among others, that (1) the flow in the tested interval is radial, and (2) the hydraulic properties of the tested interval remain constant throughout the test. In many cases, the first assumption holds good as the ratio of Kh/Kv is quite high. In order to satisfy the second assumption, especially in the fractured rocks, the fluid pressure in the tested interval should be kept low to avoid the formation of new fractures. This can be taken care of by initial pulses of 1–10 m (10–100 kPa).

Neuzil (1982) argued that the Bredehoeft–Papadopulos method does not assure the approximate equilibrium condition i.e. equal hydraulic head in the well and the formation at the beginning of the test. Therefore, the test data will not give correct values of K. To avoid this problem, Neuzil (1982) suggested a modified test procedure to ensure near-equilibrium condition at the start of the test and also the use of two packers and two transducers to monitor pressures both below the lower packer and between the packers to detect leakage.

Air-pressurized slug tests are carried out by pressurizing the air in the casing above the water-level in a well and the declining water-level in the well is noted. Afterwards the air pressure is suddenly released to monitor the rise in water-level. T and S can be estimated from the rising water-level data using type curves (Kresic 2007). Such tests are known as ‚prematurely terminated air pressurized slug tests‘.

The use of nitrogen gas, instead of water, as permeant is suggested by Kloska et al. (1989) for determining permeability of unsaturated tight formations. This is advantageous in unsaturated or less saturated low permeability rocks as nitrogen will not change the rock properties.

9.2.3 Pumping Tests

Introduction

In a pumping test, a well is pumped at a known constant or variable rate. As a result of pumping, water-level is lowered and a cone of water-table depression in an unconfined aquifer and cone of pressure relief in a confined aquifer is formed. The difference between the static (non-pumping) and pumping water-level is known as drawdown. As pumping advances, cone of depression expands until equilibrium conditions are established when the rate of inflow of water from the aquifer into the well equals the rate of pumping. The distance from the centre of the pumped well to zero drawdown point is termed as radius of cone of depression (R). In an ideal uniform, isotropic and homogeneous aquifer, the cone of depression will be symmetrical and the contours of equal drawdown will be circular or near-circular. In contrast, in a fractured aquifer, due to anisotropy, the plotted drawdown cone, based on data from a number of observation wells, will be linear, highly elongated or irregular; the longer axis will be parallel to the strike of water conducting fractures. The slope of the cone of depression and its radius (R) depend on the type of aquifer and its hydraulic characteristics. In aquifers of high transmissivity, the gradient of cone of depression is less and its radius is large, as compared with aquifers of low transmissivity (Fig. 9.10).

The speed of propagation of the cone is inversely proportional to storativity, S. In an unconfined aquifer as the storativity is high, being of the order of 10 1–10 2, corresponding to the specific yield, the spread of cone is very slow and therefore it would take a long time to reach the edges of the aquifer. In contrast to this, in confined aquifers the storativity is very small, of the order of 10 3–10 6. Hence the cone extends very rapidly in confined aquifer than in unconfined aquifer (Table 9.2).
Table 9.2

Spread of the cone of depression in metres. (After Schoeller 1959) (T = 1.25  10 3m2 s 1)

 

1 min

1 h

1 day

10 days

100 days

Unconfined

Aquifer (S = 0.2)

0.91

7.11

34.8

110

348

Confined aquifer

(S = 1 ´ 10 4)

41

318

1558

4930

15 580

In case of heterogeneous aquifers, multiple and sequential pumping tests are useful to account for spatial variation in T and S (see Sect. 9.2.6).

Planning a Pumping Test

Selection of Site

Pumping tests are expensive and therefore should be carefully planned. The cost depends on number of observation wells and duration of the test. Before conducting a pumping test, it is necessary to know the geological and hydrological conditions at the test site. Subsurface lithology, and aquifer geometry are of help in the proper interpretation of pumping test data. Existing wells in the area can provide important information about the sub-surface lithology. A preliminary estimate of the transmissivity of the aquifer can be made from subsurface lithology and aquifer thickness. At places, where subsurface geological information is not directly available, geophysical methods can be used to ascertain the lithology (Chap. 5). The presence of hydrological boundaries in the form of rivers, canals, lakes or rock discontinuities (faults and dykes etc.) should also be noted. In the absence of this information, the test data are liable to be interpreted in different ways, viz. the effect of a recharge boundary on the time-drawdown curve may almost be the same as for a leaky aquifer or an unconfined aquifer. This aspect is further elaborated in the latter part of this section.

The test site for a confined aquifer should not be close to railway track or highway to avoid effect of loading which produces water-level fluctuations. The site should be also away from existing discharging wells to avoid well interference.

Design of Pumping and Observation Wells

The well field for a pumping test consists of one pumping well and one or more observation wells. The pumping well should be at least of about 150 mm diameter so that a submersible electric pump can be installed. The diameter of the observation wells should be able to accommodate an electric depth sounder or some other water-level measuring device, viz. automatic water-level recorder.

In order to avoid entry of pumped water into the aquifer, it should be conveyed by pipes or lined drains to a distance of 200–300 m away from the discharging site.

The pumping well should tap the complete thickness of the aquifer to avoid effect of partial penetration. In thick aquifers where partial penetration cannot be avoided necessary correction in the drawdown data should be made. Observation well(s) should have the same depth and tap the same aquifer(s) as the pumping well. However, in some cases, where the interconnection between aquifers due to leakage is suspected some observation wells should also tap the overlying and underlying aquifers (Fig. 9.11a). This is also required in fractured rock aquifers to know the vertical component of flow (Fig. 9.11b).
Fig. 9.11

Layout of pumping well (PW) and observation wells (OW) in: a multilayered aquifer system, and b fractured aquifer

The length and position of the screen in the pumping well is important as this will affect the amount of drawdown. Partial penetration causes lengthening of flow lines and thereby excessive drawdown (Fig. 9.12). In unconfined aquifers, as pumping causes dewatering of the aquifer, the well screen should be put in the lower one-half or one-third of the aquifer where the flow (stream) lines will be mainly horizontal, otherwise the observation well may be located at distances one and a half times the saturated thickness of the aquifer (Fig. 9.12a). The desirable position of screen in a confined aquifer is shown in Figs. 9.12c, d und 9.13. The number of observation wells depends on the purpose of study and financial considerations. It is desirable to have at least one observation well but three observation wells at different distances from the pumping well are better. This will help in analysing the test data by both the time-drawdown and distance-drawdown methods and identify the boundary conditions. In order to assess the anisotropy and heteogeneity of the aquifer, it is necessary to have additional observation wells in other direction also. Some of the existing wells can also be used as observation wells to minimize the cost.
Fig. 9.12

Positioning of screens in observation wells, a fully penetrating pumping well in an unconfined aquifer, b partially penetrating pumping well in an unconfined aquifer, c fully penetrating pumping well in a confined aquifer, and d partially penetrating pumping well in a confined aquifer

An observation well located upstream from pumping well will provide information of storativity both for the upstream and downstream side but it cannot provide much information about the transmissivity downstream (Jiao and Zheng 1995).

Well Spacing
The spacing of observation wells from the pumping well depends on the well penetration, type of the aquifer, its hydraulic characteristics and rate and duration of pumping (Table 9.3). In unconfined aquifer, as the radius of cone of depression is comparatively small, observation well should be within a distance of 100 m from the pumping well. In a confined aquifer, this distance could be upto 200 or 250 m. For semi-confined aquifers, intermediate values may be taken. Transmissivity also influences the shape and size of the cone of depression as described earlier. In aquifers with high transmissivity, observation wells can be put farther away from the pumping well than in aquifers with low transmissivity. One observation well should be selected which is outside the cone of depression of the well field. This helps to assess the effect of atmospheric or other natural causes of water-level fluctuations so that necessary correction in the drawdown data may be made.
Table 9.3

Suggested spacing of the observation wells. (After Hamill and Bell 1986)

Aquifer and well condition

Minimum distance from pumped well to nearest observation well

General distance within which observation wells should be located

Fully penetrating well in unstratified confined or unconfined aquifer

1–1.5 times aquifer thickness

20–200 m in confined aquifer, 20–100 m in unconfined aquifer

Fully penetrating well in a thick or stratified confined or unconfined aquifer

3–5 times aquifer thickness

100–300 m in confined aquifer, 50–100 m in unconfined aquifer

Partially penetrating well (<85% open hole) in a confined or unconfined aquifer

1.5–2 times aquifer thickness

35–200 m in confined aquifer, 35–100 m in unconfined aquifer

In fractured rocks, study of structural data viz., orientation, spacing and interconnectivity of fractures is important in planning a pumping test (observation well network and pumping test duration etc.).

Installation of observation wells in fractured rocks depends on whether the fractures are discrete or interconnected and also the permeabilities of rocks on the two sides of the fractures (matrix blocks) and whether fractures are open or filled with impervious material. Multiple observation wells are preferred tapping both the pumped intervals comprised of highly transmissive fractures and the intervening less transmissive horizons. In heterogeneous aquifers, a piezometer nest is recommended for recording of water-levels at different depth horizons.

Duration of Test

The duration of pumping test depends on the type of the aquifer and the degree of accuracy required in estimating the aquifer properties. Usually the test is continued till the water-level is stabilised so that both the non-steady and steady-state methods of analysis could be used for computing aquifer parameters. The duration of constant rate pumping test depends on the type of aquifer, viz. in confined and leaky aquifers it could be about 24 h and 20 h respectively but in unconfined aquifers, as it takes longer time to reach steady state flow condition, the duration of test may be about 72 h (Hamill and Bell 1986; Kruseman and De Ridder 1990). In fractured rocks of low hydraulic conductivity, pumping test can be only of shorter duration (10–15 h).

Measurements

Two sets of measurements are taken during a pumping test: (1) groundwater levels (both during drawdown and recovery phases) in pumping and observation wells, and (2) well discharge. Water-level measurements can be made by using steel tape or electrical depth sounder. The automatic water-level recorders are better for obtaining continuous water-level records in observation wells. Fully automatic microcomputer—controlled systems are also now available for accurate recordings of water-levels.

As drawdown is faster in the observation wells close to the pumped well during early part of the test, water-level observations should be made at an interval of 1/2–1 min for the first 10 min of pumping. The time interval of measurements can later be increased to 2, 5 and 10 min. In tests of longer duration extending for a period of 1 day or more, water-level measurements for later part of the test could be taken at 1 h or even longer time intervals. After the pumping is stopped, rate of recovery in pumped well and also in observation wells is noted. As the rate of recovery is faster in the early part of recuperation, water-level measurements during this period should be taken at shorter time intervals; later the time interval can be increased. The water-levels measured during the test are likely to be also influenced by other extraneous reasons such as barometric pressure changes etc. Therefore, necessary corrections should be made to isolate the effect of such changes.

Well discharge can be measured by using different types of weirs; circular orifice weir being the most common (Driscoll 1986).

Types of Drawdown Curves

The drawdown data are plotted as time-drawdown and/or distance drawdown curves. A time-drawdown graph on a semi-logarithmic scale may show different slopes (Fig. 9.14). The earliest time response is usually due to wellbore storage as most of the water at this stage comes from the well itself. Next part of the time-drawdown curve is influenced by the contact between the well and the aquifer due to ‘skin effect’. At a later stage, the type of aquifer and its hydraulic characteristics (T and S) influence the rate of drawdown. Finally, boundary conditions affect the trend of the time-drawdown curve. There could be various possible combinations of these effects. A proper interpretation of pumping test data would therefore depend on the recognition of these effects. The duration of the test is therefore important to recognise the various components (segments) of a time-drawdown plot. The influence of various factors on time-drawdown curve is discussed below:
Fig. 9.13

Schematic section of a cluster of piezometers in a multilayered aquifer system

Fig. 9.14

Drawdown response in time sequence

Aquifer Types
A log–log time drawdown plot for an ideal unconsolidated confined aquifer tapped by a fully penetrating well will match with the Theis type curve (Fig. 9.15a) and a semi-log time-drawdown plot will be characterized by an uniform slope (Fig. 9.15a’). On the other hand, unconfined aquifers which are characterized by delayed yield, show typical S-shape on a log–log plot with three time-segments (Fig. 9.15b); the early part (first segment) conforms to the Theis type curve for confined aquifer due to instantaneous release of water from aquifer storage while the middle part (second segment) shows flattening due to vertical drainage. The last or the third segment also conforms to the Theis type curve as by that time the effect of vertical drainage becomes negligible. The semi-log plot for an unconfined aquifer is characterized by two parallel straight line slopes of segments 1 and 3 (Fig. 9.15b’).
Fig. 9.15

Comparison of log–log and semi-log plots of the theoretical time-drawdown relationship in unconsolidated aquifers, a and a confined aquifer, b and b unconfined aquifer, and c and c leaky aquifer

The leaky aquifer shows a behaviour similar to segments 1 and 2 of an unconfined aquifer as in this case additional water is transferred to aquifer by leakage through the aquitard. However, segment 3 of the time-drawdown curve for unconfined aquifer is not observed because the process of recharge by leakage continues and drawdown stabilises after sometime as is indicated by both log–log and semi-log graphs (Fig. 9.15c, c’).

Fractured Rock Aquifers

Fractured rocks show various types of drawdown and pressure build-up curves, depending on the nature of fractures and matrix blocks, boundary conditions and wellbore storage (Davis and DeWiest 1966; Streltsova 1976b; UNESCO 1979; Gringarten 1982; Kruseman and de Ridder 1990). Similar trends are also observed in geothermal and petroleum reservoirs (Grant et al. 1982; Horne 1990). In case of dewatering of fractures, due to decrease in the effective T-value, an increase in the drawdown slope with time is observed.

The response to pumping in a fractured dual-porosity aquifer, having the matrix blocks divided by fractures is similar to that of delayed yield in unconfined aquifer (Fig. 9.16a, a’). A similar trend is shown by karst aquifers (Chap. 15) consisting of dissolutional conduits intersecting a network of diffuse fractures in which the groundwater flow is slow. The first and third segments of the time-drawdown curve on a semi-log plot may have parallel slopes (Fig. 9.16a’). The first segment, with a uniform slope, is due to removal of water from storage in the fracture or in large conduits. This is also characteristic of confined aquifers. Unconfined aquifers with storage coefficient significantly less than 1 x 10 3 will also show a similar response (Kresic 2007). The second segment with a flattening character indicates additional contribution from matrix blocks or smaller fractures or fissures towards the larger fractures or dissolutional conduits in karst aquifers. A similar trend is shown by unconfined intergranular aquifers due to delayed gravity drainage (Fig. 9.16c, c’). If the pumping is carried out for longer duration, the time-drawdown curve shows a third segment with a relatively uniform slope which is characteristic of both the unconfined and dual porosity aquifers (Fig. 9.15b’ and Fig. 9.16a, a’).
Fig. 9.16

Comparison of log–log and semi-log plots of the theoretical time-drawdown relationship of fractured aquifers a and a double porosity confined aquifer, b and b single plane vertical fracture, c and c fractured permeable dyke in a massive country rock of low hydraulic conductivity. d and e Show linear flow and pseudo-radial flow respectively. (Modified after Kruseman and deRidder 1990)

The effect of pumping from a well in a single vertical fracture of infinite hydraulic conductivity in a confined homogeneous and isotropic aquifer of low permeability is also shown in (Fig. 9.16b, b’). The response at early times of pumping is characterized by a 1/2 slope straight line on the log–log plot of drawdown against time which is explained due to linear flow (Fig. 9.16d). At later time as the flow regime changes to pseudo-radial (Fig. 9.16e), the shape of the curves resembles those of (a) and (a’). Parts (c) and (c’) of Fig. 9.16 present the response of a pumping well in a fractured dyke of high permeability traversing a confined aquifer of low permeability and high storativity. The log–log time-drawdown plot in such a case is characterized by two straight line segments. The first segment has 1/2 slope like that of the well in a single vertical fracture of infinite permeability. At intermediate times, the log–log plot shows a 1/4 slope due to contribution from the host rock. At late times, the flow in the host aquifer is pseudo-radial which is reflected by a straight-line segment in the semi-log plot (Kruseman and de Ridder 1990).

In view of the above discussion, it could be concluded that results of short duration pumping tests may not be able to reflect the true aquifer characteristics and therefore the test duration should be long enough, say of few days, as the late time-drawdown data are of importance to distinguish between different aquifer types.

Some anomalous trends showing greater drawdown in wells away from the pumping well than those in the vicinity have also been reported from fractured rocks (Streltsova-Adams 1978; Gringarten 1982). Also there are field evidences of rise of water-level near the pumping well. These anomalous behaviours in literature are referred as ‘reverse water-level fluctuation’ or Noorbergum effect which can be explained by the following phenomena (Streltsova 1976b).
  1. 1.

    Delayed release of water from storage in the porous matrix of a double porosity medium in response to pumping resulting in local increase of water-level in the fracture system.

     
  2. 2.

    Decrease of the storage capacity of fractures due to their deformation as a result of pumping.

     
  3. 3.

    Additional recharge of fractures due to the recycling of pumped water discharged on the ground surface.

     
Skin Effect

The concept of skin effect or ‘skin factor’ on head loss in a well was first introduced in oil well industry, where due to mud filtrate at the well face, greater well loss was observed (Horne 1990). The idea was extended later to water-wells also. The ‘skin effect’ will be positive when the effective well bore radius and permeability is reduced due to mud filtrate. Conversely, when the effective well bore radius is increased due to acidization or other methods of well stimulation (Sect. 17.4), the skin effect will be negative. In fractured rocks, the skin effect will be negative due to good contact between the well and the water bearing fractures but when there is a clay coating on the walls of the fractures, the skin effect will be positive resulting in greater drawdown.

Method of Analyzing Test Data

Several analytical and digital techniques are available for analysing pumping test data under different geohydrological conditions. The advantage of numerical methods is that they can take care of complex geohydrological conditions. The graphical procedures are subjective and the number of type curves also become prohibitively large when the number of parameters exceeds three.

Graphical methods, although cumbersome, have an advantage as one can visualize the extent to which the field data match with the assumed conditions and the investigator can be selective in choosing the best part of the data which would give reliable results under given hydrogeological conditions. In this text only graphical methods of pumping test data analysis are described. For computer assisted methods, the reader is referred to other texts (e.g. Rushton and Redshaw 1979; Boonstra and Boehmer 1989; Boonstra and Soppe 2007).

The methods described in this section were originally developed for unconsolidated aquifers but can also be applied to fractured rocks if the fractures are interconnected.

The various commonly used methods are:
  • Thiem’s equilibrium method

  • Theis non-equilibrium type curve method

  • Jacob’s non-equilibrium straight line method

  • Theis’ recovery method

  • Walton’s type curve method for leaky confined aquifers

  • Boulton’ type curve method for unconfined aquifers

  • Neuman’s type curve method for unconfined aquifers

  • Papadopulos-Cooper type curve method for large diameter wells.

An excellent account of various methods with examples is given in Kruseman and de Ridder 1990; and Boonstra and Soppe 2007) among others. Although the above methods were primarily developed for homogeneous granular formations, but have been also used successfully for estimating hydraulic parameters in fractured basement and carbonate rocks with solution cavities (for examples see Kresic 2007).

In case of hetrogeneous aquifers, multiple and sequential pumping tests are useful to account for spatial variation in T and S (Li et al. 2007; Straface et al. 2007). In a sequential test, each well from a well field is pumped in sequence while the other wells are used as observation wells (see Sect. 9.2.6 in this chapter).

Constant Discharge Tests
Confined Aquifers

The pumping test data from a confined aquifer can be analysed both by steady state and unsteady state methods. The steady-state Eq. 7.25 for groundwater flow in a confined homogeneous aquifer is given in Sect. 7.1.3.

Thiem’s Equilibrium Method

Thiem (in Wenzel 1942) was one of the first workers to use drawdown data for estimating T from unconfined and confined aquifers. Thiem’s equation based on steady-state flow for a confined aquifer can be written as

$$ Q = \frac{{2\pi T ({{h_2} - {h_1}})}}{{ln ({r_2}}/{r_1})} $$
(9.16)
or
$$ Q = \frac{{2\pi T({{s_1}-{s_2}})}}{{2.30\,log ( {{{{r_2}}/{{r_1}}}})}} $$
where Q is well discharge in m3d 1, T is transmissivity in m2d 1, s1 and s2 are drawdowns in metres in the two observation wells located at distances r1 and r2 in metres respectively from pumping well (Fig. 9.17).
Fig. 9.17

Schematic cross section of a pumped confined aquifer

After the system has reached steady-state (equilibrium) condition, the drawdown data from different observation wells are plotted on a semi-logarithmic paper. (drawdown, s1 and s2 on arithmetic scale and distances, r1, r2… on logarithmic scale). The slope Δs of this line is determined and is substituted in Eq. 9.17 for estimating T.

$$ T = \frac{{2.30\,Q}}{{2\pi \,\Delta s}} $$
(9.17)

where Ds is the difference of drawdown per log cycle of r.

Theis Non-equilibrium Type Curve Method

The differential Eq. 7.24 for unsteady flow in plane polar coordinates is given in Sect. 7.1.3. Based on the analogy between flow of water in an aquifer and flow of heat in an equivalent thermal system, Theis (1935) obtained Eq. 9.18

$$ s = \frac{Q}{{4 \pi T}}\,\int\limits_u^\infty{\frac{{{e^ - }y}}{y}\,dy}$$
(9.18)

where,

$$ u = \frac{{{r^2}S}}{{4Tt}}$$
(9.19)

s = drawdown in metres in a piezometer or observation well at distance, r in metres from the pumping well, S is storativity of the aquifer (dimensionless); t is time since pumping started, in days; Q and T are as defined earlier.

Although the Theis equation is based on several assumptions, all of which usually do not hold good under natural field conditions, it has been very useful in solving many groundwater flow problems. The exponential integral in Eq. 9.18 is written symbolically as W(u) which is generally read as well function of u or Theis well function.

Equation 9.18 can therefore be written as

$$ s = \frac{Q}{{4\pi T}}\,W(u) $$
(9.20)

where

$$ \begin{aligned} W(u) = &- 0.5772 - ln\,u+ u \\ &- \frac{{{u^2}}}{{2.2!}} + \frac{{{u^3}}}{{3.3\,!}} - \frac{{{u^4}}}{{4.4!}}\end{aligned}$$
(9.21)

and u = \(\frac{{{r^2}S}}{{4Tt}}\) as already indicated.

Values of W(u) for various values of u are given in several standard text books on Groundwater Hydrology, e.g. Todd (1980).

From Eqs. 9.18 and 9.19 it can be shown that if s is known for one value of r and several values of t or for one value of t and several values of r, and if Q is known, then T and S can be determined.

The first step in this method is to plot on a double-log paper, the Theis type curves of W(u) vs. u, W(u) vs. √u, and W(u) vs. 1/u (Fig. 9.18). The field data curve can be matched with one of the Theis type curves as given in Table 9.4. A reverse type curve W(u) vs. 1/u can be matched with a data curve of s vs. t/r2 or if only one observation well is available then s is plotted against t. This will save computation time. If drawdown data is available for one value of t and several values of r then a distance-drawdown curve (s vs. r) is plotted which is matched with a type curve of W(u) vs. √u. As an example, the matching of the Theis type curve, W(u) vs. u with the field data curve, r2/t vs. s is shown in Fig. 9.19. An arbitrary match point is selected anywhere on the overlapping portion of the sheets and the coordinates of this point are noted. These data on substitution in Eqs. 9.20 and 9.19 will give values of T and S respectively.
Fig. 9.18

Theis type curves W(u) vs. u, W(u) vs. √u, and W(u) vs. 1/u

Fig. 9.19

Analysis of pumping test data by Theis type curve method for a confined sandstone aquifer. (After Radhakrishna et al. 1979)

Table 9.4

Various combinations of type curves and data curves in Theis method of analysis

Type curve

Data curve

W(u) vs. u

s vs. r2/t or s vs. 1/t

W(u) vs. 1/u

s vs. t/r2 or s vs. t

W(u) vs. √u

s vs. r

Jacob’s Method

For values of u < 0.01, Eq. 9.20 reduces to Eq. 9.22.

$$ s = \frac{Q}{{4\pi T}}\left( {ln\,\frac{{4Tt}}{{{r^2}S}} - 0.5772} \right) $$
(9.22)

In decimal logarithms, Eq. 9.22 reduces to

$$ s = \frac{{2.30\,Q}}{{4\pi T}}\,log\frac{{2.25Tt}}{{{r^2}S}} $$
(9.23)

which is known as the Jacob or Theis-Jacob equation.

According to Eq. 9.23, the graph of drawdown, s vs. log t will be in the form of a straight line. Equation 9.23 can be rewritten as

$$ T = \frac{{2.30\,Q\left( {log\,{{{t_2}}/{{t_1}}}} \right)}}{{4 \pi({{s_2} - {s_1}})}} $$
(9.24)

or

$$ T = \frac{{2.30\,Q}}{{4\pi \Delta s}}$$
(9.25)

where Ds is the change in drawdown over one log cycle of time.

The storativity, S can also be determined from the same semi-log plot by extending the time-drawdown curve to zero drawdown axis. Solving for storativity S, the equation in its final form becomes

$$ S = \frac{{2.25T{t_0}}}{{{r^2}}} $$
(9.26)

where t0 is the time intercept in days for s = 0.

The reliability of Theis type curve and Jacob’s straight line methods for estimating T and S in heterogeneous aquifers has been demonstrated by several workers viz. Sanchez-villa et al. 1999; Li et al. 2007; Straface 2007). However, the hydraulic connectivity of individual fractures which controls the movement of fluids and contaminants cannot be well established.

Theis Recovery Method

After pumping is stopped, the water-levels in pumping and observation wells will start rising. This is known as the recovery or recuperation phase. Theis equation for recovery phase can be written as

$$ s ^\prime = \frac{Q}{{4\pi T}}\left[ {W\!\left( u \right) - W\!\left( {u ^\prime} \right)} \right] $$
(9.27)

where

$$ u = \frac{{{r^2}S}}{{4Tt^\prime}} \quad u^\prime = \frac{{{r^2}S}}{{4Tt^\prime}}. $$
(9.28)

Q, T, S and r are defined earlier, t is the time in days since pumping started, t’ is the time in days since pumping stopped and s’ is the residual drawdown. As in Jacob’s equation, for small values of r and large values t’, Eq. 9.27 can be written as

$$ s ^\prime = \frac{{2.30Q}}{{4\pi T}}\;log\;\frac{t}{{t ^\prime}} $$
(9.29)

Therefore a plot of residual drawdown, s’ on arithmetic scale vs. t/t’ on logarithmic scale should form a straight line. Equation 9.29 can be rewritten as

$$ T = \frac{{2.30Q}}{{4\pi \Delta s ^\prime}} $$
(9.30)

where Ds’ is the change in residual drawdown per log cycle of t/t’.

Leaky (Semi-confined) Aquifers

In nature, truly confined aquifers are rare as the confining layers are not completely impermeable. Therefore, certain amount of water is contributed to the pumped aquifer due to leakage through the aquitard. Such aquifers are known as leaky or semi-confined aquifers.

Pumping from a fully penetrating well in a leaky aquifer draws water from storage in the leaky aquifer as well as from storage in the aquitard and the water-table aquifer. The flow in the leaky aquifer is radial (horizontal) towards the well but the flow through aquitard is vertical due to head difference between the two aquifers (Fig. 9.20). As pumping advances, the cone of depression expands and the head differences in between the two aquifers increases leading to greater amount of leakage. The amount of leakage also depends on the vertical hydraulic conductivity of the aquitard (K’). A steady state situation may arise when the rate of withdrawal becomes equal to the amount of water contributed from the pumped aquifer plus that contributed by leakage from the adjacent aquifer(s).
Fig. 9.20

Schematic cross section of a pumped leaky aquifer

It was earlier assumed that during pumping the hydraulic head in the unpumped aquifer remains constant and that the rate of leakage into the pumped aquifer is proportional to the hydraulic gradient across the aquitard (Hantush and Jacob 1955). The first assumption will only hold good if there is a constant source of recharge to the unpumped aquifer which may not be possible. The second assumption which ignores the effects of storage capacity of the aquitard is justified when the flow has reached steady-state condition. Under unsteady-state condition, the effect of aquitard storage cannot be neglected. Therefore, effects of aquitard storage as well as decline of hydraulic head in the unpumped aquifer were also considered by later workers (Freeze and Cherry 1979). A review of evaluation of aquifer test data in leaky aquifers is given by Walton (1979) and Kruseman and de Ridder (1990).

Walton’s Method

The unsteady state flow in a leaky aquifer without water released from storage can be given in an abbreviated form by Eq. 9.31

$$ {h_0} - h = s = \frac{Q}{4\pi T}\int\limits_u^\infty\frac{1}{y} exp \left(-y-\frac{r^2}{4B^2y}\right)dy $$
(9.31)

or

$$ s = \frac{{Q}}{{4\pi T}}\,W ({u,{r/B}})$$
(9.32)

where

$$ u{\rm{ }} = \frac{{{r^2}S}}{{4Tt}} $$
(9.33)
$$ {r/B}=\frac{r}{\sqrt{T/(K^\prime\!/b^\prime)}} $$
(9.34)

or

$$ B = {({{{Tb^\prime}/{K^\prime}}})^{\frac{1}{2}}} $$
(9.35)

where, B is the leakage factor, K’ is the vertical hydraulic conductivity of the aquitard, and b’ is the thickness of the aquitard (Fig. 9.20).

It can be seen that for large values of B, r/B would tend to be zero and therefore Eq. 9.32 will approach the Theis Eq. 9.20.

W(u.r/B) is read as the well function for leaky aquifers. Values of W(u.r/B) in terms of the practical range of u and r/B are plotted on a double-log paper to provide a family of type curves for leaky aquifers (Fig. 9.21). Pumping test data (s vs. t) is plotted on another log–log paper which is matched with one of the leaky aquifer type curves for a particular value of r/B and a match point is selected. The coordinates of the match point W(u.r/B) and l/u are read from the type curve sheet and s and t from the data curve sheet. The values of Q and match point coordinates, i.e. W(u.r/B) and s are substituted in Eq. 9.32 to calculate T and S is obtained from Eq. 9.33. The value of r/B of the type curve with which data curve was matched is substituted in Eq. 9.34 to determine K’ and also leakage, K’/b’ or its reciprocal, hydraulic resistance, C from the relation B = √TC. A straight line method for estimating hydraulic properties of leaky aquifers is given by Hantush (1956, 1964).
Fig. 9.21

Family of Walton’s type curves of W(u,r/B) vs. 1/u for different values of r/B for a leaky aquifer. (After Walton 1962)

Unconfined Aquifers
Pumping from an unconfined aquifer causes dewatering of the aquifer resulting in vertical component of flow in addition to the radial flow (Fig. 9.22). This is in contrast to a confined aquifer where there is no dewatering of the aquifer and the water is obtained due to compaction of the aquifer and expansion of water.
Fig. 9.22

Schematic cross-section of a pumped unconfined aquifer

The time-drawdown curve in response to pumping from an unconfined aquifer is characteristically S-shaped due to delayed yield as described earlier. In unconfined aquifer, if drawdown, s is small as compared to its saturated thickness, b, the vertical component of flow can be neglected and the Theis equation can be used to determine aquifer characteristics. However, when drawdowns are significant, the vertical component of flow cannot be ignored and therefore the use of Theis equation is not justified. In such conditions where gravity drainage is consisderable, methods for analysing pumping test data based on the concept of delayed yield are given by Boulton (1963) and Neuman (1975).

Boulton’s Method

Boulton (1963) gave Eq. 9.36 for drawdown in an unconfined isotropic aquifer with delayed drainage,

$$ s = \frac{Q}{{4\pi T}}\;\!W\!\left( {{U_{AY}}\frac{r}{D}} \right) $$
(9.36)

where W(Uay, r/D) is called the ‘well function of Boulton’. Under early-time conditions, describing the first segment of the time-drawdown curve, Eq. 9.36 can be written as

$$ s = \frac{Q}{{4\pi T}}\;\!W\!\left( {{U_A}\frac{r}{D}} \right) $$
(9.37)

where

$$ {U_A} = \frac{{{r^2}{S_A}}}{{4Tt}} $$
(9.38)

and SA = early time storage coefficient.

Under later-time conditions for the third segment of the time-drawdown curve, Eq. 9.36 reduces to

$$ s = \frac{Q}{{4\pi T}}\;\!W\!\left( {{U_y}\frac{r}{D}} \right) $$
(9.39)

where

$$ {U_y} = \frac{{{r^2}{S_y}}}{{4Tt}} $$
(9.40)

and, Sy = specific yield. D is called the drainage factor, which is defined as

$$ D = \sqrt {\frac{T}{{\alpha {S_y}}}}.$$
(9.41)
D has the dimensions of L, l/a is the ‘Boulton delay index’ which is an empirical constant having the dimensions of time. It is used to determine the time, t at which the delayed yield ceases to affect the drawdown. The procedure for the use of Boulton’s method involves matching of the field data curve in two steps, with one of the Boulton’s type curves (Fig. 9.23) for computing aquifer properties (Kruseman and de Ridder 1970). Prickett (1965) was among the first workers to demonstrate the applicability of Boulton’s type curve method to unconfined aquifers. An example of field data analysis using Boulton’s type curve method in a granitic aquifer is given in Fig. 9.24.
Fig. 9.23

Family of Boulton’s type curves for an unconfined aquifer. (After Prickett 1965, reproduced by permission of the journal of Ground Water)

Fig. 9.24

Analysis of pumping test data by Boulton method from a granitic aquifer in Lower Maner Basin, Andhra Pradesh. (After Radhakrishna et al. 1979)

Neuman (1972) showed that Boulton’s method can be used only for large values of pumping time. In general, the limitation of Boulton’s method becomes more severe as the ratio of horizontal permeability to vertical permeability increases, the thickness of the aquifer increases and the distance from the pumping well decreases. Neuman (1975) suggested an alternative method as described below.

Neuman’s Method

Neuman’s method for determining aquifer characteristics of anisotropic unconfined aquifer does not involve such semi-empirical quantities as Boulton’s delay index, l/a. It also takes into account aquifer anisotropy. Neuman’s drawdown equation can be written as

$$ s = \frac{Q}{{4\pi T}}\,W \!\left( {{U_{A'}}\,{U_{B'}}\,\beta }\right)$$
(9.42)

where, W(UA, UB, b) = well functions for water-table aquifer with fully penetrating wells having no storage capacity (dimensionless).

Under early time conditions Eq. 9.42 reduces to

$$ s = \frac{Q}{{4\pi T}}\,W\!\left( {{U_{A^\prime}}\,\,\beta } \right) $$
(9.43)

where

$$ {U_A} = \frac{{{r^2}{S_A}}}{{4Tt}} $$
(9.44)

SA = Elastic early-time storativity.

Under late-time conditions the third segment of the time-drawdown curve, Eq. 9.42 reduces to

$$ {U_B} = \frac{{{r^2}{S_y}}}{{4Tt}} $$
(9.45)

Neuman’s parameter b is defined as

$$ \beta = \frac{{{r^2}{K_v}}}{{{b^2}{K_h}}} $$
(9.46)

where Kv and Kh are the vertical and horizontal hydraulic conductivities of the aquifer respectively and Q, T, R, t, Sy, and b are as defined earlier.

For an isotropic aquifer Kv = Kh. Therefore,

$$ \beta = \frac{{{r^2}}}{{{b^2}}} $$
Figure 9.25 presents the family of Neuman’s type curves of W(UA, UB, b) vs. 1/UA and 1/UB for various values of b.
Fig. 9.25

Neumann’s type curves for unconfined aquifer

The aquifer characteristics can be determined by type-curve and straight-line methods (Kruseman and de Ridder 1990). Neuman (1975) showed that contrary to the assumption of Boulton, a is not a characteristic constant of the aquifer but decreases linearly with the logarithm of r.

The difference between Boulton’s theory and that of Neuman (as far as fully penetrating wells are concerned) is that the former only enables one to calculate a, whereas the latter enables one to determine the degree of anisotropy, as well as the horizontal and vertical hydraulic conductivities.

Bounded Aquifers
In the various aquifer test methods mentioned above, it was assumed that the aquifer is of infinite areal extent. However, in nature such an assumption will not always hold good as the aquifers are limited by some hydrogeological boundary. Two types of boundaries are easily identified, i.e. a recharge boundary like a river or a canal which causes replenishment of the aquifer and a barrier or impermeable boundary which is a result of either thinning or termination of an aquifer against a low-permeability formation or dyke or a fault which limits movement of water (Fig. 9.26).
Fig. 9.26

Diagram illustrating various types of boundaries. a Recharging boundary due to stream. Impermeable boundaries due to: b lateral termination, c dyke, and d fault

The method of images, commonly used in the study of heat conduction in solids, has been used for the solution of boundary problems in groundwater flow. The boundary condition is simulated by an image well or imaginary well located at the same (equivalent) distance from the boundary but on the opposite side. Depending upon the nature of the boundary, i.e. recharging or barrier type, the flow is characterized by assuming a recharging or discharging image well. Thus the real bounded system is replaced by an equivalent hydraulic system, i.e. an imaginary system of infinite areal extent. Both the recharge and discharge image wells are located perpendicular to the boundary, but on the opposite sides at equal distances from the boundary (Figs. 9.27, 9.28). In a recharging system (Fig. 9.27), the recharge image well recharges the aquifer at the same rate as the rate of withdrawal from the real well. As a result of this the build-up due to cone of impression (recharge) and the drawdown caused due to pumping from the real well exactly cancel each other at the recharging boundary resulting in a constant head along the line source. In cases where there is more than one boundary, more image wells are to be considered depending upon the configuration and nature of the boundaries. Several such geometric configurations are considered by Ferris et al. (1962) and Bear (1979), among others.
Fig. 9.27

Schematic section of: a a recharging well in a semi-infinite aquifer bounded by a perennial stream, and b of the equivalent hydraulic system in an infinite aquifer. (After Ferris et al. 1962)

Fig. 9.28

Schematic section of: a a discharging well in a semi-infinite aquifer bounded by an impermeable formation and b the equivalent hydraulic system in an infnite aquifer. (After Ferris et al. 1962)

The effect of boundary conditions on the time-drawdown trend in an observation well will depend on the nature of the boundary, the time when the cone of depression intercepts the boundary and the distance of the observation well from the pumping well. On a log–log plot, the boundary effects are observed by the deviation of the field data from the Theis curve. When time-drawdown data are plotted on semi-log paper, the recharging boundary will cause a flattening of the time-drawdown curve and a barrier (impermeable) boundary will cause further steepening of the curve (Fig. 9.29). The time of inflection of the time-drawdown curve from the normal trend (when the aquifer is of infinite areal extent) depends on the distance of the observation well from the pumping well. Note that the effect of recharging boundary on time-drawdown curve is the same as that for leaky aquifers or partially penetrating wells. Therefore, for a proper interpretation of pumping test data, necessary information about the hydrogeological system is necessary.
Fig 9.29

Effect of a recharge boundary (a and b) and impermeable boundary (c and d) on the time-drawdown curves in a confined aquifer

The recharging boundary, viz. rivers, canals, or lakes can be easily observed in the field but barrier boundaries are at times hidden as a subsurface dyke or a fault. Such boundaries can be identified and mapped by the analysis of observed time-drawdown data provided that pumping is carried out for sufficiently long time so that the cone of depression intercepts the boundary. As mentioned above, the nature of the boundary (recharging or barrier type) can first be ascertained from the trend of the time-drawdown data. The next step is to determine the distance of the image well from the observation well (ri) for which time-drawdown data from at least three observation wells is necessary. The values of ri from various observation wells can be estimated from the semi-log time drawdown plot by choosing any arbitrary value of drawdown, sw and corresponding value of time, tr, before the effect of boundary influences the time-drawdown curve (Fig. 9.30a). The time-drawdown curve is extrapolated beyond the time of inflection. The time intercept, ti of an aqual amount sw of divergence caused by the image well is read from the data curve. The distance ri, can be determined from Eq. 9.47.
Fig. 9.30

Diagrams illustrating application of image well theory to locate an unknown impermeable boundary; a time-drawdown plot showing effect of impermeable boundary, b location of the unknown impermeable boundary. OW Observation well, PW pumping well

$$ {r_i} = {r_r}\,\sqrt {{{{t_i}} \mathord{\left/ {\vphantom {{{t_i}} {{t_r}}}} \right. \kern-\nulldelimiterspace} {{t_r}}}}$$
(9.47)

where rr is the distance of the observation well from the pumping well and ri, ti and tr are as defined earlier.

After the values of ri are computed for the individual observation wells, circles are drawn with their centres at the respective observation wells and their radii equal to the respective estimated values of ri. The intersection of the arcs will give the location of image well. The perpendicular bisector of the line joining the image well and the real (pumped) well will mark the strike of the boundary (Fig. 9.30b).

Partially Penetrating Wells

One of the assumptions in the Theis and other methods as discussed earlier, is that the pumped well penetrates the entire thickness of the aquifer so that the flow towards the well is horizontal. However, under field conditions especially when aquifers are very thick, it is usually uneconomical to provide screen along the entire thickness of the aquifer. Such a well which taps only a part of the aquifer thickness is known as partially penetrating well.

In a partially penetrating well, the flow pattern differs from the radial horizontal flow which pressumably exists around a fully penetrating well. Therefore, flow towards a partially penetrating well will be three-dimensional due to vertical flow component. Accordingly, the average length of a flow line will be greater in partially penetrating well as compared with the fully penetrating well resulting in additional drawdown (Fig. 9.31b). The discharge from a partially penetrating well therefore will be less than a fully penetrating well for the same drawdown. It would also mean that if they are pumped at the same rate, the drawdown in a partially penetrating well (sp) will be greater than that in a fully penetrating well (sw). The effect of partial penetration is more near the well face and will decrease with increasing distance from the pumping well. Anisotropy causes greater effect of partial penetration on drawdown as compared to isotropic aquifers as in the former the flow is three dimensional. In an anisotropic aquifer, the effect of partial penetration will be negligible at a distance of r ³ 1.5 b √Kh/Kv and at t > Sb2/2T (Kruseman and de Ridder 1990). Methods of analyzing test data from partially penetrating wells for steady and unsteady conditions in confined, unconfined, leaky and anisotropic aquifers are described by Hantush (1964) and Kruseman and de Ridder (1990).
Fig. 9.31

a Effect of partially penetrating well on drawdown in a confined aquifer, b Effect of wells partial penetration on the time-drawdown relationship in a confined aquifer

Large Diameter Wells

The various methods described above assume that pumping well has an infinitesimal diameter which will not be valid in large diameter (dug) wells. Therefore, these methods are not applicable to large diameter wells due to the significant effect of well storage. Dugwells are common in low permeability hard rocks of India and other developing countries. The earliest method to estimate aquifer properties from large diameter fully penetrating well in a confined aquifer was given by Papadopulos and Cooper (1967). Later, several other analytical and numerical solutions of unsteady flow to large-diameter wells were developed (Boulton and Streltsova 1976; Rushton and Holt 1981; Rushton and Singh 1983; Herbert and Kitching 1981; Barker 1991; Chachadi et al. 1991; Rushton 2003).

Papadopulos–Cooper Method
Papadopulos and Cooper (1967), gave Eq. 9.48 to describe drawdown in a fully penetrating large-diameter well (with storage) in a confined aquifer (Fig. 9.32).
Fig. 9.32

Schematic cross-section of a confined aquifer pumped by a large diameter well

$$ {s_w} = \frac{Q}{{4\pi T}}\;\!F\!\left( {{U_w},\beta } \right) $$
(9.48)

where,

$$ {U_w} = \frac{{r_w^2S}}{{4Tt}} $$
(9.49)
$$ \beta = \frac{{r_w^2\,S}}{{r_c^2}} $$
(9.50)
  • rw = radius of the well screen or open well

  • rc = radius of the well casing over which the water-level is changing.

As in other type curve matching methods, the time-drawdown curve is plotted on a double-log paper which is matched with one of the type curves for large diameter wells (Fig. 9.33). A match point is selected and the values of F(Uw, b), l/uw and t are noted. The value of b of the type curve with which the observed data curve is matched is also noted. T is estimated from Eq. 9.48. Values of S can be computed by two methods (1) by substituting the values of l/uw, t, rw and T into Eq. 9.49; and (2) by substituting the values of b, rw and rc into Eq. 9.50.
Fig. 9.33

Family of a Papadopulos type curves for large-diameter wells

The main problem in curve matching is that the early part of type curves (Fig. 9.33) differs only slightly in shape from each other and therefore uncertain value of b will make significant difference in the computed value of S. Therefore, unless time of pumping is large, reliable value of S cannot be obtained but transmissivity can be computed without such a problem.

The well storage dominates the time-drawdown curve upto a time, t, given by t = (25 \({\rm{r}}_{\rm{c}}^{\rm{2}}\)/T), after which the effect of well storage will become negligible and hence reliable values of aquifer characteristics T and S can be obtained. Therefore, for obtaining a representative value of S, the well should be pumped beyond this time, t. In hard rocks, such as granites and basalts where T is about 10 m2d 1 and rc is 2 m, t will be 10 days which is impractical. Therefore, Papadopulos–Cooper type curve method may not give a correct value of S but T values can be reliable.

The Papadopulos–Cooper method although takes into account the effect of well storage but it is applicable only to fully penetrating abstraction well in a confined aquifer. A method for analysing test data from a partially penetrating well in an anisotropic unconfined aquifer is given by Boulton and Streltsova (1976). The Papadopulos–Cooper method also assumes constant rate of abstraction which is difficult to maintain when a large diameter well is discharged by a centrifugal pump. A method to maintain constant discharge during the pumping test is suggested by Athavale et al. (1983). A curve matching technique considering the falling rate of abstraction during the test is given by Rushton and Singh (1983).

The effect of seepage face which is developed in the unconfined aquifer due to differences in the water-level outside and inside a large diameter well was considered by Rushton and Holt (1981), Rushton and Singh (1987) and Sakthivadivel and Rushton (1989). The advantage of using recovery data over drawdown data is advocated by Herbert and Kitching (1981), Singh and Gupta (1986) and others. This is because during recovery: (a) well storage does not play any important role as all water is derived from the aquifer, and (b) well losses are small especially during the later stages of recovery.

The use of numerical methods for computing aquifer properties under complex geohydrological conditions are advantageous. The discrete Kernel approach suggested by Patel and Mishra (1983), Rushton and Singh (1987), and Barker (1991) is advantageous for estimating aquifer parameters from both drawdown and recovery data under varying rates of pumping. Chachadi et al. (1991) also considered storage effect both in the production and observation wells.

9.2.4 Pumping Tests in Fractured Rock Aquifers

In areas where the fractures are closely spaced and are inter-connected, conventional pumping test methods assuming confined and leaky confined models can be used. However, the double porosity model (Sect. 7.2.2) is more representative of uniformly fractured aquifers including petroleum and geothermal reservoirs (Earlougher 1977; Horne 1990; Grant et al. 1982).

Double Porosity Model

The flow characteristics in a double (dual porosity) aquifer were discussed earlier in Sect. 7.2.2. In a double porosity aquifer the rock mass is assumed to consist of a number of porous blocks as well as large number of randomly distributed, sized and oriented fractures. A method of estimating hydraulic properties of such a double porosity aquifer is described below.

Streltsova–Adam’s Method for Confined Fractured Aquifer

Streltsova–Adams (1978) assumed that a confined fractured aquifer representing a double porosity model, consists of matrix blocks and fracture units in the form of alternate horizontal slabs (Fig. 7.8a). The thickness of the matrix units is greater than that of fracture units.

The drawdown distribution in the fracture (sf) is given by Eq. 9.51

$$ {S_f} = \frac{Q}{{4\pi {T_f}}}\;\!W\!\left( {{u_f},{r /{{B_f},n}}} \right) $$
(9.51)

where

$$ {u_f} = \frac{{{r^2}{S_f}}}{{4{T_f}t}} $$
(9.52)
$$ B_f^2 = \frac{{{T_f}}}{{{\alpha _f}{S_m}}} $$
(9.53)

and

$$ n = 1 + \frac{{{S_m}}}{{{S_f}}} $$
(9.54)

Subscripts f and m represent properties of fractures and matrix blocks respectively, Bf is the drainage factor. Accordingly, the equation for drawdown in matrix block, sm is also developed.

Values of the drawdown function W(uf, r/Bf, n) computed for n = 10, 100, and 1000 and various assumed values of parameters r/Bf and 1/uf are given by Streltsova–Adams (1978).

Streltsova–Adam’s method is based on the assumptions that: (a) fractures and blocks are compressible, (b) the abstraction well is fully penetrating the fractured aquifer and receives water from it, (c) pumping is at constant rate, (d) the radius of the pumped well is vanishingly small, (e) the flow in the block is vertical, (f) flow in the fissure is horizontal, and (g) flow in both blocks and fissures obey Darcy’s law.

Based on the values of well function, and values of r/Bf and 1/uf, type curves can be plotted on double-log paper. One such set of type curves for n = 10 is given in (Fig. 9.34). Time-drawdown data is plotted on another double log paper and matched with one of the type curves. A match point is selected for which coordinate values are noted and substituted in Eqs. 9.51–9.53 to obain values of Tf, Sf, Tm, Sm and Bf. Other characteristics which can be estimated are af, and Cf. af, a characteristic of the fissure flow (days 1) is given by
Fig. 9.34

Streltsova-Adam’s type curves for double porosity confined aquifer (n = 10)

$$ {\alpha _f} = \frac{{{T_f}}}{{B_f^2\,{S_m}}} $$
(9.55)

and Cf (hydraulic diffusivity of the fissure)

$$= \frac{{{T_f}}}{{{S_f}}}\left( {{m^2}\,{d^{ - 1}}} \right)$$
(9.56)
As an example, the time-drawdown data curve from an observation well in fractured basalt is given in Fig. 9.35. The field data curve was matched with the Theis type curve, Hantush leaky aquifer type curve, and Boulton’s type curves for unconfined aquifer. However, keeping in view the hydrogeological situation and because the best match was obtained with Streltsova–Adam’s type curve (n = 10), hydraulic parameters were computed assuming double porosity model (Fig. 9.35).
Fig. 9.35

Time-drawdown plot of pumping test data from an observation well in fractured basalt, South India. (After Singhal and Singhal 1990)

The equations given by Streltsova–Adams for double-porosity aquifer, involve delayed yield from blocks to fissures. Hence Streltsova–Adam’s type curves are identical to Boulton’s type curves for unconfined aquifer involving delayed yield in unconsolidated formations. Therefore, for proper interpretation of pumping test data, hydrogeological framework of the aquifer system should be known.

Bourdet–Gringarten’s Method

The hydraulic characteristics of the fractures and the matrix block in a double porosity aquifer can also be estimated using Bourdet–Gringraten’s type curve method (Bourdet and Gringarten 1980; Gringarten 1982). The difference between Streltsova–Adam’s approach and that of Bourdet–Gringarten is that the former considered blocks and fissures of regular shape and orientation in the form of slabs, while the latter model is applicable to different geometrical shapes of blocks. Readers may refer to Kruseman and de Ridder (1990) for further details of this method.

The effect of fracture skin in a fissured double porosity aquifer is discussed by Moench (1984) and its application to estimate hydraulic parameters of fissured rock system is given by Levens et al. (1994).

Single Vertical Fracture Model

The majority of oil and gas reservoirs are hydraulically fractured to produce a single vertical fracture to augument well production. In Hot Dry Rock (HDR) experiments also, such fractures are induced for circulating water to deeper parts in the crust for tapping geothermal energy. Naturally occurring vertical fractures are also intersected during drilling for water and geothermal wells. Therefore, drawdown and pressure buildup in wells penetrating a single vertical fracture is of importance to estimate its hydraulic characteristics. The time-drawdown characteristics of a vertical fracture of high permeability was discussed earlier.

Gringarten–Witherspoon’s Method

The drawdown in an observation well as a result of pumping from a single plane, vertical fracture in a homogeneous, isotropic confined aquifer is given by Eq. 9.57.

$$ s = \frac{Q}{{4\pi T}}\;\!F\!\left( {{u_{vf}},r^\prime} \right) $$
(9.57)

where

$$ {U_{vf}} = \frac{{Tt}}{{Sx_f^2}} $$
(9.58)
$$ r^\prime = \frac{{\sqrt {\left( {{x^2} + {y^2}} \right)} }}{{{x_f}}} $$
(9.59)
  • xf = half length of the vertical fracture (m)

  • x, y = distance between observation well and pumped well, measured along the x and y axes, respectively (m).

Equations 9.57 and 9.58 indicate that the drawdown in the observation well depends on its location with respect to orientation of the fracture. The observation well could be located on the x axis or y axis or along a line at an angle of 45° to the strike of the fracture (Fig. 9.36).
Fig. 9.36

Plan view of a vertical fracture with a pumped well (PW) and observation wells (OW) at three different locations

Type curves of drawdown function F(uvf r’) for different values of Uvf and r’ and location of observation wells are given in Kruseman and de Ridder (1990). If one knows about the location of observation well with respect to the fracture, a particular set of type curves can be selected for matching the test data plot. By matching the data curve with the type curve, coordinate values of the match point are noted and T and S are calculated from Eqs. 9.57 and 9.58.

The various methods of analysing test data from fractures are primarily developed for hydraulically fractured petroleum reservoirs with the assumption that fractures have large hydraulic conductivity. Therefore, their application to natural fractures is limited as natural fractures will not have infinite hydraulic conductivity due to infilling. Moreover, the assumption of negligible storage may also not hold good as natural fracture zones may be quite wide.

Kresic (2007) compared the values of T and S obtained from karst aquifer in South Dakota, USA, estimated by Jacob’s straight line method, Neuman’s unconfined aquifer type curve method and dual porosity methods. It was noted that the values of T and S, estimated by Jacob’s solution for both the early and late time-drawdown data, are similar to those obtained using double porosity model. Further, the storativity estimated by Jacob’s method from early data is almost identical with the fracture and conduit storativity obtained from dual porosity model.

Intrusive dykes

Dykes are intrusive bodies of igneous rocks, commonly of dolerite composition which may extend to long distances of the order of several kilometres but are of limited width. They cut across different type of rocks and at places cause fracturing of the country rocks at the contact due to baking effect. The dykes are themselves fractured into a system of joints usually of columnar type which impart secondary permeability. At other places, dykes may be comparatively massive and impermeable forming barrier boundary.

Methods of analysis of pumping test data from fractured dykes are given by Boonstra and Boehmer (1986, 1989) and Boehmer (1993). It is assumed by these researchers that the dyke is of infinite length and has finite width and a finite hydraulic conductivity. The upper part of the dyke is less permeable due to weathering of the dyke rock into clayey material, the middle part is fractured and is permeable while the deeper part is massive and impervious as the fractures tend to die out with depth. Therefore, middle fractured part of the dyke forms a confined horizon which is continuous with the adjacent confined part of the aquifer in the country rock (Fig. 9.37).
Fig. 9.37

Composite dyke-aquifer system. (After Kruseman and deRidder 1990)

The time-drawdown data obtained from a pumping well in a dyke is characterized by three segments as mentioned earlier (Fig. 9.16). The hydraulic characteristics of the dyke and the country rock can be determined by analysing time-drawdown data from observation wells in a dyke using a type-curve method developed by Boonstra and Boehmer (1986). A distance-drawdown type curve method is also suggested. They have demonstrated the applicability of their method for estimating hydraulic characteristics of a dolerite dyke of the Karoo system in the Republic of South Africa (Boehmer and Boonstra 1987; Boonstra and Boehmer 1989).

9.2.5 Cross-hole Tests

As Lugeon tests, slug tests and conventional pumping tests provide information of only a part of the aquifer in the close vicinity of the well, cross–hole tests are preferred to determine three dimensional properties in both saturated and unsaturated porous and fractured rocks from an array of randomly oriented boreholes. Cross-hole tests are usually carried out for estimating relative permeabilities and interconnection between various fracture sets which are of importance in the assessment of potential repositories for the disposal of radioactive waste and solute movement.

A cross-hole project was implemented at Stripa mine, Sweden, for characterization of high level waste repository where a fan shaped array of boreholes, 200–250 m long, were drilled below the ground emanating from the end of a drift, for geophysical and hydrogeological testing (Fig. 9.38). A polar plot of directional permeabilities would indicate whether the rock mass behaves as an anisotropic continuum. An ellipsoidal configuration of directional permeabilities would support continuum assumption. In Stripa mine, Sweden, the plotted values of hydraulic diffusivity (ratio of permeability to specific storage) were not strictly ellipsoidal indicating a limited interconnection of discrete fractures (Black 1987).
Fig. 9.38

General layout of the crosshole programme borehole array at Stripa, Sweden. The fan shaped array of six boreholes (BH1BH6) lies within an orthogonal array of three existing boreholes, E1 (east), N1 (North) and V1 (vertical). (After Black and Holmes 1985)

In unsaturated fractured rocks also, single-hole and cross-hole pneumatic injection tests using air rather than water in vertical and inclined boreholes are carried out to characterise the bulk pneumatic properties and connectivity of fractures. The test holes can be either vertical or inclined depending on the geometry of fractures. A single-hole pneumatic injection test provides information for only a small volume of rock close to the borehole and therefore may fail to provide information for the rock heterogeneity. Therefore, cross-hole injection tests are preferred in which air is injected into an isolated interval within the injection well and also the other adjacent boreholes especially designed for this purpose. This facilitates the estimation of bulk pneumatic properties of larger rock volumes between observation boreholes and also the degree of interconnectivity on the scales ranging from meters to several tens of meters (Ilman and Neuman 2001).

The design and conduct of such tests and interpretation of test data by steady-state, transient type curve and asymptotic analysis from unsaturated fractured tuff at the Apache Leap Reserch Site (ALRS) near Superior, Arizona, USA are given by Ilman and Neuman (2001, 2003) and Ilman and Tartakovsky (2005). Later, Ilman and Tartakovsky (2006) suggested an asymptotic approach to analyze test data from cross-hole hydraulic tests in saturated fractured granite at the Grimsel Test Site in Switzerland.

9.2.6 Variable-Discharge (Step-drawdown) Test

In a variable-discharge (step-drawdown) test, the well is pumped in three or more steps with increase in rate of discharge, each step being of about 1 h duration. Drawdown measurements in the pumped well are taken at frequent intervals.

Step-drawdown tests are useful for the estimation of (1) aquifer transmissivity, and (2) hydraulic characteristics of wells, described in Sect. 17.5.

The drawdown in a pumping well at a given time, sw includes two components, the aquifer loss BQ and the well loss (CQn) (Fig. 9.39). Aquifer or formation loss (BQ) is due to Darcy type linear flow of water in the aquifer. Well losses (CQn) are both linear and non-linear. As the well loss is mainly due to turbulent flow it is expressed as CQn, where the value of n may vary from 1.5 to 3.5, depending on the extent of turbulence or rate of discharge. The value of n is usually taken to be 2, i.e.
Fig. 9.39

Schematic cross section of a fully penetrating well in a confined aquifer showing relation of well loss (CQ n ) and formation loss, BQ to total drawdown, S w

$$ {s_w} = BQ + C{Q^2} $$
(9.60)

where, sw = drawdown in pumping well; B = formation loss coefficient; BQ = formation loss; C = well loss coefficient; CQ2 = well loss.

Estimation of Transmissivity from Specific Capacity Data

Aquifer parameters viz. transmissivity and hydraulic conductivity can be estimated from specific capacity values (see Sect. 17.5) to avoid the cost of long duration pumping test (Walton 1962, Brown 1963, Mace 1997). Several graphs relating theoretical specific capacity with transmissivity for various values of S, T, and rw are given by Walton (1962). One such plot relating specific capacity to T and S for a given value of t and rw is illustrated in Fig. 9.40. If specific capacity has been determined from field (pumping) test, S is known a priori from either well log data or water-level fluctuations, and rw and t are known, transmissivity T can be computed by using such a graph. These graphs are based on the assumption that the well is 100% efficient and that the effective diameter is the same as the diameter of the well screen. The second assumption may hold good in consolidated rocks but in unconsolidated sediments, effective diameter is usually more than the diameter of the screen.
Fig. 9.40

Relationship between specific capacity, transmissivity and storativity based on Jacob’s non-equilibrium equation. (After Meyer 1963; Todd 1980)

The other approach to estimate T from specific capacity values is based on the Thiem’s equilibrium formula

$$ T = \frac{{2.3\,Q\;log\;{R/{{r_w}}}}}{{2\pi {s_w}}} = A({{Q/{{s_w}}}}) $$
(9.61)
where A is the dimensionless constant depending on the radius of influence, R and well radius, rw; Q/sw is the specific capacity of the well. Values of A usually range from 0.9 to 1.53 with a mean of 1.18 (Rotzoll and El-Kadi 2008). Values of A would also depend on the hydraulic characteristics of the aquifer and degree of confinement. It can also be estimated from values of rw and R. In cases where exact values of R are not available, approximate values of R can be used as given in Table 9.5 as value of A in Eq. 9.67 will not be much affected by poor estimates of R.
Table 9.5

Approximate values of radius of influence, R. (After UNESCO 1972)

Type of formation

Type of aquifer

R (m)

Fine and medium-grained sands

Confined

250–500

Unconfined

100–200

Coarse-grained sands and gravel-pebble beds

Confined

750–1500

Unconfined

300–500

Fissured rocks

Confined

1000–1500

Unconfined

500–1000

Huntley et al. (1992) have given an empirical relation Eq. 9.62 between T and specific capacity (Q/sw) for fractured rock aquifer

$$ T = A{\left( {{Q/{{s_w}}}} \right)^{0.12}} $$
(9.62)

In Eq. 9.62, both T and Q/sw are expressed in m2 d−1. Here it is assumed that well loss is negligible. This will result in the under-estimation of transmissivity in alluvial aquifers due to significant well loss.

Table 9.6 gives the summary of results obtained by several studies from fractured aquifers including methods of estimation of Sc and T. The table indicates that the values of A are site specific depending on hydrogeological characteristics of aquifer and method of determining Sc and T. Based on studies in the basalts of Hawaii islands, Rotzoll and El-Kadi (2008) concluded that correction for well loss significantly improves the correlation between T and Sc. In fractured rocks, due to larger effective radius of well and shorter duration of pumping, one will expect overestimation of transmissivity based on specific capacity values (Razack and Huntley 1991). Further, the value of T estimated from specific capacity data applies only to the aquifer adjacent to the well which is affected by drilling and well development. Therefore, values of T estimated from specific capacity data are greater than those obtained from aquifer tests.
Table 9.6

Summary of results giving empirical relationships between transmissivity and specific capacity. (Modified after Rotzoll and El-Kadi (2008))

Study (see Rotzoll and El-Kadi 2008)

Aquifer

Location

Methods to determine hydraulic parameters

Regression coefficients

Specific capacity

Transmissivity

A

Razack and Huntley (1991)

Heterogeneous alluvium

Haouz plain, Morocco

Constant-rate test (uncorrected)

Cooper-Jacob

15.30

Huntley et al. (1992)

Fractured rock

San Diego, California

Constant-rate test (uncorrected)

Cooper-Jacob (Neuman), Gringarten

0.12

Jalludin and Razack (2004)*

Sediment, fractured basalt

Djibuti, Horn of Africa

Step-drawdown test (corrected)

Cooper-Jacob, Boulton, Theis Recovery

3.64

Razack and Lasm (2006)*

Fractured rock

Man Danane, Ivory Coast

Step-drawdown test (corrected)

Theis Recovery

0.33

Eagon and Johe (1972)

Karst

NW-Ohio

Constant-rate test (corrected)

Cooper-Jacob

3.24

Mace (1997)

Karst

Edwards, Texas

All steps from step test (uncorrected)

Theis, Cooper-Jacob, Theis Recovery

0.76

Choi (1999)*

Volcanic island

Jeju, Korea

Constant-rate test (uncorrected)

Cooper-Jacob

0.45

Hamm et al. (2005)*

Volcanic island

Jeju, Korea

Constant-rate test (uncorrected)

Moench (leaky)

0.99

Adyalkar and Mani (1972)

Volcanic (Deccan trap)

India

Thiem method

 

0.37–0.63

Fernandopulle et al. (1974)

Volcanic rock

Gran Canaria Spain

  

0.39

Regression coefficients are for SC in m2/d

* For these references, see Rotzoll and El-Kadi (2008)

9.2.7 Hydraulic Tomography (HT)

HT is a sequential cross-hole hydraulic test followed by inversion of all the data to map the spatial distribution of aquifer hydraulic properties depending on the distribution and connectivity of fractures, which is important for water resources management and also for groundwater contamination, prevention and remediation (Hao et al. 2008). It provides more information than a classical pumping test by providing reasonable estimates from the same well field. HT is an application of the concept of Computerized Axial Tomography (CAT) in medical sciences and tomographic surveys in geophysics for imaging subsurface hydraulic heterogeneity.

HT involves installation of multiple wells in an aquifer which are partitioned into several intervals along the depth using packers. A sequential aquifer test at selected intervals is conducted by injecting or pumping at selected intervals and its response is noted in this well and the other observation wells. The test is repeated by pumping from another interval and also from intervals in other wells (Yeh and Liu 2000; Yeh and Lee 2007).

The results of a sequential pumping test in the alluvial formation near Naples, Italy are discussed by Straface et al. (2007). The test data were analysed by both the conventional type curve and straight line methods using distance-drawdown and time-drawdown data. They also used HT technique for characterizing the aquifer and concluded that the HT technique provides useful information about the heterogeneity pattern giving spatial distribution of hydraulic properties over a large volume of geologic material without resorting to a large number of wells.

9.3 Summary

Estimation of aquifer parameters (T and S) is one of the most difficult tasks. Laboratory methods are simple but they do not provide realistic values due to the limited sample size and formation heterogeneities in the field. Therefore field methods which include pumping test and tracer test are preferred. Field tests are expensive and therefore need proper planning about the design and location of observation wells, rate of pumping and duration of test etc. A prior knowledge of the hydrogeology of the area is necessary in planning these tests. The choice of a particular method also depends on the purpose of study.

For hydrogeologic characterisation of fractured rocks, double-porosity model is generally considered more realistic. Cross-hole tests are preferred in both saturated and unsaturated fractured rocks for ascertaining the spatial distribution of hydraulic properties. Lately, sequential pumping test (hydraulic tomography) involving pumping/injection of water at different depths and measurement of corresponding responses at various intervals is found more useful.

Further Reading

  1. Boonstra J, Soppe R (2007) Well hydraulics and aquifer Tests, in Handbook of Groundwater Engineering (Dlleur JW ed.). 2nd ed., CRC Press, Boca Raton.Google Scholar
  2. Kresic N (2007) Hydrogeology and Groundwater Modeling. 2nd ed., CRC Press, Boca Raton, FL.Google Scholar
  3. Kruseman GP, de Ridder NA (1990) Analysis and Evaluation of Pumping Test Data. 2nd ed., Intl. Inst. for Land Reclamation and Improvement, Publ. No. 47, Wageningen.Google Scholar
  4. Rushton KR (2003) Groundwater Hydrology: Conceptual and Computational Models. John Wiley & Sons, Chichester, UK.Google Scholar
  5. Schwartz FW, Zhang H (2003) Fundamentals of Ground Water. John Wiley & Sons Inc., New York.Google Scholar

Copyright information

© Springer Netherlands 2010

Authors and Affiliations

  • B. B. S. Singhal
    • 1
  • R. P. Gupta
    • 1
  1. 1.Department of Earth SciencesIndian Institute of Technology RoorkeeRoorkeeIndia

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