Applied Hydrogeology of Fractured Rocks pp 155192  Cite as
Estimation of Hydraulic Properties of Aquifers
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 TestHydraulic 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 grainsize 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 (d_{e})
where C is a coefficient based on degree of sorting (uniformity coefficient) and packing. If K is in centimetre per second and d_{e} 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.
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
where C_{T} is the pore tortuosity factor, and C_{0} is a pore shape factor; S_{sp}, 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 quasisteady 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 Quasisteady 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^{−} ^{10}m 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
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, S_{s}. 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 porewater 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
 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.
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.
Aperture size: As permeability is dependent on aperture size, an estimate from core samples obtained from deeply buried rocks will not be representative of insitu condition.
 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 insitu 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 kilometrerscale 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
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 km^{2}  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
 1.
Standard Lugeon test, which gives average hydraulic conductivity.
 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.
Crosshole 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.
where, K is hydraulic conductivity perpendicular to the axis of the bohrehole, in (m s^{−} ^{1}), Q is rate of inflow, in (m^{3} s^{−} ^{1}), L is thickness of the test zone, in (m), h_{0} and h are the piezometric heads in (m) measured in the test well and at distance (r) in the observation well, and r_{w} 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)
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
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 waterlevel conditions do not change due to injection. Errors in the evaluation of R do not affect test results very much as R >> r_{w}. Therefore,
As, lnR/r_{w} does not vary much, it can be assumed that (UNESCO 1984b)
Therefore, for numerical computations, Eq. 9.5 can be rewritten as
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, a_{eq}, can be determined from the test results using Cubic law (Novakowski et al. 2007).
The vertical distribution of T and a_{eq} can be determined by using packers at the desired levels.
 1.
The test should be carried out in a saturated zone.
 2.
Each test at each pressure is continued until steady state conditions are reached.
 3.
The test may be repeated with two or three increasing steps of pressure but care should be taken that it may not cause fracturedilation or hydrofracturing.
 4.
Pressure is measured in piezometres around the test hole.
Hydraulic conductivity, can be expressed in Lugeon unit (L_{u}). 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.
or
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, K_{f} can be expressed by
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.
Modified Lugeon Test
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.
 1.
Turbulence effect
 2.
Deformation of the medium due to high injection pressure
 3.
Influence of other fracture sets, i.e. effect of F_{2} and F_{3} when test hole is perpendicular to F_{1}
 4.
Entrance losses.
A Multifunction Bedrock Aquifer Transportable Testing Tool (BAT^{3}) has been developed at the USGS for carrying out a singlehole 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 discreteinterval groundwater samples for chemical analysis. The water should be withdrawn in case of highly transmissive fractures and injected when fractures are less transmissive. BAT^{3} 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 waterlevel with time. Alternatively a known volume of water can be withdrawn and rise of waterlevel 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 doubleporosity 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 largescale 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)
where b is dimensionless time factor, K_{f} is radial component of the hydraulic conductivity, b is effective screen length, t is the total time since the start of the test, and r_{c} 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, r_{c}.
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 waterlevel 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 timedrawdown 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
The change in waterlevel with time is given by Eq. 9.13.
where,
Values of function F(a, b) against a for different values of a and b are plotted on semilog paper as family of type curves (Fig. 9.9). The value of h_{0} is calculated from Eq. 9.12 based on the volume of water injected into or withdrawn from the well. Ratio of h_{t}/h_{0} is calculated for different values of t.
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 shutin 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} m^{2}s^{−} ^{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 insitu 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 (h_{t}/h_{0} vs. t) plotted on semilogarithmic 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 K_{h}/K_{v} 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 nearequilibrium 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.
Airpressurized slug tests are carried out by pressurizing the air in the casing above the waterlevel in a well and the declining waterlevel in the well is noted. Afterwards the air pressure is suddenly released to monitor the rise in waterlevel. T and S can be estimated from the rising waterlevel 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, waterlevel is lowered and a cone of watertable depression in an unconfined aquifer and cone of pressure relief in a confined aquifer is formed. The difference between the static (nonpumping) and pumping waterlevel 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 nearcircular. 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).
Spread of the cone of depression in metres. (After Schoeller 1959) (T = 1.25 10^{−} ^{3}m^{2} 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 subsurface 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 timedrawdown 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 waterlevel 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 waterlevel measuring device, viz. automatic waterlevel 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.
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
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 waterlevels 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 waterlevel is stabilised so that both the nonsteady and steadystate 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. Waterlevel measurements can be made by using steel tape or electrical depth sounder. The automatic waterlevel recorders are better for obtaining continuous waterlevel records in observation wells. Fully automatic microcomputer—controlled systems are also now available for accurate recordings of waterlevels.
As drawdown is faster in the observation wells close to the pumped well during early part of the test, waterlevel 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, waterlevel 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, waterlevel measurements during this period should be taken at shorter time intervals; later the time interval can be increased. The waterlevels 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
Aquifer Types
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 timedrawdown 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 semilog graphs (Fig. 9.15c, c’).
Fractured Rock Aquifers
Fractured rocks show various types of drawdown and pressure buildup 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 Tvalue, an increase in the drawdown slope with time is observed.
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 pseudoradial (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 timedrawdown 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 pseudoradial which is reflected by a straightline segment in the semilog 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 timedrawdown data are of importance to distinguish between different aquifer types.
 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 waterlevel in the fracture system.
 2.
Decrease of the storage capacity of fractures due to their deformation as a result of pumping.
 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 waterwells 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.

Thiem’s equilibrium method

Theis nonequilibrium type curve method

Jacob’s nonequilibrium 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

PapadopulosCooper 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 steadystate 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 steadystate flow for a confined aquifer can be written as
After the system has reached steadystate (equilibrium) condition, the drawdown data from different observation wells are plotted on a semilogarithmic paper. (drawdown, s_{1} and s_{2} on arithmetic scale and distances, r_{1}, r_{2}… on logarithmic scale). The slope Δs of this line is determined and is substituted in Eq. 9.17 for estimating T.
where Ds is the difference of drawdown per log cycle of r.
Theis Nonequilibrium 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
where,
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
where
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.
Various combinations of type curves and data curves in Theis method of analysis
Type curve  Data curve 

W(u) vs. u  s vs. r^{2}/t or s vs. 1/t 
W(u) vs. 1/u  s vs. t/r^{2} 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.
In decimal logarithms, Eq. 9.22 reduces to
which is known as the Jacob or TheisJacob 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
or
where Ds is the change in drawdown over one log cycle of time.
The storativity, S can also be determined from the same semilog plot by extending the timedrawdown curve to zero drawdown axis. Solving for storativity S, the equation in its final form becomes
where t_{0} 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. Sanchezvilla 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 waterlevels 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
where
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
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
where Ds’ is the change in residual drawdown per log cycle of t/t’.
Leaky (Semiconfined) 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 semiconfined aquifers.
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 steadystate condition. Under unsteadystate 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
or
where
or
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.
Unconfined Aquifers
The timedrawdown curve in response to pumping from an unconfined aquifer is characteristically Sshaped 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,
where W(U_{ay}, r/D) is called the ‘well function of Boulton’. Under earlytime conditions, describing the first segment of the timedrawdown curve, Eq. 9.36 can be written as
where
and S_{A} = early time storage coefficient.
Under latertime conditions for the third segment of the timedrawdown curve, Eq. 9.36 reduces to
where
and, S_{y} = specific yield. D is called the drainage factor, which is defined as
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 semiempirical quantities as Boulton’s delay index, l/a. It also takes into account aquifer anisotropy. Neuman’s drawdown equation can be written as
where, W(U_{A}, U_{B}, b) = well functions for watertable aquifer with fully penetrating wells having no storage capacity (dimensionless).
Under early time conditions Eq. 9.42 reduces to
where
S_{A} = Elastic earlytime storativity.
Under latetime conditions the third segment of the timedrawdown curve, Eq. 9.42 reduces to
Neuman’s parameter b is defined as
where K_{v} and K_{h} are the vertical and horizontal hydraulic conductivities of the aquifer respectively and Q, T, R, t, S_{y}, and b are as defined earlier.
For an isotropic aquifer K_{v} = K_{h}. Therefore,
The aquifer characteristics can be determined by typecurve and straightline 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
where r_{r} is the distance of the observation well from the pumping well and r_{i}, t_{i} and t_{r} are as defined earlier.
After the values of r_{i} 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 r_{i}. 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.
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 largediameter 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
where,

r_{w} = radius of the well screen or open well

r_{c} = radius of the well casing over which the waterlevel is changing.
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 timedrawdown 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 m^{2}d^{−} ^{1} and r_{c} 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 waterlevel 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 interconnected, 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 (s_{f}) is given by Eq. 9.51
where
and
Subscripts f and m represent properties of fractures and matrix blocks respectively, B_{f} is the drainage factor. Accordingly, the equation for drawdown in matrix block, s_{m} is also developed.
Values of the drawdown function W(u_{f}, r/B_{f}, n) computed for n = 10, 100, and 1000 and various assumed values of parameters r/B_{f} and 1/u_{f} 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.
and C_{f} (hydraulic diffusivity of the fissure)
The equations given by Streltsova–Adams for doubleporosity 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 timedrawdown 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.
where

x_{f} = 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).
Type curves of drawdown function F(u_{vf} r’) for different values of U_{vf} 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 timedrawdown 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.
The timedrawdown 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 timedrawdown data from observation wells in a dyke using a typecurve method developed by Boonstra and Boehmer (1986). A distancedrawdown 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 Crosshole 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. Crosshole 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.
In unsaturated fractured rocks also, singlehole and crosshole 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 singlehole 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, crosshole 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 steadystate, 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 crosshole hydraulic tests in saturated fractured granite at the Grimsel Test Site in Switzerland.
9.2.6 VariableDischarge (Stepdrawdown) Test
In a variabledischarge (stepdrawdown) 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.
Stepdrawdown tests are useful for the estimation of (1) aquifer transmissivity, and (2) hydraulic characteristics of wells, described in Sect. 17.5.
where, s_{w} = drawdown in pumping well; B = formation loss coefficient; BQ = formation loss; C = well loss coefficient; CQ^{2} = well loss.
Estimation of Transmissivity from Specific Capacity Data
The other approach to estimate T from specific capacity values is based on the Thiem’s equilibrium formula
Approximate values of radius of influence, R. (After UNESCO 1972)
Type of formation  Type of aquifer  R (m) 

Fine and mediumgrained sands  Confined  250–500 
Unconfined  100–200  
Coarsegrained sands and gravelpebble 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/s_{w}) for fractured rock aquifer
In Eq. 9.62, both T and Q/s_{w} are expressed in m^{2} d^{−1}. Here it is assumed that well loss is negligible. This will result in the underestimation of transmissivity in alluvial aquifers due to significant well loss.
Summary of results giving empirical relationships between transmissivity and specific capacity. (Modified after Rotzoll and ElKadi (2008))
Study (see Rotzoll and ElKadi 2008)  Aquifer  Location  Methods to determine hydraulic parameters  Regression coefficients  

Specific capacity  Transmissivity  A  
Razack and Huntley (1991)  Heterogeneous alluvium  Haouz plain, Morocco  Constantrate test (uncorrected)  CooperJacob  15.30 
Huntley et al. (1992)  Fractured rock  San Diego, California  Constantrate test (uncorrected)  CooperJacob (Neuman), Gringarten  0.12 
Jalludin and Razack (2004)*  Sediment, fractured basalt  Djibuti, Horn of Africa  Stepdrawdown test (corrected)  CooperJacob, Boulton, Theis Recovery  3.64 
Razack and Lasm (2006)*  Fractured rock  Man Danane, Ivory Coast  Stepdrawdown test (corrected)  Theis Recovery  0.33 
Eagon and Johe (1972)  Karst  NWOhio  Constantrate test (corrected)  CooperJacob  3.24 
Mace (1997)  Karst  Edwards, Texas  All steps from step test (uncorrected)  Theis, CooperJacob, Theis Recovery  0.76 
Choi (1999)*  Volcanic island  Jeju, Korea  Constantrate test (uncorrected)  CooperJacob  0.45 
Hamm et al. (2005)*  Volcanic island  Jeju, Korea  Constantrate 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 
9.2.7 Hydraulic Tomography (HT)
HT is a sequential crosshole 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 distancedrawdown and timedrawdown 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, doubleporosity model is generally considered more realistic. Crosshole 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
 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
 Kresic N (2007) Hydrogeology and Groundwater Modeling. 2nd ed., CRC Press, Boca Raton, FL.Google Scholar
 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
 Rushton KR (2003) Groundwater Hydrology: Conceptual and Computational Models. John Wiley & Sons, Chichester, UK.Google Scholar
 Schwartz FW, Zhang H (2003) Fundamentals of Ground Water. John Wiley & Sons Inc., New York.Google Scholar