Some analytical solutions for sensitivity of well tests to variations in storativity and transmissivity

Advances in Water Resources 28 (2005) 1057–1075 www.elsevier.com/locate/advwatres

Some analytical solutions for sensitivity of well tests to variations in storativity and transmissivity J.H. Knight

a,*

, G.J. Kluitenberg

b

a

b

CSIRO Land and Water, 120 Meiers Road, Indooroopilly, Qld 4068, Australia Department of Agronomy, 2004 Throckmorton Hall, Kansas State University, Manhattan, KS 66506-5501, USA Received 6 August 2004; accepted 17 August 2004 Available online 15 June 2005

Abstract The theory of a pumping test or a slug test to measure aquifer transmissivity or storativity assumes that the aquifer properties are uniform around the well. The response of the drawdown to small spatial variations in aquifer properties in the volume of influence is determined by spatial weighting functions or Fre´chet kernels, which in general are functions of space and time. The Fre´chet kernels determine the effective ‘‘volume of influence’’ of the measurements at any time. Under the assumption that the well is a line sink we derive explicit analytical expressions for the Fre´chet kernels for storativity and for transmissivity for both pumping and slug tests. We also derive the total sensitivity functions for uniform variations in storativity and transmissivity and show that they are the spatial integrals of the Fre´chet kernels. We consider both the case of separate pumping and observation wells and also the radially symmetric case of observations made at the pumped or slugged well. The ‘‘volume of influence’’ is symmetric with respect to the pumping or slugged well and the observation well, and far from the well the contours of equal spatial sensitivity approach the shapes of ellipses with a well at each focus, rather than circles centered on the pumping well. We use the analytical solutions to investigate the nature of the singularities in the spatial sensitivity functions around the wells, which govern the importance of inhomogeneities close to the well or observation point. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Groundwater; Pumping test; Aquifer heterogeneity; Measurement sensitivity

1. Introduction The traditional method of measuring a transport coefficients of a medium in situ is to perform a transport experiment at a known position with a known input and to measure the output. Examples of such techniques are pumping tests to measure aquifer transmissivity, heat pulse methods to measure thermal diffusivity, direct current tests to measure resistivity of the ground, and time or frequency domain electromagnetic methods for measuring soil dielectric properties and hence water content. *

Corresponding author. E-mail addresses: [email protected] (J.H. Knight), gjk@ksu. edu (G.J. Kluitenberg). 0309-1708/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2004.08.018

Under the assumption that the transport coefficient is spatially uniform, an analytical solution of the ‘‘forward problem’’ is used to infer the coefficient from the known input and measured output. It is natural to ask what is the effective measurement volume [1,2,12,19], and in the presence of inhomogeneities what averaging process does the method use to give a single value of the effective coefficient [34,31,37,49,3,35,52,14]? A traditional way to investigate this question is to derive a numerical or analytical forward solution for the output from a system with a known input and a small anomaly of known size and shape with uniform properties differing from those of the rest of the medium [50,6,7,15,16]; another is to solve the forward problem for an input a known distance from a planar boundary

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separating two uniform regions with different properties [42,27]. A more general approach is to assume a general spatial distribution of transport coefficients which differs only slightly from some other distribution (for example the uniform case) corresponding to a known output, and to use a perturbation expansion to find the resultant small change in the measured output. The small change in output is calculated using spatial convolution integrals of ‘‘spatial weighting functions’’ with the actual variations of the transport parameters [29,38,26]. The procedure for calculating these weighting functions and hence sensitivity coefficients can be given a rigorous mathematical formulation in terms of Fre´chet derivatives, with the weighting functions appearing in the integrals as Fre´chet kernels [40,41]. The drawdown in an unconfined aquifer, and the pressure in a confined aquifer or an oil reservoir satisfy a heat or diffusion equation. In this paper we will work with the equation for drawdown in an unconfined aquifer with spatially dependent storativity and transmissivity. With a minor reformulation the theory also applies to the confined flow aquifer and to the oil reservoir. An electric circuit analog of the finite difference approximation for the equations for pressure change in an oil reservoir was used [21,22] to derive a finite difference formulation of the spatial sensitivity. The diffusion equation has also been used [10] to derive an equivalent mathematical formulation for the sensitivity coefficients in a reservoir. McElwee and Yukler [34] started an extensive investigation of the sensitivity of groundwater models to variations in transmissivity and storage. Oliver [37] studied the averaging process in permeability estimation from well-test data when the observations are made at the well and storativity is taken to be uniform. For this problem he derived the Laplace transform of the spatial weighting function and expressions for the weighting function in terms of Whittaker functions and also integrals of Bessel functions. The Fre´chet derivatives can be expressed as spatial integrals over the flow domain of the product of the variations in medium properties and appropriate spatial weighting functions, which are called the Fre´chet kernels. In a landmark paper Oliver [38] extended his previous work to the influence of nonuniform transmissivity and storativity on drawdown at observation wells at different positions from the test well, and using the Theis [48] solution for drawdown he derived expressions for the Fre´chet kernels as convolution integrals in the time domain. In general these integrals must be evaluated numerically. McElwee et al. [32] investigated the sensitivity of observations at a slugged well to variations in transmissivity and storativity using a flow model with a well of finite radius, and McElwee et al. [33] studied the analogous problem for measurements at an observation well some distance from the slugged well. Jiao and Rushton

[24] studied sensitivity to parameters for large-diameter wells, and Jiao and Zheng [23,25] discussed the properties of the sensitivity functions for constant rate pumping in well tests. Sa´nchez-Vila and colleagues [45] derived higher order perturbations, and used approximations for the Laplace transforms to derive approximate forms of the Fre´chet kernels valid at large times. Romeu and colleagues [43] showed how to use Laplace transforms to calculate the permeability weighting function in a general geometry, and inverted a Laplace transform solution of [37] to find a simple formula in the time domain for the radially symmetric permeability weighting function for a pumping test with observations at the pumping well. Using Laplace transforms they solved several example problems, and recommended that the Stehfest [46,47] method be used to numerically invert any Laplace transforms lacking known analytical inverses. In this paper we will use the Laplace transform approach. Kabala [26] gave a comprehensive account of a number of approaches to sensitivity analysis, and analyzed a more complicated well problem in depth. He also worked in the Laplace transform domain, and used the more complicated but more robust and accurate de Hoog et al. algorithm [13] to numerically invert the Laplace transforms. Oliver
s method [38] with sensitivity coefficients expressed as convolution integrals in the time domain was recently used by Leven [30] to calculate the sensitivity coefficients for hydraulic tests with various configurations of wells. He also assessed the effects of anisotropy in the flow domain. The aim of this paper is to extend previous work [38,43] by deriving simple mathematical formulae in terms of well-known Bessel functions for the Fre´chet kernels for pumping tests and slug tests. These are appropriate for wells that can be treated as line sinks or sources. Unlike most previous time domain expressions for the kernels, our formulae will not need to be calculated using numerical evaluation of integrals. We will take advantage of the explicit Laplace transform inversions and use well-known properties of Bessel functions to investigate the behaviour of the Fre´chet kernels, and to explore their physical significance.

2. Spatial sensitivity functions 2.1. Well flow system Consider a fully penetrating well at position (x00 , y00 ) in an unconfined aquifer with position dependent storativity and transmissivity S(x) and T(x) respectively. The drawdown h(x, t) evolves according to the equation SðxÞ

oh ¼ r
½T ðxÞrh; ot

ð1Þ

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

where t is time and x = (x, y) is the position vector in two dimensions. In the manner of [18] we treat the well as a line sink. The sink strength q(t) may vary with time—a constant sink strength q0 corresponds to a pumping test, and a delta function sink strength q(t) = Q0d(t) corresponds to a slug test. Connell [11], for example, considered pulses of finite duration. In the case that the storativity and transmissivity have spatially uniform values S0 and T0 respectively, Eq. (1) becomes oh S 0 ¼ T 0 r2 h. ð2Þ ot The solution at (x, y) for zero initial drawdown and constant sink strength q0 at (x00 , y00 ) is the well-known Theis [48] solution,
2 q0 S 0 r1 E1 hðx; tÞ ¼ ; ð3Þ 4pT 0 4T 0 t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r1 ¼ ðx  x00 Þ2 þ ðy  y 00 Þ2 is the radial distance from the well and E1(z) is the exponential integral function with argument z [17]. In order to calculate the sensitivity of the solution of Eq. (1) to small variations in storativity and transmissivity we follow [43] and work in the Laplace transform domain. If f(t) is a function of time t we define the Laplace transform f^ ðpÞ of f by Z 1 f^ ðpÞ ¼ f ðtÞ expðptÞ dt; ð4Þ 0

where p is the Laplace transform variable. The Laplace transform of Eq. (1) for zero initial conditions is pSðxÞ^ h ¼ r
½T ðxÞr^ h; ð5Þ ^ with hðx; pÞ denoting the Laplace transform of h(x, t). When storativity and transmissivity have uniform values S0 and T0, Eq. (5) is transformed to the modified Helmholtz equation p^h ¼ Dr2 ^ h;

ð6Þ

where D = T0/S0 is the diffusivity. The solution of Eq. (6) corresponding to a line source of constant strength q0 at (x00 , y00 ) is given [9, p. 385] as
rffiffiffiffi  q0 p ^ K0 r1 ; hðr1 ; pÞ ¼ ð7Þ D 2ppT 0 where Kn(z) denotes the modified Bessel function of the second kind of order n and argument z. This is the Laplace transform of the Theis [48] solution Eq. (3), a result which can easily be obtained using formula (26) of [9, p. 495]. The solution Eq. (7) corresponds to a constant pumping rate q0 with Laplace transform q0/p; a simple generalisation gives the transform for general pumping rate q(t) with Laplace transform ^ qðpÞ as
rffiffiffiffi  ^ p qðpÞ ^ hðr1 ; pÞ ¼ K0 r1 . ð8Þ D 2pT 0

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2.2. Fre´chet derivatives To investigate the effects of the spatial distribution of inhomogeneities, we assume that the storativity and transmissivity vary with position, but differ only slightly from the uniform values S0 and T0, and that they can be expressed as SðxÞ ¼ S 0 þ eS 1 ðxÞ

and

T ðxÞ ¼ T 0 þ eT 1 ðxÞ;

ð9Þ

with e a small number. We also assume that the corresponding solution of Eq. (5) can be written as the sum of a zero order term which is a solution of Eq. (6) and a perturbation term of order e, so that hðx; tÞ ¼ h0 ðx; tÞ þ eh1 ðx; tÞ

ð10Þ

for small e. Physically, we are assuming that the pumping well sends out a drawdown signal, which spreads out radially from the well with a decreasing velocity and amplitude. A small proportion of the signal is ‘‘scattered’’ from the small nonuniformities in the storativity and transmissivity fields, and a new signal spreads out from each scatterer with a nonuniform velocity and decreasing amplitude. At any time t some of the combined scattered signals from all of the anomalies reaches an observation point as the perturbation term h1. The form of Eq. (10) assumes that we can ignore second order scattering of the previously scattered signal. In the special case that the changes in storativity and transmissivity are spatially uniform, eS1 = dS and eT1 = dT, the corresponding change in drawdown at the observation point, h1(x, t) in (10), can also be written in terms of sensitivity coefficients (oh0/oS)(x, t, S0, T0) and (oh0/oT)(x, t, S0, T0) as eh1 ðx; tÞ ¼ dS

oh0 oh0 þ dT . oS oT

ð11Þ

For the use of sensitivity coefficients see for example [51,26]. In the case of the Theis [48] well function differentiating (3) with respect to S for example, shows that the sensitivity coefficient for storativity for a pumping test is
 oh0 q0 S 0 r21 ðx; t; S 0 ; T 0 Þ ¼  exp  . ð12Þ oS 4pS 0 T 0 4T 0 t It can be seen that the sensitivity coefficient depends on the relative positions of the pumping well and observation point, and varies with time. The quantity h1(x, t) in (10) is called the Fre´chet derivative of h(x, t) with respect to the functions S(x) and T(x), the functional derivative being evaluated at the constant functions S(x) = S0 and T(x) = T0. For the properties of Fre´chet derivatives see for example [40,41]. The Fre´chet derivative h1(x, t) generalises the sensitivity coefficients for the dependence of h(x, t) on a parameter S with one degree of freedom, to a spatial weighting function which describes the dependence of

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h(x, t) on a function S1(x) of position with an uncountably infinite number of degrees of freedom. We wish to express the Fre´chet derivative as a spatial convolution integral in terms of the known functions S1(x) and T1(x) as Z S 1 ðx0 ÞF S ðx  x0 ; tÞ dx0 h1 ðx; tÞ ¼ A Z þ T 1 ðx0 ÞF T ðx  x0 ; tÞ dx0 ; ð13Þ A

depending on S1(x), T1(x) and the zero order solution ^h0 ðx; pÞ, pS 0 ^h1 ¼ T 0 r2 ^h1  pS 1 ðxÞ^h0 þ r
½T 1 ðxÞr^h0 .

ð19Þ

Because we assume that the boundary and initial conditions for h0 are the same as those for h, Eq. (19) has zero initial and boundary conditions for ^h1 . The solution of ^ 0 ; x; pÞ is (19) in terms of a Green
s function Gðx Z ^h1 ðx0 ; pÞ ¼  Gðx ^ 0 ; x; pÞfpS 1 ðxÞ^h0 ðx; pÞ A

where the area of integration A is the whole flow domain and the unknown functions FS(x, t) and FT(x, t) are called the Fre´chet kernels for storativity and transmissivity respectively. The Fre´chet kernels are the spatial weighting functions, functions of space and time that determine the weightings given to nonuniformities in storativity S and transmissivity T. It is easy to show the connection between the more traditional sensitivity coefficients and the Fre´chet kernels. We define the total sensitivities MS(x, t) and MT(x, t) at x as the spatial integrals over the whole flow region of the corresponding Fre´chet kernels, Z F S ðx  x0 ; tÞ dx0 and M S ðx; tÞ ¼ A Z M T ðx; tÞ ¼ F T ðx  x0 ; tÞ dx0 . ð14Þ A

Again considering the special case with changes in storativity and transmissivity spatially uniform, eS1 = dS and eT1 = dT, in this case Eq. (13) becomes Z Z eh1 ðx; tÞ ¼ dS F S ðx  x0 ; tÞdx0 þ dT F T ðx  x0 ; tÞdx0 . A

A

ð15Þ Comparing Eqs. (11) and (15) shows that Z oh0 ¼ M S ðx; tÞ ¼ F S ðx  x0 ; tÞ dx0 ; oS A and oh0 ¼ M T ðx; tÞ ¼ oT

Z

F T ðx  x0 ; tÞ dx0 .

ð16Þ

ð17Þ

A

The aim of this paper is to find explicit functional forms for the Fre´chet kernels in several cases of interest. We will work first in the Laplace transform domain, and then give explicit inversions of the transforms of the Fre´chet kernels. Taking the Laplace transform of Eq. (10) yields ^ hðx; pÞ ¼ ^ h0 ðx; pÞ þ e^ h1 ðx; pÞ.

ð18Þ

Substituting (9) and (18) into Eq. (5) and collecting zero order terms gives Eq. (6) satisfied by ^ h0 ðx; pÞ. Collecting first order terms in e gives the equation satisfied by ^ h1 ðx; pÞ, which is (6) with an extra ‘‘source term’’

 r
½T 1 ðxÞr^h0 ðx; pÞg dx;

ð20Þ

where the area of integration A is the whole flow do^ 0 ; x; pÞ is the main. By definition, Green
s function Gðx solution of Eq. (6) at position x 0 corresponding to a unit source at x. We wish to write Eq. (20) with the integral in the same form as in Eq. (1). A standard integration by parts using the Divergence Theorem and the properties of the divergence operator (see for example [20, p. 284]) gives Z ^h1 ðx0 ; pÞ ¼ p S 1 ðxÞGðx ^ 0 ; x; pÞ^h0 ðx; pÞ dx A Z ^ 0 ; x; pÞ
r^h0 ðx; pÞ dx.  T 1 ðxÞrGðx ð21Þ A

Taking the Laplace transform of Eq. (13) gives Z ^h1 ðx; pÞ ¼ S 1 ðx0 ÞF^ S ðx  x0 ; pÞ dx0 A Z þ T 1 ðx0 ÞF^ T ðx  x0 ; pÞ dx0 ;

ð22Þ

A

and comparing Eqs. (21) and (22) gives the Laplace transforms of the kernel for storativity F^ S ðx  x0 ; pÞ ^ 0 ; x; pÞ^ and transmissivity F^ T ðx  x0 ; pÞ as pGðx h0 ðx; pÞ 0 ^ ; x; pÞ
r^h0 ðx; pÞ respectively. In the transand rGðx form domain the kernels are products of known transforms of relatively simple functions; in the time domain these correspond to convolution integrals as used for example in [38,30]. In the time domain the storativity kernel FS is the negative of the convolution of the time derivative of the drawdown function with Green
s function, or equivalently the negative of the convolution of the drawdown function with the time derivative of Green
s function. This means that the spatial sensitivity for storativity depends on the rate of change of the imposed drawdown function. The transmissivity kernel FT is the negative of the combined convolution in the time domain and scalar product of the spatial gradient of the drawdown function with the spatial gradient of Green
s function. This means that the spatial sensitivity for transmissivity depends on the direction and magnitude of the spatial gradient of the imposed drawdown function. In this paper we will show how to derive explicit functions for a number of sensitivity kernels previously given

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

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only as convolution integrals. Without loss of generality we take the position of the well to be at x1 = (x, y) = (a/2, 0) and the observation point to be a distance a away at the position x2 = (x, y) = (a/2, 0), as shown in Fig. 1. Putting a = 0 gives the special case in which the observation point coincides with the well and the problem is radially symmetric. The Laplace transform of the drawdown at any point (x, y) due to the well at position (a/2, 0) is
rffiffiffiffi  ^ p qðpÞ ^ K0 h0 ðx; y; a=2; 0; pÞ ¼ r1 ; ð23Þ D 2pT 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where r1 ¼ ðx þ a=2Þ þ y 2 is the radial distance from

tion for the drawdown at the observation point (a/2, 0) due to an instantaneous source at any point (x, y) is
rffiffiffiffi  1 p ^ Gða=2; 0; x; y; pÞ ¼ K0 r2 ; ð24Þ 2pT 0 D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 with r2 ¼ ðx  a=2Þ þ y 2 the radial distance from the observation point. In the time domain Green
s function is of course the standard solution for the two dimensional heat Eq. (2)   1 r22 exp  Gða=2; 0; x; y; t0 ; tÞ ¼ ; 4pT 0 ðt  t0 Þ 4Dðt  t0 Þ

the well. In the Laplace transform domain Green
s func-

corresponding to the effect at position x1 = (a/2, 0) of an instantaneous source at time t 0 at position (x, y) [9, p. 258]. Substituting (23) and (24) into Eq. (21) and comparing with Eq. (22) gives expressions for the Laplace transforms of the Fre´chet kernels for a well at x1 = (a/2, 0) and an observation point at x2 = (a/2, 0). Henceforth we will understand that the Fre´chet kernels are for this geometrical configuration, without the dependence on a being explicitly indicated in the notation. For general pumping rate q(t) with Laplace transform ^qðpÞ we have the Laplace transforms of the storativity and transmissivity kernels expressed in terms of products of modified Bessel functions as
rffiffiffiffi 
rffiffiffiffi  p^q p p ^ F S ðx; y; pÞ ¼  2 2 K 0 r1 K 0 r2 ð26Þ D D 4p T 0

ð25Þ

and 
rffiffiffiffi  ^q o p K0 r1 2 2 D 4p T 0 or1 
rffiffiffiffi  o p  K0 r2 ðrr1
rr2 Þ. or2 D

F^ T ðx; y; pÞ ¼ 

Simplifying p^qðr2  a2 =4Þ F^ T ðx; y; pÞ ¼  K1 4p2 DT 20 r1 r2

ð27Þ


rffiffiffiffi 
rffiffiffiffi  p p r1 K 1 r2 D D ð28Þ

Fig. 1. Geometrical configuration of pumping well, observation point, and arbitrary point (x, y) for general and radially symmetric cases. For both cases, (x, y) represents the point at which the Fre´chet kernels FS(x, y, t) and FT(x, y, t) are calculated. The well and observation point are separated by distance a in the general case, with the well at x1 = (a/2, 0)qand the observation point at x2 = (a/2, 0). The position ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vectors r1 ¼

ðx þ a=2Þ2 þ y 2 and r2 ¼

ðx  a=2Þ2 þ y 2 define the

location of the point (x, y) relative to the pumping well and observation well respectively. The well and observation point coincide at (x, y) = (0, 0) for the radially symmetric case and the position vector pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y 2 defines the location of (x, y) relative to the origin.

withpr ffiffiffiffiffiffiffiffiffiffiffiffiffiffi standing for the radial distance from the origin r ¼ x2 þ y 2 in Eq. (28) and henceforth. If the positions of the well at (a/2, 0) and the observation point at (a/2, 0) are interchanged, then r1 and r2 must be interchanged in Eqs. (26) and (28). It is a consequence of the Reciprocity Theorem [4] that the values of the kernels are unaffected by this interchange. Recently Molz and colleagues [35] considered spatial weighting functions for the measurement of permeability during steady gas flow through porous media. They found that their spatial weighting function was proportional to the square of the gradient of the potential governing the flow, and hence to the rate of dissipation of

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energy in their system. A similar relation between weighting functions and energy dissipation was found for Time Domain Reflectometry (TDR) measurements of soil dielectric properties [29] and for TDR measurements of soil electrical conductivity [14]. In particular this relation requires that the weighting function have the same sign everywhere in the region. It was hypothesised by Molz and colleagues [35] that the proportionality between spatial weighting functions and energy dissipation rate may be a general phenomenon for spatial weighting functions. For flow around a well the energy dissipation is determined by the square of the spatial gradient of the drawdown h0 and the transmissivity. We therefore examine the form of the weighting function for transmissivity in Eqs. (21), (27) and (28). We find that when the pumping and observation wells coincide then r1 = r2 and Green
s function G and the drawdown h0 are essentially the same function. In this case the Laplace transform of the transmissivity kernel in Eqs. (21) and (27) is proportional to the square of the spatial gradient of the drawdown, and hence is related to the rate of energy dissipation. However when the pumping and observation wells do not coincide, the gradients of the drawdown and Green
s function have a more complicated relation. Their scalar product is zero on the circle r = a/2, negative inside the circle and positive outside it. As pointed out by Oliver [38], when the pumping and observation wells are not coincident this means that the transmissivity kernel is zero on the circle r = a/2 passing through both wells, positive inside the circle and negative outside. The hypothesis of [35] is not relevant for the weighting function for storativity, and holds for the weighting function for transmissivity only in the case that the observations are made at the pumping well.

3. Explicit inversions In this section we will demonstrate that the above transforms can be inverted explicitly for some pumping rate functions of practical importance. We will give explicit inversions of the transforms Eqs. (26) and (28) for three different forms of the pumping rate function q(t). The first case we will consider is a pumping test with pumping at a constant rate q0, and the second is a slug test with a delta function input Q0d(t). The third case is a slug test with constant rate pumping for a finite time t0. A comprehensive list of known Laplace transforms and their inverses has been published by [36], with later corrections [28]. We will use the result 13.97 of [36, p. 347] that the inverse of the transform expressed in our notation as
rffiffiffiffi 
rffiffiffiffi  p p ^ f ðpÞ ¼ K n r1 K n r2 ð29Þ D D

is the function
2   1 r1 þ r22 r1 r2  Kn . f ðtÞ ¼ exp  2t 4Dt 2Dt

ð30Þ

This will give simple expressions for the Fre´chet kernels in terms of products of exponential functions with modified Bessel functions of orders zero and one. In our case r21 þ r22 ¼ 2ðr2 þ a2 =4Þ and 2

2r1 r2 ¼ 2½ðr2 þ a2 =4Þ  a2 x2 

1=2

;

ð31Þ

and so the argument of the exponential function in Eq. (30) is radially symmetric but the argument of the Bessel function is not. The quantities r21 þ r22 and 2r1r2 are equal on the line x = 0. 3.1. Pumping test 3.1.1. General case Aspects of the spatial sensitivities of pumping tests have been studied by a large number of authors [5,8,37,38,7,23,24,43,45,26]. We assume that there is pumping at a constant rate q0, so that the Laplace transform of the pumping rate is ^qðpÞ ¼ q0 =p. The special case when the observation point and the well coincide was considered for spatially variable transmissivity in [37]; the general case with separate pumping and observation wells and inhomogeneities in both storativity and transmissivity was studied in [38]. For the special case he gave inversions of the Laplace transforms in terms of integrals of products of Bessel functions with exponential functions, and in the general case he derived formulae for the Fre´chet kernels as convolution integrals of Green
s functions in the time domain. All of these integrals must be evaluated numerically. For the case of observations at the pumping well a simple inversion for the Laplace transform and hence a simple form for the Fre´chet kernel for transmissivity was given in [43]. We will give explicit inversions for the general case of separate pumping and observation wells; putting a = 0 in these formulae gives the results for observations at the pumping well. The transforms of the Fre´chet kernels for storativity and transmissivity are
rffiffiffiffi 
rffiffiffiffi  q0 p p ^ F S ðx; y; pÞ ¼  2 2 K 0 r1 K 0 r2 ð32Þ D D 4p T 0 and q ðr2  a2 =4Þ F^ T ðx; y; pÞ ¼  0 2 2 K1 4p DT 0 r1 r2


rffiffiffiffi 
rffiffiffiffi  p p r1 K 1 r2 D D ð33Þ

respectively. Using Eqs. (29)–(31) with n = 0 and n = 1 gives the simple functional forms of the Fre´chet kernels as
2  r r  q r þ a2 =4 1 2 F S ðx; y; tÞ ¼  2 0 2 K 0 ð34Þ exp  2Dt 2Dt 8p T 0 t

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

for storativity and
2  q0 ðr2  a2 =4Þ r1 r2  r þ a2 =4 K1 F T ðx; y; tÞ ¼  2 2 exp  2Dt 2Dt 8p DT 0 r1 r2 t ð35Þ for transmissivity. Calculating the partial derivatives of (3) with respect to S0 and T0 respectively and using Eqs. (16) and (17) gives the total sensitivities at the observation well for storativity and transmissivity as Z 1 Z 1 F S ðx; y; tÞ dx dy M S ða=2; 0; tÞ ¼ 1 1
 q0 S 0 a2 ¼ exp  ð36Þ 4pS 0 T 0 4T 0 t and M T ða=2; 0; tÞ ¼

Z

1

Z

product of the gradients $r1 Æ $r2 = (r2  a2/4)/(r1r2) in Eqs. (27) and (28). This means that a small anomaly in transmissivity exactly on the circle has no effect; an anomaly with increased transmissivity inside the circle increases the drawdown at the observation well; and an anomaly with higher transmissivity outside the circle decreases the drawdown seen at the observation well. For this purpose, an anomaly with lower than the background transmissivity that is situated anywhere inside the circle counts as a ‘‘blockage’’ restricting flow between the two wells, and the same anomaly anywhere outside the circle acts as an external barrier restricting flow away from the system and therefore enhancing flow between the wells. To investigate the behaviour of the kernels we need the Bessel function properties given as 9.6.8, 9.6.9 and 9.7.2 in [39]. For small z,

1

F T ðx; y; tÞ dx dy 
2
 q S0a S 0 a2 ¼  0 2 E1 þ exp  . 4T 0 t 4T 0 t 4pT 0 1

1063

K 0 ðzÞ   ln z

and

K 1 ðzÞ  z1 ;

ð42Þ

1

ð37Þ The total sensitivities are well defined for a P 0 and for a > 0 respectively. We use the properties of the exponential integral function as given by 5.1.11 and 5.1.51 in [17]. For small z, E1 ðzÞ ¼  ln z  c;

ð38Þ

where c = 0.57721... is Euler
s constant, and for large z E1 ðzÞ  ð1=zÞ expðzÞ.

ð39Þ

When a > 0 both total sensitivities at the observation well are zero at time zero. As time increases both monotonically decrease (become more negative), with the total sensitivity for storativity having the finite limit q0 ð40Þ M S ða=2; 0; 1Þ ¼  4pS 0 T 0 and total sensitivity for transmissivity decreasing to 1, q M T ða=2; 0; tÞ   0 2 ln t ð41Þ 4pT 0 for large t. At all times the storativity kernel is negative everywhere, which means that a storativity anomaly anywhere in the otherwise uniform flow field causes an effect of the opposite sign to the difference in storativity; an anomaly with lower storativity causes an increased drawdown at the observation well and an anomaly with elevated storativity a decrease in observed drawdown. As discussed in [38,30], the transmissivity kernel is zero on the circle r2  a2/4 = 0, positive inside the circle, and negative outside, as determined by the sign of the scalar

and for large z, rffiffiffiffiffi p K n ðzÞ  expðzÞ. 2z

ð43Þ

At any fixed time t both of the kernels FS(x, y, t) and FT(x, y, t) considered as a functions of (x, y) have singularities at x1 = (a/2, 0) where r1 = 0 and at x2 = (a/2, 0) where r2 = 0. The functions approach zero far from the wells as r ! 1. The singularities at the pumping well and the observation well mean that the spatial sensitivities are very high near each well, and any anomalies there will have a great effect. At any fixed position (x, y) not at a singularity, the kernels FS(x, y, t) and FT(x, y, t) can be considered as functions of time t. Both of the kernels are zero at time zero. Using Eq. (43) it can be shown that for large r2/Dt (and hence for large r1 and r2 or for small time), the kernels behave like " # 2 q0 D1=2 ðr1 þ r2 Þ F S ðx; y; tÞ   3=2 2 pffiffiffiffiffiffiffiffiffi exp  ð44Þ 4Dt 8p T 0 r1 r2 t and " # 2 q0 ðr2  a2 =4Þ ðr1 þ r2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi exp  F T ðx; y; tÞ   3=2 2 4Dt 8p T 0 r1 r2 Dr1 r2 t ð45Þ respectively. The exponential factor causes the kernels to decay rapidly with distance from the wells, and at fixed time the argument of the exponential function depends only on the sum r1 + r2. A curve on which r1 + r2 is constant is an ellipse with a focus at each well, so in the x–y plane far from the wells we expect the contours of equal spatial sensitivity to approach the shapes of a family of ellipses with a focus at each well.

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J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

Using Eq. (42) it can be shown that the magnitude of the storativity kernel decreases slowly to zero at large times; for every (x, y) it has the behaviour F S ðx; y; tÞ  

q0 ln t 8p2 T 20 t

M T ðsÞ ¼

q0 ðr2  a2 =4Þ 4p2 T 20 r21 r22

ð47Þ

as t ! 1. This means that for transmissivity the sensitivity near the wells in a pumping test at large time is always very much larger than the sensitivity at a distance. Even when weighted by the radial distance r the sensitivity close to the wells will overshadow the sensitivity at a distance, so it is not possible to measure the transmissivity at a large distance away from the wells by letting the test go on for a long time. When the pumping well and observation point are not colocated, a natural length scale of the problem is the distance a between well and observation point. For a 5 0, appropriate dimensionless length and time variables are X = x/a, Y = y/a, R = r/a, R1 = r1/a, R2 = r2/a, s = Dt/a2. The pumping well and observation point are at X1 = (X, Y) = (1/2, 0) and X2 = (X, Y) = (1/2, 0) respectively, separated by a dimensionless distance of unity. Dimensionless versions of the storativity and transmissivity kernels can be defined as a2 S 0 T 0 F S ðx; y; tÞ q0

 1 R2 þ 1=4 R 1 R2 ¼  2 exp  K0 ð48Þ 8p s 2s 2s

F S ðX ; Y ; sÞ ¼

0

¼

a

Z

1

1

1

ð51Þ

40 10 4 0.1

T 20 F T ðx; y; tÞ q0

0.1 4

0.4 1

–0.02





 R2  1=4 R2 þ 1=4 R1 R2 exp  ¼ 2 K1 8p R1 R2 s 2s 2s ð49Þ

F*S (X,0,τ)

F T ðX ; Y ; sÞ

1

It must be emphasised that the dimensionless total sensitivities depend on the distance of the observation well from the pumping well, although this is not explicit in the forms of Eqs. (50) and (51) as functions only of the dimensionless variable s = T0t/(S0a2). Fig. 2 shows the behaviour of the dimensionless storativity kernel F S ðX ; 0; sÞ for Y = 0 as a function of X for various values of dimensionless time s from Eq. (48). It can be seen that at each X the magnitude of the kernel increases to a value dependent on position and then slowly decreases to zero. At the midpoint (0, 0) between the observation and pumping wells the greatest (most negative) value is reached at a time of approximately t  0.4a2/D or s  0.4. At short times the kernel has a sharp peak around the point X2 = (1/2, 0) corresponding to the observation well (and therefore also around the pumping well by the Reciprocity Principle), but at long times the kernel is approximately spatially uniform in a large area containing the well and observation point. Fig. 3 shows contours of F S ðX ; Y ; sÞ, the dimensionless kernel for storativity from Eq. (48), in the half plane X P 0 at dimensionless times s = 0.4 and s = 2. As in the previous figure the singularity at the observation point (which has a counterpart at the observation well) is evident.

for storativity and 2

Z

F T ðX ; Y ; sÞ dX dY 

 1 1 1 E1 ¼ þ exp  . 4p 4s 4s

ð46Þ

as t ! 1. However, the transmissivity kernel approaches a fixed value dependent only on position; for every (x, y) it approaches a limiting function F T ðx; y; tÞ ! 

and

1

–0.04 0.4

for transmissivity. The dimensionless forms of Eqs. (36) and (37), the total sensitivities at the observation well due to the pumping well are, for a > 0, Z 1 Z 1  M S ðsÞ ¼ F S ðX ; Y ; sÞ dX dY 1 1
 1 1 exp  ¼ ð50Þ 4p 4s

–0.06

0

0.5

1.0

1.5

X

Fig. 2. Dimensionless storativity kernel F S ðX ; 0; sÞ for a pumping test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Results are from (48) with Y = 0 for dimensionless times s = 0.1, 0.4, 1, 4, 10, and 40. At X = 0 the dimensionless storativity kernel reaches its minimum value at a time of s  0.4.

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

1065

0.4

1 τ = 0.4 −0.001

∞

–0.01

0.5

0.2

−0.0001

1 0.5

–0.02

0.2

0

F*T (X,0,τ)

Y

–0.03 −0.00001

–0.04

0.1

0

0.1 0.2 0.5 1

−0.5

∞

–0.2

−0.000001

−1 0

0.5

1.0

a

1.5

2

0

X

τ=2

0.5

0.75

1

Fig. 4. Dimensionless transmissivity kernel F T ðX ; Y ; sÞ for a pumping test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Results are from (49) with Y = 0 for dimensionless times s = 0.1, 0.2, 0.5, and 1. Also shown is the limiting value of the dimensionless transmissivity kernel for s ! 1 from (52).

−0.001 −0.003

0.5

0.25

X

1

− 0.01

at large times. Fig. 5 shows contours of F T ðX ; Y ; sÞ in the half plane XP0 at dimensionless times s = 0.4 from Eq. (49) and s = 1 from Eq. (52). This kernel is zero on the circle of dimensionless radius 1/2 but nevertheless has singularities on the circle at Y = 0 at the well and observation points. It is positive inside the circle and negative outside for all times.

−0.03

Y

–0.4

0

−0.5

−1

0

0.5

1.0

b

1.5

2

X

Fig. 3. Contour lines representing selected values (numbers on curves) of the dimensionless storativity kernel F S ðX ; Y ; sÞ for a pumping test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Contours of F S ðX ; Y ; sÞ are shown for dimensionless times s = 0.4 and s = 2. Results are from (48). The dimensionless storativity kernel has a singularity at X2 = (1/2, 0) and approaches zero far from the well and observation point as r ! 1.

Fig. 4 shows the behaviour of the corresponding dimensionless transmissivity kernel F T ðX ; 0; sÞ from Eq. (49) for Y = 0 for various values of s. The kernel is positive inside the region R < 1/2 and negative for R > 1/2, and for Y = 0 the kernel approaches the value F T ðX ; 0; 1Þ ¼ 

1 p2 ð4X 2

 1Þ

ð52Þ

3.1.2. Radially symmetric case When the observation point and the well coincide, we have a = 0, r1 = r2 = r and the Fre´chet kernels are radially symmetric. They have the simple forms

2  q0 r2 r F S ðr; tÞ ¼  2 2 exp  K0 ð53Þ 2Dt 2Dt 8p T 0 t and F T ðr; tÞ ¼ 



2  q0 r2 r exp  K1 . 2Dt 2Dt 8p2 T 20 Dt

ð54Þ

The explicit expression (54) for the radially symmetric transmissivity kernel was given in [43] in place of the integral expression of [37] which must be evaluated numerically. In this case a = 0 there is no intrinsic physical length scale and so there is no intrinsic time scale either. The time scale and the length scale occur only through the dimensionless combination r2/(Dt). The radially symmetric kernels are negative everywhere and have the same limiting behaviours as in the general case; in particular for large times

1066

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

Letting a ! 0 in (36) gives the case where the observations are made at the pumping well, and the total spatial sensitivity at the well for storativity is Z 1 q0 F S ðr; tÞr dr ¼  ð56Þ M S ðtÞ ¼ 2p 4pS 0 T 0 0

1 τ = 0.4 0

– 0.01

0.5 –0.001 0.01

–0.01 –0.1

Y

0.1

0

–0.5

–1

0

0.5

1.0

a

1.5

2

X 1

which is independent of time. However, as a ! 0 in Eq. (37) the total spatial sensitivity for transmissivity approaches 1, which means that the integral (37) does not exist for the radially symmetric case. In the non radially symmetric case there are singularities near the wells but the total sensitivities are finite so the singularities are integrable; in the radially symmetric case the singularity near the well is not integrable and so it gives an even greater emphasis to the properties of the medium around the well. For the radially symmetric storativity kernel we can calculate the part of the total spatial sensitivity that is contained within a radius r around the well. We define the cumulative radial sensitivity for storativity as Z r N S ðr; tÞ ¼ 2p F S ðr; tÞr dr with N S ð1; tÞ ¼ M S ðtÞ. 0

τ=∞

ð57Þ

–0.01

0.5

It can be verified by differentiating both sides that  
2 
2  q0 r2 r r N S ðr; tÞ ¼  K1 1  K0 4pS 0 T 0 2Dt 2Dt 2Dt
 r2  exp  . ð58Þ 2Dt

0

–0.001

0.01

Y

0.1

–0.1

0

The radially symmetric storativity kernel does not have a natural length scale, pffiffiffiffiffibut there is a natural dimensionless variable q ¼ r= Dt. The cumulative radial sensitivity function for storativity depends only on this dimensionless variable and can be written in dimensionless form as

–0.5

–1

0

N S ðqÞ ¼ S 0 T 0 N S ðr; tÞ=q0 0.5

b

1.0

1.5

2

X

Fig. 5. Contour lines representing selected values (numbers on curves) of the dimensionless transmissivity kernel F T ðX ; Y ; sÞ for a pumping test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Contours of F T ðX ; Y ; sÞ are shown for dimensionless times s = 0.4 and s = 1. Results are from (49) and (52) respectively. The dimensionless transmissivity kernel is zero on the circle X2 + Y2 = 1/4, positive inside the circle, and negative outside the circle. F T ðX ; Y ; 0.4Þ also has a singularity at X2 = (1/2, 0) and approaches zero far from the well and observation point as r ! 1.

q0 ln t ; 8p2 T 20 t q F T ðr; tÞ !  2 0 2 2 4p T 0 r

as t ! 1

ð55Þ

ð59Þ

In terms of the physical variables the cumulative weighting function preserves its shape and moves radially outwards at a rate inversely proportional to the square root of time. We can not express the Fre´chet kernel F S in terms of only the dimensionless variable q because the kernel has a factor 1/t. However if we introduce a scaled quantity F S ðqÞ ¼ T 20 tF S ðr; tÞ=q0 ¼ 

F S ðr; tÞ  

for r 6¼ 0.

1 f1  ðq2 =2Þ½K 1 ðq2 =2Þ  K 0 ðq2 =2Þ 4p  expðq2 =2Þg.

¼

1 K 0 ðq2 =2Þ expðq2 =2Þ; 8p2 ð60Þ

this scaled dimensionless quantity can be written in terms of the dimensionless variable q. Then the dimen-

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

sionless cumulative sensitivity N S ðqÞ can be expressed as a radial integral with respect to q as Z q N S ðqÞ ¼ 2p F S ðqÞq dq. ð61Þ

P T ðqÞ ¼ T 20 P T ðr; tÞ=q0 ¼ 2p

r

It can be verified by differentiation that 

2 
2  q r2 r r2 r P T ðr; tÞ ¼  0 2 1 þ K1 K0  2Dt 2Dt 2Dt 2Dt 4pT 0
 r2  exp  ð63Þ 2Dt which approaches 1 as r ! 0. By analogy with Eq. (60) we introduce the scaled quantity F T ðqÞ in dimensionless form as F T ðqÞ ¼ T 30 tF T ðr; tÞ=ðq0 S 0 Þ ¼

1 K 1 ðq2 =2Þ expðq2 =2Þ; 8p2

ð64Þ

Fig. 7 shows the dimensionless form of the transmissivity kernel F T ðqÞ from Eq. (64) and also P T ðqÞ from Eq. (66). It can be seen that the radially weighted integral from r to 1 decreases to 1 as r ! 0 and q ! 0. It is clear that the singularity in the transmissivity kernel at q = 0 is much stronger than the corresponding singularity in the storativity kernel shown in Fig. 6. 3.2. Slug test 3.2.1. General case The spatial sensitivities of slug tests have been discussed by numerous authors. Sageev [44] summarised previous work and gave detailed solutions in the Laplace transform domain for the case of wellbore storage and skin in the slugged well. He followed previous workers by implicitly assuming that the volume of influence was defined by a ‘‘radius of investigation’’ around the slugged well, even when there was a separate observation well. He used various analytical approximations for special cases, and inverted the transforms numerically using the Stehfest [46,47] method. He and others defined the ‘‘radius of investigation’’ at any given time as the distance from the well at which the perturbation in head had fallen from its value at the slugged well to a particular small fraction, for example 1% or 10%. A similar approach was used in [19] to also investigate

0

–0.02

–0.04

F*S (ρ)

N*S (ρ)

–0.06

Asymptote →

–0.08

–0.1

0

0.5

1.0

1.5

ð65Þ

1 ½ð1 þ q2 =2ÞK 0 ðq2 =2Þ  ðq2 =2ÞK 1 ðq2 =2Þ 4p  expðq2 =2Þ. ð66Þ

0

–0.04

F T ðqÞq dq;

P T ðqÞ ¼ 

F*T (ρ), P*T (ρ)

F*S (ρ), N*S (ρ)

together with the dimensionless form of its weighted integral from r to 1 from (63),

1

q

0

Fig. 6 shows the scaled dimensionless storativity kernel F S ðqÞ from Eq. (60) and its dimensionless q weighted integral N S ðqÞ from Eq. (59) as well as the limiting value 1/(4p) for the integral from 0 to 1. The (integrable) singularity in the kernel at q = 0 is evident. About 78%pof ffiffiffiffiffi the weighting is in the region 0 < q < 1 or r < Dt. For the transmissivity kernel (54) the integral from 0 to r does not exist, but we can calculate the integral from r to 1, which we will call PT(r, t), Z 1 F T ðr; tÞr dr. ð62Þ P T ðr; tÞ ¼ 2p

Z

1067

F*T (ρ)

P*T (ρ)

–0.08

–0.12

–0.16

2

ρ Fig. 6. Dimensionless storativity kernel F S ðqÞ for a pumping test with well and observation point coinciding at (X,Y) = (0,0). Results are from (60). Also shown is its dimensionless integral N S ðqÞ from (59), the cumulative radial sensitivity from 0 to q. It approaches the limiting value of 1/(4p) as q ! 1.

–0.2

0

0.5

1.0

1.5

2

ρ Fig. 7. Dimensionless transmissivity kernel F T ðqÞ for a pumping test with well and observation point coinciding at (X,Y) = (0,0). Results are from (64). Also shown is its dimensionless integral P T ðqÞ from (66), the cumulative radial sensitivity from q to 1. It increases in magnitude without limit as q ! 0.

1068

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

the effect of a linear or radial no-flow boundary. Spatial sensitivities to variations in aquifer properties around wells of finite radius were studied in [32] for the measurements taken at the slugged well and in [33] for measurements taken at an observation well. An analytical slug-test model with two annular zones with different properties was considered in [3]. We will present a simple analytical solution for the Fre´chet kernels for a slug test in a well that is assumed to be a line sink or source. This idealised case with a simple solution acts as a guide for the properties of the more realistic case of a finite size well, and can be used as a special case to test numerical solutions of more complicated cases. In Section 3.3 we will consider pulse tests of finite duration; here we assume that the pumping rate q(t) is a delta function of size Q0 at time zero, so that its Laplace transform is ^ qðpÞ ¼ Q0 . For the slug test the transform of the drawdown at a point x due to a well at x1 is
rffiffiffiffi  p ^h0 ðx; pÞ ¼ Q0 K 0 r1 ð67Þ D 2pT 0

for the storativity kernel and r r  Q0 ðr2  a2 =4Þ h 1 2 r r K 1 2 0 2 2 2Dt 16p2 T 0 D r1 r2 t3
2  r r i r þ a2 =4 1 2 2 2 þ ðr þ a =4ÞK 1 exp  2Dt 2Dt

F T ðx; y; tÞ ¼ 

ð72Þ for the transmissivity kernel. Calculating the partial derivatives of Eq. (68) with respect to S0 and T0 respectively, setting r1 = a and using Eqs. (16) and (17) gives the total sensitivities at the observation well for storativity and transmissivity as Z 1 Z 1 M S ða=2; 0; tÞ ¼ F S ðx; y; tÞ dx dy 1 1
 Q0 a2 S 0 a2 ¼ exp  ð73Þ 4T 0 t 16pT 20 t2 and M T ða=2; 0; tÞ ¼

and Q pðr2  a2 =4Þ F^ T ðx; y; pÞ ¼  0 2 2 K1 4p T 0 Dr1 r2


rffiffiffiffi 
rffiffiffiffi  p p r1 K 1 r2 D D ð70Þ

respectively. Apart from constant factors these are the transforms from the previous section multiplied by p. Since the functions all take the value zero at time zero, this corresponds to the operation of differentiation in the time domain. Differentiating Eqs. (34) and (35) with respect to time gives the inversions of Eqs. (69) and (70) as "
  Q0 r2 þ a2 =4 r1 r2  F S ðx; y; tÞ ¼  2 2 2  1 K0 2Dt 2Dt 8p T 0 t #
 r 1 r 2 r 1 r 2  r2 þ a2 =4 K1 þ ð71Þ exp  2Dt 2Dt 2Dt

1

Z

1

F T ðx; y; tÞ dx dy

 Q0 S 0 a2 S 0 a2 1 ¼ exp  . 4T 0 t 4T 0 t 4pT 20 t 1

corresponding in the time domain to the function
 Q0 S 0 r21 exp  h0 ðx; tÞ ¼ . ð68Þ 4pT 0 t 4T 0 t The imposed drawdown at a point a distance r1 from the slugged well increases to a maximum value at a time t ¼ r21 =ð4DÞ and then decreases. At the observation well the maximum value of the drawdown occurs at time t = a2/(4D) or s = 1/4. The transforms of the Fre´chet kernels for storativity and transmissivity are
rffiffiffiffi 
rffiffiffiffi  Q0 p p p ^ F S ðx; y; pÞ ¼  2 2 K 0 r1 K 0 r2 ð69Þ D D 4p T 0

Z

1

ð74Þ The total sensitivities at the observation well are well defined for all a P 0 and t P 0. We can define dimensionless forms of the storativity and transmissivity kernels as F S ðX ; Y ; sÞ ¼ a4 S 20 F S ðx; y; tÞ=Q0 

2  1 R þ 1=4 R 1 R2  1 K0 ¼ 2 2 8p s 2s 2s

 2 R1 R 2 R1 R 2 R þ 1=4 K1 þ exp  2s 2s 2s ð75Þ and F T ðX ; Y ; sÞ ¼ a4 S 0 T 0 F T ðx; y; tÞ=Q0 
 ðR2  1=4Þ R 1 R2 ¼ R R K 1 2 0 16p2 R1 R2 s3 2s

 R1 R 2 R2 þ 1=4 þðR2 þ 1=4ÞK 1 exp  2s 2s ð76Þ respectively. For a > 0 we can define dimensionless forms of the total sensitivities at the observation well as
 Z 1Z 1 1 1 M S ðsÞ ¼ F S ðX ;Y ;sÞdX dY ¼  exp  16ps2 4s 1 1 ð77Þ

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

and M T ðsÞ ¼

Z

1

Z

1

F T ðX ; Y ; sÞ dX dY

 1 1 1 1 ¼ exp  . 4ps 4s 4s 1

1

ð78Þ

The spatial sensitivity for storativity depends on the time rate of change of the imposed drawdown function. As the time rate of change of the drawdown changes sign at the observation well at time t = a2/(4D) or s = 1/4, we expect the properties of the storativity kernel to change at this value of the dimensionless time. As in the previous section both kernels have singularities at the positions x1 = (a/2, 0) and x2 = (a/2, 0) or X1 = (1/2, 0) and X2 = (1/2, 0). Elsewhere both kernels are zero at time zero. The values of the storativity kernel around the singularities are initially negative. At the time t = a2/(4D) or s = 1/4 we can substitute s = 1/4 in the dimensionless storativity kernel (75) to get 
 2 1  2 F S ðX ; Y ; 1=4Þ ¼  2 2R  K 0 ð2R1 R2 Þ p 2 
 1 2 þ 2R1 R2 K 1 ð2R1 R2 Þ exp 2R  . 2 ð79Þ Setting R1 = 1 in Eq. (79) and taking the limit as R2 ! 0 gives the limiting behaviour near the observation well. Using Eq. (42) gives the limit as F S ð1=2; 0; 1=4Þ ¼ 2=ðp2 eÞ, so the physical storativity kernel has the finite value 2Q0 =ðp2 eS 20 a4 Þ at (a/2, 0) and (a/2, 0). For s > 1/4 the singularities change from negative to positive, and the storativity kernel becomes positive around each well in a region that grows with time. As r2/(D t) ! 1, which corresponds to small times or large distances r, the kernels behave like " # 2 2 Q0 ðr1 þ r2 Þ ðr1 þ r2 Þ exp  F S ðx; y; tÞ   1=2 4Dt 32T 20 ðp3 Dr1 r2 t5 Þ

At a fixed position the storativity kernel starts off as zero, becomes negative for small time, reaches a most negative value, then increases through zero and becomes positive, reaches a maximum value, and then decreases towards zero at long times. At times slightly greater than a2/(4D) the contour in the (x, y) plane on which FS(x, y, t) takes the value zero consists of two small closed curves around the two singularities, with positive values of the kernel inside and negative values outside the curves. As time increases these expand and then connect at (0, 0) to become a ‘‘figure of eight’’. This curve then expands to form a single closed curve enclosing both singularities, which approaches the shape of an ellipse at long times. Fig. 8 shows the dimensionless storativity kernel F S ðX ; 0; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0) from (75) with Y = 0, for various dimensionless times s = 0.05, 0.1, 0.2, 0.25, 0.5, and 2. It can be seen that the curve for s = 0.25 is the special case (79) for which there are no singularities at the well or observation point. As can be seen, the singularity is negative at earlier dimensionless times and positive for s > 0.25. On the y axis r1 = r2 and the quantities r21 þ r22 ¼ 2ðr2 þ a2 =4Þ and 2r1r2 in Eq. (71) are equal. Let the time at which the ‘‘figure of eight’’ contour forms be t*, and for t P t* let (0, y*) be the point where the contour crosses the y axis, such that FS(0, y*, t) = 0. Let r*(t) be the distance from either singularity to the point (0, y*). At that point r1 = r2 = r*, and so
2 
2 
2  r r r2 r  1 K0 þ  K1 ¼ 0. ð84Þ 2Dt 2Dt 2Dt 2Dt If a is the unique solution of ð1  aÞK 0 ðaÞ ¼ aK 1 ðaÞ;

F T ðx; y; tÞ  

2

2

"

Q0 ðr1 þ r2 Þ ðr  a =4Þ ðr1 þ r2 Þ

1=2 exp  2 3 3 3 4Dt 32T 0 p3 D r1 r2 t5

2

#

ð81Þ for transmissivity. As in the previous case, far from the wells the contours of equal spatial sensitivity approach the shapes of a family of ellipses with a focus at each well. At long times the kernels behave like Q ln t F S ðx; y; tÞ  02 2 2 ð82Þ 8p T 0 t and F T ðx; y; tÞ  

Q0 ðr4  a4 =16Þ . 8p2 T 20 Dr21 r22 t2

ð83Þ

2

0.5

0 0.25

F*S (X,0,τ)

2

ð85Þ

0.1

ð80Þ for storativity and

1069

0.05

0.2

–0.1

0.2

0.05

–0.2 0.1

0.1

–0.3

–0.4

0

0.25

0.5

0.75

1.0

X

Fig. 8. Dimensionless storativity kernel F S ðX ; 0; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Results are from (75) with Y = 0 for dimensionless times s = 0.05, 0.1, 0.2, 0.25, 0.5, and 2. The curve for s = 0.25 is the special case (79) for which there are no singularities at the well or observation point.

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J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

pffiffiffiffiffiffiffiffiffiffi then a  0.31466 and r ðtÞ ¼ 2aDt with r* P a/2. The condition r*(t*) = a/2 gives t* = a2/(8aD) or s* = 1/(8a). Fig. 9 shows contour lines of the dimensionless storativity kernel F S ðX ; Y ; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Contours are shown for dimensionless times s = 0.25 and s = 1/(8a) = 0.3973 from Eqs. (79) and (75) respectively. The dimensionless storativity kernel is negative every-

0.75 τ = 0.25 0.50

0.25

x2 y2 þ ¼1 2aDt 2aDt  a2 =4

–0.06

–0.05

– 0.07 Y

where at time s = 0.25 and has no singularity at X2 = (1/2, 0). Its maximum lies on the X axis at a point slightly to the right of the observation well. The kernel has a positive singularity at X2 at time s = 1/(8a) = 0.3973, and it changes sign from positive to negative with increasing distance from the observation point. The ‘‘figure of eight’’ shape of the contour at value zero is evident. The ellipse with foci at the singularities (a/2, 0) and (a/2, 0) and passing through the point (0, y*) for t P t* satisfies the equation pffiffiffiffiffiffiffiffiffiffi ð86Þ r1 þ r2 ¼ 2 2aDt. pffiffiffiffiffiffiffiffiffiffi The ellipse has major semi-axis 2aDt and minor semipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi axis 2aDt  a2 =4 for t P t*, and so an equivalent form of its equation is

0

or

X2 Y2 1 þ ¼ . 8as 8as  1 4 ð87Þ

– 0.08 –0.25 –0.05 –0.50

– 0.04

– 0.01 –0.03 –0.02

–0.75 0

–0.001 –0.0001

0.5

1

1.5

X

a

Fig. 10 compares the contour lines on which the dimensionless storativity kernel F S ðX ; Y ; sÞ from Eq. (75) is zero at various dimensionless times from 0.35 to 2.5, with the corresponding ellipses from Eq. (87). It can be seen that the region where the storativity kernel is positive is always larger than the expanding ellipse, and the ellipse becomes a better approximation as time increases. The ‘‘figure of eight’’ zero contour at

0.75 1

τ = 1/(8α) – 0.02

– 0.02 0.25

–0.01

1.5

0.8

– 0.01

1.25

1

0.6

0.01

Y

Y

0

0

2.5

2.0

0.50

0.75

–0.25 0.4

–0.50

0.5

0.2

–0.75 0

b

0.5

1

1.5 0.3973

X 0

Fig. 9. Contour lines representing selected values (numbers on curves) of the dimensionless storativity kernel F S ðX ; Y ; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Contours of F S ðX ; Y ; sÞ are shown for dimensionless times s = 0.25 and s = 1/ (8a), where a  0.31466. Results are from (79) and (75) respectively. The dimensionless storativity kernel is negative everywhere at time s = 0.25 and has no singularity at X2. The kernel has a positive singularity at X2 at time s = 1/(8a), and it changes sign from positive to negative with increasing distance from the observation point.

0

0.2

0.35

0.4

0.6

0.8

1

X Fig. 10. Contour lines showing the behavior of the dimensionless storativity kernel F S ðX ; Y ; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). The solid lines, obtained from (75), represent contours along which F S ðX ; Y ; sÞ is zero for various dimensionless times ranging from s = 0.35 to s = 2.5. The dashed lines are ellipses from (87), which are approximations to the exact contours.

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075 0.75

τ = 0.25

–0.01

0.50

0 0.02 0.05

0.25

Y

s = 0.3973 is the same ‘‘figure of eight’’ contour as shown in Fig. 9. As for the pumping test, the transmissivity kernel for the slug test is zero on the circle r2 = a2/4, positive inside, and negative outside, and this behaviour does not change with time. Unlike the pumping test case, the transmissivity kernel initially grows negatively or positively away from zero, reaches a minimum or maximum and then approaches zero as time goes to infinity. Fig. 11 shows the dimensionless transmissivity kernel F T ðX ; Y ; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0) from Eq. (76) with Y = 0, for dimensionless times s = 0.05, 0.1, 0.25, 0.5, and 1. The corresponding contour lines of F T ðX ; Y ; sÞ for dimensionless time s = 0.25 are shown in Fig. 12. The kernel is zero on the circle X2 + Y2 = 1/4, positive inside the circle, and negative outside the circle. It has a singularity on the circle at X2 = (1/2, 0).

1071

0.1

0

0.2

–0.2 –0.1 –0.05

–0.25

–0.02 –0.01

–0.50

–0.75

0

0.5

1

1.5

X

3.2.2. Radially symmetric case The radially symmetric kernels with observations at the slugged well are obtained by setting a = 0 in Eqs. (71) and (72) to give 
2 
2 
2  Q0 r r r2 r  1 K0 þ K1 F S ðr;tÞ ¼  2 2 2 2Dt 2Dt 2Dt 2Dt 8p T 0 t
 r2 ð88Þ  exp  2Dt for storativity and

0.5

0.25 0.05 0.5

F*T (X,0,τ)


2 
2  Q0 r2 r r F T ðr; tÞ ¼  K0 þ K1 2 2 3 2 2Dt 2Dt 16p T 0 D t
 2 r  exp  2Dt

ð89Þ

for transmissivity. They inherit their behaviour from the general case, and the storativity kernel FS(r, t) is zero pffiffiffiffiffiffiffiffiffiffi when r ¼ r ¼ p 2aDt above, ffiffiffiffiffiffiffiffiffiffiwith a satisfying Eq. (85) pffiffiffiffiffiffiffiffiffiffi positive for r < 2aDt, and negative for r > 2aDt. It has a singularity at r = 0. On the other hand, the radially symmetric transmissivity kernel FT(r, t) is negative everywhere, and at r = 0 at the well has the finite value

0.1

0.25

Fig. 12. Contour lines representing selected values (numbers on curves) of the dimensionless transmissivity kernel F T ðX ; Y ; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Contours of F T ðX ; Y ; sÞ are shown for dimensionless time s = 0.25. Results are from (76). The dimensionless transmissivity kernel is zero on the circle X2 + Y2 = 1/4, positive inside the circle, and negative outside the circle. F T ðX ; Y ; 0.4Þ also has a singularity at X2 and approaches zero far from the well and observation point as r ! 1.

1 0.05

F T ð0; tÞ ¼ 

0

1

ð90Þ

.

Putting a = 0 in Eqs. (73) and (74) gives the total spatial sensitivities at the well for storativity and for transmissivity as Z 1 M S ðtÞ ¼ 2p F S ðr; tÞr dr ¼ 0 ð91Þ

0.25

–0.25

Q0 2 8p T 20 Dt2

0.5 0.1

0

and –0.5

0

0.25

0.5

0.75

1

X Fig. 11. Dimensionless transmissivity kernel F T ðX ; 0; sÞ for a slug test with well at X1 = (1/2, 0) and observation point at X2 = (1/2, 0). Results are from (76) with Y = 0 for dimensionless times s = 0.05, 0.1, 0.25, 0.5, and 1.

M T ðtÞ ¼ 2p

Z 0

1

F T ðr; tÞr dr ¼ 

Q0 . 4pT 20 t

ð92Þ

The total sensitivities for the radially symmetric storativity and transmissivity kernels are zero and negative respectively for all time. This means that the integral

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

pffiffiffiffiffiffiffiffiffiffi from r = 0 to r ¼ 2aDt of the storativity kernel, weighted by the radius r, where the kernel is positive pffiffiffiffiffiffiffiffiffiffiexactly balances the weighted integral from r ¼ 2aDt to infinity where it is negative. The total sensitivity for transmissivity is undefined at time zero and increases monotonically from 1 to its limit of zero for large time. For each radially symmetric kernel we can calculate the part of the total spatial sensitivity that is contained within a radius r around the well. We define the cumulative radial sensitivities for storativity and transmissivity as Z r N S ðr; tÞ ¼ 2p F S ðr; tÞr dr with N S ð1; tÞ ¼ M S ðtÞ 0

ð93Þ and N T ðr; tÞ ¼ 2p

Z

r

F T ðr; tÞr dr

with N T ð1; tÞ ¼ M T ðtÞ.

0

ð94Þ Integrating Eqs. (88) and (89) gives N S ðr; tÞ ¼ 


2 
2 
 Q0 r r r2 K0 exp  4pS 0 T 0 t 2Dt 2Dt 2Dt ð95Þ

and N T ðr; tÞ ¼ 


2 
 Q0 r2 r r2 1  K exp  . 1 2Dt 2Dt 2Dt 4pT 20 t ð96Þ

The radially symmetric kernels (88) and (89) can not be expressed in dimensionless form in terms of the variable q because they each have a factor 1/t2. As in the previous section, we can scale the kernels and define scaled dimensionless kernels as F S ðqÞ ¼ T 20 t2 F S ðr; tÞ=Q0 ;

ð97Þ

F T ðqÞ

ð98Þ

¼

T 30 t2 F T ðr; tÞ=ðS 0 Q0 Þ;

or 1 F S ðqÞ ¼  2 ½ðq2 =2  1ÞK 0 ðq2 =2Þ þ ðq2 =2ÞK 1 ðq2 =2Þ 8p  expðq2 =2Þ; ð99Þ

The cumulative radial sensitivity functions have a factor (1/t). Scaled dimensionless forms of the cumulative radial sensitivity functions can be defined in dimensionless form as N S ðqÞ ¼ S 0 T 0 tN S ðr; tÞ=Q0 ¼ 

q2 K 0 ðq2 =2Þ expðq2 =2Þ 8p ð101Þ

and N T ðqÞ ¼ T 20 tN T ðr; tÞ=Q0 ¼

1 ½1  ðq2 =2ÞK 1 ðq2 =2Þ expðq2 =2Þ. 4p

ð102Þ

The scaled cumulative weighting function for storativity has the limit of 0 as q ! 1. The scaled cumulative weighting functions move radially outwards at a rate inversely proportional to the square root of time. For the radial transmissivity kernel aboutp50% ffiffiffiffiffi of the weighting is in the region 0 < q < 1 or r < Dt. Fig. 13 shows the scaled dimensionless weighting function F S ðqÞ for storativity from Eq. (99) and the scaled dimensionless cumulative weighting function N S ðqÞ from Eq. (101) for storativity. When weighted by the dimensionless radius q the integral of the positive part of the kernel is equal to the weighted integral of the negative part, and the cumulative weighting function approaches zero for large dimensionless radius. The singularity in the kernel at q = 0 is evident. Fig. 14 shows the dimensionless weighting function F T ðqÞ for transmissivity from Eq. (100) and the dimensionless cumulative weighting function N T ðqÞ from Eq. (102) for transmissivity, with the limiting value 1/(4p) as q approaches 1. The kernel has the finite value 1/(8p2) at q = 0.

0.08

0.06 F*S (ρ), N*S (ρ)

1072

0.04 F*S (ρ)

0.02

N*S (ρ)

0 2

F T ðqÞ ¼ 

q ½K 0 ðq2 =2Þ þ K 1 ðq2 =2Þ expðq2 =2Þ. 16p2 ð100Þ

With this scaling, the dimensionless storativity kernel F S ðqÞ is zero when q2 = 2a or q = 0.7933 from (85), and the dimensionless transmissivity kernel F T ðqÞ has the limiting value 1/(8p2) when q = 0.

–0.02 0

0.5

1.0

1.5

2

ρ Fig. 13. Dimensionless storativity kernel F S ðqÞ for a slug test with well and observation point coinciding at (X, Y) = (0, 0). Results are from (99). Also shown is its dimensionless integral N S ðqÞ from (101), the cumulative radial sensitivity from 0 to q. It approaches zero as q ! 1.

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

and for transmissivity

0


2  Q0 ðr2  a2 =4Þ r1 r2  r þ a2 =4 K exp  1 2Dt 2Dt 8p2 DT 20 r1 r2 t0 t   2 2 Q H ðt  t0 Þðr  a =4Þ r1 r2 þ 0 2 2 K1 2Dðt  t0 Þ 8p DT 0 r1 r2 t0 ðt  t0 Þ  2  2 r þ a =4  exp  . ð105Þ 2Dðt  t0 Þ

F*T (ρ)

F T ðx;y;t;t0 Þ ¼ 

F*T (ρ), N*T (ρ)

–0.02 N*T (ρ)

–0.04

–0.06

Asymptote →

–0.08

–0.1 0

1073

0.5

1.0

1.5

2

ρ Fig. 14. Dimensionless transmissivity kernel F T ðqÞ for a slug test with well and observation point coinciding at (X, Y) = (0, 0). Results are from (100). Also shown is its dimensionless integral N T ðqÞ from (102), the cumulative radial sensitivity from 0 to q. It approaches the limiting value of 1/(4p) as q ! 1.

a4 S 20 F S ðx; y; t; t0 Þ Q0

 1 R1 R2 R2 þ 1=4 ¼  2 K0 exp  8p ss0 2s 2s   H ðs  s0 Þ R 1 R2 K0 þ 2 8p ðs  s0 Þs0 2ðs  s0 Þ   R2 þ 1=4  exp  ð106Þ 2ðs  s0 Þ

F S ðX ; Y ; s; s0 Þ ¼

3.3. Slug test of finite duration The simple theory of the slug test assumes that the pumping rate is a delta function of zero duration. Connell [11] pointed out that it in some circumstances it may be important to consider the finite duration t0 of a slug test. He found a Laplace transform solution for the drawdown in a well of finite diameter when pumping proceeds for a finite time, and used the Stehfest [46,47] method to numerically invert the Laplace transforms. We will consider a slug test for a well of zero diameter with pumping for a time t0. For comparison with the results of the slug test with pumping of zero duration we assume that the constant pumping rate q0 satisfies q0 = Q0/t0. The pumping rate as a function of time can be expressed as qðtÞ ¼ ðQ0 =t0 ÞH ðtÞ  ðQ0 =t0 ÞH ðt  t0 Þ;

In the limit as t0 approaches zero the finite differences between the values of the pumping functions (34) and (35) evaluated at t and t  t0, divided by the interval t0, become derivatives with respect to time, and Eqs. (104) and (105) approach the slug test functions (71) and (72) respectively with q0 = Q0/t0. For a > 0 a dimensionless time lag variable is s0 = Dt0/a2. Dimensionless versions of the storativity and transmissivity kernels (104) and (105) for dimensionless time lag s0 can be defined as

for storativity and a4 S 0 T 0 F T ðx;y;t;t0 Þ Q0
2 
 2 ðR  1=4Þ R1 R2 R þ 1=4 exp  ¼ 2 K1 8p R1 R2 ss0 2s 2s   2 H ðs  s0 ÞðR  1=4Þ R1 R2 þ K1 2 8p R1 R2 ðs  s0 Þs0 2ðs  s0 Þ  2  R þ 1=4  exp  ð107Þ 2ðs  s0 Þ

F T ðX ;Y ;s;s0 Þ ¼

ð103Þ

where H(z) is the Heaviside function which is zero when z is negative and unity for positive z. This means that the formulae (34) and (35) for the pumping test, shifted in time by an amount t0, can be used to derive the Fre´chet kernels for the slug test of finite duration. For storativity the result is
2  r r  Q0 r þ a2 =4 1 2 F S ðx; y; t; t0 Þ ¼  2 2 K 0 exp  2Dt 2Dt 8p T 0 t0 t   Q H ðt  t0 Þ r1 r2 K0 þ 20 2 2Dðt  t0 Þ 8p T 0 t0 ðt  t0 Þ   r2 þ a2 =4  exp  ð104Þ 2Dðt  t0 Þ

for transmissivity. Fig. 15 shows the effect on the dimensionless storativity kernel F S ð0; 0; s; s0 Þ at the origin (X, Y) = (0, 0) of a finite pumping duration, with dimensionless durations of s0 = 0.025, 0.05, 0.075 and 0.1 from Eq. (106). The curve of zero duration s0 = 0 from Eq. (75) is also shown. The corresponding radially symmetric forms with observations at the slugged well are for storativity
2 
 Q0 r r2 exp  K 0 2Dt 2Dt 8p2 T 20 t0 t     2 Q H ðt  t0 Þ r r2 exp  þ 20 2 K0 2Dðt  t0 Þ 2Dðt  t0 Þ 8p T 0 t0 ðt  t0 Þ

F S ðr;t;t0 Þ ¼ 

ð108Þ

1074

J.H. Knight, G.J. Kluitenberg / Advances in Water Resources 28 (2005) 1057–1075

0

–0.10 0.1 5 0.07 5 0.0 0.025 0

F*S (0,0,τ,τ0)

–0.05

–0.15 –0.20 –0.25 0

0.1

0.2

τ

0.3

0.4

0.5

Fig. 15. Dimensionless storativity kernel F S ð0; 0; s; s0 Þ for a slug test with finite pumping duration. The well is at X1 = (1/2, 0) and the observation point is at X2 = (1/2, 0). Results are from (106) for (X, Y) = (0, 0) and dimensionless pumping durations of s0 = 0.025, 0.05, 0.075 and 0.1. The curve labeled s0 = 0 is the storativity kernel F S ð0; 0; sÞ for the ‘‘instantaneous’’ slug test as computed from (75).

and for transmissivity
2 
 Q0 r r2 K exp  1 2Dt 2Dt 8p2 T 20 Dt0 t   2 Q H ðt  t0 Þ r K1 þ 2 02 2Dðt  t0 Þ 8p T 0 Dt0 ðt  t0 Þ   2 r  exp  . 2Dðt  t0 Þ

F T ðr; t; t0 Þ ¼ 

ð109Þ

The radially symmetric functions (108) and (109) for finite duration pumping can not be written as functions of only the dimensionless combination r2/(Dt), unlike the corresponding radially symmetric functions (71) and (72).

4. Discussion Using the Theis [48] approximation of a well as a line source we have derived explicit expressions for the Fre´chet kernels or spatial sensitivity functions for variations in storativity and in transmissivity during both pumping and slug tests. We have also derived the total sensitivity functions which are the spatial integrals of the Fre´chet kernels. These analytical solutions are for an idealised case, but yield considerable insight into the properties of spatial weighting functions. Our solutions show clearly that the storativity spatial sensitivities depend on the interaction of the time rate of change of the drawdown centered on the pumped well with a Green
s function centered on the observation well. The transmissivity spatial sensitivities depend on the scalar product of the spatial gradient of the drawdown centered on the pumped well with the gradient of a Green
s function

centered on the observation well. The kernel for storativity for the pumping test is always negative, whereas for the slug test it is initially negative everywhere and then becomes positive in an increasing region around the wells. The kernels for transmissivity for both tests are zero on the circle r = a/2 passing through both wells, positive inside the circle and negative outside. In the radially symmetric case, at all times the total sensitivity for transmissivity is infinite for the pumping test and the total sensitivity for storativity is zero for the slug test, so in these cases we can not normalise the spatial weighting functions to have unit spatial integrals. There has been a ‘‘conventional wisdom’’ that because the drawdown is radially symmetrical about a tested well, the spatial sensitivity has the same property. Our results show clearly that in the case that the pumped or slugged well and the observation well are not colocated, the two wells have the same importance with respect to spatial sensitivity. Far from the wells the contours of equal spatial sensitivity approach the shapes of ellipses with foci at each well. The analytical solutions for Fre´chet kernels for well testing can be generalised to multiple wells and to measurement techniques for systems governed by the heat or diffusion equation in other contexts and geometrical configurations.

Acknowledgments Most of this work was done while the first author was a Visiting Professor in the Department of Agronomy at Kansas State University, supported by a grant from the Kansas State University President
s Faculty Development Awards Program. This work was also supported by funding from the National Aeronautics and Space Administration (Grant NAG 9-1399). Contribution number 04-204-J from the Kansas Agricultural Experiment Station, Manhattan, KS.

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