Stochastic Environmental Research and Risk Assessment 16 (2002) 399–424 Ó Springer-Verlag 2002 DOI 10.1007/s00477-002-0113-5

Sensitivity analysis of a no-crossflow model for the transient flowmeter test Z. J. Kabala, H. K. El-Sayegh, H. P. Gavin

399 Abstract. Logarithmic sensitivities and plausible relative errors are studied in a simple no-crossflow model of a transient flowmeter test (TFMT). This model is identical to the model of a constant-rate pumping test conducted on a fully penetrating well with wellbore storage, surrounded by a thick skin zone, and situated in a homogeneous confined aquifer. The sensitivities of wellbore drawdown and wellface flowrate to aquifer and skin parameters are independent of the pumping rate. However, the plausible relative errors in the aquifer and skin parameters estimated from drawdown and wellface flowrate data can be proportionally decreased by increasing the pumping rate. The plausible relative errors vary by many orders of magnitude from the beginning of the TFMT. The practically important flowrate and drawdown measurements in this test, for which the plausible relative errors vary by less than one order of magnitude from the minimum plausible relative errors, can begin approximately when the dimensionless wellface flowrate exceeds qD ¼ q=Q  0:4. During most of this stage of the test, the plausible relative errors in aquifer hydraulic conductivity (Ka ) are generally an order of magnitude smaller than those in aquifer specific storativity. The plausible relative errors in the skin hydraulic conductivity (Ks ) are generally larger than the plausible relative errors in the aquifer specific storativity when the thick skin is normal (Ks > Ka ) and smaller when the thick skin is damaged (Ks < Ka ). The specific storativity of the skin zone would be so biased that one should not even attempt to estimate it from the TFMT. Keywords: Ground water, Transient flowmeter test, Sensitivity analysis

1 Introduction 1.1 The flowmeter test The flowmeter test is the simplest and most convenient test for estimating the downhole distribution of aquifer parameters. As opposed to the other

Z. J. Kabala (&), H. K. El-Sayegh, H. P. Gavin Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708-0287, USA We acknowledge Wiebe H. van der Molen for recommending the De Hoog algorithm and sharing his code. This research was partially supported by the US Geological Survey, USGS Agreement #1434-HQ-96-GR-02689 and North Carolina Water Resources Research Institute, WRRI Project #70165.

400

single-borehole tests suitable for this purpose (the multilevel slug test, the drillstem test, and the dipole-flow test), the flowmeter test requires only one packer. Interpretation of the flowmeter test is based on the layered-aquifer conceptualization. During the test a flowmeter is placed in the well at a (real or conceptual) layer junction. After the pump is lowered into the well and turned on, the flowrate and pressure are recorded by the flowmeter and a pressure transducer, respectively. The flowmeter is then moved to a different layer junction and the measurements are repeated. The vertical distribution of layer parameters can be calculated from these measurements. Depending on the type of data collected, El-Sayegh and Kabala (2001) classified the flowmeter test into quasi-steady-state and transient tests. The traditional, quasi-steady-state flowmeter test (QFMT) (Molz et al., 1989, 1990; Rehfeldt et al., 1989; Ruud and Kabala, 1996, 1997a, b; Ruud et al., 1999) requires pumping the well until a quasi-steady state is reached before taking the measurements. Although the QFMT has been used extensively for aquifer characterization (Hada, 1977; Keys and Sullivan, 1978; Schimschal, 1981; Hufschmied, 1983; Hess et al., 1986; Sudicky, 1986; Paillet et al., 1987; Morin et al., 1988; Rehfeldt et al., 1989; Paillet, 1993, 1995; Young, 1995; and others), its interpretation methodology yields only the downhole distribution of the horizontal hydraulic conductivity. The transient flowmeter test (TFMT) collects the transient drawdown and downhole flowrate data in times significantly shorter than it takes to reach the quasi-steady state. This makes the TFMT particularly advantageous for characterization of highly contaminated aquifers, where extracted water must be treated at considerable expense as hazardous waste. The TFMT data could provide information not only about the downhole distribution of the aquifer horizontal hydraulic conductivity, as does QFMT, but also about distributions of the aquifer specific storativity and the skin properties. Kabala and El-Sayegh (2002) reviewed the TFMT models used in petroleum engineering and hydrogeology. Based on whether or not the inter-layer crossflow, i.e., the vertical flux between adjacent aquifer layers, is accounted for, they classified the TFMT models into the crossflow models and no-crossflow models. Although the former are physically more appropriate than the latter, they are computationally significantly less robust. Kabala and El-Sayegh (2002) noted that the crossflow models suffer from two problems: non-uniqueness (due to the local minima trapping) and feasibility (due to too many parameters being estimated). The non-uniqueness problem in the parameter estimates, discussed and illustrated in detail by Kabala and El-Sayegh (2002) and also reported by Bourdet (1985), Shah et al. (1988), and Rapach et al. (1990), is caused by the existence of a number of local minima in a TFMT inverse model’s objective function in the nonlinear least squares algorithm. This is true for the semi-analytic TFMT crossflow models (Bourdet, 1985; Ehlig-Economides and Joseph, 1987; Hatzignatiou et al., 1987; Anbarci et al., 1989; Gomes and Ambastha, 1993, 1994; Kabala and El-Sayegh, 2002) as well as for the numerical crossflow models (Shah et al., 1988; and others). Whereas for a convex function least squares algorithms find the global minimum, for a non-convex function, such as a TFMT inverse model’s objective function, they generally identify only one of the local minima near the initial guess of the parameter vector. If a crossflow TFMT model is thus to be used in interpreting TFMT data, it is imperative that good initial guesses of the layered system parameters be found first so that the local minima trapping could be mitigated or avoided altogether. This can be done via the no-crossflow models.

The feasibility problem arises when crossflow models are to be used to interpret TFMTs conducted informations with a large number of layers. For a nonconvex objective function, computationally expensive search among the local minima is necessary to find the parameter estimates, i.e., the global minimum vector. This search becomes infeasibly expensive, i.e., the least squares algorithm practically breaks down, when more than 20–30 parameters are being estimated. This limits the number of layers that can be practically considered in TFMT interpretations with a crossflow model to about 5 to 10, depending on parametrization. In formations with more layers, an interpretation of a TFMT with a no-crossflow model may be the only feasible alternative. Since the crossflow is important in layered formations only in the early times and it becomes negligible in the late times, as demonstrated by Javandel and Witherspoon (1969) in their numerical simulations, the no-crossflow models may actually provide reasonable estimates.

1.2 Sensitivity analysis A well-aquifer system can be analyzed as an input-output model that relates the output information (wellbore drawdown and layer flowrate) to the input parameters characterizing this system (layer and skin hydraulic conductivities and specific storativities). Since sensitivity analysis was developed to study how errors (uncertainties) in input parameters influence outputs of a system, it can and has been used to study hydrogeologic systems. Kabala (2001) recently reviewed the available approaches to sensitivity analysis. Given a model with output Oðx; t; P 1 ; . . . ; P n Þ that is a function of the spatial location x, time t, and parameters P i , the total differential of the output (also as a function of x and t) dO ¼

n X oO i¼1

oP i

dP i

ð1Þ

defines the traditional sensitivity, oO=oP i , of the output to the model parameter P i . Although this concept is often used in sensitivity analysis (Bishop et al., 1976; McElwee and Yukler, 1978; Morita and Gray, 1981; Cob et al., 1982; Butt and McElwee, 1985; Knopman and Voss, 1987; Shah et al., 1988; Datta-Gupta et al., 1997; Vasco and Datta-Gupta, 1999), it generally does not allow for comparison of the output sensitivities to parameters with different physical units (e.g., hydraulic conductivity and specific storativity) or the comparison of sensitivities of different outputs (e.g., drawdown and flowrate) to even the same parameter. The dimensionless concept of the logarithmic sensitivity

P i oO o lnðOÞ ¼ O oP i o lnðP i Þ

ð2Þ

overcomes these disadvantages. It arises naturally when the total differential, (1), is rearranged for a non-zero output as follows (Kabala, 2001)

 n  dO X P i oO dP i ¼ O oP i P i O i¼1

ð3Þ

401

402

Each logarithmic sensitivity can thus be interpreted as a transfer coefficient between the relative error in the input parameter and the relative error this parameter alone would induce in the output. Although powerful, this concept has been used in surprisingly few hydrologic studies (Zhou and Stone, 1993; Chen et al., 1997; Munoz-Carpena et al., 1999; Tebes-Stevens and Valocchi, 2000; Kabala, 2001). Kabala (2001) discussed the role of sensitivities in inverse modeling and sampling design. He also noted that there are two different perspectives of sensitivity analysis present in the literature: the parameter-estimation perspective and the sensitivity-only perspective. Although the former involves calculation of sensitivities, it focuses on parameter estimation (McElwee and Yukler, 1978; Cob et al., 1982; McElwee, 1987; Butler and Liu, 1991, 1993; McElwee et al., 1995a, b) and/or on formulation of an optimal sampling design (Knopman and Voss, 1987, 1988, 1989; Knopman et al., 1991). The latter focuses solely on the sensitivities (Koda etal., 1979; Reuven et al., 1987; Rabitz, 1987; Kabala and Milly, 1990, 1991a, b; Chen et al., 1997) and provides information about which model parameters are likely to be estimated accurately and which measurement points (in space or time) should be considered as candidates for the sampling design. The sensitivityonly perspective is adopted for this paper.

1.3 Imprecise measures of information content in the system output Whenever a number (larger than one) of parameters are estimated simultaneously, their accuracy depends not only on the measurement noise and the magnitude of the parameter sensitivities, but also on the deterministic correlation (or co-variance) between the parameters. Not to be confused with the statistical correlation of random variables, this correlation is fully determined by sensitivities. Indeed, consider how an error dP j in P j affects P i given perfect measurements of all the other parameters. The deterministic logarithmic correlation of the parameter P i with the parameter P j follows from (3) as the ratio of logarithmic sensitivities (Kabala, 2001) log

RPi Pj ¼ 

dP i =P i ðP j =OÞoO=oP j ¼ dP j =P j ðP i =OÞoO=oP i

ð4Þ

This correlation is also dimensionless. If its magnitude is large, small relative errors in P j will result in large relative errors in P i and vice versa. Thus large correlations of a parameter with other parameters lead to large parameter uncertainties, even when absolute values of the sensitivities are high. Kabala (2001) pointed out the obvious: the logarithmic sensitivity does not solely define the accuracy of a parameter estimate. Therefore, the maximum sensitivity principle for selecting measurement points, used widely in the literature (Knopman and Voss, 1987; McElwee, 1987; Sim and Chrysikopoulos, 1996; Jiao and Zheng, 1997) cannot be true in general. He noted that minimizing the plausible relative errors, rather than maximizing the corresponding sensitivities, and avoiding as much as possible measurements with high parameter correlations should serve as a heuristic guide to identifying the measurements most useful for parameter estimation or as candidates for optimal sampling design. Kabala (2001) defined the plausible relative error as

  dP i P i oO ¼ f ðOÞ Pi O oP i

ð5Þ

where f ðOÞ is the known calibration function for the output measuring device; for example, for a given absolute measurement error DO this function is

f ðOÞ ¼

DO O

ð6Þ

(Obviously, in field measurements a field calibration curve is preferable to the laboratory calibration curve.) The plausible relative error combines the effect of the logarithmic sensitivity with that of the relative measurement error. It can be interpreted as a relative error in the input parameter that one would obtain from the relative measurement error in the output, f ðOÞ, given perfect knowledge of all the other parameters. Kabala (2001) applied the discussed concepts to a semi-analytic model for the pumping test in a fully penetrating pumping well and a fully penetrating observation well situated in a confined homogeneous aquifer with an infinitesimal skin and wellbore storage effects. He found that the plausible relative errors in transmissivity are the smallest, and are an order of magnitude smaller than the plausible relative errors in storativity. Furthermore, he showed that the plausible relative errors in the parameters from wellbore drawdown data are proportional to the pumping rate, and hence can be reduced by an arbitrary factor through increasing the pumping rate by the same factor. No corresponding analysis exists for the case with a thick skin surrounding the well.

1.4 Objectives The objectives of this paper are to study the logarithmic sensitivities, plausible relative errors, and deterministic parameter correlations in a simple semi-analytic no-crossflow model of the transient flowmeter test (TFMT) that accounts for a thick skin around the well. These quantities should provide insight into the ease with which the aquifer and skin parameters could be estimated and thus should shed light on the quality of the no-crossflow interpretation of the TFMT. 2 Semi-analytic model of well response and its sensitivities 2.1 Semi-analytic model As mentioned earlier, no-crossflow models of the TFMT tend to be more robust than crossflow models because they are mathematically simpler. This is especially true when the transient wellface discharge from a given layer can be calculated from the measurements. This information, along with the measured transient wellbore drawdown and the no-crossflow assumption, allows one to treat each layer as a unit independent of all the other layers and thus to simplify the model to that of a single-layer aquifer. This situation arises when the transient wellbore velocities are measured simultaneously by two flowmeters placed at the top and the bottom of the layer (Ehilg-Economides et al., 1986) or when the transient wellbore velocities are measured by one flowmeter sequentially from bottom layer up and are then converted to wellface discharges from the bottom layer of sequentially increasing thickness (Rapach et al., 1990). For our no-crossflow model we consider a single-layer homogeneous aquifer with a fully penetrating well of radius rw , situated in a confined aquifer of thickness b, and surrounded by a thick skin of radius rs . The aquifer is isotropic

403

and has a hydraulic conductivity Ka , and specific storativity Ssa . The thick skin is also isotropic and has a hydraulic conductivity Ks , and specific storativity Sss . Figure 1 shows the schematic. The well is pumped with the total constant pumping rate, Q. The initial-boundary value problem (IBVP) describing the flow towards the well in this system is

 o2 s 1 os for rs < r < 1 þ or2 r or  2  Sss oss o ss 1 oss ¼ for rw < r < rs þ Ks ot or2 r or Ssa os ¼ Ka ot

404



ð7Þ

ð8Þ

sðr; t ¼ 0Þ ¼ 0

ð9Þ

ss ðr; t ¼ 0Þ ¼ 0

ð10Þ

sðr ¼ 1; tÞ ¼ 0

ð11Þ

 oss  dsw Q ¼ 2prw Ks b  þ Cw or rw dt

ð12Þ

ss ðrs ; tÞ ¼ sðrs ; tÞ   oss  os  ¼ Ka  Ks or rs or rs sw ðtÞ ¼ ss ðrw ; tÞ

ð13Þ ð14Þ ð15Þ

where sðr; tÞ and ss ðr; tÞ are the drawdowns in the aquifer and the skin, respectively, at the radial coordinate, r, and time t, and Cw ¼ prc2 is the wellbore storage coefficient, where rc is the casing radius. Equations (7) and (8) govern the drawdown in the aquifer and skin, respectively. Equations

Fig. 1. Diagram of a fully penetrating well surrounded by a thick skin zone and situated in a homogeneous confined aquifer

(9) and (10) specify initial conditions, and (11) entails the assumption that the aquifer is of infinite extent. Wellface boundary condition (12) states that the total pumping rate is equal to the sum of the wellface flowrate from the aquifer and the contribution from wellbore storage. Through wellbore storage, this condition makes the wellface flowrate transient, even when the total pumping rate is constant. The drawdown and flux are continuous at the skinaquifer interface as stated by (13) and (14), respectively. Also, (15) specifies that the drawdown in the skin zone at the wellface is equal to the wellbore drawdown. Following Kabala (2001), we non-dimensionalize the IBVP using the following parameters

sD ¼

s ; rw

qD ¼

q Q

rD ¼

r ; rw

tD ¼

Ka t Ssa rw2

a¼

ssD ¼

ss ; rw

swD ¼

sw rw

ð16Þ ð17Þ

rsD ¼

rw2 Ssa b; rc2

rs rw

ð18Þ ð19Þ

c¼

2prw Ka b ; Q

j¼

Ks ; Ka

k¼

Sss Ssa

ð20Þ

The dimensionless analog of the IBVP (7)–(15) becomes

osD o2 sD 1 osD ¼ 2 þ otD orD rD orD

for rsD < rD < 1

k ossD o2 ssD 1 ossD ¼ 2 þ orD j otD rD orD

for 1 < rD < rsD

ð21Þ

ð22Þ

sD ðrD ; tD ¼ 0Þ ¼ 0

ð23Þ

ssD ðrD ; tD ¼ 0Þ ¼ 0

ð24Þ

sD ðrD ¼ 1; tD Þ ¼ 0  1 ossD  1 oswD ¼ j þ orD rD ¼1 2a otD c

ð25Þ

ssD ðrsD ; tD Þ ¼ sD ðrsD ; tD Þ   ossD  osD  ¼ j orD rsD orD rsD ssD ð1; tD Þ ¼ swD

ð26Þ ð27Þ ð28Þ ð29Þ

405

R1 We apply the Laplace transform, fðpÞ ¼ Lff ðtD Þ; tD ! pg ¼ 0 f ðtD ÞeptD dtD , to the IBVP (21)–(29) and find its semi-analytic solution (in the Laplace domain) in the form

406

p 1 sðrD ; pÞ ¼ AKo ðrD pÞ c i 1 p p 1h swD ðpÞ ¼ CKo ð pk=jÞ þ DIo ð pk=jÞ ¼ swD jc¼1 c c i 1 p p 1h ss ðrD ; pÞ ¼ CKo ðrD pk=jÞ þ DIo ðrD pk=jÞ ¼ ssD jc¼1 c c

ð30Þ ð31Þ ð32Þ

where Io and Ko are the modified Bessel functions of the first and second kind, respectively. In the above two equations the brackets define ssD jc¼1 and swD jc¼1 as independent of c. The coefficients A, C, and D follow from solving the system of three linear equations formed by substituting (30) and (32) into the Laplace-transformed boundary conditions (26)–(28). Since these conditions depend on the dimensionless parameters in (20), so do the semi-analytic solutions (30) and (32). The solution for the wellface flowrate follows from (32) and is also independent of c

 p p p p osD  ¼ C pkjK1 ð pk=jÞ  D pkjI1 ð pk=jÞ qD ðpÞ ¼ cj  orD rD ¼1

ð33Þ

where I1 and K1 are also the modified Bessel functions of the first and second kind, respectively.

2.2 Sensitivities of the transient flowmeter test responses It follows from the non-dimensional expressions for wellbore drawdown, (16), and wellface flowrate, (17), and the solutions (31) and (33) that sw ðt; Ka ; Ssa ; Ks ; Sss ; rs Þ ¼ rw swD ðtD ðt; Ka ; Ssa Þ; cðKa Þ; aðSsa Þ; jðKa ; Ks Þ; kðSsa ; Sss Þ; rsD ðrs ; rw ÞÞ ¼

rw swD ðtD ðt; Ka ; Ssa Þ; 1; aðSsa Þ; jðKa ; Ks Þ; kðSsa ; Sss Þ; rsD ðrs ; rw ÞÞ cðKa Þ

qðt; Ka ; Ssa ; Ks ; Sss ; rs Þ ¼ QqD ðtD ðt; Ka ; Ssa Þ; aðSsa Þ; jðKa ; Ks Þ; kðSsa ; Sss Þ; rsD ðrs ; rw ÞÞ

ð34Þ

ð35Þ

Expressions for the logarithmic sensitivities, (2), follow via the chain rule from (34) and (35). For example, the sensitivity of wellbore drawdown to aquifer hydraulic conductivity is



Ka osw Ka oswD otD oswD oc oswD oj ¼ þ þ sw oKa swD otD oKa oc oKa oj oKa tD oswD jc¼1 j oswD jc¼1 1 ¼ swD jc¼1 otD swD jc¼1 oj

ð36Þ

where we used the relations otD =oKa ¼ tD =Ka , oc=oKa ¼ c=Ka , and oj=oKa ¼ j=Ka that follow from (19) and (20). The other logarithmic sensitivities are derived analogously

Ssa osw tD oswD jc¼1 a oswD jc¼1 k oswD jc¼1 ¼ þ  sw oSsa swD jc¼1 otD swD jc¼1 oa swD jc¼1 ok

ð37Þ

Ks osw j oswD jc¼1 ¼ sw oKs swD jc¼1 oj

ð38Þ

Sss osw k oswD jc¼1 ¼ sw oSss swD jc¼1 ok

ð39Þ

rs osw rsD oswD jc¼1 ¼ sw ors swD jc¼1 orsD

ð40Þ

Ka oq tD oqD j oqD ¼  q oKa qD otD qD oj

ð41Þ

Ssa oq tD oqD a oqD k oqD ¼ þ  q oSsa qD otD qD oa qD ok

ð42Þ

Ks oq j oqD ¼ q oKs qD oj

ð43Þ

Sss oq k oqD ¼ q oSss qD ok

ð44Þ

rs oqD rsD oqD ¼ qD ors qD orsD

ð45Þ

We note that the logarithmic sensitivities as functions of dimensionless time, tD , are independent of c. In calculating the logarithmic sensitivities, we invert the solution into the dimensionless time domain numerically using the Stehfest (1970) or De Hoog et al. (1982) algorithm in the Mathematica environment (Wolfram, 1996).

3.3 Measurement domains Following Kabala (2001), we define the meaningful measurement domains for the wellbore drawdown and wellface flowrate. Given a pressure transducer, the useful wellbore drawdown measurements using this device are within the range Dsmax  sw  smax

ð46Þ

407

where Dsmax is the maximum absolute measurement error, and smax is the maximum allowable measurement with the device. The measurement range (46) can be expressed in the dimensionless form

cs  swD jc¼1  cns

ð47Þ

where

s ¼ 408

Dsmax rw

and

ns ¼

smax rw

ð48Þ

Similarly, for an electromagnetic flowmeter the useful measurements are in the range

Dqmax  q  qmax ) q  qD  nq

ð49Þ

where

q ¼

Dqmax Q

and

nq ¼

qmax Q

ð50Þ

Dqmax is the maximum absolute measurement error, and qmax is the maximum allowable flowrate measurement with the device.

2.4 Plausible relative errors For a constant absolute error in the measurements, the plausible relative errors may be calculated by substituting the logarithmic sensitivities, (2), and calibration function, (6), into (5). For plausible relative errors in the aquifer hydraulic conductivity contained in wellbore drawdown measurements, we use additionally (36) along with (34) and (48) to obtain   dKa Dsmax Ka osw ¼ Ka sw sw oKa !   1 oswD jc¼1 oswD jc¼1 dKa  ¼ cs ¼ cs tD ð51Þ  swD jc¼1 j otD oj Ka cs ¼1 In a similar manner we obtain

!    oswD jc¼1 oswD jc¼1 oswD jc¼1 1 dSsa dSsa  k ¼ cs tD þa ¼ cs Ssa otD oa ok Ssa cs ¼1 !    oswD jc¼1 1 dKs dKs  ¼ cs j ¼ cs Ks oj Ks cs ¼1 !    oswD jc¼1 1 dSss dSss  ¼ cs k ¼ cs Sss ok Sss cs ¼1 !    oswD jc¼1 1 drs drsD  ¼ cs rsD ¼ cs rs orsD rsD cs ¼1

ð52Þ ð53Þ ð54Þ ð55Þ

Analogous expressions for the plausible relative errors in the parameter estimates from wellface flowrate measurements are

 !   dKa oqD oqD 1 dKa  ¼ q tD j ¼ q Ka otD oj Ka q ¼1

ð56Þ

!    dSsa oqD oqD oqD 1 dSsa  k ¼ q tD þa ¼ q Ssa otD oa ok Ssa q ¼1

ð57Þ

 !   dKs oqD 1 dKs  ¼ q j ¼ q Ks oj Ks q ¼1

ð58Þ

!    dSss oqD 1 dSss  ¼ q k ¼ q Sss ok Sss q ¼1

ð59Þ

 !   drs oqD 1 drsD  ¼ q rsD ¼ q rs orsD rsD q ¼1

ð60Þ

409

The relations (51)–(55) demonstrate that the plausible relative errors in the aquifer and skin parameters from drawdown data are proportional to c and s , whereas the relations (56)–(60) demonstrate that the plausible relative errors in the flowrate data are proportional to q : Hence, these plausible relative errors can be decreased by half if the pumping rate is doubled. Of course another way to reduce the plausible errors is to use more accurate instruments with smaller Dsmax and Dqmax . These findings are analogous to those of Kabala (2001).

2.5 Validation We validated the derived model for wellbore drawdown and wellface flowrate and their sensitivities by comparing them to the corresponding quantities in the Papadopulos and Cooper (1967) model for a homogeneous aquifer with wellbore storage. All quantities were matched. 3 Results and discussion In this section we consider three representative synthetic aquifers. In Aquifer I the properties of the thick skin are identical to those of the aquifer, i.e., the aquifer is homogeneous with no thick skin (Ks ¼ Ka ). In Aquifer II, the well is surrounded by a normal thick skin (Ks > Ka ), and the flow towards the well is thus controlled by the aquifer. In Aquifer III, the well is surrounded by a damaged thick skin (Ks < Ka ), and the flow towards the well is thus controlled by the skin. Table 1 shows the properties of these three aquifer-well systems. In the accompanying figures, we plot for these systems the logarithmic sensitivities, (36)–(45), of wellbore drawdown and wellface flowrate to the aquifer parameters (Ka and Ssa ) and the skin parameters (Ks , Sss , and rs ), the corresponding plausible relative errors (51)–(60), and the deterministic logarithmic correlations, (4), of the aquifer parameter Ka with the other parameters (Ssa , Ks , Sss and rs ). As a reference, we also plot in these figures the dimensionless flowrate,

Table 1. Parameters of the three synthetic aquifer-well systems Aquifer Ka (m/s) I II III

410

Ssa (1/m)

Ks (m/s)

Sss (1/m)

b rw rs Q c (m) (m) (m) (m3 /h)

1  105 1  106 1  105 1  106 10 1  105 1  106 5  105 1  106 10 1  105 1  106 2  106 1  106 10

0.1 0.1 0.1

0.2 0.2 0.2

5 5 5

a

j

k

0.04524 105 1 1 0.04524 105 5 1 0.04524 105 0.2 1

qD , to indicate the duration of the wellbore storage phase (which ends when qD  1) and the dimensionless drawdown, swD jc¼1 , to help identify the dimensionless-time domain during which the measurements are feasible. As is apparent from the accompanying figures, the plausible relative errors vary by many orders of magnitude over the meaningful measurement domains, defined for wellbore drawdown and wellface flowrate by (47) and (49), respectively. It is thus convenient to define within these domains the corresponding range of practically important measurements as the time domain over which the plausible relative errors vary by less than one order of magnitude from the minimum plausible relative errors. The extent of the meaningful measurement domains and the corresponding practical measurement domains are marked in the plots of the plausible relative errors by circles and squares, respectively. The practically important flowrate and drawdown measurements in the flowmeter test begin approximately when the dimensionless wellface flowrate exceeds qD  0:4.

3.1 Aquifer I: no thick skin (Ks = Ka ) As follows from Fig. 2, the sensitivities of wellbore drawdown to aquifer parameters, and skin hydraulic conductivity, increase by about an order of magnitude per log cycle during the wellbore storage phase. After that, the sensitivity to aquifer hydraulic conductivity reaches a constant value while the sensitivities to aquifer specific storativity and skin hydraulic conductivity decrease by about 10% per log cycle. The corresponding plausible relative errors in aquifer parameters and skin hydraulic conductivity from the wellbore drawdown measurements decrease by about two orders of magnitude per log cycle during the wellbore storage phase. After that, the plausible relative errors in aquifer hydraulic conductivity decrease by about 30% per log cycle while the plausible relative errors in aquifer specific storativity and skin hydraulic conductivity remain constant at their minimum values. The sensitivity of wellbore drawdown to skin specific storativity stays low throughout most of the wellbore storage phase and rapidly decreases at its end. Even when the flowrates are still barely measurable (qD  0:01) this sensitivity is already orders of magnitude smaller than the sensitivity of wellbore drawdown to the other parameters. The plausible relative errors in skin specific storativity from the wellbore drawdown measurements are orders of magnitude larger than those in the other parameters. This makes the skin specific storativity practically impossible to estimate from the wellbore drawdown measurements. Therefore, the skin specific storativity should be treated instead as a known quantity in interpretation of the TFMT. Indeed, just from the visual inspection of a drilling core, an experienced hydrologist should be able to provide an estimate of the skin-zone specific storativity within 2 orders of magnitude, which is much better than could be estimated from the TFMT data.

411

Fig. 2a–d. Sensitivities of well responses to aquifer hydraulic conductivity, Ka , aquifer specific storativity, Ssa , skin radius, rs , skin hydraulic conductivity, Ks , and skin specific storativity, Sss , along with plausible relative errors in these parameters for Aquifer I (a ¼ Ssa brw2 = rc2 ¼ 105 and j ¼ Ks =Ka ¼ 1). a Sensitivities of wellbore drawdown, b plausible relative errors from wellbore drawdown, c sensitivities of wellface flowrate, and d plausible relative errors from wellface flowrate. The extent of the meaningful and practical measurement domains are marked with circles and squares, respectively

The sensitivity of wellface flowrate to aquifer hydraulic conductivity increases by about 100% per log cycle and reaches a maximum during the part of the wellbore storage phase when qD < 0:1. It decreases rapidly in the later stages of the wellbore storage phase. The sensitivities of wellface flowrate to aquifer specific storativity and skin hydraulic conductivity decrease by about 30% per log cycle during most of the wellbore storage phase and much more rapidly toward its end. The corresponding plausible relative errors in aquifer parameters and skin hydraulic conductivity from the flowrate measurements decrease by about an order of magnitude per log cycle during the wellbore storage phase and reach their minimum values when qD  0:6. After that, the plausible relative errors in aquifer hydraulic conductivity increase by about one and a half orders of magnitude per log cycle while the plausible relative errors in aquifer specific storativity and skin hydraulic conductivity increase even more rapidly. The sensitivity of flowrate to skin specific storativity decreases rapidly throughout the wellbore storage phase. It, too, is orders of magnitude smaller

412

than the flowrate sensitivities to the other parameters even when the flowrates are still barely measurable (qD  0:01). The plausible relative errors in skin specific storativity remain practically constant during the wellbore storage phase up to the time when qD  0:6 and then increase rapidly. They are orders of magnitude larger than those of the other parameters. This makes the skin specific storativity practically impossible to estimate from the flowrate measurements as well. When all the plausible relative errors are around their minimal values, i.e., when qD  0:6, the plausible relative errors in aquifer hydraulic conductivity are an order of magnitude smaller than the plausible relative errors in aquifer specific storativity and skin hydraulic conductivity and about five orders of magnitude smaller than the plausible relative errors in skin specific storativity. This is the case for both the wellbore drawdown and wellface flowrate measurements. As Kabala (2001) pointed out, minimizing the plausible relative errors rather than maximizing the corresponding sensitivities should serve as a guide for identifying the measurements most useful for parameter estimation or as candidate measurements for optimal sampling. The plots in Fig. 2 illustrate the fallacy of the principle of maximum sensitivity for selecting measurements discussed earlier (Kabala, 2001). Indeed, the maximum sensitivity of wellface flowrate to aquifer hydraulic conductivity occurs when qD  0:06, i.e., at a different time than the minimum in the corresponding plausible relative error (when qD  0:6). This is also the case for the sensitivity of wellface flowrate to skin hydraulic conductivity, which decreases continuously through the wellbore storage phase, while the corresponding plausible relative error reaches a minimum when qD  0:6. We note in passing that El-Sayegh (2000) considered in his dissertation a number of related cases with different parameter values. He noted that although the duration of the wellbore storage phase may change, the minimum plausible errors still occur when qD  0:6. He also considered linearly and exponentially variable pumping rates and investigated their effect on sensitivities and plausible errors. He found that qualitatively they are similar to those for the case of a constant pumping rate. In Fig. 3, we plot the deterministic logarithmic correlations, (60), of aquifer hydraulic conductivity with skin hydraulic conductivity, aquifer specific storativity, and skin specific storativity. These correlations do not cross each other for both wellbore drawdown and wellface flowrate measurements and are smaller than one in the time domain of practical flowrate measurements (when qD > 0:04). They thus demonstrate that the parameters estimated from the TFMT conducted in a well surrounded by a thick skin of similar properties to those of the aquifer can be ordered from the most to the least accurate as follows: the aquifer hydraulic conductivity, the skin hydraulic conductivity, the aquifer specific storativity, and the skin specific storativity. However, the last parameter would be so biased that one should not even attempt to estimate it from the TFMT.

3.2 Aquifer II: normal thick skin (Ks > Ka ) From the comparison of sensitivities and plausible errors for the homogeneous aquifer with a well surrounded by a normal thick skin (Fig. 4) to those for the homogeneous aquifer with no thick skin (Fig. 2) it is apparent that a normal thick skin has no significant effect on the logarithmic sensitivities of wellbore drawdown and wellface flowrate to the aquifer parameters and the skin specific storativity, nor on the corresponding plausible relative errors. However, the

413

Fig. 3a, b. Logarithmic correlations, or ratios of logarithmic sensitivities, for Aquifer I (a ¼ 105 and j ¼ 1). a From wellbore drawdown, b from wellface flowrate

Fig. 4a–d. Sensitivities of well responses to aquifer hydraulic conductivity, Ka , aquifer specific storativity, Ssa , skin radius, rs , skin hydraulic conductivity, Ks , and skin specific storativity, Sss , along with plausible relative errors in these parameters for Aquifer II (a ¼ Ssa brw2 = rc2 ¼ 105 and j ¼ Ks =Ka ¼ 5). a Sensitivities of wellbore drawdown, b plausible relative errors from wellbore drawdown, c sensitivities of wellface flowrate, and d plausible relative errors from wellface flowrate. The extent of the meaningful and practical measurement domains are marked with circles and squares, respectively

414

sensitivities of wellbore drawdown and wellface flowrate to the skin hydraulic conductivity decrease by half an order of magnitude, whereas the corresponding plausible relative errors increase by the same factor. The logarithmic sensitivity of wellbore drawdown to the skin radius rs , shown also in Fig. 4, is the second largest sensitivity; it is only smaller than that to the aquifer hydraulic conductivity. The corresponding plausible relative error in the skin radius is the second smallest; it is only larger than that in the aquifer hydraulic conductivity. Analogous relations hold for logarithmic sensitivity of wellface flowrate to the skin radius rs and the corresponding plausible relative error in this parameter. We note that at the end of the wellbore storage phase (qD  1), the Stehfest (1970) algorithm for the numerical inversion of the Laplace transforms produces oscillations in the logarithmic sensitivity of the wellface flowrate to the skin specific storativity and in the corresponding plausible relative errors. The oscillations are artifacts of this numerical inversion algorithm that requires the inverted function to be monotonic, which is clearly not the case here. When recalculated with the De Hoog et al. (1982) algorithm, the oscillations disappear as expected. In Fig. 5 we present the deterministic logarithmic correlations, (4), of aquifer hydraulic conductivity with skin hydraulic conductivity, aquifer specific storativity, skin radius, and skin specific storativity. For practical wellbore drawdown measurements, with the exception of the correlation of the aquifer hydraulic conductivity with the skin radius, these correlations do not cross each other and are smaller than one. They thus demonstrate that the parameters estimated from wellbore drawdowns measured in a TFMT conducted in a well surrounded by a normal thick skin (Ks > Ka ) can be ordered from the most to the least accurate as follows: the aquifer hydraulic conductivity, the aquifer specific storativity, the skin hydraulic conductivity, and the skin specific storativity. We note that at the end of wellbore storage phase, the correlation of the aquifer hydraulic conductivity with the skin radius drops below the correlations of the aquifer hydraulic conductivity with the aquifer specific storativity and with the skin hydraulic conductivity. This indicates that the estimate of the skin radius from wellbore drawdowns measured during the wellbore storage phase would be even more accurate than the estimate of the aquifer hydraulic conductivity, whereas the estimate of the skin radius from wellbore

Fig. 5a, b. Logarithmic correlations, or ratios of logarithmic sensitivities, for Aquifer II (a ¼ 105 and j ¼ 5). a From wellbore drawdown, b from wellface flowrate

drawdowns measured after the wellbore storage phase would be more accurate only than the estimate of the skin specific storativity. For practical wellface flowrate measurements, none of the correlations cross each other and all are smaller than one. They thus demonstrate that the parameters estimated from the TFMT conducted in a well surrounded by a normal thick skin can be ordered from the most to the least accurate as follows: the skin radius, the aquifer hydraulic conductivity, the skin hydraulic conductivity, the aquifer specific storativity, and the skin specific storativity. We note again that the last parameter would be so biased that one should not even attempt to estimate it from the TFMT. 415

3.3 Aquifer III: damaged thick skin (Ks < Ka ) From the comparison of sensitivities and plausible errors for the homogeneous aquifer with a well surrounded by a damaged thick skin (Fig. 6) to those for the

Fig. 6a–d. Sensitivities of well responses to aquifer hydraulic conductivity, Ka , aquifer specific storativity, Ssa , skin radius, rs , skin hydraulic conductivity, Ks , and skin specific storativity, Sss , along with plausible relative errors in these parameters for Aquifer III (a ¼ Ssa brw2 = rc2 ¼ 105 and j ¼ Ks =Ka ¼ 0:2). a Sensitivities of wellbore drawdown, b plausible relative errors from wellbore drawdown, c sensitivities of wellface flowrate, and d plausible relative errors from wellface flowrate. The extent of the meaningful and practical measurement domains are marked with circles and squares, respectively

416

homogeneous aquifer with no thick skin (Fig. 2) it is apparent that a damaged thick skin lessens the logarithmic sensitivities of wellbore drawdown and wellface flowrate to the aquifer parameters and the skin specific storativity, and it increases the corresponding plausible relative errors. However, the damaged thick skin increases sharply – within a factor of three – the sensitivities of wellbore drawdown and wellface flowrate to the skin hydraulic conductivity and decreases the corresponding plausible relative errors within the same factor. Thus, the damaged skin, which controls the flow towards the well, increases the information content about the skin hydraulic conductivity in the wellbore drawdown and wellface flowrate measurements. In the presence of the damaged thick skin, as in the presence of the normal thick skin, the logarithmic sensitivity of wellbore drawdown to the skin radius rs , shown also in Fig. 6, follows closely the logarithmic sensitivity of wellbore drawdown to the skin hydraulic conductivity. The corresponding plausible relative error in the skin radius follows closely the plausible relative error in the skin hydraulic conductivity. Analogous relations hold for the logarithmic sensitivity of wellface flowrate to the skin radius rs and the corresponding plausible relative error in the skin radius. Again, to avoid oscillations at the end of the wellbore storage phase (qD  1) in the logarithmic sensitivity of the wellface flowrate to the skin specific storativity and in the corresponding plausible relative error the numerical inversion of the Laplace transform is calculated via the De Hoog et al. (1982) algorithm rather than the Stehfest (1970) algorithm. In Fig. 7 we present the deterministic logarithmic correlations, (4), of aquifer hydraulic conductivity with skin hydraulic conductivity, aquifer specific storativity, skin radius, and skin specific storativity. For wellbore drawdown measurements, the deterministic logarithmic correlations of the aquifer hydraulic conductivity with the skin hydraulic conductivity and the skin radius are above one until qD  0:06, whereas for wellface flowrate measurements they are above one until qD  0:05. Although this implies that the early enough measurements could yield more accurate estimates of the skin hydraulic conductivity than those of the aquifer hydraulic conductivity, these estimates would be biased by the plausible errors that are orders of magnitude larger than the plausible errors in estimates from the later measurements.

Fig. 7a, b. Logarithmic correlations, or ratios of logarithmic sensitivities, for Aquifer III (a ¼ 105 and j ¼ 0:2). a From wellbore drawdown, b from wellface flowrate

We note in passing that the crossing of correlation of the aquifer hydraulic conductivity with the skin hydraulic conductivity is consistent with the earlier results of Butler (1988). Although he did not account for wellbore storage or used logarithmic sensitivities, as we do, he found that in sufficiently early times, the sensitivity of the wellbore drawdown measurements to the skin hydraulic conductivity should exceed the corresponding sensitivity to the aquifer hydraulic conductivity. For practical wellbore drawdown measurements, with the exception of the correlation of the aquifer hydraulic conductivity with the skin radius, these correlations do not cross each other and are smaller than one. They thus demonstrate that the parameters estimated from wellbore drawdowns measured in a TFMT conducted in a well surrounded by a damaged thick skin (Ks < Ka ) can be ordered from the most to the least accurate as follows: the skin hydraulic conductivity, the aquifer hydraulic conductivity, the aquifer specific storativity, and the skin specific storativity. At the end of wellbore storage phase, the correlation of the aquifer hydraulic conductivity with the skin radius drops below the correlations of the aquifer hydraulic conductivity with the aquifer specific storativity and with the skin hydraulic conductivity. This indicates that the estimate of the skin radius from wellbore drawdowns measured during the wellbore storage phase would be as accurate as the estimate of the skin hydraulic conductivity, whereas the estimate of the skin radius from wellbore drawdowns measured after the wellbore storage phase would be more accurate only than the estimate of the skin specific storativity. For practical wellface flowrate measurements, none of the correlations cross each other and all are smaller than one. They thus demonstrate that the parameters estimated from the TFMT conducted in a well surrounded by a normal thick skin can be ordered from the most to the least accurate as follows: the skin radius, the skin hydraulic conductivity, the aquifer hydraulic conductivity, the aquifer specific storativity, and the skin specific storativity. We note again that the last parameter would be so biased that one should not even attempt to estimate it from the TFMT.

4 Synthetic examples Consider performing the transient flowmeter test in a fully penetrating well situated in two confined aquifers with the following equipment:  a 15 psig pressure transducer with Dsmax ¼ 0:026 m and smax ¼ 21 m  a flowmeter with Dqmax ¼ 0:2 m3 /h and qmax ¼ 10 m3 /h We select the total pumping rate of Q = 5 m3 /h. Assuming that the model presented in this paper perfectly represents the test, we calculate the plausible relative errors in the aquifer and skin parameters estimated from the measurements produced by the TFMT conducted in two aquifers.

4.1 Example 1 – aquifer controlled flow, normal thick skin (Ks > Ka ) Assume that from the interpretation of our TFMT we get the following parameters Ka ¼ 105 m=s; Ks ¼ 5  105 m=s;

Ssa ¼ 106 m1 ; and Sss ¼ 106 m1

which are identical to those of Aquifer II (see Table 1).

417

From (20), (48), and (50) we then obtain the following dimensionless parameters

c ¼ 0:04524; q ¼ 0:04;

418

a ¼ 105 ; nq ¼ 2;

s ¼ 0:26;

j ¼ 5;

ns ¼ 210;

k¼1

Since the relevant type curves of sensitivities and plausible relative errors are already generated in Fig. 4 for this case, we use them in our subsequent analysis. From (47) follows the range of meaningful drawdown measurements, cs ¼ 0:01176  swD jc¼1  cns ¼ 9:5004. As is apparent from Fig. 4 this range corresponds to the range of dimensionless times 0:63  103  tD  0:25  109 . Next, we read off this figure, developed for cs ¼ 1, the values of minimum plausible relative error for each parameter. These values are then scaled by cs  0:01176. For the wellbore drawdown measurements, we thus obtain the actual minimum plausible relative errors in the parameters

dKa  1:325  103 ¼ 0:13%; Ka dKs  0:085 ¼ 8:5%; Ks

dSsa  0:024 ¼ 2:4% Ssa

dSss  194:3  1:9  104 %; Sss

drs  0:015 ¼ 1:5% rs

In an analogous way we read off the plausible relative errors for q ¼ 1 from Fig. 4 and scale them by q ¼ 0:04. From (49) follows the range of meaningful wellface flowrate measurements, q ¼ 0:04  qD  nq ¼ 2. This range corresponds to the range of dimensionless times 0:79  104  tD  1. For the wellface flowrate measurements, we thus obtain the actual minimum plausible relative errors in the parameters

dKa  0:130 ¼ 13:0%; Ka dKs  4:970 ¼ 497:0%; Ks

dSsa  1:378 ¼ 137:8% Ssa dSss  1:5  103 ¼ 1:5  105 %; Sss

drs  0:861 ¼ 86:1% rs

We also obtain the deterministic correlations (or the relative error ratios), (4), of aquifer hydraulic conductivity to aquifer specific storativity and skin parameters. For wellbore drawdown measurements at points of minimum plausible relative errors in aquifer hydraulic conductivity we have

dKa =Ka  0:090; drs =rs

dKa =Ka  0:056; dSsa =Ssa

dKa =Ka  0:016; dKs =Ks

dKa =Ka  6:82  106 dSss =Sss

Similarly, for wellface flowrate measurements at points of minimum plausible relative errors in aquifer hydraulic conductivity we get

dKa =Ka  0:151; drs =rs dKa =Ka  0:026; dKs =Ks

dKa =Ka  0:094; dSsa =Ssa dKa =Ka  8:67  105 dSss =Sss

The above results confirm the described earlier relations between the plausible relative errors for the parameters. In particular, it is clear that one should not even attempt to estimate the skin specific storativity from the TFMT.

4.2 Example 2 – skin controlled flow, damaged thick skin (Ks < Ka ) Assume now that from the interpretation of our TFMT we get the following parameters Ka ¼ 1  105 m=s; 6

Sss ¼ 10

Ssa ¼ 106 m1 ;

Ks ¼ 2  106 m=s;

and

1

m

which are identical to those of Aquifer II (see Table 1). From (20), (48), and (50) we then obtain the following dimensionless parameters

c ¼ 0:04524; q ¼ 0:04;

a ¼ 105 ; nq ¼ 2;

s ¼ 0:26;

j ¼ 0:2;

ns ¼ 210;

k¼1

Again, since the relevant type curves of sensitivities and plausible relative errors are already generated in Fig. 6 for this case, we use them in our subsequent analysis. We follow the procedure from Example 1. The range of meaningful drawdown measurements, cs ¼ 0:01176  swD jc¼1  cns ¼ 9:5004, follows from (47), whereas the range of the corresponding dimensionless times 0:63  103  tD  0:16  107 follows from Fig. 6. From this figure, developed for cs ¼ 1, we read off the values of minimum plausible relative error in each parameter and scale them by cs  0:01176. For the wellbore drawdown measurements, we obtain the following minimum plausible relative errors in the parameters

dKa  2:45  103  0:25%; Ka

dSsa  0:029 ¼ 2:9% Ssa

dKs  4:22  103  0:42%; Ks drs  3:67  103  0:37% rs

dSss  188:2  1:9  104 %; Sss

In an analogous way, from Fig. 6 developed for q ¼ 1 we read off the minimum plausible relative errors and scale them by q ¼ 0:04. The range of meaningful wellface flowrate measurements, q ¼ 0:04  qD  nq ¼ 2, follows from (49) and the corresponding dimensionless times 0:16  105  tD  1 follows from Fig. 6. For the wellface flowrate measurements, we thus obtain the following minimum plausible relative errors in the parameters

419

dKa  0:193 ¼ 19:3%; Ka

dSsa  2:148 ¼ 214:8% Ssa

dKs  0:310 ¼ 31:0%; Ks

dSss  0:026  105 ¼ 2:6  105 %; Sss

drs  0:269 ¼ 26:9% rs 420

As in Example 1, we also obtain the correlations or the relative error ratios, (4), in aquifer hydraulic conductivity to aquifer specific storativity and skin parameters. For wellbore drawdown measurements at points of minimum plausible relative errors in aquifer hydraulic conductivity we have

dKa =Ka  0:67; drs =rs

dKa =Ka  0:58; dKs =Ks

dKa =Ka  0:085; dSsa =Ssa

dKa =Ka  1:30  105 dSss =Sss

Similarly, for wellface flowrate measurements at points of minimum plausible relative errors in aquifer hydraulic conductivity we get

dKa =Ka  0:72; drs =rs

dKa =Ka  0:62; dKs =Ks

dKa =Ka  0:09; dSsa =Ssa

dKa =Ka  7:42  105 dSss =Sss

The above results confirm again the described earlier relations between the plausible relative errors for the parameters. We again see clearly that one should not attempt to estimate the skin specific storativity from the TFMT.

5 Conclusions A simple no-crossflow model of a transient flowmeter test (TFMT) conducted on a well surrounded by a thick skin and situated in a confined aquifer of thickness b is considered. It accounts for the wellbore storage. The model is identical to that of a pumping test conducted on a fully penetrating well with the total constant pumping rate, Q. Logarithmic sensitivities of wellbore drawdown, sw , and wellface flowrate, q, with respect to the aquifer parameters (hydraulic conductivity, Ka , and specific storativity, Ssa ) and skin parameters (skin hydraulic conductivity, Ks , skin specific storativity, Ssa , and skin radius, rs ) are calculated along with the corresponding plausible relative errors and the deterministic logarithmic correlations. We find that the shapes of the logarithmic sensitivities and plausible relative errors in the model considered here are similar to the corresponding shapes in the model for the case of a pumping test on a well with an infinitesimal skin situated in a homogeneous aquifer considered by Kabala (2001). Furthermore, analogous to his conclusions we find that the sensitivities of wellbore drawdown and wellface flowrate to aquifer and skin parameters, (36)–(45), as functions of the dimensionless time tD ¼ Ka =ðSsa rw2 tÞ, are independent of the dimensionless parameter c ¼ 2prw Ka b=Q, i.e., are independent of the pumping rate. As is apparent from (51)–(60), the plausible relative errors in the aquifer and skin parameters from drawdown data are proportional to c, whereas the plausible relative errors in wellface flowrate are proportional to q ¼ Dqmax =Q. Thus they all can be decreased by increasing proportionally the pumping rate.

All sensitivities and plausible relative errors are dependent on the dimensionless wellbore storage a ¼ Ssa brw2 =rc2 and skin-to-aquifer parameter contrasts j ¼ Ks =Ka and k ¼ Sss =Ssa . The plausible relative errors vary by many orders of magnitude over the meaningful measurement domains, defined by (47) and (49). Within these domains, we define the time domains of practically important measurements over which the plausible relative errors vary by less than one order of magnitude from the minimum plausible relative errors. The practically important flowrate and drawdown measurements in the flowmeter test begin approximately when the dimensionless wellface flowrate exceeds qD ¼ q=Q  0:4. We also confirm that minimizing the plausible relative errors rather than maximizing the corresponding sensitivities should serve as a guide for identifying the measurements most useful for parameter estimation or as candidate measurements for optimal sampling. The plots in Fig. 2 illustrate the fallacy of the principle of maximum sensitivity for selecting measurements discussed earlier (Kabala, 2001). Indeed, the maximum sensitivity of wellface flowrate to aquifer hydraulic conductivity occurs at a different time than the minimum in the corresponding plausible relative error. This is also the case for the sensitivity of wellface flowrate to skin hydraulic conductivity and the corresponding plausible relative error. Particular conclusions from this research include: 1. For both wellbore drawdown and wellface flowrate measurements, the plausible relative errors in aquifer hydraulic conductivity are generally significantly smaller than the plausible relative errors in the other parameters. This is especially true for homogeneous aquifers with no thick skin (Ks ¼ Ka ) or with a normal thick skin (Ks > Ka ). This is also true, however, for homogeneous aquifers with damaged skin (Ks < Ka ) for most of the practically important time domain (qD > 0:04). This means that of all the parameters, the aquifer hydraulic conductivity is most accurately estimated from the TFMT. 2. For both wellbore drawdown and wellface flowrate measurements, the plausible relative errors in aquifer specific storativity are generally an order of magnitude larger than the plausible relative errors in the aquifer hydraulic conductivity. This is consistent with the quality of the aquifer parameter estimates from the slug test (McElwee et al., 1995). 3. For both wellbore drawdown and wellface flowrate measurements, the plausible relative errors in the skin hydraulic conductivity are  larger than the plausible relative errors in the aquifer specific storativity when the thick skin is normal (Ks > Ka ),  approximately equal to the plausible relative errors in the aquifer specific storativity when the thick skin is identical to the aquifer (Ks ¼ Ka ), and  smaller than the plausible relative errors in the aquifer specific storativity when the thick skin is damaged (Ks < Ka ). For the TFMT, this suggests an analogous relation between the actual estimation errors of the skin hydraulic conductivity and the aquifer specific storativity. 4. For both wellbore drawdown and wellface flowrate measurements taken in the practically important time domain (qD > 0:04), the plausible relative errors in the skin specific storativity are generally more than 4–5 orders of magnitude larger than the plausible relative errors in the aquifer hydraulic conductivity. Therefore, the specific storativity of the skin zone from the TFMT would be so

421

422

biased that one should not even attempt to estimate it. Instead, it should be treated as a known quantity in interpretation of the test. An experienced hydrologist should be able to provide an estimate of the skin-zone specific storativity within 2 orders of magnitude from the visual inspection of the drilling core, which is most likely much better than could be estimated from the TFMT data. 5. The plausible relative errors in the skin radius are generally larger than the plausible relative errors in aquifer hydraulic conductivity but smaller than those in the other parameters or close to the plausible relative errors in the skin hydraulic conductivity of damaged skins. This suggests that obtaining reasonably accurate estimates of the skin radius from the TFMT may be possible.

References Anbarci K, Grader AS, Ertekin T (1989) Determination of front locations in multilayer composite reservoir, paper SPE 19799 presented at the SPE Annual Meeting, San Antonio, TX Bishop KA, Brett VS, Green DW, McElhiney JE (1976) 51st Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, New Orleans Bourdet D (1985) Pressure Behavior of Layered Reservoirs with Crossflow, paper SPE 13628 presented at the SPE California Regional Meeting, Bakersfield Butler JJ (1988) Pumping tests in nonuniform aquifers – the radially symmetric case. J. Hydrol. 101: 15–30 Butler JJ, Liu WZZ (1991) Pumping tests in nonuniform aquifers – the linear strip case. J. Hydrol. 128(1–4): 69–99 Butler JJ, Liu WZZ (1993) Pumping tests in nonuniform aquifers – the radially asymmetric case. Water Resour. Res. 29(2): 259–269 Butt MA, McElwee CD (1985) Aquifer-parameter evaluation from variable-rate pumping tests using convolution and sensitivity analysis. Ground Water 23(2): 212–219 Chen L, Rabitz H, Considine DB, Jackman CH, Shorter JA (1997) Chemical reaction rate sensitivity and uncertainty in a two-dimensional middle atmospheric ozone model. J. Geophys. Res.-Atmos. 102(D13): 16201–16214 Cob PM, McElwee CD, Butt MA (1982) Analysis of leaky aquifer pumping test data: an automated numerical solution using sensitivity analysis. Ground Water 20(3): 325–333 Datta-Gupta A, Vasco DW, Long JCS (1997) On the sensitivity and spatial resolution of transient pressure and tracer data for heterogeneity characterization. SPE Formation Evaluation 137–144 De Hoog FR, Knight JH, Stokes AN (1982) An improved method for numerical inversion of Laplace transforms. SIAM J. Stat. Comput. 3: 357–366 Deif AS (1996) Sensitivity Analysis in Linear Systems, Springer-Verlag, New York Ehlig-Economides C, Joseph J, Erba M, Vik SA (1986) Evaluation of single-layered Transients in multilayered Systems. SPE European Petroleum, held in London Ehlig-Economides C, Joseph J (1987) A new test for determination of individual layer properties in a multilayered reservoir. SPE Formation Evaluation El-Sayegh HK (2000) The Transient Flowmeter Test for Aquifer Characterization, dissertation, Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina Kabala ZJ, El-Sayegh HK (2002) Transient flowmeter test: semi-analytic crossflow model. Adv. Water Res. 25: 25(1): 103–121 Gomes E, Ambastha AK (1993) An Analytical Pressure-Transient Model for Multilayered, Composite Reservoirs With Pseudosteady-State Formation Crossflow, SPE Western Regional Meeting held in Anchorage, Alaska, USA Gomes E, Ambastha AK (1994) An analytical pressure-transient model for complex reservoir scenarios. Computers and Geosciences 20(7–8): 57–69 Hada S (1977) Utilization and interpretation of micro flowmeter. Engineering Geology (Japan) 18: 26–37

Hamilton DC, Watts DG (1985) Quadratic design criterion for precise estimation in nonlinear regression models. Technometrics 27(3): 241–250 Hatzignatiou DG, Ogbe DO, Dehghani K, Economides MJ (1987) Interference Pressure Behavior in Multilayered Composite Reservoirs 62nd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dallas, TX Hess AE (1986) Identifying hydraulic conductive fractures with a slow-velocity borehole flowmeter. Canadian Geotechnical Journal 23: 69–78 Hueni GB, Jones RE, Fitzgerald KM, Kohlhaas CA (1990) 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA Hufschmied P (1983) Ermittlung der Durchlassigkeit von Lockergesteins-Grundwasserleitern, eine Vergleichende Untersuchung verschiedener Feldmethoden, Doctoral Dissertation No. 7397, ETH, Zurich, Switzerland Javandel I, Witherspoon PA (1969) A method of analyzing transient fluid flow in multilayered aquifers. Water Resour. Res. 5: 856–869 Jiao JJ, Zheng C (1997) The different characteristics of aquifer parameters and their implications on pumping-test analysis. Ground Water 35(1): 25–29 Kabala ZJ (2001) Sensitivity analysis of a pumping test on a well with wellbore storage and skin. Adv. Water. Res. 24(5): 483–504 Kabala ZJ, Milly PCD (1990) Sensitivity analysis of flow in unsaturated heterogeneous porous media: theory and numerical model. Water Resour. Res. 26(4): 593–610 Kabala ZJ, Milly PCD (1991a) Sensitivity analysis of infiltration, evaporation and redistribution in unsaturated Miller-similar porous media. Water Resour. Res. 27(10): 2655–2666 Kabala ZJ, Milly PCD (1991b) Sensitivity analysis of partial differential equations: a case for the functional sensitivity approach. Numer. Meth. Partial Differential Equations 7(2): 101–112 Keys WS, Sullivan JK (1978) Role of borehole geophysics in defining the physical characteristics of the Raft River geothermal reservoir, Idaho. Geophysics 44: 1116–1141 Knopman DS, Voss CI (1987) Behavior of sensitivities in the one-dimensional advectiondispersion equation: implications for parameter estimation and sampling data. Water Resour. Res. 23(2): 253–272 Knopman DS, Voss CI (1988) Further comments on sensitivities, parameter estimation, and sampling design. Water Resour. Res. 24(2): 225–238 Knopman DS, Voss CI (1989) Multiobjective sampling design for parameter estimation and model discrimination in groundwater solute transport. Water Resour. Res. 25(10): 2245–2258 Knopman DS, Voss CI, Garabedian SP (1991) Sampling design for groundwater solute transport: tests of methods and analysis of Cape Cod tracer test data. Water Resour. Res. 27(5): 925–949 Koda M, Dogru AH, Seinfeld JH (1979) Sensitivity analysis of partial differential equations with applications to reaction and diffusion processes. J. Comput. Phys. 30: 259–282 McElwee CD (1987) Sensitivity analysis of groundwater models. In: Advances in Transport Phenomena in Porous Media, Bear J, Corapcioglu MY (eds), NATO Advanced Study Institute, Ser. E. 128, pp. 751–817 McElwee CD, Bohling GC, Butler JJ (1995a) Sensitivity analysis of slug tests. 1. The slugged well. 164(1–4): 53–67 McElwee CD, Butler JJ, Bohling GC, Liu W (1995b) Sensitivity analysis of slug tests. 2. The Observation well. J. Hydrol. 164(1–4): 69–87 McElwee CD, Yukler MA (1978) Sensitivity of groundwater models with respect to variations in transmissivity and storage. Water Resour. Res. 14(3): 451–459 Molz FJ, Morin H, Hess AE, Melville JG, Gu¨ ven O (1989) The impeller meter for measuring aquifer permeability variations: Evaluation and comparison with other tests. Water Resour. Res. 25: 1677–1683 Molz FJ, Gu¨ ven O, Melville JG (1990) A new approach and methodologies for characterizing the hydrogeologic properties of aquifers, Report EPA/600/2-90/002, US Environmental Protection Agency, Ada, OK Morita N, Gray KE (1981) Type curve matching techniques with error sensitivity numbers. SPE Formation Evaluation

423

424

Munoz-Carpena R, Parsons JE, Gilliam JW (1999) Modeling hydrology and sediment transport in vegetative filter strips. 214(1–4): 111–129 Paillet FL, Hess AE, Cheng CH, Hardin E (1987) Characterization of fracture permeability with high-resolution vertical flow measurements during borehole pumping. Ground Water 25(1): 28–40 Paillet FL (1993) Using borehole geophysics and cross-borehole flow testing to define hydraulic connections between fracture zones in bedrock aquifers. J. Appl. Geophysics 30: 261–279 Paillet FL (1995) Using borehole flow logging to optimize hydraulic-test procedures in heterogeneous fractured aquifers. Hydrogeology Journal 3(3): 4–20 Papadopulos IS, Cooper HH. Jr (1967) Drawdown in a well of large diameter. Water Resour. Res. 3: 241–244 Rapach JM, Jefferies R, Guedroudj AH, Westaway PJ (1990) Practical transient multilayer test design, implementation, and analysis of gas wells in the North Sea, Southern Basin, SPE Paper #20534, 65th SPE Annual Technical Conference and Exhibition, held in New Orleans, LA Rabitz H (1987) Chemical dynamics and kinetics phenomena as revealed by sensitivity analysis techniques. Chem. Rev. 87: 101–112 Rehfeldt KR, Hufschmied P, Gelhar LW, Schaefer ME (1989) The borehole flowmeter technique for measuring hydraulic conductivity variability, Report No. EN 6511, Electric Power Research Institute, Palo Alto, CA Reuven Y, Smooke MD, Rabitz H (1987) Sensitivity analysis of one-dimensional mixed initial-boundary value problems: application to freely propagating premixed laminar flames. J. Sci. Comput. 2(4): 345–370 Ruud NC, Kabala ZJ (1996) Numerical evaluation of flowmeter test interpretation methodologies. Water Resour. Res. 32(4): 845–852 Ruud NC, Kabala ZJ (1997a) Response of a partially penetrating well in a heterogeneous aquifer: integrated well-face flux versus uniform well-face flux boundary conditions. J. Hydrol. 194(1–4): 76–94 Ruud NC, Kabala ZJ (1997b) Numerical evaluation of the flowmeter test in a layered aquifer with a skin zone. J. Hydrol. 203(1–4): 101–108 Ruud NC, Kabala ZJ, Molz FJ (1999) Numerical evaluation of head-loss effects in the flowmeter test. J. Hydrol. 224: 55–63 Schimschal U (1981) Flowmeter analysis at Raft River, Idaho. Ground Water 19: 93–97 Shah PC, Karakas M, Kuchuk F, Ayestaran LC (1988) Estimation of the permeabilities and skin factors in layered reservoirs with downhole rate and pressure data. SPE Formation Evaluation Sim Y, Chrysikopoulos CV (1996) One-dimensional virus transport in porous media with time-dependent inactivation rate coefficients. Water Resour. Res. 32(8): 2607–2611 Stehfest H (1970) Numerical inversion of Laplace transforms. Commun. ACM 13: 47–49 Sudicky EA (1986) A natural gradient experiment on solute transport in a sand aquifer: spatial variability of hydraulic conductivity and its role in the dispersion process. Water Resour. Res. 22: 2069–2082 Tebes-Stevens CL, Valocchi AJ (2000) Calculation of reaction parameter sensitivity coefficients in multicomponent subsurface transport models. Adv. Water Res. 23(6): 591–611 Vasco DW, Datta-Gupta A (1999) Asymptotic solutions for solute transport: a formalism for tracer tomography. Water Resour. Res. 35(1): 1–16 Wolfram S (1996) The Mathematica Book, 3rd edn., Wolfram Media/Cambridge University Press Young SC (1995) Characterization of high-K pathways by borehole flowmeter and tracer tests. Ground Water 33: 311–318 Zhou ST, Stone PH (1993) The role of large-scale eddies in the climate equilibrium. 2. Variable static stability. J. Climate 6(10): 1871–1881