Sensitivity analysis of a pumping test on a well with wellbore storage and skin

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Abstract

This paper reviews the basic concepts of sensitivity analysis and points out their limitations. A case is then made for logarithmic sensitivity. The magnitude of logarithmic sensitivity alone does not determine the accuracy of an aquifer parameter estimate, especially when the relative measurement errors are not uniform throughout space and time. Deterministic parameter correlations and plausible relative errors in parameter estimates are introduced as imperfect measures of information content in measurements. A plausible relative error in the parameter estimate combines the effect of logarithmic sensitivity with that of relative measurement error. Minimizing the plausible relative errors rather than maximizing the corresponding sensitivities should serve as a guide to identifying the measurements most useful for parameter estimation or as candidate measurements for optimal sampling. Furthermore, avoiding among them measurements with high parameter correlations as much as possible may help ensure that the sensitivity matrix X (or XTX) is well-conditioned and, thus, that the parameter estimates are accurate.

The discussed concepts are then applied to a model of a pumping test conducted on a fully penetrating well situated in a confined aquifer. The model accounts for the wellbore storage and an infinitesimal skin. In contrast to the traditional and normalized sensitivities, the logarithmic sensitivities of the drawdown in the pumping well, the drawdown in an observation well, and the wellface flowrate to transmissivity, T, storativity, S, and the skin factor, η, depend on a small number of parameters. They can thus be represented by a single type curve or a family of a relatively few type curves. The plausible relative errors in T, S, and η estimated from wellbore drawdown rapidly decrease during the wellbore storage phase and reach a plateau or slowly decrease outside the wellbore storage phase. The plausible relative errors from the wellface flowrate rapidly decrease during the wellbore storage phase before reaching a minimum (around the time when the wellface flowrate is equal to about half the pumping rate) and then rapidly increase. This means that transient flowmeter test measurements of drawdown and wellface flowrate should not be made during the early times of the wellbore storage phase.

Introduction

Dimensionless sensitivity analysis via logarithmic sensitivities has been widely used in sciences because it allows one to compare the sensitivity of one output with respect to one parameter with the sensitivity of another output with respect to yet another parameter. Although very attractive, this approach has been underutilized in hydrology.

The objectives of this paper are: (i) to review the basic concepts of sensitivity analysis and their limitations, (ii) to introduce the concept of a plausible relative error in a parameter estimate, and (iii) to apply logarithmic sensitivity, deterministic correlation, and plausible relative error to a model of a pumping test conducted on a fully penetrating well, with wellbore storage and infinitesimal skin, situated in a confined aquifer.

Section snippets

General sensitivity and error analysis

Most engineering, physical, chemical, and biological systems can be viewed as input–output models that relate the output information to the appropriate input parameters. If the model parameters were known perfectly, the output could be calculated. Since this is rarely the case, a question arises: How do the errors in the input parameters influence the outputs, or, in particular, does a small input perturbation cause a large change in the output? Sensitivity analysis was devised to study such

Selected applications of sensitivity analysis in hydrology

Sensitivity analysis has been applied in solving inverse problems and estimating model parameters, in finding optimal sampling designs, as well as in studying sensitivities for their own sake. The large-perturbation approach and traditional sensitivities have been widely used in hydrologic problems, whereas normalized sensitivities and, especially, logarithmic sensitivities are used less often.

A case for logarithmic sensitivities and a new measure of information content

The main drawback of the normalized sensitivity, (3), and traditional sensitivity, (2), is their dimensional character. As mentioned earlier, neither concept is applicable when two different outputs are measured, such as wellbore drawdown and wellface flowrate in the flowmeter test. The dimensionless logarithmic sensitivity, (4), however, is useful in such cases. In fact, it has an appealing interpretation.

Consider again an evolution of a system O(x,t;P1,…,Pn). For a nonzero output, its total

Semi-analytic pumping test model

Consider a fully penetrating well situated in a confined homogeneous aquifer [31] of horizontally infinite extent. The initial boundary value problem (IBVP) for the well response that accounts for wellbore storage and infinitesimal skin is2sr2+1rsr=STst,s|t=0=0,s|r=∞=0,sw=s|r=rw−ηrwsrr=rw,q=2πrwTsrr=rw=−Q−πrc2dswdt,where s(r,t) is the drawdown at the distance r from the pumping well and time t, S aquifer storativity, T=Kb aquifer transmissivity, K aquifer hydraulic conductivity, b

Drawdown measurements

Consider drawdown measurements s(r,t) in an observation well with a pressure transducer characterized by the maximum absolute measurement error Δsmax and the maximum allowable drawdown smax. The useful measurements with this instrument are those that fulfillΔsmax⩽s⩽smax.

Now, let us introduce two additional dimensionless numbersϵs=Δsmaxrwandξs=smaxrw.It follows from , , , (32), and (45) thatγϵs⩽sD|γ=1⩽γξs.

For drawdown measurements sw(t) in the pumping well an analogous inequality arisesγϵs⩽swD|

Sensitivity analysis of the pumping test data

It follows from , and (36), and , , thatsw(t;T,S,η)=rwswD(τ(t;T,S),γ(T),α(S),η),s(r,t;T,S,η)=rwsD(ρ,τ(t;T,S),γ(T),α(S),η),q(t;T,S,η)=QqD(τ(t;T,S),α(S),η).

, , and (38) imply thatτT=τT,τS=−τS,γT=γT,αS=αS.

The logarithmic sensitivity, (4), of drawdown with respect to transmissivity follows via the chain rule from , , , , and (60)TssT=TsDsDττT+sDγγT=τsD|γ=1sD|γ=1τ+γsD|γ=1γ1γsD|γ=1=τsD|γ=1sD|γ=1τ−1.

Similarly, chain rule and , , , and (61) imply thatSssS=−τsD|γ=1sD|γ=1τ+

Pumping test plausible relative errors under the fixed absolute measurement error model

Although the plausible relative errors in aquifer parameter estimates for the simple measurement model of (25) can be calculated from (26), we obtain them directly from (23) utilizing the already calculated logarithmic sensitivities. Thus it follows from these equations and from (62) thatdTT=ΔsmaxsTssT=ΔsmaxrwsD|γ=1τsD|γ=1sD|γ=1τ−1ordTT=γϵsτsD|γ=1τ−sD|γ=1−1=γϵsdTTγϵs=1.Similarly, , , and , lead todSS=γϵs−τsD|γ=1τsD|γ=1α−1=γϵsdSSγϵs=1,dηη=γϵsηsD|γ=1η−1=γϵsdηηγϵs=1.The plausible

Results

Although logarithmic sensitivities define the logarithmic correlations between parameters, (22), and plausible relative errors, (23), it is still instructive to plot all of them rather than just the sensitivities.

In Fig. 1, Fig. 2, we plot the logarithmic sensitivities of wellbore drawdown and wellface flowrate to transmissivity, storativity, and skin factor; the corresponding plausible relative errors in the three parameters; and the corresponding logarithmic correlations between these

Synthetic example 1

Consider a hypothetical pumping test with a constant total pumping rate of Q=5 m3/h=0.001389 m3/s conducted on a well of radius rw=rc=0.1 m situated in a sandy confined aquifer. Assume that the test lasted tmax=6.4 h and that the data were collected in the pumping well using

  • a 30-psig pressure transducer with Δsmax=0.03 m and smax=21 m, and

  • a flowmeter with Δqmax=0.25 m3/h and qmax=20 m3/h.

Note that the flowmeter accuracy is identical to that of the state-of-the-art Tisco electromagnetic

Synthetic example 2

Consider the same pumping test with the same instruments as in the previous example. Assume again that the test can be described exactly by the pumping test model for a confined aquifer; and this time, assume that a type-curve or least-squares fitting of the collected wellbore drawdown data (or wellface flowrate data) yields the following estimates:T=4.37×10−4m2/s,η=3,andα=S=10−5.The related dimensionless numbers are the same as in , . Since the estimated η and α are identical to those used to

Extension to the model for a leaky aquifer

An extension to the pumping test in a leaky aquifer is straightforward [31]. Eq. (27) would be replaced by2sr2+1rsrsB2=STst,where B2=Kb/(K/b), K is the aquitard conductivity and b is the aquitard thickness; one more dimensionless number would be introduced,β=rwB,all p in , , would be replaced by p+β2, and expressions in , , would depend on B or β. However, the relations , , , , , , , , would remain unchanged.

Conclusions

The logarithmic sensitivity, (4), naturally arises in a normalized total differential. It can be interpreted as a transfer coefficient between the relative error in an input parameter and the relative error this input parameter alone induces in the output. All logarithmic sensitivities are dimensionless and thus can be compared to one another, as opposed to traditional sensitivities of an output to parameters of different dimensions and as opposed to normalized sensitivities of outputs of

Acknowledgements

I acknowledge Clifford I. Voss and four anonymous reviewers for their critical in-depth comments on the manuscript. I am grateful to Casey Miller, the editor, for his timely and exemplary handling of the review process. I am also grateful to 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 #

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