The estimation of aquifer parameters from pumping test data is tra-ditionally based on a trial-and-error graphical approach in whichpumping test data is plotted on a logarithmic or semilogarithmicscale, and parameters from a hydraulic model are varied to achievea best fit to the data. Before the advent of the personal computer thiswas done by using type curves and the match point method of Theis(1935), or the Cooper-Jacob method, for example, although itis now commonplace to automate this process and find a best fitin the least squares sense.
In practice, pumping test data are noisy, which means that evenunder ideal conditions no two pump tests will result in the samedrawdown traces. As a result, different pumping tests will resultin different estimates of aquifer parameters. Conversely, whenmeasurement and model errors are taken into account, a singlepumping test will be consistent with a range of aquifer parameters.
The focus of this paper is the uncertainty quantification ofaquifer parameters from pumping test data. Whereas least squareswill give a single ‘point’ estimate of aquifer parameters for a giventest, the range of parameter values or predictions that are consistentwith the measurement set need to be quantified.
There are several sources of uncertainty which may be broadlycategorized as (1) model uncertainty and (2) measurement uncer-tainty. Model uncertainty results from several sources of errorincluding
• Model error due to an overly simplistic/incomplete descriptionof the aquifer,
• Barometric efficiency modeling error,• Base trend modeling error, and• Pump rate control error.
In this paper, uncertainty quantification is carried out in theBayesian framework in which all uncertainties are modeled asprobability distributions (Kaipio and Somersalo 2004; Calvettiand Somersalo 2007). Bayesian analysis is a methodology in whichmeasurement and a priori knowledge about unknown parametersare combined to quantify their uncertainty. The main product ofBayesian inference is the posterior distribution over parameters,which quantifies their uncertainty relative to the data and the modelused to interpret the data, and from which estimates of parameters,their uncertainty, or predictive inference are made.
Bayesian analysis has been applied to groundwater problems ina geostatistical setting in which unknown spatially distributedparameters are modeled as random fields (Rubin 2003; Hoeksemaand Kitanidis 1985). Kitanidis (1986) provides a general accountof the application of Bayesian analysis to quantifying uncertainty ofspatially distributed parameters. The main difficultly with uncer-tainty quantification for groundwater problems is the significantcomputational burden because it is normally necessary to use arandom sampling technique, such as Markov chain Monte Carlo(MCMC), to explore the posterior distribution and hence, quantifyuncertainty. In a more recent paper, Lee et al. (2002) used MCMCto sample the posterior distribution for a geostatistical model of anaquifer with unknown permeabilities. In this paper, flow is inducedin the aquifer by simultaneous injection and extraction untilequilibrium is achieved; a tracer is then injected, and concentra-tions and travel times measured at the extraction wells. Fu andGómez-Hernández (2009) used MCMC to carry out an uncertaintyanalysis to quantify the effects of using different data sets (conduc-tivity, head, and travel time data) to estimate a permeability field.These geostatistical models are characterized by their relative highdimensionality: the underlying permeability/transmissivity fieldis pixelated, and the corresponding discretised equations solved
computationally. Tractability is achieved for these authors bythe assumption of a correlation structure (e.g., kriging) betweenthe discrete spatially distributed permeabilities/transmissivities.All these papers focus on identifying permeability/transmissivityfields. However, of equal importance to groundwater engineeringis storativity/yield and leakage and surface water depletion.
This paper lies at the opposite end of the spectrum because verylow dimensional models (≤ 5 parameters) are considered. Thepaper is of interest to groundwater engineers on several accounts.First, the models considered have practical use and encapsulatethe essential physics of more complete aquifer models. Second,the focus is on pumping tests and the aquifer parameters that typ-ically affect groundwater engineering decisions. Third, uncertaintycan be quite significant for these models and is not just relevant formore complex aquifer models. Fourth, they provide a simple con-text for discussing uncertainty, which is not obfuscated by statis-tical machinery or computational discretization.
The aim of this paper is to describe the inverse probability, orBayesian, approach to the estimation of parameters for hydraulicmodels of groundwater flow in the context of pumping tests. Tofacilitate the exposition the fundamentals of Bayesian inferenceare presented. Basic facts about least squares estimation arereviewed, and an overview of Bayesian inference applied to pump-ing test analysis is given. Uncertainty quantification is then carriedout on three examples. First, synthetic data from the Theis (1935)model is analyzed. Then, synthetic and field data for the Boultondelayed yield model are analyzed, and finally, an analysis ofsynthetic data from the Hunt-Scott multilayer model (Hunt andScott 2007) is presented.
Forward and Inverse Problems: MathematicalFormulation
Let s ¼ sðr; tÞ be the drawdown at a distance r from the pumpingwell at time t. These true drawdowns are assumed to satisfy amathematical model, generally a simplification of the groundwaterflow equation. The models considered in this paper are parame-trized by transmissivity, storage, and leakage parameters. Ifθ denotes the set of possible parameters for the model (the param-eter space), s ¼ sðr; t; θÞ is written to express the functional rela-tionship between the parameter space and the drawdown. For thefixed values of r and t at which measurements are made, the func-tional dependence of s on θ defines the forward map between theparameter space and the noise-free drawdown. For brevity, theforward map is written as sðθÞ. The identification of aquifer param-eters from a pumping test is then the inverse problem of estimatingθ from noisy measurements of drawdown s.
Consider, for example, the classical Theis model (1935). Thenthe drawdown s ¼ sðr; tÞ solves the Theis equation
S∂s∂t ¼ T∇2sþ QδðrÞ ð1Þ
where ∇2 = Laplacian; S = storage coefficient; T = transmissivity;Q = constant pumping rate; and δð·Þ = radial Dirac δ-function,subject to zero initial drawdown and zero drawdown at infinitedistance. Hence, the unknown aquifer parameters are θ ¼ ðS;TÞ.For the Theis model, the forward map is given exactly by theformula
sðT ;SÞ ¼ � Q4πT
Ei
�� Sr2
4Tt
�ð2Þ
where Eið·Þ = exponential integral. For more complex models thattake into account the spatial distribution of aquifer parameters, the
forward map is usually approximated by a numerical schemeintroducing a further source of uncertainty when computationalerror is significant.
This paper is confined to a discussion of measurement errorand it is assumed that noise-perturbed measurements ~s of the truedrawdown s in a monitoring bore are observed
~s ¼ sþ noise ð3Þ
where the noise = additive Gaussian white noise; that is, observa-tions are generated from a normal distribution with mean 0 andstandard deviation σ. Fig. 1 shows synthetic pumping test datafor the Theis model in an observation 50 m from a well pumpedover 3 days at Q ¼ 1;000 m3∕day. The underlying true transmis-sivity is T ¼ 1;000 m2∕day and storage coefficient S ¼ 1 × 10�4.The synthetic data was generated by perturbing the noise-free datawith additive Gaussian noise with a standard deviation of 0:01 m.
Least Squares Estimation
Least squares techniques have almost universal application inengineering and sciences to problems in which there is a require-ment to fit models to data. This is, in part, caused by the simplenotions that underlie the idea of best-fit and the fact that black-box gradient-based optimizers for computing least squares fitsare extremely easy to implement and use, the best-knownexample is the PEST software (Doherty et al. 2010).
In practice, the most common and well-known impediments toleast squares fitting to data are (1) sensitivity to outlier data, whichmeans in the context of pump test analysis, extraneous data mayneed to be removed to get a realistic fit to the data and, in general,nonlinear least squares algorithms, (2) compute a local best-fit todata, and (3) do not perform well when multimodality is an issue.The best-fit to data is not usually best-fit to parameters (Kaipio andFox 2011).
Another aspect of least squares estimation alluded to previouslybut which is not always recognized in applications is that under theassumption of random noise, there is a natural variability in leastsquares estimation in the sense that different pump tests will,in general, result in different least squares estimates of aquiferparameters.
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draw
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Theis solutionnoisy data
Fig. 1. Synthetic pump test data for the Theis model
Fig. 2 shows the distribution of least squares estimates oftransmissivity and storage derived from 10,000 realizations of apump test using noise-perturbed drawdown measurements inan observation well 50 m from a well pumped over 5 days atQ ¼ 1;000 m3∕day, the true storage S ¼ 1 × 10�4 and transmis-sivity T ¼ 1;000m2∕day; the standard deviation of the noise is0:05 m. Although the range of parameters is small, it shows thatvariability does occur between different pump tests. This variabilityis often used as a measure of error in the least squares estimatederived from a single pump test. However, the actual error isgenerally larger (Kaipio and Fox 2011).
Bayesian Inference
This section provides a brief overview of the Bayesian approachto inverse problems. Further details can be found in Kaipio andSomersalo (2004) and Calvetti and Somersalo (2007).
In the Bayesian formulation, all uncertainties are modeled byusing probability distributions, with the distribution over unknownparameters relative to the measurement data of central interest.So, whereas the least squares estimate of parameters minimizesthe sum of the square of the residuals, in a Bayesian formulation,the noise process is explicitly modeled. Under the assumption ofadditive measurement error, the measurement process, Eq. (3), iswritten as
~s ¼ sðθÞ þ n ð4Þ
where the noise n is distributed according to some probability dis-tribution πn. Modeling of this noise distribution is usually a matterof fitting a simple distributional model to a histogram of measurederrors, although the model may also include an outlier process toautomatically recognize and exclude erroneous measurements.In applications, such as pumping tests with a few parametersbut many measurements, characteristics of the noise, such as thestandard deviation, are easily determined from measurements.
In view of Eq. (4), the conditional distribution over measure-ments ~s given parameters θ is
πð~sjθÞ ¼ πn½~s� sðθÞ� ð5Þ
which gives the probability of observing ~s for a given set of trueparameters. As a function of θ, this is called the likelihood functionand codes all the information that the measurements ~s containabout the unknown model parameters θ. For the Theis modelthe likelihood is given by
πð~sjT ;SÞ ¼Yni¼1
1ffiffiffiffiffiffiffiffiffi2πs2
p exp
�� ½~sðtiÞ � sðti; T;SÞ�2
2s2
�ð6Þ
where the product is taken over the number of data points; ~sðtiÞ =observed drawdown at time ti, sðti; T;SÞ is given by the Theisformula, Eq. (2); and s = standard deviation of the drawdownnoise. Fig. 3 shows the likelihood function for the synthetic Theispumping test data in Fig. 1.
The focus of Bayesian inference applied to parameter estimationis the posterior probability distribution over parameters given theobservations, denoted by
πðθj~sÞ ð7Þ
The posterior distribution is then the solution to the inverseproblem, and any set of parameters belonging to the numericalsupport (> 0) of the posterior distribution is said to be consistentwith the data and a feasible solution to the inverse problem.
Bayes’ theorem allows the posterior distribution to be related tothe likelihood
πðθj~sÞ ∝ πð~sjθÞπðθÞ ð8Þ
The reason for the ∝ instead of = in Eq. (8) is that there is anormalizing constant to ensure that that πðθj~sÞ is a probabilitydistribution. In general, this constant can be written in closedform only in simple cases.
The distribution πðθÞ is the prior distribution over parameters,that is the distribution of the parameter space which is assumed apriori to any measurements. Parameter estimation for pumpingtests typically involves few parameters and many measurements,in which case the posterior distribution is dominated by the shapeof the likelihood function, so the prior distribution has little roleunless strong prior information is to be asserted. In the absenceof further information, the simple assumption of a uniform distri-bution on the parameter space is made; that is, an equi-probabilityassumption is imposed, commonly known as a flat prior. Thisand other simple distributions, such as restricting parameters to
be positive, are examples of improper priors that pose technicalproblems, although not any practical difficulty (Hartigan 1983;Berger 1985; Robert 2007).
In the Bayesian context, parameter estimation then consists ofproviding summary statistical measures of the parameters sup-ported by the posterior distribution, with a typical estimate asthe mean of the distribution, whereas the posterior standarddeviation gives a suitable measure of uncertainty, and the posteriorcorrelation gives a measure of dependency. Under the assumptionof Gaussian white noise and flat priors, the posterior mean reducesto the maximum likelihood estimate which in turn is given by theleast squares estimate. However, the Bayesian formulation providesa framework for uncertainty quantification.
Even for a model as simple as the Theis model, it is not possibleto write down an analytic formula for the mean, standard deviation,or correlation coefficient of S and T . It is therefore necessary tocompute these statistics numerically. For the Theis model andlow dimensional models (≤ 5 parameters), it is possible to usea quadrature rule, although it may be difficult to implement. How-ever, random sampling in the form of Markov chain MonteCarlo of the posterior distribution provides a flexible and intuitiveapproach and gives a simple and effective way of visualizing theposterior distribution by using histograms.
Markov Chain Monte Carlo
Drawing random samples from the posterior distribution isachieved by using Markov chain Monte Carlo (MCMC) algorithms(Grimmett and Stirzaker 2001). The MCMC methods describe asampling methodology that generates a random walk that in thelong term sample from the posterior distribution. The only require-ment is that the nonnormalized posterior distribution in Eq. (8) canbe evaluated, with one benefit that it is not necessary to evaluate thenormalizing constant. The main computational burden is evaluatingthe forward map in each iteration of the MCMC and that standardMCMC algorithms require many iterations. Furthermore, mostMCMC algorithms require tuning of the proposal distribution(see Liu 2001).
However, MCMC algorithms give a guaranteed method ofdrawing samples from the posterior distribution, hence allowingevaluation of any statistic that provides the means to parameterestimation and prediction with quantified uncertainties. Onceone has created a reasonable set of samples from the posteriordistribution one can then summarize statistics of the posterior dis-tribution. There is considerable leeway here; in the case thatthe posterior distribution is peaked around a single value, the dis-tribution is well-summarized by giving that value. When theposterior distribution has several modes, reporting each of thosemodes may be the best summary.
In this work, a black-box sampler is used, the t-walk (Christenand Fox 2010), which requires the user to supply a computerroutine that evaluates the target distribution Eq. (5) only. The coderequires no further user intervention (such as tuning parameters) toproduce correct samples. The ease of use is somewhat off-set by thelong computational time, but this is analogous to using black-boxoptimization methods that use function values only. The t-walkis initialized by choosing two initial values for the aquifer param-eters and then proceeds to construct two coupled random walks,each of which, after some initial burn-in period, are correctlydistributed as the posterior distribution. The ergodic propertymeans that posterior estimates of quantities can be made as aver-ages over the chain.
Although it is possible to develop a better tuned algorithm,taking into account scale and correlation structure between theparameters, the t-walk allows the investigator to focus on data
analysis rather than MCMC algorithms. Formulating efficientMCMC algorithms is an area of currently active research, includingthe search for methods that effectively use gradient information,as in modern optimization algorithms. In this paper the concernis more with using the output of the MCMC to quantify posterioruncertainty in pumping tests.
Results
Theis Model
In the first example it is supposed that noise-perturbed measure-ments of drawdown in an observation well 50 m from a wellpumped over 3 days atQ ¼ 1;000 m3∕day is given. The underlyingtrue transmissivity is T ¼ 1;000 m2∕day and storage S ¼1 × 10�4. Synthetic data was generated by perturbing noise-freedata with additive Gaussian noise with a standard deviation of0:01 m (Fig. 1). Fig. 3 shows the likelihood for this data set.For comparison, summary statistics were computed by using quad-rature and MCMC and given in Table 1. Although the agreement isgood for the means and standard deviations, there is a larger dis-crepancy between the estimates for the correlation coefficient ρTSbecause of the fact that the random samples are not statisticallyindependent.
The t-walk was initialized with T0 ¼ 500 m2∕day andS0 ¼ 1 × 10�3, and a second value was constructed by taking asmall random perturbation. To explore the posterior distributionthe t-walk was run for 250,000 steps. Fig. 4 shows traces of param-eter values for the initial 1,000 samples. An initial burn-in isevident, in which the parameter values show significant trendtoward final values, after which the t-walk appears to sample froma small interval around 1,000 for T and around 1 × 10�4 for S.The samples in burn-in are discarded, and the resulting randomsample provides an estimate of the distribution over T and S. Fig. 5shows the MCMC random sampling with the initial 20,000 burn-insamples removed.
To give a simplified explanation of MCMC for any initial start-ing values for transmissivity and storage, the t-walk and MCMCsamplers, in general, will converge to a set of samples from theposterior distribution, the initial value is forgotten after a largenumber of steps, and the resulting sample approximates a randomsample from the posterior distribution.
Although the mean values T and S of T and Sin Table 1 give the most likely values of the transmissivity andstorage, one can equally correctly use other parameter values fromthe histograms in Fig. 5 to fit the data. However, one cannot simplysample independently from these histograms and expect to get agood model fit because the joint distribution for the model param-eters T and S is important, which is generated by the random sam-pling in Fig. 6, with integrals over the joint distribution required togive the individual parameter distributions plotted in Fig. 5.The negative correlation ρTS between transmissivity and storage
Table 1. Summary Statistics for the Theis Model Fit to the Synthetic PumpTest Data in Fig. 1
displayed by the joint distribution is consistent with trial-and-errorfitting of pumping test data.
The fit shown in Fig. 7 uses the mean values T and S in Table 1,whereas the envelope shows the range of fits that are consistentwith the data.
Boulton Model
In a seminal series of papers (Boulton 1954, 1955, 1963, 1973),Boulton studied the concept of a delayed yield aquifer. His deriva-tion of the delayed yield equation was on the basis of an empiricalrelationship, which received criticism on account of not having a
clear hydraulic interpretation. In the original formulation, Boultonconsidered the delay effect in a phreatic aquifer, although he laterextended the idea to a semiconfined situation. Cooley and Case(1973) showed that Boulton’s delay constant can be interpretedin the vertical hydraulic conductivity of an overlying water-tableaquitard containing a free surface whose equation was linearizedto allow only vertical motion; the same situation was consideredlater (Boulton 1973). Fig. 8 shows a schematic of the aquifersystem modeled by Boulton’s equation. As a result, Boulton’sdelay equation can be uncoupled and written
S1∂s1∂t ¼ �K 0
B0 ðs1 � s2Þ ð9Þ
S2∂s2∂t ¼ T2∇2s2 þ
K 0
B0 ðs1 � s2Þ þ QδðrÞ ð10Þ
in which s1 and s2 = drawdowns in the overlying aquitard andpumped aquifer, respectively; ∇2 = two dimensional Laplacian,the aquitard yield = S1; the thickness and vertical hydraulicconductivity for the aquitard = B0 and K 0, respectively; T2 andS2 = transmissivity and storativity, respectively for the pumpedaquifer; and Q = pumping rate.
Analysis of Synthetic Boulton Data
This section repeats the theme presented in the section on theTheis model. It is supposed that noise-perturbed measurementsof drawdown at 1 min intervals in an observation well 50 m froma well pumped over 120 h at Q ¼ 175 m3∕h are given. The under-lying true transmissivity is T2 ¼ 100 m2∕h and storageS2 ¼ 1 × 10�4, leakage K 0∕B0 ¼ 5 × 10�41∕h, and aquitard yieldS1 ¼ 0:1. The data were perturbed by additive Gaussian noise witha standard deviation of 0:005 m, Fig. 9.
The t-walk was initialized with T0 ¼ 500, S0 ¼ 1 × 10�3,λ ¼ 1 × 10�2, and σ0 ¼ 10�1, and a second value was constructed
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MCMC sampling of storage
Fig. 4. MCMC simulation of transmissivity and storage for the Theismodel using t-walk for initial 1,000 samples showing burn-in
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Fig. 5. MCMC simulation of transmissivity and storage for the Theis model using t-walk with burn-in removed
by taking a small random perturbation and t-walk was run for500,000 steps, with the initial 100,000 steps removed as burn-in.The posterior distributions of aquifer parameters in Fig. 10 for theBoulton model show two noteworthy features: first the histogramsover aquifer parameters show a small off-set and do not includethe true values, which means that certain realizations of the noise pro-cess can be interpreted in terms of the model. Second, the posteriordistributions are evidently multimodal, which has two consequences:the first is that t-walk or any other MCMC sampler may need to berun a large number of times to escape a local mode and explore theposterior distribution. The second is that it complicates how one sum-marizes the posterior distribution to find a model fit; in particular, aleast squares algorithm will at best only find a local best-fit.
It is interesting to compare the Bayesian approach with leastsquares estimation. Table 2 shows the least squares parameterestimates of aquifer parameters for the Boulton model by usingthe same initial parameter values and the Bayesian estimates, whichare simply the average values of the posterior distributions inFig. 10. The least squares estimates are very poor estimates; thesame spurious least squares parameter estimates are obtained evenwhen the true parameter values are taken to initialize Matlab’snonlinear least squares algorithm.
Figs. 11 and 12, respectively show Bayesian and least squares fitto the synthetic pump test data. Clearly, the least squares fit is ratherpoor. Although t-walk needs to be run for a large number of timesto generate the posterior distribution, it gives better estimates and acomprehensive picture of what is going on.
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−4 Joint distribution of S and T
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Fig. 6. Joint distribution of transmissivity and storage
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)
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Fig. 7. Model fit to synthetic pumping test data using mean transmis-sivity and storage
Fig. 13 shows pumping test data for a test carried out in Lough(2004). The pumped aquifer is overlain by a water-table aquitardand reasonably described by Boulton’s model. The test well waspumped over a period of approximately 5 days and monitoredin a well 62 m away, and the recovery monitored for a further3 days. There was an evident decrease in pumping rates overthe test period. The assumed pump rates are given in Table 3.To test the modeling assumption Eq. (3), polynomials were fittedto different parts of the data and the residuals tested for normality.The normal quantile plot of residuals in Fig. 14 shows that theassumption of normality is not unreasonable. The standarddeviation of noise was taken to be s ¼ 7 × 10�3 m.
For the reported simulation, burn-in occurred quite quickly andthe first 20,000 samples removed, although 500,000 samples weremade to comprehensively investigate the possibility of multimodal-ity of the posterior distributions. Fig. 15 shows the posteriordistributions of aquifer parameters for the Boulton model; the meanvalues are given in Table 4. The correlation structure of the aquiferparameters is shown in Fig. 16. In this case, the least squaresestimates are in agreement with the Bayesian estimates. Fig. 17shows the fit to the data by using the parameters in Table 4.
Hunt-Scott Model
As a final example, a multilayer aquifer model of Hunt andScott (2007) is considered, shown in Fig. 18 and modeled bythe following equations:
S1∂s1∂t ¼ T1∇2s1 �
K 0
B0 ðs1 � s2Þ ð11Þ
S2∂s2∂t ¼ T2∇2s2 þ
K 0
B0 ðs1 � s2Þ þ QδðrÞ ð12Þ
in which s1 and s2 = drawdowns in the phreatic and pumpedsemiconfined aquifers, respectively; ∇2 = the two dimensionalLaplacian; Q = constant well discharge rate; T1 and S1 = transmis-sivity and specific yield, respectively for the phreatic aquifer;
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Boulton solutionnoisy data
Fig. 9. Synthetic pump test data for the Boulton model
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Fig. 10. Posterior distributions of aquifer parameters for synthetic pump test data in Fig. 9
Table 2. Parameter Estimates for the Boulton Model Fit to the SyntheticPump Test Data in Fig. 9
T2 and S2 = transmissivity and storativity, respectively for thepumped aquifer; the thickness and hydraulic conductivity for theaquitard = B0 and K 0, respectively.
Synthetic drawdown data in Fig. 19 were measured in a mon-itoring well screened in the pumped aquifer over 5 days, 100 mfrom a well pumped at 5;000 m3∕day. The data was generated withunderlying parameters T1 ¼ 500 m2∕day, T2 ¼ 3;000 m2∕day,S1 ¼ 0:1, S2 ¼ 1 × 10�4, K 0∕B0 ¼ 0:031∕day, and standarddeviation of noise s ¼ 0:01 m.
Fig. 20 shows the posterior distributions of aquifer parameters.The main feature of interest is the large range of the transmissivity
T1 of the phreatic aquifer, and although this does not have a greateffect on the possible fits to data in the pumped aquifer, theenvelope of possible fits for the phreatic aquifer is large, Fig. 21,which means that the predictive uncertainty of drawdown of thewater-table following sustained pumping is significant.
Discussion
The synthetic Theis data example showed the basic aspects of thesample-based approach to Bayesian inference. For the Theis model,
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Fig. 11. Model fit to pump test data by using Bayesian parameter estimates in Table 2
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Fig. 12. Model fit to pump test data by using least square parameter estimates in Table 2
the Bayesian approach is consistent with least squares parameterestimation in that the mean of the posterior distribution of trans-missivity and storage is numerically equivalent to the least squaresestimates. However, the posterior distribution provides rangesof correlated values that are consistent with the data and whoseuncertainty can be measured.
Analysis of Boulton’s delayed yield aquifer model presents amore interesting situation: first, the posterior distribution of aquiferparameters may be multimodal; in this case, least squares maymake a poor and wholly unreasonable fit to data, as seen in Fig. 12.This occurred even when the underlying true values were used to
initialize the nonlinear least squares algorithm. Although nonlinearleast squares model fitting may or may not work, Bayesian infer-ence provides us with the posterior distribution, which summarizeseverything that can be known relative to the data, and thereforegives a more complete picture of reality relative to the data.
Secondly, for certain realizations of noise, the posterior distri-bution of aquifer parameters may not include the true parameters,which means that noise may be interpreted in the model. The reasonthis occurs is that the synthetic data were sampled linearly at 1 minintervals so that on a log time scale there are relatively few datapoints during the initial period of pumping when the drawdowncurve is largely determined by T2 and S2. This is significantbecause it suggests that multimodality may occur because thereis relatively little data on the time scale when the effect of a param-eter is most significant.
Parameters for the Theis and Boulton models were shown to bewell-determined by the data in that the level of uncertainty is quitesmall. The strong correlations between parameters seen in Figs. 6and 16 is a feature of groundwater models with few parametersand is consistent with trial-and-error fitting to pumping test data.
Uncertainty quantification is more informative when quantify-ing parameters that are not well-determined by the data, as seen inthe Hunt-Scott example. The practical consequence of this exampleis that it is hard to capture the effect of the transmissivity T1 ofthe phreatic aquifer from the pumping test data and differentiatebetween the Hunt-Scott model and the Boulton model. However,the actual effect of T1 after sustained pumping may be quitesignificant (Ward and Lough 2011). The effect is further magnifiedwhen stream depletion as a result of pumping is ofinterest, which is the subject of a future study. Although theuncertainty for this particular model may be reduced by havingmore vertically distributed observation wells, this may not befeasible in practice. Furthermore, this example highlights a genericproblem in computational groundwater modeling.
Although the basic assumption of Gaussian white noise is rea-sonable in many situations, pumping test data were examinedwhere the distribution of residuals, and hence, noise, althoughsymmetric is somewhat heavier tailed than the normal distribution.It is important to carry out an analysis of the data and use anappropriate distribution for the noise.
In all the examples, it is necessary to use MCMC to carry outthe uncertainty quantification. The main issue with MCMC is thatcompared with nonlinear least squares, computations take muchmore time and, in general, demand some intervention on the partof the user. The t-walk has the merit of requiring no user interven-tion and although relatively slow, will sample the posterior distri-bution and is therefore analogous to a black-box least squaresoptimizer. Hence, it is a useful tool for investigating and quantify-ing uncertainty. Although any starting values may be used to ini-tialize a MCMC random sampler, burn-in time may be significantlyreduced by initializing with a least squares estimate.
In this paper known pumping rates and known noise (estimatedfrom the data) were assumed. Because of the difficulty of actuallyachieving a constant pump rate during a pump test and the fact thatpump rates are often measured with instrumentation producing anoise corrupted time series, Q and s could themselves be assumedto be unknown and expressed in the posterior distributions. Further-more, the model uncertainties listed in the introduction can alsobe incorporated into the same probability framework. One ofthe merits of Bayesian inference is that the possibilities can beexplored.
In practice, one usually observes drawdown in a number ofobservation wells. In pumping test analysis, it is standard practiceto fit an analytic model to each observation well, and the results
−8 −6 −4 −2 0 2 4 6 8x 10
−3
0.001
0.01
0.05
0.25
0.50
0.75
0.95
0.99
0.999
Data
Pro
babi
lity
Normal Probability Plot
Fig. 14. Normal quantile plot of residuals
10−1
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101
102
103
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105
0
0.05
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0.15
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time (minutes)
draw
dow
n (m
)
Fig. 13. Field pumping test data
Table 3. Pumping Rates Assumed for Field Data in Fig. 13
averaged to obtain a single model representation of the pumpedaquifer. Often there can be significant differences between param-eters from fitting each observation well, separately reflecting aqui-fer heterogeneity. To overcome this inexacitude and model the
spatial distribution of parameters, the aquifer domain may be pixe-lated and the underlying equation solved computationally. The sol-ution to the inverse problem for this scenario generates the posteriordistribution over aquifer parameters for each pixel and therefore
Fig. 16. Correlation structure of aquifer parameters for Boulton model using field pumping test data in Fig. 13
0.85 0.86 0.87 0.88 0.89 0.9 0.910
0.5
1
1.5
2
2.5
3x 10
4
T2
freq
uenc
y
Posterior distribution of transmissivity
0.95 1 1.05 1.1 1.15 1.2 1.25
x 10−3
0
0.5
1
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2
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3
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S2
freq
uenc
y
Posterior distribution of storage
2.6 2.65 2.7 2.75 2.8 2.85
x 10−5
0
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1
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2
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3
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4
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freq
uenc
y
Posterior distribution of leakage
0.024 0.025 0.026 0.027 0.028 0.029 0.030
0.5
1
1.5
2
2.5
3
3.5x 10
4
S1
freq
uenc
y
Posterior distribution of yield
Fig. 15. Posterior distributions of aquifer parameters for Boulton model using the field pumping test data in Fig. 13
has many more dimensions than considered here. Cui (2010), forexample, develops MCMC algorithms to calibrate a geothermalmodel with 104 parameters. Although this might seem excessive,experience with using more realistic models is that the summarystatistics and predictive capabilities of a calibrated model are farmore representative of reality.
Conclusions
The main aim of this paper was to describe the inverse probability,or Bayesian approach, to the uncertainty quantification of aquiferparameters for groundwater models with few parameters in thecontext of pumping tests, in which it was assumed that pumpingtest data are corrupted by Gaussian white noise. The focus of thisapproach is the posterior distribution over parameters, conditionedon the measured data, from which estimates of parameters, theiruncertainty, or predictive inference are made.
Whereas least squares methodology provides a best fit to data,the posterior distribution quantifies uncertainty of parametersrelative to the data and hence, the range of parameters that areconsistent with the data.
Aquiclude
screen s
2, T
2, S
2
Semi−confined aquifer
AquitardK′, B′
Water table
s1, T
1, S
1Unconfined aquifer
bore
Ground surface
Fig. 18. Cross section of Hunt-Scott model
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101
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draw
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)
Hunt−Scott solutionnoisy data
Fig. 19. Synthetic pumping test data for Hunt-Scott model
0 2000 4000 6000 8000 10000 120000
0.1
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awdo
wn
(m)
Model fit
estimatedata
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Model fit
estimatedata
Fig. 17. Model fit to pump test data by using mean parameters given in Table 4
Table 4. Parameter Estimates for the Boulton Model Fit to the Pump TestData in Fig. 13
The generic MCMC sampler, the t-walk, is an effective toolthat does not require specialist knowledge or user interventionfor sampling the posterior distribution to quantify uncertaintyand dependency between aquifer parameters.
The models and examples in this paper exhibit many of thefeatures typically encountered in computational groundwatermodeling, such as multimodality, failure of least squares optimizersto give reasonable answers, and poorly determined parameters.Statistical analysis and visualization of the posterior distribution
provides a comprehensive picture of the relationship betweenmodel and data.
Acknowledgments
In conclusion, the authors would like to extend their gratitude toHilary Lough for sharing pumping test data from her thesis andto Tiangang Cui for some very helpful discussions. Matlab code
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drawdown in water table
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time since start of pumping (days)
draw
dow
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)
drawdown in semi−confined aquifer
datamean fitenvelope of possible fits
Fig. 21. Model fit to pumping test data in Fig. 13 showing upper and lower envelope of fits consistent with the measurement data
0 500 1000 15000
1
2x 10
4
T1
freq
uenc
y
2600 2800 3000 32000
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2x 10
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0.05 0.1 0.150
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y0.95 1 1.05 1.1
x 10−4
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0.025 0.03 0.035 0.040
1
2x 10
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freq
uenc
y
Fig. 20. Posterior distributions of aquifer parameters for synthetic pumping test data in Fig. 13
for the examples considered here may be downloaded from www.ocmo.co.nz.
Notation
The following physical variables are used in this paper:B0i = thickness of aquitard ½L�;
K 0i = hydraulic conductivity in aquitard ½L�∕½T�;Q = pumping rate ½L3�∕½T�;r = radial coordinate ½L�;
S1 = specific yield in water table aquifer/aquitard;S2 = storativity in semiconfined aquifer;
si ¼ siðr; tÞ, i ¼ 1; 2 = drawdown ½L�;Ti, i ¼ 1; 2; 3 = transmissivity ½L2�∕½T �; and
t = time ½T � ;statistical estimates of aquifer parameters aredenoted with a hat, e.g., T , standard deviation for aparameter is denoted by σ, e.g., σT , and correlation by ρ,e.g., ρT ;S is the correlation between T and S for the Theismodel.
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