Journal of Hydrology (2006) 328, 538–547

ava i lab le at www.sc iencedi rec t . com

journal homepage: www.elsevier .com/ locate / jhydro l

Stochastic modeling of transient stream–aquiferinteraction with the nonlinear Boussinesq equation

Kirti Srivastava a,1, Sergio E. Serrano b,*, S.R. Workman c

a National Geophysical Research Institute, Uppal Road, Hyderabad 500007, Indiab Civil and Environmental Engineering Department, Temple University, 1947 N 12th Street, Philadelphia, PA 19122, USAc Department of Biosystems and Agricultural Engineering, University of Kentucky, Lexington, KY 40546, USA

Received 21 January 2005; received in revised form 21 December 2005; accepted 27 December 2005

Summary In this article the effect of highly fluctuating stream stage on the adjacent alluvialvalley aquifer is studied with a new analytical solution to the nonlinear transient groundwaterflow equation subject to stochastic conductivity and time varying boundary conditions. A ran-dom conductivity field with known correlation structure represents uncertain heterogeneity.The resulting nonlinear stochastic Boussinesq equation is solved with the decompositionmethod. New expressions for the mean of the hydraulic head and its variance distributionare given. The procedure allows for the calculation of the mean head and error bounds in realsituations when a limited sample allows the estimation of the conductivity mean and correla-tion structure only. Under these circumstances, the usual assumptions of a specific conductivityprobability distribution, logarithmic transformation, small perturbation, discretization, orMonte Carlo simulations are not possible. The solution is verified via an application to the SciotoRiver aquifer in Ohio, which suffers from periodic large fluctuations in river stage from seasonalflooding. Predicted head statistics are compared with observed heads at different monitoringwells across the aquifer. Results show that the observed transient water table elevation inthe observation well lies in the predicted mean plus or minus one standard deviation bounds.The magnitude of uncertainty in predicted head depends on the statistical properties of theconductivity field, as described by its coefficient of variability and its correlation length scale.ª 2006 Elsevier B.V. All rights reserved.

KEYWORDSUnconfined aquifer;Nonlinear equation;Stream–aquiferinteraction;Stochastic model;Analytical solution

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022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reservedoi:10.1016/j.jhydrol.2005.12.035

* Corresponding author. Tel.: +1 215 204 6164.E-mail addresses: kirti_ngri@rediffmail.com (K. Srivastava),

serrano@temple.edu (S.E. Serrano), sworkman@bae.uky.eduS.R. Workman).1 Present address: Civil and Environmental Engineering Depart-ent, Temple University, 1947 N 12th Street, Philadelphia, PA9122, USA.

Introduction

The study of stream–aquifer hydraulics is of great interestas several flow and contaminant problems can be modeled,understood and quantified. The quantification of thehydraulics of the stream–aquifer in an alluvial valley re-quire a good knowledge of the controlling input hydrogeo-

.

Stochastic modeling of transient stream–aquifer interaction with the nonlinear Boussinesq equation 539

logical parameters, such as hydraulic conductivity, specificyield, recharge, as well as boundary effects that are associ-ated with the stream. Small changes in the stream elevationcan cause a large variation in the groundwater elevation inthe aquifer. The spread of contaminants in stream–aquifersystems from the river to the aquifer or from the aquifer tothe river is also of concern. The hydraulics of such systemshas been studied in the literature from both a deterministicpoint of view as well a stochastic point of view.

Stream–aquifer systems can be quantitatively studiedusing Laplace’s equation subject to nonlinear free surfaceboundary conditions and time dependent river boundaryconditions. Strack (1989) has shown that when Dupuitassumptions of negligible vertical flow, as compared to hor-izontal, are valid the nonlinear Boussinesq equation is a via-ble alternative to Laplace’s equation. For the Boussinesqequation the vertical coordinate does not exist and the freesurface boundary condition is not needed, and hence timedependent boundary conditions can be easily incorporatedinto the analysis. The hydraulics of stream–aquifer systemshave been studied by several researchers when the inputcontrolling parameters are known with certainty using bothanalytical and numerical methods (Polubarinova-Kochina,1962; Kirkham, 1966). The solution to Laplace’s equationwith free surface boundary conditions was compared withthe solution of the linearized Boussinesq equation for thecase of sudden drawdown in the river levels and for uniformand nonuniform rainfall (Van De Giesen et al., 1994). Tabi-dian et al., 1992 studied groundwater level fluctuations forchanges in the stage levels in the associated streams. Hus-sein and Schwartz (2003) have studied the coupled ground-water–surface water problems and quantified the expectedtransport in both the aquifer and the stream. The coupledcanal flow–groundwater flow equation has been solved ana-lytically and the results are compared with the numericalmodel, MODFLOW, for small changes in the water level dis-turbances (Lal, 2000). Moench and Barlow (2000) havesolved analytically the one-dimensional flow equation inconfined and leaky aquifers and the two-dimensional flowequation in a plane perpendicular to the stream in phreaticaquifers where the stream is assumed to penetrate the fullthickness of the aquifer.

Most stream–aquifer systems can be viewed as heteroge-neous geological media with the governing flow equationsbeing described as stochastic equations. When the aquiferparameters and the boundary conditions are not known withcertainty, the preferred way to model the system has beenstochastically. Using the stochastic approach, stream–aqui-fer models have been solved both analytically as well asnumerically. Weissmann and Fogg (1999) and Weissmannet al. (1999) have studied the alluvial fan system from bothdeterministic as well as stochastic approaches, by modelingthe large scale features deterministically and the intermedi-ate heterogeneity using transition probability geostatistics.Hantush and MariZo (1997, 2002) have also studied the flowequations from a stochastic point of view.

The stream–aquifer interaction problem with time vary-ing boundaries has been solved by Workman et al., 1997)using the deterministic linearized Boussinesq equation. Ina later study, Serrano and Workman (1998) solved the sameproblem using the nonlinear Boussinesq equation and Ado-mian’s method of decomposition. Decomposition is now

being used to solve deterministic, stochastic, linear or non-linear equations in various branches of science and engi-neering (Adomian, 1991, 1994; Srivastava and Singh, 1999;Biazar et al., 2003; Wazwaz, 2000; Wazwaz and Gorguis,2004; Srivastava, 2005). In groundwater flow problems, thismethod has been extensively used (Serrano, 1992, 1995a,b,2003; Serrano and Unny, 1987; Serrano and Adomian, 1996;Adomian and Serrano, 1998), to obtain analytical, andsometimes closed-form, solutions to linear, nonlinear andstochastic problems. The method has been shown to be sys-tematic, robust, and sometimes capable of handling largevariances in the controlling hydrogeological parameters.

In this paper the work of Serrano and Workman (1998)has been extended to incorporate heterogeneity in thehydraulic conductivity represented stochastically. The non-linear transient groundwater flow equation subject to sto-chastic conductivity and time varying boundary conditionsis solved using decomposition. A random conductivity fieldwith known correlation structure represents uncertain het-erogeneity. Since decomposition does not require theassumptions of normality or smallness, we consider thepractical situation when only the mean and correlation ofthe hydraulic conductivity are given based on a limited setof field samples. This is a common scenario in hydrologicpractice. From the practical point of view, a modeling pro-cedure that limits its application to Gaussian or any otherconductivity fields is restrictive, since there is usually notenough information to ascertain the underlying probabilitydistribution. A more common scenario in applications isone when the modeler possesses a limited sample fromwhich only the first two moments can be estimated. Thus,a modeling procedure that allows the inclusion of thesetwo measures, regardless of the underlying density func-tion, appears practical. In this paper the solution and itssecond-order statistics are verified via an application tothe Scioto River aquifer in Ohio. Predicted head statisticsare compared with observed heads at different monitoringwells across the aquifer.

Analytical solution to the stochastic transientgroundwater flow equation in unconfinedaquifers

The groundwater flow-equation in a horizontal unconfinedaquifer of length lx with Dupuit assumptions given by Bear(1979) as

oh

ot� 1

S

o

oxKh

oh

ox

� �¼ R

S0 6 x 6 lx and 0 < t ð1Þ

where h(x, t) is the hydraulic head (m), K is the hydraulicconductivity (m/day), R is the recharge (m/day), lx is thelength of the aquifer (m), x is the spatial coordinate (m),and t is the time coordinate (day).

The boundary conditions imposed on (1) are

hð0; tÞ ¼ H1ðtÞhðlx; tÞ ¼ H2ðtÞhðx; 0Þ ¼ H0ðxÞ

ð2Þ

where H1(t) and H2(t) are time fluctuating heads at the leftand the right boundaries (m). H0(x) is the initial head acrossthe aquifer (m).

540 K. Srivastava et al.

The spatial variability in the hydraulic conductivity isrepresented by

KðxÞ ¼ K þ K 0ðxÞ ð3Þ

where K is the mean and K 0(x) is the fluctuation part.Common representations for the fluctuating part in the

hydraulic conductivity assume the log conductivity as a nor-mally distributed random variable. In the present work, wedo not adopt a specific distribution and instead assume therandom conductivity has known first and second-order sta-tistics. From the practical hydrologic point of view, limitedfield data bases often offer sufficient information to inferthe first two moments only. For the present exercise weconsider the common case of an exponential two-point cor-relation function. Thus, the random component of thehydraulic conductivity is represented by

EfK 0ðxÞg ¼ 0

EfK 0ðx1ÞK 0ðx2Þg ¼ r2Ke�qjx1�x2 j

ð4Þ

where E{ } denotes the expectation operator, r2K is the var-

iance in the hydraulic conductivity (m/day)2, q is the corre-lation decay parameter (1/m), or (1/q) is the correlationlength scale (m).

Substituting (3) into (1) we get the stochastic flow equa-tion as

oh

ot¼ R

Sþ 1

S

o

oxðK þ K 0Þh oh

ox

� �0 6 x 6 lx and 0 < t ð5Þ

Following the procedure given by Serrano and Workman(1998), the flow system (5) with associated boundary condi-tions (2) is solved and the t-decomposition solution to theproblem is obtained. For an introduction to the method ofdecomposition in hydrology the reader is referred to Serrano(2001, 1997).

Let us define the operator Lt ¼ oot. Applying the inverse

operator L�1t to (5) and rearranging, we obtain

h ¼ H0ðxÞ þRt

Sþ L�1t NðhÞ ð6Þ

where the nonlinear operator is

NðhÞ ¼ 1

SðK þ K 0Þh o2h

ox2þ ðK þ K 0Þ oh

ox

� �2

þ hoK 0

ox

oh

ox

( )

Using decomposition (Adomian, 1991, 1994), the solution toh may be written as the series

h ¼ h0 þ h1 þ h2 þ � � � ð7Þ

The first term in the series h0 satisfies the first term in theright-hand side of (5) and the solution is written as

h0 ¼ H0ðxÞ þRt

Sð8Þ

where H0(x) is the initial condition. The initial condition iscrucial in obtaining the solution to the problem. The initialcondition can be expressed as a second order polynomial inx as

H0ðx; tÞ ¼ AðtÞx2 þ BðtÞx þ CðtÞ ð9Þ

where the constants A(t), B(t), and C(t) should be obtainedfrom well data across the aquifer, as is done in the presentstudy. In cases where sufficient well data are not available,

the solution to the homogeneous groundwater flow equationwith Dupuit assumptions and a constant transmissivitywould be an appropriate approximation to an initial condi-tion. This is a second order polynomial in x with the con-stants given by (Serrano, 1997)

A ¼ � R

2T

B ¼ H2ð0Þ � H1ð0Þlx

þ Rlx2T

� �C ¼ H1ð0Þ

ð10Þ

Subsequent terms in the decomposition series (7) can be ex-presses as

h1 ¼ L�1t A0

h2 ¼ L�1t A1

..

.

hnþ1 ¼ L�1t An

ð11Þ

The Adomian polynomials are given by

A0 ¼ Nðh0Þ

A1 ¼ h1dNðh0Þdh0

A2 ¼ h2dNðh0Þdh0

þ h21

2!

d2Nðh0Þdh2

0

A3 ¼ h3dNðh0Þdh0

þ h1h2d2Nðh0Þdh2

0

þ h31

3!

d3Nðh0Þdh3

0

..

.

ð12Þ

Thus, from (6) and (12) the second term in the series is ex-pressed as

h1 ¼ L�1t

1

S

� 2KAðtÞðAðtÞx2 þ BðtÞxþ CðtÞÞ þ Kð2AðtÞxþ BðtÞÞ� �þ L�1t

2KAðtÞRtS2

þ L�1t

1

S

� K 0fð2AðtÞxþ BðtÞÞ2 þ 2AðtÞh0gþ ð2AðtÞxþ BðtÞÞh0oK 0

ox

� �ð13Þ

With two terms in the series, the solution to the nonlinearproblem (5) is

hðx; tÞ ¼ AðtÞx2 þ BðtÞxþ CðtÞ þ Rt

Sþ L�1t

1

S

�f2KAðtÞðAðtÞx2 þ BðtÞxþ CðtÞÞ þ Kð2AðtÞxþ BðtÞÞg

þ L�1t

2KAðtÞRtS2

þ L�1t

1

S

� K 0fð2AðtÞxþ BðtÞÞ2 þ 2AðtÞh0gþ ð2AðtÞxþ BðtÞÞh0oK 0

ox

� �ð14Þ

Gabet (1993, 1994) and Abbaoui and Cherruault (1994) haveshown that when decomposition series converge, they do sovery rapidly and only a few terms in the series are requiredfor an accurate solution. Serrano (1998) has shown the rapidconvergence of the decomposition series for groundwaterflow problems. The inclusion of additional terms in the

Stochastic modeling of transient stream–aquifer interaction with the nonlinear Boussinesq equation 541

solution would require the availability of higher-order mo-ments in the hydraulic conductivity.

Taking expectation on both sides of (14) we obtain themean head as

Efhðx;tÞg ¼ Ax2þBxþCþRt

Sþ L�1t

1

S

�f2KAðtÞðAðtÞx2þBðtÞxþCðtÞÞþKð2AðtÞxþBðtÞÞg

þ L�1t

2KAðtÞRtS2

ð15Þ

The fluctuating part in Eq. (14) is

vðx; tÞ ¼ c1ðtÞK 0ðxÞ þ c2ðtÞoK 0ðxÞ

oxð16Þ

where

c1ðtÞ ¼t

Sfð2AðtÞx þ BðtÞÞ2 þ 2AðtÞðAðtÞx2 þ BðtÞx þ CðtÞÞg

� �

þ AðtÞRt2

S2

!

c2ðtÞ ¼t

Sð2AðtÞx þ BðtÞÞðAðtÞx2 þ BðtÞx þ CðtÞÞ

� �

þ Rt2

2S2ð2AðtÞx þ BðtÞÞ

!

The two-point covariance function is given by

Efv1ðx1; t1Þv2ðx2; t2Þg ¼ E c1ðt1ÞK 0ðx1Þ þ c2ðt1ÞoK 0ðx1Þ

ox1

� ��

� c1ðt2ÞK 0ðx2Þ þ c2ðt2ÞoK 0ðx2Þ

ox2

� ��ð17Þ

By setting x1 = x2 = x and t1 = t2 = t the variance is obtainedas

r2ðx; tÞ ¼ C2kK

2ðc21 þ 2qc1c2 þ q2c22Þ ð18Þ

Model verification

The methodology developed above has been applied to eval-uate the effect of a highly fluctuating river stage on the adja-cent aquifer flow. The study area is an alluvial valley aquiferin South Central Ohio, OMSEA (Ohio Management SystemEvaluation Area) (Fig. 1). The Scioto River drains much ofcentral Ohio and forms the western boundary of the OMSEA

Figure 1 Location of the Scioto River Aquifer in Ohio.

site and a stream gage at Higby, Ohio (approximately21 km upstream from the OMSEA) has monitored flow inthe 13,290 km2 watershed for 60 years (Nortz et al., 1994).The change in river stage between maximum and minimumflows was 7.4 m. The National Weather Service (NWS) oper-ates a wire-weight gage at Piketon, Ohio. Jagucki et al.(1995) used the Scioto River elevations adjacent to the OM-SEA site to develop a gradient correction factor of 1.52 m be-tween the NWS gage site and the OMSEA site.

The alluvial valley aquifer is approximately 18–20 mthick and about 2 km wide (Fig. 2). The aquifer is composedof various types of sand and gravel with high transmissivityvalues. The US Geological Survey (USGS) conducted 13 pumptests in the aquifer over a 25-year period to determinehydrologic properties and water resource potential for aDepartment of Energy project (Norris and Fidler, 1969; Nor-ris, 1983a,b; Nortz et al., 1994; Jagucki et al., 1995). Atime-drawdown analysis of the data indicated the hydraulicconductivity of the regional aquifer to vary between 122 and170 m/day with a mean value of 142 m/day. The specificyield has been estimated to be around 0.2 m3/m3.

Eleven water-table wells were constructed over a 260 haarea surrounding the OMSEA site. The wells were con-structed with 152 mm diameter PVC casing. A 6.1-m long,2.54-mm slotted, PVC screen was positioned to bracketthe highest and lowest expected water-table elevations ineach well. All of the water-table wells were instrumentedwith shaft encoders and electronic data loggers that re-corded hourly water levels.

Transient stage at the Scioto River constituted the leftboundary condition for the model, and well R1 the rightone (Fig. 2). The three wells (R5, R4, and S10) shown inFig. 2 lie on the flow path from the eastern edge of the OM-SEA site to the Scioto River. These three monitoring wellsare located R5-215 m, R4-975 m, and S10-1525 m, respec-tively, from the Scioto River where measurements weremade. Big Beaver Creek, located at the right boundary ofthe flow domain, drains an area composed predominantlyof the poorly permeable bedrock uplands from which runoffwas rapid and bank storage was minimal. These conditionsproduce rapid stage fluctuations in the creek during stormevents but only intermittent flow during the dry summermonths. The transient nature of the boundary conditionsin the river and the well is depicted in Fig. 3, which showsthat the fluctuations with respect to time at the rightboundary are very small as compared to those of the SciotoRiver on the left side. Fig. 3 shows the total change in stagerecorded during the period of October 1991 to September1992 was approximately 4.9 m. The seasonal fluctuationsin the water level of the Scioto River were important andappeared to directly affect the ground-water levels in areasnear the river.

The assumptions specified for the development of themathematical model were met with the aquifer underlyingthe OMSEA site. Researchers have observed that the regio-nal groundwater flow is predominantly in the horizontaldirection, the water table elevations exhibited mild slopes,and the aquifer thickness is small compared to the horizon-tal length. These aquifer and flow characteristics justify theDupuit assumptions. When the Dupuit assumptions are valid,the linearized Boussinesq equation appears to be a viablealternative to Laplace’s equation. With the Boussinesq

Figure 2 Cross section of the Scioto River alluvial valley aquifer.

Figure 3 Daily elevation of the Scioto River at the rightboundary and well R1 at the left boundary of the OMSEA aquiferfrom October 1991 to September 1992.

542 K. Srivastava et al.

equation, the vertical coordinate is eliminated, and thefree-surface boundary condition is not needed. The resultis a simplified model where the effect of time-dependentriver boundary conditions can be easily incorporated inthe analysis.

Earlier investigations in the region have shown that theperformance of the nonlinear model in predicting stream–aquifer interaction is similar to that of the linearized model.Workman et al. (1997) have used these data and modeledthe water table elevations across the aquifer using the line-arized Boussinesq equation. In their study they simulatedthe stream–aquifer interaction for linear models. Theyhave obtained the solution by combining the solution ofthe steady state component and the transient component.In a later study, Serrano and Workman (1998) derived a non-linear solution to the Boussinesq equation with transientboundary conditions using the decomposition method. Theyhave shown that the results of the nonlinear model and thelinearized model are very similar in predicting the stream–aquifer interaction at the OMSEA research site. This is basi-cally because the aquifer thickness was large compared tothe changes in the river elevation. When the saturatedthickness is not large compared to the boundary conditionsSerrano and Workman (1998) have shown that the nonlinear

model predicts the water table response more accurately.Also the nonlinear model better represents the dynamic re-sponse of the water table elevation when a large change inthe transmissivity field occurs. In the same study, no re-charge was simulated in the aquifer for the purposes ofdetermining the direct influence of the river on ground-water levels.

Workman and Serrano (1999) studied the effects of re-charge on the aquifer to determine the separate influencesof river stage and recharge over a 5-year period. When theriver stage reaches a height of 166 m above mean sea level,the floodplain becomes inundated and water is rechargeddirectly to the aquifer. Over 2 m of recharge was computedduring the 5-year period with most recharge events occur-ring during these flood events. The period of record chosenfor this study (1991–1992) does not include any dates ofoverbank flow and allows the study of the river influenceon hydraulic heads. Appendix A in the present paper consistsof a sensitivity analysis of the effect of recharge relative tothat of a highly fluctuating left boundary condition on themagnitude of the aquifer head. It is seen that the effectof recharge in this particular scenario is so small that theassumption of no recharge is reasonable. We remark thatthe methodology presented could include recharge esti-mates if it is found to be more important or as importantas the effect of the boundaries.

The distance of influence of the river flood wave into theaquifer is dependent on the transmissivity. Since the aquiferexhibits a large variation in the hydraulic conductivityparameter a stochastic model has been developed to quan-tify the errors in the head variation due to errors in thehydraulic conductivity.

Results and discussion

The proposed methodology has been tested on these wellsthat lie along the flow path and the theoretical values have

Table 1 Controlling hydrogeological parameters for thealluvial valley aquifer

Aquifer length 2000 mMean hydraulic conductivity 142 m/daySpecific yield 0.2 m3/m3

Coefficient of variability inhydraulic conductivity (CK)

0.4

Correlation length scale 500 m

Stochastic modeling of transient stream–aquifer interaction with the nonlinear Boussinesq equation 543

been computed with its error bounds and compared with theobserved heads. A portion of the data from October 1991 toSeptember 1992 (Jagucki et al., 1995) has been used toquantify the water table elevations at the three wells inthe study domain. The controlling aquifer parameters aretabulated in Table 1. Using these parameters the transientmean water table elevation is computed from (15), andthe standard deviation from (18). Fig. 4 shows the plot ofthe mean and mean plus or minus one standard deviationat Well R5, approximately 215 m from the Scioto River.The well exhibits a large variation in the water level overthe period of 1 year from October 1991 to September

Figure 4 Observed, mean, and mean plus and minus onestandard deviation of the water table elevation at Well R5(215 m), at Well R4 (975 m) and Well S10 (1525 m) and the dailyelevations at the Scioto River.

1992. The mean and the error bounds have been computedat the well with a coefficient of variability in the hydraulicconductivity CK = 0.4 and a correlation length scale of500 m. The mean head is in reasonable agreement withthe observed head (Fig. 4). Fig. 4, the error bounds aredependent on time as well as on distance from the riverboundary. Fig. 4 also shows the mean plus minus one stan-dard deviation, and the observed head at wells R4, located975 m, and S10, located 1525 m, respectively, from the Sci-oto River. From the well hydrographs in Fig. 4, we observethat the head standard deviation is seen to slightly increasewith distance. This confirms prior theoretical studies (e.g.,Serrano, 1995a) that suggest that the head variance in-creases with distance from a known head boundary condi-tion, where it is zero. However, the variance appears tobe stationary in time. Quantification of the effect of randomconductivity indicate that the observations lie in the plusminus one standard deviation limit for a given variation inthe hydraulic conductivity. However, it is observed thatthere is a slight over prediction or under prediction of themodeled head for some months at the wells. This is becausewe are considering the mean behavior of the system. We re-mark that the observed head is in fact a sample functionfrom the ensemble. Individual sample functions may ormay not lie within the bounds of the standard deviation.In Fig. 4 the months observed levels of June and July area bit beyond these bounds. Theoretically, an appropriatecomparison between model and prototype should be doneby comparing mean of prototype, based on many samples,with the corresponding mean of the model; then comparingthe variance of the prototype with that of the model. How-ever, in this case only one sample over time is available andthe comparison is qualitative. A quantitative assessment ofthe differences between a measured hydrograph and a sim-ulated one is only possible in a traditional deterministicanalysis that assumes that the measured hydrograph is the‘‘true,’’ error free, one. In the present analysis, the mea-sured hydrograph is subject to errors too (i.e., is a samplefrom an unknown random process), while the statistics formthe simulated hydrograph (e.g., the mean hydrograph) rep-resents the ensemble average of all samples in the model. Ina stochastic analysis sense, a uniform comparison would bepossible if both moments of prototype and model wereavailable. In other words, if for example the mean proto-type head is compared to the mean model head. However,only a limited measured series is available.

Next we study the effect of uncertainty in the hydraulicconductivity, as measured by its variance and correlationlength scale, on the predicted heads. The sensitivity analy-sis on the effect of the random components of the hydraulicconductivity on the head statistics at the wells R5, R4 andS10 has been carried out (Fig. 5). It is clear from the figurethat the errors increase with distance. The head standarddeviation was computed for various values of the coefficientof variability in hydraulic conductivity and its correlationlength scale . In this Figure we see that the maximum stan-dard deviation is about 0.05 m and the minimum is about0.015 m at well R5 located at 215 m from the river, the max-imum standard deviation is 0.09 m and the minimum is0.03 m at well R4 located at 975 m from the river and themaximum standard deviation is 0.165 m and the minimumstandard deviation is about 0.03 at well S10 located at

Figure 5 Standard deviation as a function of coefficient of variability in the hydraulic conductivity and its correlation length scaleat well R5, R4 and S10 for November 1991.

544 K. Srivastava et al.

1525 m from the river. At well R5 we also observe that forsmaller correlation length scale and larger coefficient ofvariability in hydraulic conductivity the errors behave in anonlinear pattern. It is also observed that the errors arelarge for smaller correlation length scale than for largerlengths. Hence if we consider the aquifer to be highly het-erogeneous we need to consider smaller correlation lengthscale. Depending on the uncertain behavior of the hydraulicconductivity the coefficient of variability in the conductivityfield can be used. Ideally, the correlation length should bededuced from a fitted serial correlation coefficient if a suf-ficiently large sample of spatial conductivity is available.Fig. 5 further demonstrates the errors on the water tableheights for a range of the controlling random componentsof the hydraulic conductivity. This shows that the errorsare highly controlled by these parameters and an appropri-ate choice depending on the nature of the aquifer should beused. Ideally it should be calculated from the availableinformation in the field. Since sufficient data on the hydrau-

Figure 6 Standard deviation as a function of coefficient of variabiat well R4 for October 1991, April 1992 and July 1992.

lic conductivity data are not available a 40% error in thehydraulic conductivity i.e CK = 0.4 has been considered.From Fig. 5 we observe that for a correlation length scaleof greater than 1/4th the errors stabilize more or less.Hence in this investigation a correlation length scale ofabout 500 m has been considered. The error propagationwith time has also been investigated and it is observed thatit is stationary with respect to time (Fig. 6). For this we haveconsidered well R4 and the error statistics for October 1991,April, 1992 and July 1992 have been computed. It is clearthat there is no change in the errors in the three monthsat the given well location.

Summary and conclusions

Using decomposition, the nonlinear Boussinesq equationwithrandom hydraulic conductivity and time varying boundaryconditions has been solved to simulate the stream–aquifer

lity in the hydraulic conductivity and its correlation length scale

Stochastic modeling of transient stream–aquifer interaction with the nonlinear Boussinesq equation 545

interaction in an alluvial valley. A practical scenario hasbeen considered when limited sampling on the hydraulicconductivity allows the estimation of its first and second-order statistics only, and thus assumptions of a specificprobability distribution are not reasonable. Simple analyti-cal expressions to the mean and standard deviation in thewater table elevation have been obtained. The model hasbeen applied to simulate the mean and standard deviationof the water table elevation in different wells across the2 km wide aquifer, which is subject to large periodic fluc-tuations in river stage. Statistics of the water table eleva-tion as a function of those of the hydraulic conductivityhave been quantified. It is seen that the magnitude ofuncertainty in the predicted variable depend on distance,time, coefficient of variability in the hydraulic conductiv-ity, and on the conductivity correlation length scale. Theuncertainty on the water table elevation at a particularlocation and time has been demonstrated for different val-ues of the coefficient of variability and correlation lengthscale. It is observed that for smaller correlation lengthscales the errors are high and are also seen to increasewith an increase in the coefficient of variability of hydrau-lic conductivity. For highly heterogeneous aquifers a smal-ler correlation length scale can be taken. The variability inhydraulic conductivity in such situations is also known tobe high. Hence, the quantification of these uncertaintieshelps predict the error bounds in the dependent variableand in corollary variables, such as aquifer dispersion coef-ficient and seepage velocity.

Appendix A. Sensitivity analysis of the effectof recharge

One of the reviewers of this paper suggested the investiga-tion of the effect of recharge relative to that of a fluctu-ating boundary condition on the overall magnitude of thehydraulic head. Intuitively, one may assume that a fluctu-ating river boundary condition exhibiting several meters inamplitude is more important than that of recharge fromrainfall, which usually amounts to much less than a meter.In this section we verify this hypothesis quantitatively bycomparing the magnitude of aquifer head in an aquiferwith a highly fluctuating boundary and recharge with onewithout recharge. For simplicity we consider the linearizedequation (1) for a hypothetical aquifer subject to a peri-odic left boundary condition and a no flow right boundarycondition:

oh

ot� T

S

o2h

ox2¼ R

S0 6 x 6 lx and 0 < t ðA1Þ

where T is the mean transmissivity (m2/day). (A1) is subjectto

hð0; tÞ ¼ H1ðtÞ ¼ H0 þ ASin2ptB

� �ohðlx; tÞ

ox¼ 0; hðx; 0Þ ¼ H0

ðA2Þ

where A and B are constants. The solution of (A1) may beexpressed as the summation of two components (Serranoand Unny, 1987)

hðx; tÞ ¼ Vðx; tÞ þWðx; tÞ ðA3Þ

Substituting Eq. (A3) into Eq. (A1) we get

oW

ot� T

S

o2W

ox2¼ R

S� oV

ot� T

S

o2V

ox2

!ðA4Þ

subject to

Wð0; tÞ ¼ H1ðtÞ � Vð0; tÞ ¼ W0 ðA5ÞoWðlx; tÞ

ox¼ � oVðlx; tÞ

oxðA6Þ

Wðx; 0Þ ¼ H0ðxÞ � Vðx; 0Þ ðA7Þ

We choose a smooth function V(x, t) such that the boundaryconditions in Eqs. (A5) and (A6) become zero:

Vðx; tÞ ¼ H1ðtÞ ðA8Þ

The solution to Eq. (A4) is (Serrano and Unny, 1987)

Wðx; tÞ ¼Z t

0

Jt�sR

Sds�

Z t

0

Jt�soH1ðsÞ

osds ðA9Þ

where the semigroup operator associated with Eq. (A4), Jt,is given as

Jtf ¼X1n¼1

bnðfÞ/nðxÞMnðtÞ ðA10Þ

the Fourier coefficients, bn, are given as

bnðfÞ ¼2

lx

Z lx

0

f sinðknnÞdn ðA11Þ

the eigenvalues, kn, are

kn ¼2n� 1

2lx

� �p n ¼ 1; 2; 3; . . . ðA12Þ

the basis function, /n(x), is given as

/nðxÞ ¼ sinðknxÞ ðA13Þ

and

MnðtÞ ¼ e�k2nTt=S ðA14Þ

Since the recharge and the specific yield are assumed con-stant, the solution of Eq. (A4) is obtained as

Wðx; tÞ ¼ R

S

X1n¼1

bnð1Þ/nðxÞInðtÞ �X1n¼1

bnð1Þ/nðxÞLnðtÞ ðA15Þ

where

InðtÞ ¼Z t

0

Mnðt� sÞds ¼ S

k2nTð1� e�k2nTtÞ ðA16Þ

and

LnðtÞ ¼Z t

0

Mnðt� sÞ oH1ðsÞos

ds ¼ 2pASk4nT

2B2þ4p2S2

� k2nTBe

� k2nTtS � k2

nTB cos2ptB

� �� 2pS sin

2ptB

� �� ðA17Þ

From (A3) and (A8), the solution reduces to

hRðx; tÞ ¼ H1ðtÞ þR

S

X1n¼1

bnð1Þ/nðxÞInðtÞ �X1n¼1

bnð1Þ/nðxÞLnðtÞ

ðA18Þ

546 K. Srivastava et al.

where hR(x,t) is the hydraulic head including recharge, and

hðx; tÞ ¼ H1ðtÞ �X1n¼1

bnð1Þ/nðxÞLnðtÞ ðA19Þ

where h(x, t) is the head without recharge. Now we adoptparameter values reflecting those of the Scioto River,K = 142 m/day, S = 0.2, lx = 2000 m, a large recharge of0.3 m/year, and average aquifer thickness b = 19 m,A = 5 m, B = 120, and a mean transmissivity T � Kb =2698 m2/day. Fig. A1 shows the typical aquifer head spatialdistribution at t = 120 days with and without recharge,according to Eqs. (A18) and (A19), respectively. The effectof recharge increases with distance and it is maximum atthe right boundary. Recharge appears to increase a fractionof the magnitude of the hydraulic head. To assess this quan-titatively we consider a measure of the relative error, E, as

E ¼ ðhRðx; tÞ � hðx; tÞÞhðx; tÞ � 100 ðA20Þ

Figure A1 Aquifer head spatial distribution at t = 120 dayswith and without recharge.

Figure A2 Maximum relative error due to recharge at theright boundary.

Numerical simulations indicated that the maximum relativeerror due to recharge occurs at the right boundary. This er-ror increases with time and reaches an asymptotic value ofabout 0.35%. Thus, the results from this exercise suggestthat the effect of a highly fluctuating river boundary condi-tion appears to be significantly more important than re-charge in determining the magnitude of the aquiferhydraulic head. This supports the assumption of neglectingrecharge in the present problem (Fig. A2).

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