Sensitivity of stream?aquifer seepage to spatial variability of the saturated hydraulic conductivity of the aquifer

Sensitivity of stream–aquifer seepage to spatial variability

of the saturated hydraulic conductivity of the aquifer

Michael P. Bruen*, Yassin Z. Osman

Centre for Water Resources Research, Department of Civil Engineering, University College Dublin, Earlsfort Terrace, Dublin 2, Ireland

Received 23 December 2002; revised 2 February 2004; accepted 6 February 2004

Abstract

In this paper, the sensitivities of stream–aquifer seepage flow and the state of connection or disconnection between stream

and aquifer, to spatial variability in the aquifer-saturated hydraulic conductivity, are studied with the variably saturated

modelling programme, SWMS_2D, using a Monte Carlo technique. A simple stream–aquifer flow system is used with a

rectangular stream, with a clogging layer on its perimeter, partially penetrating an alluvial aquifer. A 2D nearest-neighbour

method is used to generate fields of log normally distributed saturated conductivity for the alluvial aquifer for a specific set of

degree of heterogeneity and the nearest-neighbour autoregressive parameters. Seven different cases are studied. Five of these

investigate the effect of the nearest-neighbour parameters, and three cases investigate the effect of aquifer water table level. The

results show that the nearest-neighbour parameters and the aquifer degree of heterogeneity significantly affect the modelling of

stream seepage, pressure heads and the state of connection/disconnection between stream and aquifer, particularly when the

water table is some distance below the streambed. Moreover, comparisons of these results with a purely deterministic model

show good agreement when the stream and the aquifer are connected or disconnected but with a high water table. However,

there are significant differences between the deterministic and stochastic models when the water table is well below the

streambed.

q 2004 Elsevier B.V. All rights reserved.

Keywords: River; Aquifer; Interaction; Model; Monte Carlo; SWMS_2D

1. Introduction

A recent paper (Osman and Bruen, 2002), showed

that current modelling practice could seriously under-

estimate seepage from streams to alluvial aquifers

when the water table is below the streambed and

proposed and tested an improved method for estimat-

ing such seepage. They based their conclusions on a

deterministic numerical study, using a variably

saturated computer code, which assumed a homo-

geneous distribution of hydraulic properties in the

aquifer and streambed. Their proposed method also

predicts the water table elevation at which ‘discon-

nection’ of the aquifer takes place, i.e. no saturated

connection between aquifer and stream. The question

of how sensitive the deterministic results of that paper

and its proposed new method are to the more complex,

but more realistic, case of spatially distributed

hydraulic properties is important and is addressed in

this paper.

0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.jhydrol.2004.02.003

Journal of Hydrology 293 (2004) 289–302

www.elsevier.com/locate/jhydrol

* Corresponding author. Fax: þ353-1-7167399.

E-mail address: [email protected] (M.P. Bruen).

http://www.elsevier.com/locate/jhydrol

It is only since the mid- to late-1970s that

subsurface hydrologists realised that it is often best

to address subsurface modelling within a stochastic

framework rather than with the traditional determi-

nistic framework (Freeze, 1975; Smith and Freeze,

1979a; Gutjahr and Bras, 1993; Dagan and Bresler,

1983).

In stream–aquifer interactions, three different

types of flow are involved; free surface flow in the

stream, saturated groundwater flow in the underlying

aquifer and unsaturated flow in the vadose zone. If

there is a clay or silt ‘clogging’ or ‘impeding’ layer in

the bed of the stream with hydraulic properties

different from the aquifer then this further complicates

the modelling problem. A stream can either fully or

partially penetrate an unconfined aquifer. A stream

fully penetrates if its bed is at or below the lower

boundary of the aquifer. The stream partially

penetrates when its bed is above the lower boundary.

When a stream fully penetrates an unconfined aquifer

there are two types of relationships between the

stream and the aquifer. The first, called a connected

gaining stream, is when the water table is higher than

the water level in the stream and water flows from the

aquifer to the stream. The second, called a connected

losing stream, is when the water table is below the

stream water level and water flows from the stream

into the aquifer. In both cases the flow is predomi-

nantly through a saturated medium. When the stream

partially penetrates an unconfined aquifer four types

of relationships exist, Fig. 1. As in the fully

penetrating stream, type (a) is a connected gaining

stream and type (b) is a connected loosing stream.

Types (c) and (d) occur when the water table in the

unconfined aquifer falls below the streambed level

and there is no longer a direct connection of saturated

medium between aquifer and stream. The water table

is said to disconnect from the stream base. Type (c),

called a disconnected stream with shallow water table,

is when the water table is not far below the streambed

and can influence the seepage from the stream. Type

(d), called a disconnected stream with deep water

table, is when the water table falls far below the

streambed and any further drop does not affect the

seepage from the stream, Osman and Bruen (2002).

The distinction between the shallow and deep cases

depends on the configuration of stream, clogging layer

and aquifer and their hydraulic and geometric

properties.

There is a major difference in the flow system

behaviour between rivers having a clogging layer on

their bed and banks and those devoid of it (Spalding

and Khaleel, 1991). Clogging layers on beds and

banks of rivers consist of fine-grained clay or silt soils

or biologically degraded organic matter. They usually

have lower permeability than the underlying aquifer.

The seepage between the stream and the aquifer in the

two disconnected cases is unsaturated in the aquifer

below the streambed. So, to accurately model the

interactions in stream–aquifer flow system, a com-

bined saturated–unsaturated flow model is required.

However, before the model results can be reliably

used in decision-making some sensitivity/uncertainty

analyses should be conducted. This can be done with a

Monte Carlo approach, which can assess the uncer-

tainty associated with deterministic model predictions

and can determine how sensitive they are to the

inherent heterogeneity in the hydraulic properties of

the porous media and to our uncertainty as to the exact

Fig. 1. Stream–aquifer relationships (partially penetrating stream).

M.P. Bruen, Y.Z. Osman / Journal of Hydrology 293 (2004) 289–302290

nature of these heterogeneities (Freeze, 1975). Here,

the hydraulic properties are considered to be random

variables with an associated probability density

function at each point of the flow system. The spatial

dependence between neighbouring values of the

random variables is defined in terms of a spatial

correlation. The Monte Carlo method produces

probability distributions for the output variables

(e.g. hydraulic head, flow velocity, or concentration)

which reflect the uncertainty arising from the

heterogeneity of the porous media.

The specific objectives of the paper are to: (1) see if

aquifer heterogeneity causes substantial differences in

the state of connection; (2) assess the uncertainty in

the seepage due to the uncertainty in the saturated

hydraulic conductivity for different types and degrees

of spatial correlation; (3) assess the uncertainty in the

interconnected flow due to uncertainty in the saturated

hydraulic conductivity for different water table levels;

and (4) compare the estimates of seepage for different

water table levels.

This paper is organised as follows: first, a brief

account of stochastic methods used in subsurface

hydrology is given, with particular emphasis on

unsaturated flow studies. Second, the methodology

for this study is described. Third, the results of the

investigation are presented and discussed. Finally, the

conclusions are described.

2. Review of previous work

Four techniques have been used in stochastic

modelling of Richard’s equation for unsaturated

flow and transport. These techniques are: (i) analytical

solutions for simplifications of the problem (Anders-

son and Shapiro, 1983; Dagan and Bresler, 1983; and

Bresler and Dagan, 1983); (ii) analytical–spectral

approaches (Yeh et al., 1985a,b,c; Mantoglou and

Gelhar, 1987a,b,c), (iii) numerical-perturbation

approaches (Hoeksema and Kitanidis, 1984; Proto-

papas, 1988; Protopapas and Bras, 1988, 1990), and

(iv) Monte Carlo simulations, reviewed in the

following paragraphs.

Pioneering work in stochastic modelling in satu-

rated flow was done by Freeze (1975). Using the

Monte Carlo technique, his findings considered one-

dimensional (1D) saturated flow only, and neglected

spatial dependence of the saturated hydraulic con-

ductivity, which was taken as a random variable. Later

work by Smith and Freeze (1979a,b) and Bakr et al.

(1978) incorporated a spatial correlation structure in

the generated hydraulic conductivity field. Bakr et al.

(1978) used the spectral approach to solve the

perturbed forms of the stochastic differential equation

describing flow through porous media with a pre-

determined covariance of the randomly varying

hydraulic conductivity field.

Studies of spatial variability in unsaturated flows

were done by Warrick et al. (1977), Warrick and

Amoozegar-Fard (1979), Peck et al. (1977), and

Hopmans et al. (1988). They applied the scaling

theory based upon the similarity concept (Miller and

Miller, 1956; Warrick et al., 1977; Hopmans, 1987)

to express the variability of soil hydraulic properties

in terms of a characteristic length scale, which

is assumed random and statistically independent in

the horizontal. Andersson and Shapiro (1983)

modelled an autocorrelated unsaturated hydraulic

conductivity.

Stochastic analysis of unsaturated flow in more

than one dimension was introduced in a series of

papers by Yeh et al. (1985a,b,c). The analysis is based

upon spectral solutions of a perturbation approxi-

mation of the stochastic equation. Analytical

expressions were derived to describe the variances

of soil water pressure head, flux, and the effective

hydraulic conductivity as a function of the statistical

properties of the soil medium and the flow character-

istics. A similar study was published by Mantoglou

and Gelhar (1987a,b,c). Both studies stressed that

anisotropy and heterogeneity of the saturated con-

ductivity may have a significant effect on the

contribution of the lateral flow component and that

the traditional approach of infiltration modelling may

underestimate the magnitude of horizontal flow.

Hopmans et al. (1988) examined the stochastic nature

of soil water pressure head and vertical and horizontal

flux density under a two-dimensional (2D) unsatu-

rated steady state flow conditions. Ababou (1988)

used a Monte Carlo simulation approach to derive

pressure and moisture content distributions that result

when the parameters of unsaturated hydraulic con-

ductivity are allowed to vary continuously throughout

a three-dimensional (3D) domain.

M.P. Bruen, Y.Z. Osman / Journal of Hydrology 293 (2004) 289–302 291

3. Methodology of this study

This study uses the Monte Carlo method with the

nearest-neighbour technique to investigate the effect

of spatial heterogeneity of hydraulic properties on

stream–aquifer seepage.

3.1. Stream–aquifer model

The modelling framework is the same as that used

in the previous deterministic study (Osman and

Bruen, 2002) (Fig. 2). A simplified 2D (vertical

plane) groundwater flow system is used. It consists of

a section through a rectangular stream channel (with a

clogging layer on its bank and bed) which partially

penetrates an alluvial unconfined aquifer, which itself

overlies a regional aquifer as depicted in Fig. 2.

Infiltration and evapotranspiration from the ground

surface are ignored (i.e. no-flow boundary on the

ground surface), with the two lateral boundaries are

taken either as no-flow or fixed head boundaries and

the bottom boundary is taken as a constant head

boundary (known hereafter as the underlying head).

When the lateral boundaries are no-flow boundaries,

the only exchange of water within the model domain

is through the streambed and banks, and through the

bottom boundary of the domain where water can drain

to or rise from the lower part of the unconfined

aquifer. The dimensions of the flow domain are taken

here in such a way to reduce the computational efforts.

The materials of the alluvial aquifer and the clogging

layer are assumed to be medium sand and clay loam,

respectively. Since the flow system in Fig. 2 is

symmetrical, only half of this domain is simulated.

The simulation domain used in the Monte Carlo

simulation is discretised into 1352 nodes and 1257

square elements of 0.6 m sides.

3.2. Numerical model code

The computer code SWMS_2D (Simunek et al.,

1994) is used here for the variably saturated

simulations. It uses a Galerkin linear finite element

formulation to solve a 2D form of Richards’ equation

for water flow with a sink term, coupled with a

convection–dispersion equation for solute transport.

The unsaturated soil hydraulic properties formulation

used is that of Vogel and Cislerova (1988), which

under certain conditions becomes the van Genuchten

(1980) model. Flow and transport can occur in a

vertical or a horizontal plane, or in a 3D region

exhibiting radial symmetry about a vertical axis.

Verifications of the code by comparison with field

data or the output of other variably saturated codes

e.g. UNSAT2, SWATRE are well-documented in the

SWMS-2D user’s manual and are not discussed here.

Fig. 2. Simple stream–aquifer flow system.


Some minor modifications have been made to the

code to suit the present problem.

3.3. Distribution of the saturated conductivity

The log normal distribution has been used to

describe the random nature of values of saturated

conductivity in an alluvial aquifer (Law, 1944;

Bulnes, 1946; Warren et al., 1961; Bennion and

Griffiths, 1966; Freeze, 1975; Smith and Freeze, 1979;

Dagan and Bresler, 1983; Hopmans et al., 1988). For

unsaturated conditions, the hydraulic conductivity is

represented by van Genuchten’s (1980) equation

uwðwÞ ¼ ur þus 2 ur

½1 þ lawln�mð1:aÞ

KðwÞ ¼ KsS1=2e ½1 2 ð1 2 S1=m

e Þ�2 ð1:bÞ

with

Se ¼uw 2 ur

us 2 ur

ð1:cÞ

where us is the saturated moisture content [dimen-

sionless]; ur is the residual moisture content [dimen-

sionless]; a is a parameter in the soil retention

function [L]; n is an exponent in the soil retention

function ðn . 1Þ [dimensionless]; m ¼ 1 2 1=n

[dimensionless] and Ks is the soil saturated hydraulic

conductivity, [LT21].

Table 1 shows typical soil parameters for van

Genuchten’s (1980) model for the soils used in this

study. Note Eq. (1.b) indicates that spatial variation in

the saturated hydraulic conductivity also influences

the unsaturated conductivity.

For log normally distributed hydraulic conduc-

tivity K; a normally distributed parameter Y ¼

N½my;sy�; can be defined such that Y ¼ log10ðKÞ:

This new parameter has a mean of my and a standard

deviation of sy. Freeze (1975) summarised the results

of a number of studies (Bennion and Griffiths, 1966;

Law, 1944; McMillan, 1966, and Willardson and

Hurst, 1965), indicating that the standard deviation,

sy; of Y (when K is in cm/s), varies between 0.2 and 2.

3.4. The nearest-neighbour stochastic model (NNSM)

The nearest-neighbour model (NNM) is designed

to model spatial variations in a statistically homo-

geneous random field in which the stochastic

dependence is local (i.e. depends only on distance

between points). It can be formulated in terms of a

Markovian conditional probability or a set of

simultaneous joint probabilities (Brook, 1964; Smith

and Freeze, 1979b). Its interpretation can be either

completely spatial or linked to a model describing the

time sequence of geologic deposition.

Consider a porous medium with a log normally

distributed saturated hydraulic conductivity, K: A

first-order nearest-neighbour auto-regressive relation

in two dimensions ðx; zÞ can be written as

Yi;j¼axðYi21;jþYiþ1;jÞþazðYi;j21þYi;jþ1Þþei;j ð2Þ

where

Yi;j; the random variable satisfying the nearest-

neighbour relation;

ei;j; normal random variable uncorrelated with any

other ei;j;

ax; an autoregressive parameter expressing the

degree of spatial dependence of Yi;j on its two

neighbouring values in the x-direction, Yi21;j and

Yiþ1;j ðlaxl # 1Þ;

az; an autoregressive parameter expressing the

degree of spatial dependence of Yi;j on its two

neighbouring values in the z-direction, Yi;j21 and

Yi;jþ1 ðlazl # 1Þ;

Table 1

Soil types and characteristics according to van Genuchten’s model

Domain Unit Soil Type hc us ur Ks a n

Aquifer Medium sand 0 0.34 0.05 30.00 0.18 3.07

Clogging layer Clay loam 1 0.44 0.30 0.02 0.04 3.37

hc, air entry value (m); us, saturated moisture content; ur, residual moisture content; Ks, saturated hydraulic conductivity (m d21);

a; parameter in the soil water retention function (m); n; exponent in the soil water retention function ðn . 1Þ:


Fig. 3 illustrates the above process equation for a

discrete block model. A single value Yi;j applies

everywhere within the block i; j: The coordinate

systems of the stochastic process model and the flow

domain are aligned. The scale of the model is defined

on the discrete interval Dx ¼ xi;j 2 xi21;j; and

Dz ¼ zi;j 2 zi;j21: Here, it will generally be assumed

that Dx and Dz are chosen equal.

In a statistically homogeneous medium, Eq. (2)

holds for every block within the medium. If ax is

equal to az; the medium has a statistically isotropic

covariance structure; that is, the statistical dependence

between neighbouring conductivity values is inde-

pendent of their orientation. If ax is not equal to az;

the medium has a statistically anisotropic covariance

structure, and the covariance between conductivity

values depends on their orientation within the

medium.

Therefore, for a 2D system of conductivity blocks

within the flow domain, the stochastic process model

for the entire set of p blocks (m rows, n columns) can

be written as a system of p linear equations, each

based on Eq. (2), as:

{Y} ¼ ½W�{Y} þ {e} ð3Þ

where the matrix ½W� is a spatial lag operator of scaled

weights wkl: These scaled weights are defined by

wkl ¼wpkl=r for k ¼ 1;…;p; l¼ 1;…;p and k– l ð4Þ

where wpkl ¼ ax if blocks k and l are contiguous in the

x-direction, wpkl ¼ az if blocks k and l are contiguous

in the z-direction, and wpkl ¼ 0 otherwise. The scaling

factor, r; is the total number of contiguous blocks

surrounding block k and is required to preserve the

statistical homogeneity in the generated sequence. For

points in the interior of the block, r will be 4, on an

edge it will be 3 and in a corner 2. The ½W� matrix

indicates which conductivity values in the block

system are linearly related to each other.

Each element of the normally distributed random

vector {e} is independent of the others, has a mean me

and a standard deviation se: Similarly, each member

of the {Y} sequence will have a mean my; and standard

deviation sy: Generally, the standard deviation sy will

be related to that of the {e} sequence through the

weight matrix ½W� and the values of the autoregres-

sive parameters. However, in generating synthetic

sequences, it is often required to simulate a pre-

determined standard deviation sy: Starting from the

normally distributed, independent, random sequence

{1} with a standard deviation of one, this vector can

be pre-multiplied by an appropriate factor to yield a

desired value of sy: This factor, denoted here by h; is

applied to the nearest-neighbour system of Eq. (3) to

yield

{Y} ¼ ½W�{Y} þ h{1} ð5Þ

Solution of Eq. (5) for the vector {Y} can be

written as

{Y} ¼ ð½I�2 ½W�Þ21·h{1} ð6Þ

where ½I � is an identity matrix.

Therefore, the matrix inverse, ð½I �2 ½W�Þ21; can

be regarded as a filter operating on a random vector

{1} to yield an output vector {Y} with an internal

correlation which depends upon the form of the spatial

lag operator ½W� and the parameters ax and az: A

similar set of equations will apply for a 1D set of

blocks; only the structure of the spatial lag operator

½W� changes.

The nearest-neighbour filter in Eq. (6) requires

that both {1} and {Y} sequences have a mean of

zero. Vectors, Y* ; with any required non-zero mean

can be produced, without affecting the covariance,

by adding the required mean my to each element of

the {Y} calculated from Eq. (6). It is assumed that

the mean is independent of the covariance of the

process.

In summary, using the nearest-neighbour model to

generate a 2D discrete block of conductivity realis-

ations with a known spatial covariance structure

Fig. 3. Schematic illustration of a nearest-neighbour grid, first-order

model.


requires the following steps: (i) generate the uncorre-

lated {1} using a normal distribution random number

generator with mean zero and standard deviation one

ðN½0; 1�Þ; (ii) using, Eq. (5), calculate the internally

correlated output sequence {Y}; which should be

N½0;sy�; (iii) add the mean my to each member of {Y};

yielding {Y*} as N½my;sy�; (iv) calculate the log

normally distributed conductivity values {K} by

applying the exponential transform

Ki ¼ 10Yi* ð7Þ

yielding the distribution LN½mk; sk� of K; (v) insert the

conductivity values {K} into their appropriate blocks

within the flow domain; and (vi) solve the resulting

deterministic equations, using SWMS_2D.

The correlation scale l is an important property of

the nearest-neighbour field. Given that the generated

field is statistically homogeneous, that is to say, it

satisfies equation (8) below:

E½Yðx; zÞ� ¼ my R½Yðx1; z1Þ;Yðx2; z2Þ� ¼ F½j�

where j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 2 x2Þ

2 þ ðz1 2 z2Þ2

q ð8Þ

where

R; covariance function

z; distance between points ðx1; z1Þ and ðx2; z2Þ:

Then the correlation scale l is defined as

l ¼1

Fð0Þ

ð1

0FðzÞdz ð9Þ

It is a measure of the mean distance over which

points are positively correlated in each direction. If

lx ¼ lz; the medium has a statistically isotropic

covariance function. If lx – lz; the covariance

function is statistically anisotropic. Integral scales

for each conductivity realisation in each direction are

approximated here by summing up the area under the

autocorrelation function taken along each direction

until it first crosses the zero-axis. Because the sample

autocorrelation function is different for each realis-

ation, so are the integral scales. Therefore the integral

scales that characterise a certain nearest-neighbour

process are found by taking the mean of the integral

scales obtained over all of the realisations in a set of

Monte Carlo simulations for specified sy; ax; az:

As can be seen from the nearest-neighbour

generator in Eq. (2), the values at the boundaries

can have reduced correlations with the inner values

and this could perturb the generated distribution of K:

For this reason, the nearest-neighbour domain is

normally taken larger than the actual simulated

domain in order to reduce any boundary effects.

The 2D nearest-neighbour domain used here is

slightly larger in both width and depth than the

simulated domain, described in Section 3.1 and Fig. 2,

above. A depth of 10.2 m and a width of 4.62 m are

used with a block size of 0.6 m. There are 1309

conductivity blocks in which 77 are in the horizontal

direction and 17 in the vertical direction. The

stochastic characteristics of the nearest-neighbour

domain require specifying appropriate values for the

factor h for given values of ax; az; my and sy and for a

specified number, m; of Monte Carlo simulations.

This is done here by trial and error using a computer

program prepared for this purpose.

Seven cases are reported here. Cases 1–5 are used

for comparing the simulated outputs for a discon-

nected loosing-stream with shallow water table

(underlying head 19.08 m) for different stochastic

characteristics. The remaining two cases (cases 6

and 7), in addition to case 1, are used to compare the

Monte Carlo outputs, with the corresponding homo-

geneous model, for different water table levels. The

five nearest-neighbour aquifer fields considered are:

(i) 2D statistically homogeneous and anisotropic

ððax ¼ 0:85Þ . ðaz ¼ 0:5ÞÞ; (ii) 2D statistically

homogeneous and isotropic ðax ¼ az ¼ 0:85Þ; (iii)

2D statistically homogeneous and anisotropic ððax ¼

0:5Þ , ðaz ¼ 0:85ÞÞ; (iv) 1D statistically homo-

geneous for one layer in horizontal direction

(ax ¼ 0:85; az ¼ 1:0); and (v) 2D uncorrelated

conductivity ðax ¼ az ¼ 0:0Þ media. The values for

the autoregressive parameters are similar to those

used by Smith and Freeze (1979a,b). The heterogen-

eity in all the cases is expressed in terms of the

standard deviation of Log10ðKÞ; i.e. sy: Seven degrees

of heterogeneity from sy ¼ 0:25 to 1.75, spanning

most of the range reported by Freeze (1975), are used

in each case.

The five different nearest-neighbour fields are

simulated with three different stream–aquifer con-

ditions. The underlying aquifer head at the bottom

boundary of the flow domain controls these


relationships. In this study, an underlying head of

19.14 m is used for the connected losing stream

conditions, an underlying head of 19.08 m is used for

the losing stream with shallow water table conditions,

and an underlying head of 18.00 m is used for the

losing stream with deep water table conditions.

Corresponding deterministic solutions for these three

cases are found with use of the SWMS_2D code.

The number of realisations in each Monte Carlo

experiment was determined by examining the con-

vergence of the statistics of the simulation results as

the number of simulations increased. As a reasonable

compromise between computational effort and con-

vergence of resulting statistics 300 was chosen.

4. Results and discussion

The simulations are analysed by comparing the

statistical behaviour of the stream–aquifer seepage

and examining the distribution of connected and

disconnected simulations.

4.1. Different nearest-neighbour scenarios

Fig. 4 shows how the ensemble statistics

(mean, maximum, minimum, standard deviation

and coefficient of variation) of seepage from the

stream ðQÞ; varies with degree of heterogeneity in the

aquifer for an underlying head of 19.08 m (discon-

nected loosing stream with shallow water table) for

case 1. The ensemble mean and minimum of Q

decrease with increasing heterogeneity in the aquifer,

while the ensemble maximum, standard deviation and

coefficient of variation increase with increase in

aquifer heterogeneity. Their responses to the degree

of heterogeneity are approximately exponential. An

increase in the aquifer heterogeneity implies an

increase in the variability of the model outputs. Note

that the minimum seepage drops at a much faster rate

that the increase in the maximum, indicating a highly

skewed response. The distribution of the individual

values for each simulation is strongly negatively

skewed. Fig. 5 shows the frequency histogram for the

290 successful simulations for the case where sy is 1.5.

The mode of the distribution is close to the maximum,

as is the mean in Fig. 4, but a small number of very

much lower values occur. Fig. 6 shows the ensemble

standard deviation of seepage, sQ; as a function of

aquifer heterogeneity for different nearest-neighbour

scenarios (cases 1–5, in Table 2). The standard

deviation, and hence the degree of uncertainty, of the

seepage increases with increasing aquifer hetero-

geneity. In the graph of case 5, the uncorrelated

Fig. 4. Behaviour of the stream seepage statistics, case 1.


conductivity field, ðax ¼ az ¼ 0Þ; falls at the bottom of

the figure and the graph of case 4 (a 1D layer, ax ¼

0:85; az ¼ 0) lies at the top of the figure. Thus, for a

given aquifer heterogeneity, uncertainty in stream

seepage flow increases with increase in the value of the

autoregressive parameters and particularly with that of

the horizontal correlation, ax:

Some stream–aquifer systems, which are classified

as disconnected when analysed with a homogeneous

model, can exhibit some connected conditions as

aquifer heterogeneity increases. For case 1, which the

homogeneous solution indicates is hydraulically

unconnected, Fig. 7 shows the percentage of the

Monte Carlo simulations which became hydraulically

Fig. 5. Statistical distribution of simulation results.

Fig. 6. Standard deviation of seepage flow versus aquifer heterogeneity.


connected as a result of aquifer heterogeneity. This

happens because some parts of the aquifer can possess

sufficiently low conductivity values to locally impede

the downward flow and hence cause a locally elevated

water table which connects with the stream. The

different curves for each simulation case show that

this effect depends on the degree of aquifer hetero-

geneity. For a given degree of aquifer heterogeneity

the percentage of connected cases depends on the

value of the autoregressive parameters, with horizon-

tal correlation (case 4) having the most influence.

4.2. Different stream–aquifer relationships

Fig. 8 shows how the ensemble standard deviation

of the seepage increases as the aquifer heterogeneity

increases for case 1 (disconnected loosing stream

with shallow water table conditions, case 6 (connec-

ted loosing stream) and case 7 (disconnected

loosing stream with deep water table conditions).

The uncertainty in seepage estimates increases as

the water table drops. This is despite the seepage

usually being considered, by the conventional homo-

geneous models, to be independent of the water table

for case 7.

Fig. 9 shows graphs of the percentages of

connected cases as functions of aquifer heterogeneity

for cases 1, 6 and 7. A stream, which is indicated as

connected to its underlying aquifer (case 6) for the

homogeneous case, tends to remain connected as

aquifer heterogeneity increases. However, for

streams, which are indicated as disconnected from

Table 2

Definition of simulated cases and their nearest-neighbour characteristics

Case no. ax az lx

(m)

lz

(m)

Underlying

head (m)

Scaling factor, h; for different aquifer heterogeneities

sy ¼ 0:25 sy ¼ 0:50 sy ¼ 0:75 sy ¼ 1:00 sy ¼ 1:25 sy ¼ 1:50 sy ¼ 1:75

1 0.85 0.50 1.82 0.34 19.08 0.191 0.382 0.573 0.765 0.956 1.147 1.338

2 0.50 0.85 0.77 0.58 19.08 0.194 0.388 0.582 0.775 0.969 1.163 1.357

3 0.85 0.85 2.45 0.79 19.08 0.145 0.289 0.434 0.579 0.723 0.868 1.013

4 0.85 0.00 5.42 0.00 19.08 0.098 0.195 0.293 0.390 0.488 0.585 0.683

5 0.00 0.00 0.11 0.07 19.08 0.250 0.500 0.749 0.999 1.249 1.499 1.749

6 0.85 0.50 1.82 0.34 19.14 0.191 0.382 0.573 0.765 0.956 1.147 1.338

7 0.85 0.50 1.82 0.34 18.00 0.191 0.382 0.573 0.765 0.956 1.147 1.338

Fig. 7. Percentage of connected cases for different nearest-neighbour scenarios.


their aquifer (cases 1 and 7) for the homogeneous

case, the chances of re-connection increase with both

increasing aquifer heterogeneity and rising water

table.

4.3. Monte Carlo versus homogeneous approaches

Fig. 10 compares the ensemble mean of the

seepage flow from the Monte Carlo simulations with

Fig. 8. Ensemble standard deviation of the seepage flow for different stream–aquifer relationships.

Fig. 9. Percentage of connected cases for different stream–aquifer relationships.


the seepage flow obtained by the homogeneous

model, for cases 1, 6 and 7. In these graphs,

MCase1 stands for the Monte Carlo case and

HCase1 stands for the homogeneous estimates of the

seepage flow in case 1. For cases 1 and 6, there is

almost no difference for the different modelling

approaches for most degrees of the aquifer hetero-

geneity. For example, the seepage flow from the

homogeneous solution for case 1 is 0.801 m2/d,

whereas the Monte Carlo solution for all possible

range of aquifer heterogeneity varies between 0.7794

and 0.8028 m2/d.

However, there is a significant difference

between the two approaches when the water table

falls far below the stream bottom as in case 7. For

this, the homogeneous solution is 2.295 m2/d

whereas for the Monte Carlo solution with the

lowest tested sy value, 0.25, the mean drops to

2.286 m2/d, (individual values vary between 2.169

and 2.3643 m2/d). The mean of the Monte Carlo

results decreases even further as the spatial varia-

bility increases. For instance, for sy equal to 1.75,

the mean of the Monte Carlo results drops to

2.0709 m2/d, with individual values varying from

0.9448 to 2.5075. Table 3 shows these and results

for intermediate levels of variability. Not only does

the mean decrease consistently as the variability

increases, but also the standard deviation and spread

of the individual results increase.

5. Conclusions

This paper examines the effects of spatial variability

of the aquifer hydraulic conductivity on estimates of

seepage from a stream to an aquifer. It is basically a

sensitivity study of the homogeneous, effective layer,

approach to modelling a stream–aquifer system.

Uncertainties in the values of stream seepage flows

(measured as their ensemble standard deviation) are

Fig. 10. Monte Carlo and homogeneous estimations of the seepage flow for different stream–aquifer relationships.

Table 3

Deviation of stochastic results from deterministic solution (case 7)

sy Q (m2/d)

Mean Std

Deviation

Max.

value

Min.

value

0.25 2.29 0.03 2.36 2.17

0.50 2.26 0.08 2.41 1.90

0.75 2.22 0.14 2.44 1.57

1.00 2.18 0.19 2.47 1.31

1.25 2.13 0.24 2.49 1.14

1.50 2.09 0.27 2.50 1.02

1.75 2.07 0.28 2.51 0.94


found to increase generally with increase in the degree

of aquifer heterogeneity, particularly in the horizontal

direction.

Homogeneous approaches closely approximate the

Monte Carlo results in estimating the seepage when

the stream and the aquifer are connected or when they

are disconnected but the water table is close to the

streambed.

However, Monte Carlo modelling of the system

can produce, depending on the water table elevation in

the aquifer, situations of connection between the

stream and the aquifer when the homogeneous models

indicate disconnected conditions, but not vice versa.

The effects of spatial variability in the aquifer-

saturated hydraulic conductivity are important when

modelling stream–aquifer seepage in alluvial aquifers

where the water table falls substantially below the

riverbed. The seepage estimates can be quite different

from those given by homogeneous models.

Acknowledgements

The authors would like to thank Dr Jirka Simunek

of the US Salinity Laboratory for providing the

SWMS_2D code. The second author, in particular,

would like to acknowledge the financial support he

received from the Centre for Water Resources

Research, UCD, during the course of his PhD.

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Sensitivity of stream?aquifer seepage to spatial variability of the saturated hydraulic conductivity of the aquifer