A Method for Setting the Artificial Boundary Conditions of Groundwater Model

doi:10.4236/ojg.2013.32B012

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Open Journal of Geology, 2013, 3, 50-54

doi:10.4236/ojg.2013.32B012 Published Online April 2013 (http://www.scirp.org/journal/ojg)

A Method for Setting the Artificial Boundary

Conditions of Groundwater Model

Yipeng Zhou1,2, Zhaoli Shen1, Weijun Shi2, Jinhui L iu 2, Yajie Liu2

1School of Water Resources and Environment, China University of Geosciences, Beijing, China

2Department of Civil and Envir onmental Engineerin g , East China Institute of Technology, Nanchang, China

Email: [email protected], [email protected]

Received 2013

ABSTRACT

Numerical simulation technolog y is nowadays an important means for groundwater issues because of its efficiency and

economical advantages. But in case of natural hydrogeological boundaries are not within the interest area, it may be a

big trouble to set boundary conditions of the model artificially without enough field investigation information. This pa-

per introduced a method for solving such problem applying field pumping test and recovery test. The method was ap-

plied to build an in-situ leaching of uranium model. Results show ed that the model boundary conditions can be set sat-

isfactorily, and also the calculated heads matched the observed data well in both two models.

Keywords: Groundwater; Numerical Model; Artificial Bound ary Conditions

1. Introduction

Since the 1960s, with the development of computer

technology, the method of numerical simulation had been

widely used to solve groundwater flow and solute trans-

port problems because of its effectiveness, flexibility and

relatively economical with spend, and gradually become

an important method for groundwater issues [1-7].

However, although lots of models have been built in

various applications, few people care about the real ef-

fects of those models in practices [8]. One of important

factors influencing the reliability of the groundwater

model is geology and hydrogeology investigation; and

usually making reasonable understandings on boundary

conditions is a big challenge [9]. Once the boundary

conditions are distorted to the truth, it is bound to lead to

significant deviation of the model calibration parameters

from actual values, and then serious impact on the reli-

ability of the mo del would not be avoided.

Model boundary conditions are usually set according

to field investigations. When the interest area is small

that the model boundaries are far away from natural hy-

drogeological boundaries, artificial model boundary con-

ditions need to be set according to a long-term observa-

tion of groundwater at those boundaries. However, in

many cases, the required observation data are often un-

available; in this dilemma, one alternative way is to ex-

pand model extent so that the groundwater can be as-

sumed not to be affected by human activities (such as

pumping test) taken placed in the interest area; then the

boundary conditions of first type or of second type can

be set at model’s boundaries [9]. Bu t this kind of so lu tion

also has shortcomings, one of which is that to estab lish of

model hydrogeology configuration beyond the interest

area without supplementary geology investigation infor-

mation may bring unexpected serious error to the simula-

tion results [10,11]. In this pap er, in order to build a flow

model of groundwater and leaching solution during in-

situ leaching of uranium process, a method has been em-

ployed to set artificial model boundaries by combining

theoretical calculation according groundwater unsteady

flow theory and the model iterative calibration using ob-

servation data of pumping test and recovery test inde-

pendently.

2. Methods

2.1. Basic Principles

The basic principle is using field pumping test and re-

covery test to calibrate the model parameters and bound -

ary conditions simultaneously. First, get the head draw-

down function derived from Jacob formula of ground-

water unsteady flow at the boundaries located within the

cone depression; Then set initial heads generalized from

the head drawdown function of the model boundaries for

model building and calibration; finally, calibrate model

parameters and boundary conditions iteratively using the

observation data of pumping test and recovery test inde-

pendently and then make the results fit the facts to the

most degree.

Y. P. ZHOU ET AL. 51

2.2. Functions of the Model Boundary Heads

During pumping test in the confined aquifer, draw down

lead to the formation of the cone of depression of pres-

sure head. The range of cone expands continuously with

pumping, and gradually achieves a relative stable state.

When the whole model is located within the cone, heads

at model boundaries would vary with time; so obviously,

model boundary conditions need to be set according to

head changes. Therefore, the function of head variation

must be got at first.

According to the theory of confined water’s unsteady

flow towards to fully penetrating well, the variation of

the head drawdown within pumping influence scope can

be approximately described by Jacob Formula [12], as in

2.25

sTr





 (1)

where s is the head drawdown; Q is the pumping rate; T

is the coefficient of transmissibility; t is the pumping

time; r is the distance to pumping well; μ * is the coeffi-

cient of storage.

Thus, the live head can be calculated via Equation (2):

(,)(, )(,)

rtHrt srt (2)

where (,)

rt is the head at the point with the dis-

tance of r to the pumping well and at the time of t;

(, )

rt is the initial head at the point with the distance

of r to the pumping well; (,)

rt is the head drawdown

at the point with the distance of r to the pumping well

and at the time of t.

Since in Equation (2) the head is a continuity function

of time, it can not to be applied to set model boundary

conditions yet; it need to be temporally discretized to n

periods, and in each period the head is a constant, thus,

the variation of the head drawdown can be described by

Equation (3):

10 1

21 2

(,) ......

nn n

ttt

srt

ttt



















(3)

So, the live head anywhere within the cone of depres-

sion during pumping test can be given by the piecewise

constant function as Equ a tion (4):

01 01

02 12

(, ),

(,) ......

(, ),

nn n

Hrtsttt

Hrtstt t

Hrt

Hrtstt t



















(4)

Divide the simulation time to n stress periods, and in

each period set the model boundary head according to the

corresponding constant value of each definition domain

of the function shown in Equation (4).

2.3. Calibrations of the Model Boundary

Conditions

After initial boundary conditions being set through the

theoretical calculation mention ed above, run the model to

calibrate boundary conditions and parameters iteratively

using observation data of the pumping test and recovery

test independently. The flow chart of the calibration

process is shown in Figure 1.

First, build groundwater flow model of the pumping

test and set the model initial boundary conditions for

iteration according to Equation (4).

Then, run the model built in the former step to cali-

brate model parameters and boundary conditions by the

observation data of the pumpin g test. If the stand ard error

of estimate (S.E.E) exceeds 5%, the hydrogeology pa-

rameters and boundary conditions would be adjusted

slightly and then the calibration repeats. When the S.E.E

is below 5%, the calibration process goes to the next

step.

Finally, build groundwater model of the recovery test

applying the calibration results of the second step as ini-

tial state; run it to calibrate model parameters and

boundary conditions again using the observation data of

the recovery test. The S.E.E of 5% also is applied as

calibration error criterion; if results meet the criterion,

the calibration process ends and the model parameters

and boundary conditions are fixed; otherwise, return to

the first step, modify the model parameters and boundary

conditions and then the whole process repeats again.

3. Application Example

3.1. Backgrounds and Model Overview

The study was conducted at the piedmont alluvial slope

in the southern region of the Turpan-Hami basin. The

Figure 1. Flow chart of the model calibration process.

Y. P. ZHOU ET AL.

interest confined groundwater system has stable imper-

vious roof and base; the groundwater flew from south-

west to northeast with a hydraulic gradient of 0.02, and

was mainly recharged indirectly by the Quaternary

phreatic water from the southern mountainous bedrock

fissure. The studied issue was about simulation of

groundwater and leaching solution flow in the ore-bear-

ing aquifer at an in-situ leaching of uranium site. There

are five wells (Figure 2); the well CK1 was for extrac-

tion, and the rests were for leaching solution injection.

The hydrogeology characteristic of the aquifer within

the mining scope is as shown in Figure 3. The average

thickness of the aquifer is about 40m; the stable imper-

vious roof and base are mainly of mudstone and silty

mudstone (in gray); in the aquifer (in blue) there are four

discontinuous interlayer, one is of silty mudsto ne (shown

in gray) with the thickness of 1 - 3 meters, and the three

others are of calcareous sandstone with the thickness of

0.3 - 0.9 m (in white).

Field pumping test and recovery test were conducted

employing well CK1 for pumping, well ZK1 and well

ZK3 for observation, the test results are shown in Table

Figure 2. The plan view of well distribution.

Figure 3. Cross section of the aquifer.

Table 1 field tests and the hydraulic parameters of the aq-

uifer.

Field test Test

time

(min)

Pumping

rate

(m3/h)

Coefficient of

transmissibility

(m2/d)

coefficient

of storage

Pumping test2900 7.2

Recovery test2770 － 0.57 1.95 × 10-4

Based on those field investigations, the conceptual

model of the groundwater flow during the pumping test

was built as Equation (5):

()()()

,( ,,),0

(,,,)(,,)(,,)

(,,,) (,,,)(,,)

xxyy zz

HHH

xxyyzz

SxyzDt

HxyzoH xyzxyzD

xyztf xyztxyzS































(5)

where Kxx, Kyy and Kzz are respectively the conductivities

in x, y and z direction in the three-dimension space; H is

the confined water head; W is the flux rate per unit vol-

ume, which is used to describe the flow rate of wells; Ss

is the specific storativity; t is the time; H0(x,y,z) is the

initial head at position with the coordinate (x, y, z);

(,,,)

xyzt is the head at the model boundary. The

model area denoted D and boundary S.

The initial head of H0(x, y, z) was set according to

static water level observed at the beginning of the pump-

ing test. Since the modeling area was small, its edges are

far away from the natural hydraulic boundaries known,

hence artificial boundaries were necessary. A circle sur-

rounding the well CK1 and with radius of 100 meters

was set as the boundary of the model.

3.2. Head Functions of the Model Boundary

In case of the coefficient of transmissibility is 5.7 m2/d,

the coefficient of storage is 1.95 × 10 – 4 and the pumping

rate is 7.2 m3/h, the radius of influence of the pumping

test is more than 800 meters; clearly, the heads at the

pumping test model boundaries which were 100 meters

away from the pumping well must to be varying with

time. Calculation according to Jacob formula showed

that head drawdown started to take place at the model

boundaries after pumping for 3.65 hours. The drawdown

function can be derived from Equation (1), as in

0,0 3.65

2.34ln2.95, 3.6550

stt







 (6)

Temporally discretized the continuity function of the

drawdown to a piecewise constant one, as shown in Fig-

ure 4. A broken line of discrete function ()

 was used

Y. P. ZHOU ET AL. 53

to approximately replace the curve of the original func-

tion s(t), making sure the difference between the two

adjacent constants of the function ()

 was not greater

than 0.5 m. The function values are listed in Table 2.

And then, divided the whole simulation time of the

pumping test into 15 periods according to the function

()

, in each period initial b oundary heads of the model

were set as a corresponding constant value, which could

be get from the Equation (4) by replacing the values of

function ()

 listed in Table 2 for the drawdown. Then,

the iterative calibration process started.

3.3. Results

After correcting the model parameters and boundary

conditions repeatedly via the calibration processes of the

pumping test model and recovery test model, the model

parameters and boundary conditions were fixed on. Re-

sults showed that the calculated heads matched the ob-

served data satisfactorily in both two models (Figure 5

and Figure 6); the mean absolute error between the cal-

culated heads and observed data of pumping test simula-

tion was 0.694 m, and recovery test simulation 0.655 m;

both variances were less than 5%. Furthermore, the coef-

ficient of transmissibility was 5.3 m2/d; it was close to

the results of field tests (5.7 m2/d).

Figure 4. The curves of function s (t) and function s'(t).

Table 2. The values of function s'(t).

(h) )(' ts

(m)

(h) )(' ts

(m)

(h) )(' ts

(m)

0 - 4 0 8 - 10 2.08 22 - 26 4.39

4 - 5 0.5 10 - 12 2.56 26 - 30 4.75

5 - 6 0.91 12 - 14 2.95 30 - 34 5.07

6 - 7 1.32 14 - 18 3.44 34 - 42 5.47

7 - 8 1.66 18 - 22 3.96 42 - 50 5.91

Figure 5. The results of calculated heads matched to ob-

served heads of the pumping test model.

Figure 6. The results of calculated heads matched to ob-

served heads of the recovery test model.

4. Conclusions

The boundary condition is one of key factors influencing

the reliability of groundwater model. In case of artificial

boundary conditions are needed, they should be set rea-

sonably using as much field investigation data as we can

get, otherwise, it is prone to cause great distortion to the

truth and make the model worthless. Study results show

that the method intro duced in this pap er can be a feasible

choice to set artificial boundary conditions of the

groundwater model.

5. Acknowledgements

Thanks for the fund of the Major State Basic Research

Development Program of China (973 Program) (No.

2012CB723101).

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