Open Journal of Geology, 2013, 3, 50-54
doi:10.4236/ojg.2013.32B012 Published Online April 2013 (
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
Received 2013
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-
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.
Copyright © 2013 SciRes. OJG
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
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
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
Hrtstt t
Hrtstt t
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
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
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.
Copyright © 2013 SciRes. OJG
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-
Field test Test
Coefficient of
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
HxyzoH xyzxyzD
xyztf xyztxyzS
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
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
Copyright © 2013 SciRes. OJG
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
(h) )(' ts
(h) )(' ts
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.
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