Full text of DOI:10.1007/BF02875398

Vol. 44 No. 11
SCIENCE IN CHINA (Series D)
November 2001
A research on grey numerical imitation and modeling of
groundwater seepage system
WU Qiang (武 强)1,2, LI Shuwen (李树文)3, WANG Yinao (王义闹)3,
DONG Donglin (董东林)2 & WU Xiong (武 雄)2
1. Institute of Hydrogeology and Environmental Geology, Chinese Academy of Geological Science, Shijiazhuang
050803, China;
2. Institute of Water Resource and Environment Engineering, China University of Mining & Technology, Beijing
100083, China;
3. Department of Environment Engineering, Hebei Institute of Architectural Science & Technology, Handan 056038,
China
Correspondence should be addressed to Wu Qiang (email: wuq@cumtb.edu.cn)
Received May 28, 2001
Abstract Based on grey set, grey numbers and their operation properties, the grey numerical
model of groundwater seepage system was set up for the first time, the whole grey solving method
of the model was given and it was proved that the common solving method of the model was
only a special case of the grey solving methods. At the same time, the grey solving method was
compared widely with common solving method, classical numerical method. The study shows that
the grey solving method is better in depicting the procedure of transporting grey data of ground-
water system. On the basis of the theoretical study, two basic kinds of cases about groundwater
seepage were selected: the prediction of pit yield and the evaluation of groundwater resources on
a groundwater basin. In the cases, systematical analyses were made for generalization and grey-
lization of the hydrogeologic conditions, setting up of the grey model, identification and correction
of the model as well as its prediction and evaluation. It was pointed out that when the grey nu-
merical model is used to predict pit yield, the upper limit of the “grey band” of groundwater level
cannot be higher than planed safe groundwater level, when evaluating the groundwater resource,
the lower limit of the “grey band” of groundwater level cannot be lower than controlled level of
groundwater.
Keywords: grey set, seepage system, grey numerical model, grey solving method, generalization and greylization.
1
The rising of problem
In order to correctly predict and scientifically manage the underground water, oil and gas re-
sources, and to protect ecological environment, researchers have put a lot of human and material
resources to study the numerical models of seepage system and got plenty of gratifying achieve-
ments. But it cannot be denied that the complication of the actual seepage system and the limita-
tion of the system information grasp brought about a lot of difficulties in treating the uncertain
information such as inner construction of system, hydrogeological parameters, boundary condition
and spacial variation of the aquifer system for further development of the classical numerical
model. Such problems cannot make the classical numerical model of seepage system get the satis-

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factory modeling results in many conditions. Although the researchers tried to improve these nu-
merical methods unceasingly, they could not break off the thinking way of classical maths. Based
on grey mathematical theory, this paper will put forward a new method of grey numerical model-
ing method for groundwater seepage system analog, and according to grey mathematical operation
rules testify that the common numerical model is only a special case of grey numerical model. The
feasibility and validity of the grey numerical model and the rationality in resolving the actual
problem are illustrated by means of anylysis of actual cases. Compared with the general numerical
model, the grey numerical model can consider the input and output information as grey values and
calculate them according to the grey mathematical operation rules. Thus, the problems of uncer-
tainty and incompleteness about the information of the inner construction of system, parameter,
boundary conditions and so on can be resolved well, and the development and decision of the wa-
ter resources will be more scientific and reasonable.
The research in grey models for groundwater seepage system started in the late 1980s. At that
time, researchers noticed the grey information in groundwater seepage system, but they focused
on the research of grey concentration parameter model[1, 2]. Compared with it, the distribution pa-
rameter model is better to depict the characters of groundwater seepage field in time and space.
Therefore in 1990s, people began to research the modeling problem of seepage system of distrib-
uted parameter[3, 4], but the research work was far from enough. The research results[3,4] marks the
beginning of the grey distribution parameter model research on seepage system. The former[3] in-
quired into the fundamental character of model output data when the parameter and input data of
the grey numerical model were all the grey data and analysed the feasibility of application of the
grey distribution parameter model; the latter[4] considered only the parameters as grey data while
the other data in the input was certain. The two researches have not adopt the fundamental opera-
tion rules of grey maths to set up finite element equation of discretized node, but they used com-
mon (traditional) operation method in solving procedure of the grey numerical model. Thus it
cannot be avoided that the information is omitted and the application of model is limited. So the
paper, based on the grey set and grey mathematical theory, has proposed the grey numerical ana-
log theory and method of groundwater seepage system and developed a new field to evaluate the
groundwater resource scientifically and reasonably.
2
Concept of grey set and grey number operation properties
2.1
Concept of grey set
As pointed in ref. [5], “part known, part unknown” grey information, which is known only in
a rough range , can be described with defined grey set as follows[5,6]
:
Definition 1. That G is called a grey subset of U means that two mappings are given from
U to [0,1]:
μG : U →[0 ,1] , u → μG (u)∈[0 ,1],

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GREY NUMERICAL IMITATION AND MODELING OF GROUNDWATER
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μ : U →[0 ,1] , u → μ (u)∈[0 ,1],
G
G
where μG (u) ≥μ (u) , and called μG and μ upper and low membership grade function
G
G
versus G respectively; μG (u) and μ (u) are called upper and low membership grade respec-
G
tively, u versus G.
This definition tells us that the so called grey set is the set determined by a couple of
characteristic functions μG (u) and μ (u) whose domain of definition is U, range of values is
G
[0,1]. ∀u ∈U , its creditable degree is in the interval [ μ (u), μG (u) ] whose endpoints are μ (u)
G
G
and μG (u) .
2.2 Definition of grey number
Difinition 2. When grey set G’s domain U = R, G is called grey number and often ex-
pressed as ⊕G.
Definition 3. Suppose a ,b∈ R and a≤b,
1
0
when x∈[a , b]
when x∉[a , b]
⎧
⎨
⎩
(x)
μG
=
, μ （x) = 0 , x∈ R ,
G
so grey number G is called information type grey number, also called DengShi grey number or
interval type grey number, signed as [a, b].
In the definition, because μG (x) ≠ μ (x) , x∈[a , b], grey domain of grey number G is
G
[a, b]. This type of grey number is often met in engineering.
2.3 Operation rules and characters of grey number
(i) Addition operation law is
[a , b]+[c , d]Δ[a + c , b + d];
(ii) Subtraction operation law is
[a , b]−[c , d]Δ[a − d , b − c] ;
(iii) Multiplication operation law is
{
[a , b]×[c , d]Δ[min ac, ad,bc,bd
}
, max
{
ac, ad,bc,bd ];
}
(iv) Division operation law is that when 0∉[c , d] , stipulate
1 1
⎡
⎤
[a , b]÷[c , d]Δ[a , b]×
,
.
⎢
⎣
⎥
⎦
d
c
Definition 4. Suppose G1 ,G2 ∈ g(I) . If real number k exists and makes G1 = [k, k]G2,
then G1 is k times as much as G2. Interval type grey number’s operation is of the following prop-
erties:

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Suppose G1 , G2 , G3 ∈ g(I) , G0 = [0 , 0] , G1 = [1,1] , then
Character 1 G1 + G2 = G2 + G1 ;
Character 2 (G1 + G2 ) + G3 = G1 + (G2 + G3 ) ;
Character 3 G1 + G0 = G1 ;
Character 4 G1 ×G2 = G2 ×G1 ;
Character 5 (G1 ×G2 )×G3 = G1 ×(G2 ×G3 ) ;
Character 6 G1 ÷G1 = G1
Note 1. Multiplication distributive law versus addition is untenable. For example [−2 , 2]×
{[−1, 2] + [−2 ,1]} = [−6 , 6] , [−2 , 2]×[−1, 2]+[−2 , 2]×[−2 , 1] = [−8 , 3] . But when disserted
grey numbers are positive interval type grey numbers, we have
Definition 5. Suppose that G1 , G2 , G3 are positive interval type grey numbers, then
G1 ×(G2 + G3 ) = G1 ×G2 + G1 ×G3 .
Note 2. Deadening law is untenable. For example [−2 , 2]×[−1, 2] = [−2 , 2]×[−2 , 1] ,
but [−1, 2] ≠ [−2 ,1].
3
Grey numerical model and grey solving method of groundwater seepage system
3.1 Fundamental grey numerical model
Based on grey information analysis the differential equation of groundwater seepage system
can be expressed as
⎡
⎤
∂
∂x
∂(⊗H)
∂x
∂
∂y
∂(⊗H)
∂y
⎡
⎤
⊗ Tx
+
⊗ Ty
− [(⊗Qw )δ (x − xw , y − yw ) + (⊗ε)]
⎢
⎣
⎥
⎦
⎢
⎥
⎣
⎦
∂(⊗H)
∂t
(1)
− (⊗μ)
= 0.
By Galerkin’s finite element method, multiplying basic function with eq. (1) and integrating
result in the following equation:
⎡
⎢
⎣
⎤
⎥
⎦
⎡
∫∫⎣
⎤
⎡
⎤
∂
∂x
∂(⊗H)
∂x
∂
∂y
∂(⊗H)
∂y
⎡
⎤
⊗ Tx
+
⊗ Ty
•
(⊗ε)ϕi (x, y)dxdy
ϕ (x, y)dxdy + ∫∫
i
⎢
⎥
⎦
⎢
⎣
⎥
⎦
⎢
⎥
⎣
⎦
Di
Di
∂(⊗H)
∂t
⎡
⎤
− ∫∫[
∫∫⎣
(⊗Qw )δ (x − xw , y − yw )
]
•
ϕi (x, y)dxdy −
(⊗μ)
•
ϕi (x, y)dxdy
⎢
⎥
⎦
Di
Di
⎧
⎪
−
(⊗ql ) •ϕi (x, y)dl,
in the second typeof the boundary,
in other types of the boundary.
∫
Γi
=
(2)
⎨
⎪
⎩
0,
Eq. (2) is further discretized and all grey numbers in it are signified as interval grey numbers,
then the grey finite element equation belonging to any node i can be gained:

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GREY NUMERICAL IMITATION AND MODELING OF GROUNDWATER
1047
[Ta ,Tb ]
4Δβ
Δβ
3
{αi [Hia , Hib ] + α j [H ja , H jb ] + αk [Hka , Hkb ]} + [Qa ,Qb ] −
[εa,εb ]
∑
∑
β
β
1
⎧
⎨
⎩
ΔH
Δt
{liu [qua, , qub ] + liv [qva, , qvb ]},
+
β Δβ[ [μa , μb ]
=
2
∑
0
at nodes of the second type of the boundary,
at nodes of other types of the boundary,
(3)
i = 1, 2, …, m,
where αi ,α j ,αk are numbers about coordinates. Eq. (3) is called basic grey numerical model.
3.2 Solving method analysis of grey numerical model
The basic grey numerical model above contains many interval type grey numbers, so solution
to the basic grey numerical models should start from grey set theory and keep to grey number op-
eration properties in order to result in real “grey results”. In previous study of the problem, the
solution method of “white” equation is used for solving the basic grey numerical model (called
common solving method in this paper). The authors think that the method lacks necessary theo-
retic basis. So the grey solving method of grey numerical model is studied deeply and compared
with the common solving method.
3.2.1 Grey solving method of grey numerical model. Eq.(3) is simply signed as
I1 + I2 − I3 + I4 = I5 .
The above equation can be solved term by term with the use of grey number operation rules.
For solutiom to term I1, we should notice the symbols αi ,α j ,αk especially . Because αi is
always greater than zero but α j ,αk are double possible positive number or negative, we have to
discuss them in different cases:
[Ta ,Tb ]
I1 = I1−1
I1 = I1−2
I1 = I1−3
I1 = I1−4
=
=
=
=
[αi Hia + α j H ja + αk Hka ,αiHib + α j H jb +αk Hkb ] ,
∑
β
4Δβ
when α j ,αk > 0 ;
(4)
(5)
(6)
[Ta ,Tb ]
4Δβ
[αi Hia + α j H jb +αk Hkb ,αiHib + α j H ja + αk Hka ] ,
∑
∑
β
β
when α j ,αk < 0 ;
[Ta ,Tb ]
4Δβ
[αiHia +α j H ja +αk Hkb ,αiHib + α j H jb +αk Hka ],
when α j > 0,αk < 0;
[Ta ,Tb ]
4Δβ
[αiHia +α j H jb +αk Hka ,αiHib +α j H ja +αk Hkb ] ,
∑
β

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Vol. 44
when α j < 0,αk > 0.
(7)
Other terms are
I2 =[Qa ,Qb ],
(8)
(9)
Δβ
I3 =
[εa ,εb ],
∑
β
3
Δβ
t+1
t+1
I4 =
[μa , μb ]{[H , H ] − [Hita , Hitb ]}
ia ib
∑
∑
β
3Δt
Δβ
β 3Δt
=
[μa (Hita+1 − Hitb ),μb (Hitb+1 − Hita )],
(10)
(11)
1 [liu qua + liv qva ,liu qub + liv qvb ]
⎧
⎨
⎩
2
I =
.
5
0
Putting eqs. (4)—(11) in eq. (3), we can gain grey finite element equations of node i in four cases.
The first case: When α j ,αk > 0 , the grey finite element equation of a node is as follows:
⎡
⎢
Δβ εb Δβ μa
+
Ta
t+1
t+1
t+1
t+1
(αi H + α j H + αk H ) + Qa −
(H − Hitb ) ,
ia
ja
ka
ia
∑
β
4Δβ
3
3Δt
⎢
⎣
⎤
Δβ εa Δβ μb
+
Tb
t+1
t+1
t+1
(αi H + α j H + αk H ) + Qb −
(H t+1 − H t )
⎥
ib
jb
kb
ib
ia
4Δβ
3
3Δt
⎥
⎦
12 [liu qua + liv qva ,liu qub + liv qvb ], at nodes of the second typeof the boundary,
⎧
⎨
⎩
=
(12)
0,
at nodes of other types of the boundary,
i = 1, 2, …, m.
In turn we can give grey finite element equations of a node in other three cases.
To practical seepage numerical problem, we can certify that so long as the three angles in
each triangular element are neither cuspidal nor blunt, among αi ,α j ,αk , concerning coordinates
in each triangular element around any node, αi is greater than zero, α j and αk are less than
zero. So eq. (12) can be further changed into the following matrix form:
a H
a12H2a a13H3a ... a1mHma
f
1a
⎡
⎢
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎡
⎢
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎥
11 1b
a21H1a a22H2b a23H3a ... a2mHma
a31H1a a32H2a a33H3b ... a3mHma
f2a
f3a
...
=
...
...
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
am1H1a am2H2a am3H3a ... ammHmb
fma
……………………………………………………………
(13)

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GREY NUMERICAL IMITATION AND MODELING OF GROUNDWATER
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a H
a12 H2b a13 H3b ...
a
a
a
1m Hmb
2m Hmb
3m Hmb
f
1b
⎡
⎢
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎡
⎢
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎥
11 1a
a21H1b a22 H2a a23 H3b ...
a31H3b a32 H2b a33 H3a ...
f2b
f3b
...
=
...
...
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
am1H1b am2 H 2b am3 H3b ... amm H ma
fmb
The above equation is of special structure and can be solved by special method. The steps are:
first to solve Hib from the former m equations (1—m), then to substitute them for the latter m
equations to solve and get the lower limit value Hia of grey water level, and to calculate Hib from
former m equations by back replacement[7].
3.2.2 Common solving method of grey numerical model. This method is to make all lower limit
values of grey parameters and grey variables correspond to those of the grey water level function
to form white linear equations to get lower limit value of the grey water level (Hia,i = 1, 2, …m);
and make all upper limit values correspond to each other to form another white linear equations to
get upper limit value of grey water level. In short, it is a method of “lower to lower, upper to
upper”, namely to get upper limit and lower limit values of grey water level by solving two inde-
pendent linear equations.
3.3 Comparison and analysis of calculated results
Suppose an artesian aquifer which, in accordance to Theis’s conditions, can be divided into
30 triangle units with 23 nodes. In comparison of numerical results, all calculated results by three
numerical method, grey solving, common solving and classical numerical method, are taken into
account at the same time. In terms of input information, the relation between transmissibility coef-
ficient T, [Ta, Tb] values and output values of water level H, [Ha, Hb] is considered because
transmissibility coefficient is more difficult to treat in grey solving method and also is the key to
set up grey numerical model. In the following, nodes 4 and 8 are selected to compare their process
curves of grey water level (figs. 1, 2). Node 4 is of less water level drop during modeling time, but
node 8 (on a well) is of a greater drop during the time. In concrete calculation, T =1000m2/d was
used for classical numerical model, and [Ta,Tb]= [800,1200] m2/d was used for grey numerical
model.
The comparison of process curves, the “grey band”, of two solving methods of grey numeri-
cal model shows that the difference in grey degree belween grey solving method and common
solving method is obvious at the node with a greater water level drop during the modeling time,
but less obvious at the node of less water level drop during the time. It is possible that the former
should include the latter ones. At other nodes of less water level drops, the difference is minor.
The reason is that the greater the water level drops, the more uncertain the system becomes.
Therefore it can be believed that the grey solving method is a more complete and correct method
for transporting the grey information.

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Fig. 1. Comparing curves of different solving methods on node 4.
Fig. 2. Comparing curves of different solving methods on node 8.
4
Analysis of two actual cases
4.1 Application of grey numerical model in prediction of pit yield
4.1.1 General situation of the mine. We took the Gaotai mine in Hebei Province as an example.
To resolve the exploitation problem of the Xiasanzu coal, a lot of drilling and drainage experi-
ments were done. Based on the analysis of grey information in the pit yield system, we set up the
grey numerical model for modeling the drainage process and predicting the drainage capacity
from Daqing limestone aquifer in the mine production.
4.1.2 Establishment and identification of the grey numerical model. According to the
hydrgeological conditions of the pit, the grey numerical model of the yield system can be ex-
pressed as

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GREY NUMERICAL IMITATION AND MODELING OF GROUNDWATER
1051
∂2 (⊗H) ∂2 (⊗H)
(⊗u* ) ∂(⊗H)
(⊗KM ) ∂ t
+
+ (⊗Q)δ (x − x0 , y − y0 ) =
(x,y∈Ω, t > 0),
∂X 2
∂Y 2
H(x ,y, 0) = H0(x, y) (x,y∈Ω),
∂H
|r = 0 (t >0),
∂n
⊗H| r1 = [120,123] (t > 0).
The test and identification of grey numerical model is different from that of classical nu-
merical model. The identification of the classical numerical model is to fit the output curve with
the actual curve, but for grey numerical model it is to fit the “grey band” out put by the model
with the actual curve. In other words, the former is to coincide the smooth curve output by the
model with the actual variegated curve, and the latter is to include the actual curve in the “grey
band” output by the model. According to the rule of information compound, it is regarded by grey
system theory that the information compound promoting increase of neg-entropy in the system is
the upper limit of the system development, while that promoting decrease of neg-entropy is the
lower limit of the system development. Because they form a grey closed interval, all the grey fac-
tors are definitely included in it. On this point, the identification of the grey numerical model is to
adjust the grey variables in the model, including grey parameters, grey boundary value and the
grey drainage capacity, to make grey band include actual data measured[8].
According to the above rules, the research area is divided into 219 units with 140 nodes (fig.
3), among which 14 nodes are in the observed holes. Figs. 4 and 5 show the fitting results of water
level in holes 5 and 12.
Fig. 3. Discretization in modeling area.
From the figures, we can see that the curves of grey water level from grey numerical model,
“grey band”, contain the variegated curve of the actual water level. The “grey band” results from
the information “known in part and unknown in part” of the seepage system. The determined grey
ranges of model parameters are listed in table 1.

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Fig. 4. Fitting curves of the water level in hole 5.
Fig. 5. Fitting curves of the water level in hole 12.
Table 1 Grey parameter values of the grey model
No.
1
35
2
41
3
46
4
52
5
62
6
7
98
8
K
114
145
b
K/m·d−1
Ka
μb
μa
30
37
42
44
56
102
90
133
μ∗
0.0001
0.00005
0.0001
0.00005
0.0001
0.00005
0.0001
0.00005
0.0001
0.00005
0.0001
0.00005
0.0001
0.00005
0.0001
0.00005
4.1.3 Prediction of pit yield. Prediction of pit yield using grey numerical model is to adjust the
water level in each drainage hole until the level is lower than the designed safe water level of the
pit and the determined drainage capacity is the wanted pit yield. It should be noticed that in terms
of the grey water level, upper limit of the “grey band” must be lower than the designed safe water
level. In that case, the whole “grey band” is below the safe water level.
Fig. 6 shows the contour lines of the groundwater flow field after 35 days of drainage when
the total discharge reaches 17566 m3/d. In calculation, except parameter value which was selected
within the grey range, the border water level was also given within some interval according to the
actual data of many years:
[Ha, Hb]| Γ1 = [120, 123]m (t >0).
Fig. 6. Contour lines of upper limit in 35 days of pumping.

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4.2 Application of grey numerical model to evaluation of water resource
4.2.1 General conditions of the studied area. The Kangtao groundwater basin in Hebei Prov-
ince was selected in the study. The aquifer consists of the Quaternary sand-gravel and is of inho-
mogeneous permeability. At the same time, the hydrogeological information of the studied area is
very limited. So the grey numerical analog method is used for evaluating the permitted develop-
ment quantity of groundwater resource.
4.2.2 Establishment and identification of grey numerical model. From the hydrogeological
conditions of the area, the grey mathematical model can be expressed as
⎡
⎤
∂
∂x
∂(⊗H)
∂x
∂
∂y
∂(⊗H)
∂y
⎡
⎤
(⊗Kx h)
+
(⊗K y h)
− [(⊗Qw )δ (x − xw , y − yw )
⎢
⎣
⎥
⎦
⎢
⎥
⎣
⎦
∂(⊗H)
∂t
+ (⊗ε)] − (⊗μ)
= 0
(x, y ∈Ω, t > 0 ),
H(x, y, 0) =H0 (x, y) (x, y∈Ω),
∂(⊗H)
= [qa , qb ] (t > 0).
∂n
Γ
2
The area is divided into 479 triangle units forming 300 nodes (fig. 7). According to the properties
of the aquifer, the area is divided into 9 parameter areas.
Fig. 7. Discretization in studied area.
The fitting process of the model is to adjust the input data of system such as parameters until
a reasonable result comes out. For the grey numerical model, “the better imitation” means that the
grey band output by grey numerical model can contain the observed data. The analyses of
groundwater levels in hole 5 and hole 13 are shown in figs. 8 and 9, and the determined hydro-
geologic parameters are listed in table 2.

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Fig. 8. Comparing curves in hole 5.
Fig. 9. Comparing curves in hole 13.
Table 2 Determined parameters of the grey model
No.
1
2
3
4
5
6
7
8
9
Ka
Kb
μa
μb
50
114
141
0.10
0.12
202
222
0.33
0.37
35
90
56
12
23
22
K/m·d−1
62
44
113
0.30
0.34
66
17
28
28
0.26
0.30
0.10
0.12
0.10
0.12
0.23
0.27
0.24
0.28
0.23
0.27
μ
4.2.3 Grey numerical prediction and evaluation of permitted development quantity in the basin.
Since the movement of groundwater is closely related to the atmospheric rainfall in the basin, we
adopted a typical year of low rainfall to predict the permitted development quantity. Such data as
boundary conditions, supply quantity of the atmospheric rainfall are given grey interval value in
the predicting model because it is difficult to obtain them exactly and completely.
Evaluating the permitted development quantity with the grey numerical model is to repeat-
edly adjust the pumping quantity in each hole until the lower limit of the grey band of water level
in each hole is not lower than the controlled water level——“permitted dropping depth” when the
permitted development quantity of groundwater resource in the basin is determined. As long as the
lower limit of the grey band of water level is not lower than the controlled level of groundwater,
the water levels in the holes are all above the controlled level of groundwater definitely. It is just
contrary to the problem of mining drainage.
The predicting curve of exploiting proce-
dure of groundwater——“grey band” (fig. 10)
shows that the first part of curve is the forming
phase of the trough of the lower water level, the
next is the approximate steady phase of the
trough, the last one is the rising phase of water
level. The whole curve illustrates an adjustable
and balanced dynamic character in the exploiting
Fig. 10. Predicted curves in exploiting.
procedure.
The predicted contour lines in low water period (fig. 11) shows the greatest water level drop
of 2.21 m in the central part of the basin, which has little effect on other water supply wells. The

No. 11
GREY NUMERICAL IMITATION AND MODELING OF GROUNDWATER
1055
hydraulic gradient of groundwater is J= 3/1000, which can meet the designation. Under this hy-
draulic gradient, pollution of the river on the aquifer can be controlled and will not surpass the
environmental capacity.
Fig. 11. Predicted contour lines in exploiting.
5
Conclusions
(1) The grey, in fact, is “ part known, part unknown” not only for the system but also for the
number.
(2) Solving the fundamental grey numerical model of the seepage system must be based on
grey set and gery mathematical theory, so as to make sure that grey information transmission is
correct and the “grey solution” is reliable.
(3) Compared with the classical model, the grey numerical model can accept the grey interval
numbers and output a “grey band” resulting from the effect of many grey information of the seep-
age system with a certain range by the model operation. It is difficult to determine such values as
aquifer parameters, boundary conditional value, supply and drainage quantity.
(4) Using the grey numerical model to predict the pit yield is to adjust the drainage quantity
in the holes to make the upper limit of “grey band” of groundwater lever be not higher than the
planed safe groundwater level. As long as the upper limit of the “grey band” is not higher than
the planed level, the “grey band” of groundwater lever will be all below the planed safe ground-
water level.
(5) Using the grey numerical model to predict the submitted groundwater resource in a
groundwater basin is to adjust the pumping quantity in the holes to make the lower limit of the
“grey band” of groundwater level be not lower than the controlled groundwater level, namely
“permitted depth”. As long as the lower limit of the “grey band” is not lower than the controlled

1056
SCIENCE IN CHINA (Series D)
Vol. 44
groundwater level, the “grey band” will be all above the controlled groundwater level.
Acknowledgements This study was supported by the Ministry of National Lands and Resources (Grant No.
200010301-66).
References
1. Jiao Jiujiu, Grey hydrogeologic system analysis and time series model, Survey Science and Technology (in Chinese), 1987,
(10): 39—43.
2. Li Shuwen, Wang Baolai, Xiao Guoqiang, A compound model of grey and periodic scrape and its application in ground-
water prediction, Journal of Hebei Institute of Architectural Science & Technology (in Chinese), 1992, (3): 246—251.
3. Wang Qingyin, Li Shuwen, Grey distributed parameter model and groundwater analog, Journal of Hebei Institute of Ar-
chitectural Science & Technology (in Chinese), 1992, (3): 66—70.
4. Guo Chunqing, Xia Riyuan, Liu Zhenglin, Gray Systematic Theory and Methodological Study of Krast Groundwater Re-
sources Evaluation (in Chinese), Beijing: Geological Publishing House, 1993, 3—60.
5. Wang Qingyin, Liu Kaidi, The Mathematical Method of Grey Systematic Theory and Its Application (in Chinese),
Chengdu: Publishing House of Southwestern China University of Communication, 1990, 23—27.
6. Wang Qingyin, Wu Heqing, The concept of grey number and its property, in Proceedings of NAFIPS'98, USA, 1998,
45—49.
7. Givoli, D., Doukhovni, I., Finite element programming approach for contact problems with geometrical nonlinearity,
Computers and Structures, 1996, (8): 31—41.
8. Li Shuwen, Wang Zhiqiang, Wu Qiang, The superiority of storage-centered finite element method in solving seepage
problem, Coal Geology and Exploration (in Chinese), 1999, (5): 46—49.