Application of grey numerical model to groundwater resource evaluation

Abstract

Based on interval grey number theory, a grey numerical model was developed to simulate groundwater flow in two-dimensional porous-medium aquifer. The solution method proposed in this paper ensures the integrity and validity of the grey data transportation in the calculation processes. Because most of the parameters characterizing groundwater flow are of grey nature, a grey model can better describe the groundwater flow than the traditional “white” model. The method was applied to evaluate the sustainable groundwater resource in a groundwater basin of north China. The results from the case study help develop the most appropriate approaches to utilize the water resources in the area.

Appendices

Appendix

Interval grey number and its operations

Definition of interval grey numbers (Deng 1987)

Assume a, b∈R, and a≤b

$$ \overline \mu _G (x) = \left\{ {\begin{array}{*{20}c} 1 & {{\text{when }}x \in [a,b]} \\ 0 & {{\text{when }}x \notin [a,b]} \\ \end{array} } \right.\quad \underline \mu _G (x) = 0,x \in R $$

where a and b are any real numbers; R represents a group of real numbers; $\in$ and $\notin$ stand for “belong to” and “not belong to”, respectively; x is a variable; $\overline \mu_G \{x)$ and $\underline \mu_G \{x)$ are upper and lower functions of a grey number G, respectively. Grey number G is called information type grey number, also called Deng grey number (Deng 1987; Wang and Wu 1998) or interval type grey number. It can be written as G =[a, b]. When a≥0, [a, b] and [b, a] are called positive interval type grey number.

Operation rules of interval grey numbers (Wang and Wu 1998)

Addition:

$$\left[{a,b} \right] + \left[{c,d} \right]\underline{\underline \Delta} \left[{a + c,b + d} \right]$$

(5)

where “ $\underline{\underline \Delta}$” means “being defined as,” a, b, c,and d are real numbers, [a, b] are interval grey numbers.

Subtraction:

$$\left[{a,b} \right] - \left[{c,d} \right]\underline{\underline \Delta} \left[{a - d,b - c} \right]$$

(6)

Multiplication:

$$\left[{a,b} \right] \times \left[{c,d} \right]\underline{\underline \Delta} \left[{\min \left\{{ac,ad,bc,bd} \right\},\max \left\{{ac,ad,bc,bd} \right\}} \right]$$

(7)

Division when $0 \notin \left[ {c,d} \right]:$

$$ \left[ {a,b} \right] \div \left[ {c,d} \right]\underline{\underline \Delta } \left[ {a,b} \right] \times \left[ {\frac{1} {d},\frac{1} {c}} \right] $$

(8)

Operation properties of interval grey numbers

Assume G1, G2∈g(I), if a real number k exists and G1=[k, k]G2,then G1 is k times as much as G2. Interval type grey number’s operation has the following properties if G1, G2, G3∈g(I), in which g(I) is grey number group and G⁰ =[0,0], G¹ =[1,1] (Wang and Liu 1990):

• 1. G1 + G2 = G2 + G1

• 2. (G1 + G2) + G3 = G1 + (G2 + G3)

• 3. G1 + G⁰ = G1

• 4. G1 × G2 = G2 × G1

• 5. (G1 × G2) × G3 = G1 × (G2 × G3)

• 6. ${\text{G}}1 \div {\text{G}}^1 = {\text{G}}1$