For the surface rivers in Heihe river basin, it can be assumed to be one-dimensional unsteady gradual flow. When the river bed slope is very small (i.e. i=sinθ≈tgθ), there are water (or water diversion) on both banks, and there is a weak permeable layer at the bottom of the river bed, so there are conditions for mutual transformation between river water and groundwater, so the river water movement model can be described by two basic equations, namely, the continuous equation and the motion equation:

Water Cycle and Evolution Model of Groundwater in Heihe River Basin

Where: B—— width of river section (m);

Z—— River water level (m);

Q—— River cross-section flow (m2/d);

X—— River section spacing (m);

QL—— Inflow (positive) or outflow (negative) per unit river length along both banks, in m2/d;

K-flow coefficient;

C- Xie Cai coefficient;

A—— River cross-sectional area (m2);

R—— hydraulic radius (m);

E—— the vertical exchange capacity (m3/d) between river water and groundwater per unit length in the direction of river flow, which is positive when the river discharges groundwater and negative when the river supplies groundwater;

Gravity acceleration.

There are three vertical exchange modes between river and aquifer system: jacking drainage, pressure leakage and leaching leakage.

The groundwater exploitation along the river is maintained at a low level, the difference between the river water level and the groundwater level is small, and the water exchange between the surface water body and the groundwater body is continuous; When the groundwater level is higher than the river level, the groundwater is discharged into the river by jacking; When the groundwater level is lower than that of the river, and there is little difference between the groundwater level and the river level (that is, it does not "cross the line"), the river replenishes the groundwater by pressure leakage; When the river and groundwater are not in a straight line, the leakage water flows vertically downwards, which belongs to one-dimensional flow. Under the action of gravity, the hydraulic gradient of natural vertical infiltration has reached the maximum, and the river replenishes groundwater through leaching, and the leakage reaches the limit leakage amount Em, which has nothing to do with the water level difference. The leakage recharge intensity (leakage intensity) of this reach should not be greater than the overflow intensity qu above this reach, nor should it exceed the ultimate leakage intensity Em. Before "breaking off diplomatic relations", it is approximately considered that the vertical exchange capacity e between river water and groundwater has a linear relationship with the difference between river water level and groundwater level (which can be regarded as overflow), that is, it obeys Darcy's law (Liu,1998; Jiang, 1994, 1999), the mathematical expression of the water exchange model at this time is:

Water Cycle and Evolution Model of Groundwater in Heihe River Basin

Where: CPS is the recharge coefficient of the restricted layer at the bottom of the surface river bed, CPS = ks/bs;

Ks—— the permeability coefficient of the pressure bearing layer (fine silt with weak water permeability) at the bottom of the river bed;

Bs—— the thickness of the river bottom restraint layer (m);

B—— average width of river section (m);

H1-water level of diving system (m);

Z- water level of surface river (m).

Second, the mathematical model of groundwater flow in multi-layer aquifer system

For multi-layer groundwater system, discrete kernel method can be used to simulate. Assuming that each aquifer is regarded as a plane two-dimensional flow, the seepage relationship between aquifers is established by Nenuman and Witherspoon's seepage theory. Then Maddock( 1974) and others are used to study the multi-layer aquifer system, that is, an M-layer groundwater aquifer system can be composed of a diving system and several lower confined aquifers, and the groundwater in each aquifer mainly flows horizontally, and the water exchange between aquifers is carried out through vertical overflow, thus generalizing the multi-aquifer system into a heterogeneous, isotropic and quasi-three-dimensional groundwater flow system.

Water Cycle and Evolution Model of Groundwater in Heihe River Basin

After simplification, it can be described by the above problem of definite solution of differential equation (Xue Yuqun et al.,1980; Zhu et al.,1990; Xu Guangquan, 2000):

Water Cycle and Evolution Model of Groundwater in Heihe River Basin

Where: (x, y)- space coordinate (m);

QM——m artificial mining intensity of m aquifer (m3/d);

Mining intensity. Well in i-M aquifer (m3/d);

MP-number of production wells;

Yi —— the coordinate of the first well (m);

δ(x-xi, y-yi)- the value of two-dimensional Dirac δ function at (xi, yi);

Hm0—— initial water level elevation of each aquifer (m);

Tm—— the uneven permeability coefficient (m2/d) of the mth aquifer, which is t1= k (h1-z) in the phreatic layer, and z is the bottom elevation of the phreatic system;

CPAM-overflow coefficient of the m-th weak permeable layer (d-1);

ε—— vertical recharge intensity except surface river recharge, overflow recharge and artificial exploitation (m3/d);

Hm—— water level elevation of the mth aquifer (m);

Sm—— the water storage coefficient of the mth aquifer, which is the specific yield of the aquifer under pressure and the elastic water storage coefficient under pressure;

γ 2 —— the second boundary of each aquifer;

γ1-a boundary of aquifers;

Q(x, y, t) —— the discharge per unit width of the second boundary of m aquifer (m2/d);

Hm 1—— boundary water level of the mth aquifer (m);

—— the outer normal direction of each boundary;

E—— the vertical exchange capacity (m/d) of river water and groundwater per unit length in the direction of river flow, in which the groundwater discharged by the river is negative and the groundwater replenished by the river is positive;

Ω —— Calculated area of aquifer.

The vertical exchange capacity e between river water and groundwater system can be treated according to the model method of surface river water movement. Equation (9-3) describes the groundwater movement, initial conditions and boundary conditions of each aquifer in the multi-layer aquifer system of diving system and confined water system respectively. The simultaneous equations (9- 1) and (9-3) can obtain the coupling simulation model of river and groundwater flow in areas with surface rivers, water exchange between groundwater and surface rivers, and multi-layer aquifer system of groundwater aquifer.