The response matrix method firstly uses the groundwater system simulation model to determine the response relationship between the system output and the system input-unit impulse response function, and forms the set of its function values-response matrix. This input-output function response relationship reflects the inherent physical laws of the groundwater system itself, mainly the principle of water balance and energy conversion conservation, which is embodied in the quantitative relationship between the state variables and controllable input variables of the system [11]. This quantitative relationship is applied to the optimization model, and the controllable input variables are used to represent the state variables that need to be controlled, and then other factors are comprehensively considered to complete the construction of the management model.

The groundwater numerical simulation model composed of partial differential equations and their definite solution conditions generally belongs to nonlinear structure, but the two sub-models can be separated. First, there is no controllable input, but the natural water level and the corresponding natural flow field formed by initial conditions, boundary conditions and uncontrollable input belong to a nonlinear system. The second is the water level drop and the corresponding artificial flow field caused only by controllable input. In this sub-model, the initial conditions and boundary conditions are zero, which belongs to a linear system. Response matrix method can only use the submodel of linear system to construct management model.

In recent years, response matrix method has been widely used in the study of groundwater management model, but there is little research on the value of unit pulse (unit pumping flow) in response matrix method [1 12]. Many researchers directly establish the management model in units of 100000m3/d and1000m3/d.. Theoretically, the number of unit pulses is arbitrary, but for the practical field of groundwater management, the unit pulse function describes the input-output conversion process in the groundwater system, excluding the influence of external input on the system. It is only related to the structure, scale and water-bearing medium properties of the system, and is actually a means to artificially calibrate the internal properties of the system [1 13]. For groundwater management models with different scales and scales, the response of the same unit pulse to groundwater system and its influence on the boundary are very different. The value of unit pulse is generally based on the fact that all water level control points in the whole groundwater system have obvious response values and will not have great influence on the system boundary, and the specific hydrogeological conditions and management objectives have an optimal value [1 14]. In order to determine this value, a typical time period with known initial and boundary conditions and groundwater recharge and discharge conditions should be selected to simulate the unit pulse value.

3.3.2. 1. 1 calculation method of unit impulse response function

There are two ways to solve the unit impulse response function β(i, j, n-k+ 1): ① Pump water only at the j-th node (j= 1, 2, …, m) and calculate the i-th node (i= 1, 2, …, m). (2) Pump water continuously at the j-th node at the unit flow rate of each cycle, and calculate the difference between the cumulative water level drop at the end of the k-th cycle and the end of the k- 1 th cycle of the i-th node, which is the unit impulse response function [11]. This book uses the first method to solve the unit impulse response function.

3.3.2. 1.2 Discussion on the Suitable Value of Unit Pulse

Taking the optimal allocation of unstable pumping capacity as an example, this paper discusses the influence of unit pulse value on the optimal result when the response matrix method is applied. There is a rectangular phreatic aquifer (3L×L, L= 1000m), the river on the west side is a constant head boundary, with an annual average water level of H0=40m, and other boundaries are water-resisting boundaries (Figure 3. 1). The annual average rainfall infiltration intensity N=0.0004m/d, the aquifer permeability coefficient t can be regarded as a constant, T= 1000m2/d, and the water supply μ=0.3. The aquifer is divided into three square units, all of which can be pumped. At present, the aquifer has not been developed. The planning requirements are: how to rationally exploit groundwater and maximize the sum of the water levels of the three units at the end of the second year under the conditions that the water supply in the first year and the second year meet the requirements of P 1=5000m3/d and P2=7500m3/d respectively, and at the same time meet the water level of the unit at the end of the second year 1 = 39m.

Figure 3. 1 Three-unit rectangular phreatic aquifer

Taking one year as a time period, * * * is divided into two time periods, the controllable input variable is the pumping flow of each unit in each time period, and the state variable is the intermediate water level of each unit, where I = 1, 2,3 is the unit serial number, and k = 1, 2 is the time period serial number.

According to the known conditions, the stable water level in the natural state when the aquifer is undeveloped is =40.6m, =4 1.4m, = 41.8 m. According to the meaning of the problem, the simulation model of unsteady flow after pumping is composed of the water balance equations of the first and second units. By substituting the known data into these water balance equations, a set of equality constraints can be obtained.

Calculation results of (1) embedding method to construct management model

The objective function is:

Study on groundwater dynamic planning management model with covariates

Constraints include:

Study on groundwater dynamic planning management model with covariates

Lingo software is a software package specially used to solve mathematical programming problems, mainly used to solve linear programming, nonlinear programming, quadratic programming and integer programming, in which the optimal distribution of pumping capacity of phreatic aquifer can be determined. See table 3. 1 for the calculation results.

Table 3. Calculation Results of1Embedding Method

(2) Calculation results of response matrix method under different unit pulses: According to the idea of response matrix method and unsteady flow simulation model, first, given a unit pumping flow, the unit pulse response functions of three units in two periods are obtained, then the state variables are expressed by controllable input variables, and then the management model is constructed by combining other constraints.

The objective function is:

Study on groundwater dynamic planning management model with covariates

Constraints include:

Study on groundwater dynamic planning management model with covariates

The calculation results of this management mode under different unit pulses are shown in Table 3.2- Table 3.4.

Table 3.2 Calculation Results of Unit Pulse Q= 100000 m3/d Management Mode

Table 3.3 Calculation Results of Unit Pulse Q= 1000 m3/d Management Mode

Table 3.4 Calculation Results of Unit Pulse Q= 100 m3/d Optimal Management Model

In this case, although the difference of management target values obtained under different unit pulses is not too big, and the total pumping capacity of each unit meets the planning requirements, these differences will be obvious in large-scale multi-objective management problems, and there is also a pumping cost problem in practical problems. Sometimes the pumping cost of different units is different, so the allocation of pumping capacity is very important.

Embedding method is suitable for groundwater management problems of steady flow and unsteady flow, and it is simple and convenient to deal with small-scale problems without destroying the original groundwater simulation model structure, thus making the calculation results more accurate. Response matrix method is suitable for large-scale and multi-stage unstable groundwater management, and will show its superiority in dealing with complex hydrogeological conditions and large-scale problems. It has strict modeling premise and harsh application conditions, and when the boundary position is close to the manually controllable input point, it will produce large errors. Considering the small scale of this example, this example uses embedding method to calculate, and the result is more accurate. By comparing with the optimization results of embedding method, it is more accurate to choose 100000 m3/d as the unit pulse value when the management model is constructed by response matrix method, which is the appropriate unit pulse value under this specific hydrogeological condition.

It can be seen from this example that the value of unit pulse needs to be tested and compared repeatedly. To determine the unit pulse value, we should consider the scale of the model (the size of the simulation area), the order of magnitude of mining, the basic hydrogeological conditions and generalization of the model, the accuracy control standard of the optimization software and other factors. According to the specific hydrogeological conditions and management objectives, it is necessary to simulate the unit pulse value to find the most suitable value, which is a necessary prerequisite for doing a good job in groundwater management.

Construction method of 3.3.2.2 covariant management model

When the response matrix method is used to construct the groundwater management model with covariates, it is necessary to consider the relationship between artificial exploitation (recharge), covariates and groundwater level, and ensure that the covariates fed back to each other remain unchanged in the calculation process. Therefore, it is necessary to add additional constraints to the optimization model, and combine other constraints and objective functions to form a groundwater management model with covariates. By solving this groundwater management model, we can get the optimal amount of artificial exploitation (recharge) and the change of covariate, then get the optimal groundwater level through the response matrix, and then get the covariate under the influence of artificial exploitation (recharge) through further calculation, so as to get the optimal artificial exploitation (recharge), groundwater level and covariate. Please refer to the reference [100] for the specific calculation process, which is not repeated here.