I. Overview of groundwater management mode
(A) Mathematical expression of groundwater management model
The mathematical expression of groundwater management model generally consists of objective function and constraint conditions:
objective function
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Constraint condition
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Equation (10- 1) is an objective function, and the management objective is expressed by a mathematical expression, which can take the maximum value or the minimum value. When k= 1, it is a single-objective management problem; When k > 1, it is a multi-objective management problem.
XJ (j = 1, 2, ..., n) is the decision variable in the above two formulas.
Equations (10- 1) and (10-2) indicate that the value of each decision variable xj of the objective function is obtained and the objective function is maximized (or minimized) under the condition of satisfying the constraints in equation (10-2).
(B) the composition of groundwater management model
1. decision variable
By manipulating controllable variables, the groundwater system is deployed, and finally the system goal is optimized. This controllable variable is called decision variable. For groundwater management, the possible decision variables are:
(1) Spatio-temporal distribution of pumping capacity;
(2) Distribution of artificial replenishment in space and time;
(3) the water level of the surface water body related to the aquifer;
(4) the quality of the mined groundwater;
(5) Water quality of artificial groundwater recharge source;
(6) The drainage capacity and replenishment capacity, location and service time of newly added wells and artificial replenishment facilities.
In practical work, decision variables can be selected according to specific management objectives and local hydrogeological conditions.
2. The objective function
In the management of groundwater resources, every problem has a clear goal. This goal is expressed by the function of the decision variable, which is called the objective function. Different goals have different objective functions. The solution to the problem can be to maximize or minimize the objective function. The most commonly used objective functions are as follows:
(1) Maximize the total net profit of water supply system;
(2) Minimize the cost of water supply per unit water volume (or investment in developing water resources);
(3) Maximize the total industrial and agricultural output value of some departments closely related to the utilization of water resources;
(4) In order to control the continuous decline of the groundwater level, the minimum value of the objective function of the total water level decline of each node in the management area is calculated, or the maximum value of the objective function of the groundwater level rise in this area when artificial recharge is adopted;
(5) The fitting degree between the actual water level and the specified water level is usually expressed by the sum of the squares of the water level difference or the sum of the absolute values of the water level difference, which is used as the objective function to find the minimum value;
(6) Under the condition of specified depth reduction, the maximum water quantity is obtained;
(7) Find the maximum value of the objective function of reclaimed water discharge after sewage treatment and formation self-purification, or the minimum value of the objective function of aquifer groundwater pollution, and so on.
Step 3 limit
In groundwater management, solving every problem is constrained by certain conditions. It is usually expressed by mathematical expressions of decision variables, which are called constraints. Usually, there is more than one constraint. In linear programming model, a group of decision variables are usually expressed by linear equality or inequality. Common limiting factors are:
(1) equilibrium constraint: the groundwater flow state equation or the combined groundwater solute transport equation is often used as the equality constraint condition of water equilibrium constraint. Its purpose is to ensure that the groundwater management model must obey the objective law of groundwater movement in the optimization process.
(2) Water quantity constraint: if the sum of water consumption in the management area is not greater than the local total water supply index (including water imported from other areas); The sum of groundwater recharge should not be greater than the total recharge available and the recharge capacity of recharge facilities; Pumping capacity shall not exceed the water output of pumping equipment; The exploitation amount in the management area should ensure that it can meet the local industrial, agricultural and domestic water demand.
(3) Water quality constraints: including the constraints that some chemical components and physical properties of the extracted groundwater should not exceed the corresponding water quality standards for different water supply purposes (artificial recharge) and the water quality constraints of sewage discharge.
(4) Water level constraint: it includes the restriction on the fluctuation of groundwater level, so as to prevent, control and improve various environmental geological problems in the management area, such as continuous decline of water level, drainage of aquifer, decrease or interruption of spring water, land subsidence, seawater intrusion, deterioration of water quality, soil salinization and salinization.
(5) Economic constraints: For example, the distribution of water resources in the region must ensure that the total industrial and agricultural output value meets the planning requirements, and the investment in developing water resources does not exceed the budget standard.
According to different specific circumstances, the relevant items in the above items can be used as constraints. In addition, there may be social and economic constraints, which are not listed here.
(3) Classification of groundwater management modes
At present, the commonly used groundwater management models are as follows.
1. According to the parameter distribution form of groundwater system.
(1) Centralized parameter system management model-mainly used for macro planning and control of groundwater system.
(2) Distributed parameter system management model-mainly used in areas with high degree of hydrogeological research to allocate and manage specific groundwater resources.
2. According to the relationship between system state and time.
(1) Steady-state management model-the state variables of the model do not change with time.
(2) Unstable management model-the state variables of the model are functions that change with time.
3. According to the management purpose of the system.
(1) hydraulic management model-a management model with the hydraulic elements of groundwater and surface water as the main state variables and decision variables, which is mainly used to solve the problems of water allocation and water level control.
(2) Water quality management mode-a management mode mainly used to solve the problems of groundwater quality management and pollution control. Usually, hydraulic model is an important part of it.
(3) Economic management mode-more consideration should be given to relevant economic factors (such as completion cost, cost of building surface or underground reservoirs, depreciation cost of equipment and facilities, etc.). ), and hydraulic model and water quality model are often part or sub-model of this model.
4. Displays the target number of system management issues.
(1) Single-objective management model-a management model established when the goal pursued in the process of water resources optimization decision-making is a single goal.
(2) Multi-objective management model-a management model established when the objectives pursued in the decision-making process of water resources optimization are multi-objectives.
In practical work, we can choose the type of management mode according to the tasks and objectives of the research topic and the research conditions provided by the actual data.
Groundwater management model is a coupling model, which is often a coupling model of groundwater flow or solute transport simulation model and optimal management model. According to the type of its model, the corresponding optimization planning method can be used to solve it.
There are many methods to solve the groundwater management model. The most commonly used optimization planning methods are linear programming, dynamic programming, nonlinear programming, multi-objective programming and genetic algorithm. The essence of these methods, you can refer to the relevant reference books, so I won't go into details here.
This paper takes the most widely used groundwater distribution parameter system management model as an example.
Second, groundwater management model of distributed parameter system
Because the groundwater distribution parameter model can simulate the groundwater state at different time and space points in the system, the accuracy and practicability of the management model are greatly improved. As mentioned above, the groundwater management model is a coupling model. Groundwater management model of distributed parameter system uses distributed parameters to describe groundwater flow (or solute distribution) in groundwater system, which is coupled with optimization model as a set of constraints, and finally forms groundwater management model. At present, there are two methods to realize coupling, namely response matrix method and embedding method. The following are introduced separately.
(A) response matrix method
According to the principle of linear system, groundwater level decline is the convolution integral of mining volume and response coefficient of groundwater level decline. The relationship between pulse (exploitation or water injection) and response (groundwater level fluctuation) in groundwater system is expressed. Among them, the response coefficient is often expressed in matrix form, which is a set of constraints in the optimization management model to realize the coupling with the optimization model, so it is called response matrix method.
1. superposition principle.
The superposition principle is that if φ 1=φ 1(x, y, t) and φ2=φ2(x, y, t) are the general solutions of the inhomogeneous linear partial differential equation L 1(φ)=f 1 and L2(φ)=f2, respectively.
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(where C 1 and C2 are arbitrary constants) is also the solution of the inhomogeneous linear partial differential equation L(φ)=f 1+f2.
L stands for linear operator. For example, the linear operator in a two-dimensional confined aquifer system can be written as:
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For a definite solution problem with non-homogeneous initial boundary conditions, for example:
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Then the superposition principle can divide this problem into two problems:
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and
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The sum of solution φ 1 of equation (10-4) and solution φ2 of equation (10-5) φ=φ 1+φ2, which is the solution of equation (10-3). Where h is the confined water head; T 1, T2 is the hydraulic conductivity; S is the water storage coefficient; г 1 is a boundary.
Therefore, using the superposition principle, a problem can be decomposed into several simple sub-problems to solve. In the problem of groundwater flow, several scattered sources or sinks act on a certain point (or a certain area) independently at the same time and the effect equivalent to the simultaneous joint action of all sources or sinks; The sum of the influences of several single boundaries on the groundwater level in the calculation area is equivalent to the comprehensive influence of the total boundary.
If the mathematical equation describing aquifer groundwater flow is linear, it is a linear groundwater system and satisfies the superposition principle. Groundwater system is a nonlinear system, and the equation must be linearized before applying the superposition principle to solve the groundwater flow problem.
2. Establishment of response function
For a linear groundwater flow system with homogeneous initial boundary conditions, the total water level response of a point in the system after applying pulses to some sources or sinks can be obtained by the algebraic sum of the responses generated by applying pulses to that point by each source or sink separately. Here, pulse refers to the input added to stimulate the state change in a groundwater system, that is, the pumping (or injection) amount of groundwater; Response is the output produced by the change of system state after the pulse acts on the system, that is, the change of groundwater level (water level drops or rises, etc.). ).
In linear systems, the relationship between response and pulse can be expressed in the form of convolution integral;
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Where: S(u, uj, t)- response value [l] generated at point u when pulse Q(uj, τ) is applied at point u before time t;
Q(uj, τ) —— groundwater pulse amount acting on uj point during τ period [L3/t];
β(u, uj, t-τ)—— the response coefficient of unit pulse, which indicates the response value [t/L3] at time t to point u when the unit pulse value acts on point uj in the τ period;
U, UJ-two-dimensional space coordinates (X, Y) and (xj, YJ) [L] respectively;
T- time coordinate [t].
Equation (10-6) is called groundwater response function, and some people call it algebraic technical function.
In order to solve the groundwater level response function, the equation (10-6) is discretized to obtain the following equation:
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Where: S(i, J, N) —— the water level response at point I at the end of the nth cycle caused by the pulse at point J [L];
β(i, j, n-k+ 1)—— the average response of the I-th observation well (point) at the end of the n-period, that is, the response coefficient of unit pulse [t/l2];
Q(j, k)—— the average pulse quantity of Well J (point) in the k-th cycle [L3/t].
Equation (10-7) expresses the relationship between impulse and response in groundwater system in the form of linear equation.
If there are n sources (or sinks) pumping water (or injecting water) at the same time in the seepage field, the response coefficient of M×N unit pulses will be formed in a period of time, which is called response matrix. If there are l time periods, a three-dimensional response matrix of M×N×L is formed. At this time, the total response (falling or rising) of water level to point I at the end of the nth time period is equal to the total result (sum) of each well acting alone before this time period. That is:
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Where: S(i, n) is the last moment of the nth cycle, and the water level at point I is always responsive, and other symbols have the same meanings as before.
Therefore, if the response matrix is obtained, the formula (10-8) can be used to obtain the total drawdown or total increase of water level produced by each pumping well or water injection well at any observation point (or subdivision node) at any time.
From the above analysis, it can be seen that the response matrix of groundwater level reflects the characteristics of groundwater system itself, including aquifer type, internal structure, boundary nature and shape, hydrogeological parameters, spatial distribution position of source and sink points, etc. It has nothing to do with the input and output of groundwater.
3. Determination of unit impulse response coefficient
Unit impulse response coefficient is the response value of groundwater system when unit impulse is applied to groundwater system based on groundwater flow simulation model.
At present, in production practice, numerical methods are mainly used to establish groundwater flow simulation models to simulate large-scale heterogeneous anisotropic aquifers and groundwater flows with irregular boundary conditions. Therefore, numerical method is usually used to determine the response coefficient.
Determination of response coefficient under (1) homogeneous initial-boundary value condition: If the initial-boundary value condition of seepage field is homogeneous, that is, the initial condition and boundary condition are zero, the solution steps of unit impulse response coefficient are as follows:
① Divide seepage field in space and management period in time.
(2) On the basis of subdivision, numerical simulation models are established by using certain principles and methods, such as confined water (or diving), one-dimensional (or two-dimensional, three-dimensional) flow, homogeneous (or heterogeneous), stable (or unstable) simulation models.
(3) In the first period, the first well pumps water at the unit flow rate, and other wells do not pump water, thus calculating the water level drawdown value of each observation well (node); Then the second well pumps water at the unit flow rate, and the rest wells do not pump water, and then calculate the depth of water level decline of each observation well ... and so on until all N pumping wells are circulated. Through certain transformation, the response coefficients of water level on m observation wells or nodes can be obtained, and finally M×N response values of water level drop depth can be obtained, forming an M×N order response matrix.
And (4) solving each time period circularly according to the step (3) until all l time periods are calculated to obtain m * n * l response coefficients and form an m * n * l order three-dimensional response matrix.
For the response coefficient under the condition of steady flow, the solution steps are basically the same as above, except that there is no need to divide the time and cycle time (see step ④), and the obtained response matrix is an M×N order two-dimensional response matrix.
(2) Determination of response coefficient under the condition of non-homogeneous initial-boundary value:
The plane two-dimensional flow problem in confined aquifer system can be described as the definite solution problem of the following partial differential equations:
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Where: h(x, y, t)- water level distribution;
T, μ * —— hydraulic conductivity and water storage coefficient respectively;
γ 1, γ 2-the first boundary and the second boundary respectively;
G0(x, y)—— initial water level drawdown value;
G 1(x, y, t)- a boundary value;
G2(x, y, t)- boundary value of the second kind;
P(x, y, t) —— controllable pulse quantity (such as pumping quantity and reinjection quantity) at time point (x, y);
ε(x, y, t)- uncontrollable pulse quantity (such as rainfall infiltration quantity);
(x, y)- plane position coordinates;
D—— Seepage area of groundwater.
If the initial boundary condition is non-homogeneous, that is, g(x, y, t)≠0 or the uncontrollable pulse quantity of the groundwater system is not zero, the groundwater system is not a linear system, and the response coefficient of the unit pulse quantity cannot be directly determined by applying the superposition principle. In order to solve this problem, the definite solution equation (10-9) can be decomposed into the following two definite solution problems:
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and
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According to the principle of mathematical equations, there are:
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Where: h is the actual water level determined by the definite solution problem (10-9); H is uncontrollable pulse, which is only the water level distribution formed by initial flow field, boundary conditions and natural recharge, that is, the natural water level determined by formula (10- 10); S is a homogeneous initial boundary value condition, which is not affected by uncontrollable pulse, and the water level is only decreased (or increased) by controllable pulse (pumping or injecting water), that is, the artificially influenced water level drop determined by formula (10-1) indicates the water level drop determined by water level response function, and the response coefficient of unit pulse can be determined by applying superposition principle.
Subtract the initial water level H0 from both sides of the equal sign of the formula (10- 12):
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According to the formula (10-8), the formula (10- 13) can be transformed into the formula of water level depth drop at point I in the nth cycle:
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Or write:
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The formula (10- 15) shows that there are n pumping wells acting on the groundwater system; St(i, n) represents the actual water level drop value; S0(i, n) represents the additional drawdown value of water level determined only by initial boundary conditions and natural compensation, diameter and drainage conditions.
Therefore, for the nonlinear groundwater system with non-homogeneous initial-boundary value (uncontrollable pulse quantity) of natural recharge or discharge, in order to determine the response coefficient, the influence of natural recharge, discharge, initial flow field and boundary conditions on the actual seepage field St(i, n) can be simulated by decomposing the seepage field, and then the non-homogeneous initial-boundary value can be conditioned to be homogeneous.
4. Composition of groundwater management model
As mentioned earlier, the groundwater management model includes two main parts: objective function and constraint conditions. The objective function represents the purpose of management problems in groundwater system. Generally, it can be the optimal value of water quantity and water quality of groundwater system (such as maximum total groundwater exploitation or minimum total drainage) or the optimal value of water level under certain constraints (such as minimum water level drawdown). You can also seek the optimal value of system economic benefits (such as minimum mining cost and maximum total income, etc.). ) Under certain water quantity, water quality and water level restrictions; It can also be the pursuit of the optimal value of environmental benefits or social benefits related to groundwater system (often implied in finding the optimal value of groundwater level or water quality). Constraints, as mentioned above, are restrictions on decision-making variables in terms of technology, economy, environment and society for specific management objectives.
The composition of the model is illustrated by taking the establishment of the management goal of seeking the maximum exploitation of groundwater system as an example. The objective function is to maximize the total groundwater exploitation of n wells in L planning stages during the management period;
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Constraints include the following aspects:
(1) Limit of water production capacity of each well in each planning stage:
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(2) Constraints of total groundwater exploitation index (resource constraints):
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(3) Limit constraint of water level decline:
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(4) State constraint of groundwater flow. In order to express the state equation of the relationship between water quantity and water level in groundwater system, this constraint can be realized by the relationship between pumping quantity and water level drop depth related to response matrix:
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(5) Non-negative constraints, which means that all decision variables are non-negative:
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Where: I = 1, 2, ..., n;
j= 1，2，…，N；
n= 1，2，…，L
N- the number of observation logs or nodes in the system;
Q(j, n)—— the pumping capacity of the j well in the nth planning stage;
S(i, n)—— the water level drawdown value of the ith observation well or node from the beginning of pumping to the end of the nth stage;
Qt- groundwater demand index;
Q0—— water production capacity of each well;
S0-the limit of groundwater level decline.
If the response matrix of the nth stage is expressed as:
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The corresponding water level drop depth and pumping capacity are also expressed in matrix as follows:
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Then the fourth equation of the constraint can be expressed as:
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That is, through the response matrix [R], the continuity equation of groundwater flow enters the optimization model as an equality constraint, thus forming the groundwater management model.
(2) Embedding method
The embedding method of establishing groundwater management model is also called nesting method or mosaic method.
The basic principle of the embedding method is to take the groundwater flow simulation model as a part of the constraints of the optimal management model and enter the management model to realize the coupling between the groundwater flow simulation model and the management model. Bredehoeft and Young first proposed this method in 1974. Together, they solved the finite difference flow simulation model and linear programming model. After that, Aguad and Remson further discretized the groundwater flow equation in 1974 by using the finite difference method, and took the linear algebraic equations as a set of constraints of the optimal management model to form a linear programming model. Under the condition of meeting certain water supply requirements, the optimal pumping quantity distribution and head distribution are determined with the highest head at a specific position in the aquifer as the goal. Alley, Aguad and Remson gradually established a series of management models of unsteady flow problems in 1976, which made the embedding method mature and applied to a certain extent.
The management model, groundwater flow simulation model and management model established by embedding method run simultaneously and are completed in one step. For the groundwater flow simulation model established by numerical method, the linear algebraic equations formed by numerical discretization should be "embedded" into the management model in the form of constraints.
Embedding method is effective for stable flow problems and some unstable flow problems with short management period, short time period and small calculation area. However, the management mode is difficult to solve the problems of regional and multi-period planning management, which limits the practical application of embedded methods.
Comparing the above two methods and steps of establishing groundwater management model of distributed parameter system, it has the following characteristics:
The (1) response matrix method divides the establishment of groundwater management model into two steps, that is, first, the response matrix is calculated through the groundwater state simulation model, and then the relationship between water level or concentration and water quantity represented by the response matrix is used as a constraint condition to enter the management model. The embedding rule is one-step method, that is, the whole set of discrete groundwater flow equations is directly regarded as the equality constraint condition of the optimization model, and the groundwater state model and groundwater management model run simultaneously and complete in one step. The principle of the method is relatively simple.
(2) The response matrix method can constrain the variables to some key areas or time periods in the management area for specific groundwater management problems, and it is not necessary to constrain all points in the whole area, thus avoiding the difficulty of large-scale calculation.
(3) The response matrix method is especially suitable for large-scale and multi-stage groundwater management. Embedding rule is suitable for the management of small area steady flow groundwater.
(4) The optimal decision obtained by response matrix method in modeling and management operation only contains decision variables (pumping capacity) or state variables (water level and concentration), and other variables are obtained by flow simulation model or response matrix. Embedding rules also give various variable results of the scheme.
The optimization technique used in the above-mentioned modeling process is mainly linear programming. In fact, dynamic programming, nonlinear programming and integer programming can also be widely used to solve groundwater management problems.