There are two mathematical models. A model that contains one or more random variables in a mathematical relationship is called a random model. If there is a strictly definite relationship between variables in a mathematical model, it is called deterministic model. This book mainly discusses the latter.
When describing the actual groundwater flow with deterministic model, as mentioned above, the following conditions must be met: (1) There is a partial differential equation (or a set of equations) that can describe the movement law of this kind of groundwater; At the same time, the range and shape of the corresponding seepage zone and various parameter values appearing in the equation are determined. (2) The corresponding definite solution conditions are given. But the problem does not end here, because at this time we are not sure whether the model established through the above steps can truly represent the geological body under study; The parameters appearing in the model cannot be given accurately at this time. Therefore, it is necessary to test the established model, that is, to compare the predicted results of the model with the actual observation results or long-term observation data of groundwater dynamics in a region after pumping test or other tests have exerted certain influence on the aquifer to see if they are consistent. If they are inconsistent, the model should be corrected, that is, conditions (1) and (2) should be corrected until it is satisfactorily fitted. This step is called identifying the model or correcting the model.
The corrected model can represent the studied geological body or a replica of the actual water flow system, so it can be calculated or predicted as needed, such as predicting the water inflow and groundwater pollution when the deposit is drained.
In addition, the mathematical model for simulating practical problems should also meet the following basic conditions: (1) the solution (that is, the solution satisfying the conditions 1 and 2) exists; (2) The solution is unique (uniqueness); (3) This solution continuously depends on the original data (stability). It goes without saying that the solution of the problem is required to exist and be unique. The third condition, that is, the requirement of stability, means that when the parameters or definite solution conditions change slightly, the change of the solution is also very small. Only with this guarantee, when there are some errors in the data of parameters and definite solution conditions, the obtained solution can still be close to the true solution; Otherwise, the solution is not credible, and the mathematical model at this time should be considered as faulty. In practical work, there are inevitably some errors in the original data, so this condition is very important. A problem that satisfies the above three conditions is called a well-posed problem, and as long as one of them is not satisfied, it is a well-posed problem. The questions mentioned in this book are all appropriate.
Here are some examples to illustrate how to use mathematical models to describe groundwater flow.
Example 1 See Figure 1-37 for the geological conditions of the study area. Let W(x, y, t) represent the vertical recharge per unit time and unit area, and P(x, y, t) represent the pumping flow per unit area in the planned mining area, and try to write its mathematical model.
Figure 1-37 Schematic diagram of a research area
(according to J.Bear)
Boundary BC is a natural water-blocking boundary. The river cuts the whole aquifer, and there is a close hydraulic connection between them. Therefore, the boundary AD can be regarded as the first boundary. In addition, there is no natural boundary in both directions, and the aquifer extends far away. How to deal with it? One method is to artificially draw a boundary in the area far away from the mining area, which is actually not affected by pumping. In this section, according to the relevant data, choose a connection line composed of several boreholes with dynamic observation data, or choose an isohead line or streamline as the boundary. In Figure 1-37, two streamline lines (BA and CD) are used as the boundary (in fact, the other two methods may be better). At this time, the calculation area consists of the area surrounded by ABCDA. The boundaries BA and CD are artificially defined on the premise that they are not actually affected by the pump. Obviously, the validity of this hypothesis needs to be tested. Another method is to move the boundary further after calculation and repeat the calculation. If the depth of water level drop actually has little effect, the boundary selection is reasonable; Otherwise, the boundary should be moved away from the mining area until the water head near the boundary has no obvious influence.
According to the given conditions, the equation describing this phreatic flow should be (1-95). The seepage area is ABCDA, marked D, and the boundaries BA and CD are equivalent to the water-resisting boundary. The mathematical model is as follows:
Groundwater dynamics (second edition)
Where H0 and f are known functions, and f is the river water level at different times; Z(x, y) is the elevation of the aquiclude.
In some rivers, there is no direct hydraulic connection between rivers and groundwater because of the weak permeable layer at the bottom of the river, but groundwater is replenished through the weak permeable layer. This reach cannot be treated as the first kind of boundary, but should be treated as an overflow term. If other conditions are assumed to be similar to the example 1, the equation should be rewritten as:
Groundwater dynamics (second edition)
If the pumping treatment is (1-108), the left end p of (1-120) is removed and the boundary condition is added:
Groundwater dynamics (second edition)
In the above formula, Kz and dz are the vertical permeability coefficient and thickness of the weak permeable layer at the river bottom, respectively, and Hz is the river water level treated as an overflow term, both of which are known values; Rwj and Qj are the radius and flow of the J-th well respectively; N is the number of wells.
With the mathematical model, if the hydrogeological parameters (t, μ, etc. ) and the definite solution conditions, the head h can be obtained. This kind of problem is usually called forward problem or head prediction problem. If the hydrogeological parameters are inversed according to dynamic observation data or pumping test data, then this kind of problem is the inverse problem of the former or the inverse problem of parameters.