1. analysis method

That is, mathematical and physical methods (separation of variables, Laplace transform, Fourier transform, Hangel transform, etc. ) is used to solve the mathematical model and get the analytical expression of the variation law of some variables, that is, analytical solution or analytical solution. Because the necessary assumptions of this method are limited (such as homogeneous aquifer and regular geometric boundary), it is difficult to solve the mathematical model, which limits the application of this method.

2. Number solution

Mainly the finite difference method and the finite element method. The basic steps are as follows:

1) According to the conditions, the seepage area is divided into multiple units (unit homogeneity, boundary rules), and a node (point element) is defined on the unit as needed, and the continuous head distribution in the seepage area is dispersed into an array composed of multiple numbers on all nodes.

2) On the basis of discretization, the simultaneous boundary conditions of partial differential equations are transformed into linear algebraic equations.

3) Solving the linear algebraic equations to obtain the head distribution. In the case of unsteady flow, the equation should be solved several times according to the initial head distribution to obtain the head distribution at each moment.

When the finite difference method transforms the differential equation into a linear algebraic equation, the difference quotient is used instead of the derivative. Finite element rules are discretized by linear or high-order interpolation functions, and then partial differential equations are transformed into linear algebraic equations by variational or other mathematical methods. With the development of computer, numerical methods become more and more important for solving the mathematical model of groundwater movement.

summary

This chapter requires understanding and mastering the following basic concepts and principles: permeability and seepage, permeability coefficient and permeability, water storage coefficient and water storage rate, steady flow and unsteady flow, pressure flow and free surface flow, one-dimensional flow, two-dimensional flow and three-dimensional flow, Darcy's law and expressions of seepage refraction law.

Review thinking questions

1. What are the commonly used methods to study seepage, and why?

2. In groundwater dynamics, why can the total pressure head be replaced by the pressure head?

3. What are the expressions of hydraulic gradient? What are the conditions for using different methods?

4. Why can't Darcy's law be called laminar flow law?

5. What is the difference between permeability coefficient and permeability? Under what conditions can they replace each other?

6. What are the homogeneity and heterogeneity, isotropy and anisotropy of water medium?