In order to illustrate the characteristics of stochastic equations, we first discuss the famous P.Lanngevin equation in physics (Zhang, 2007). Langevin equation is a stochastic equation used to describe Brownian motion. 1905, Einstein published a paper on Brownian motion, put forward the diffusion equation of the probability distribution function of Brownian motion particles, and obtained the formula of the average displacement changing with time. On this basis, Langevin put forward the random force hypothesis, arguing that the dynamic behavior of Brownian motion particles can be described by the following equation (Zhang, 2007):
Groundwater motion equation
Where: x is the displacement of the particle; M is the mass of the particle; γ is the retardation coefficient corresponding to the viscosity of the fluid; F(t) is a random force generated by random collision of molecules (atoms). Equation (5. 1) is a one-dimensional Langevin equation based on Newton's law, which describes the dynamic process of Brownian motion of a single particle in the form of a stochastic equation. As long as the statistical characteristics of random force F(t) are known, the statistical characteristics of Brownian motion trajectory can be obtained by mathematical solution using the stochastic equation above.
The randomness of groundwater dynamic process can also be described by adding random source and sink terms in the form of Langevin equation (Gelher et al., 1974). The phreatic aquifer is most vulnerable to the random influence of recharge processes such as precipitation infiltration. In order to analyze this random effect, we might as well linearize Boussinesq equation of one-dimensional movement of diving surface and rewrite it as
Groundwater motion equation
Where: h is the height of groundwater level (relative to horizontal ground); Ha is the reference groundwater level height; W(t) is a source-sink term with random characteristics; L is the length of the aquifer. Equation (5.67) is a simplified stochastic equation describing the dynamic change of groundwater level.
Now study a phreatic aquifer as shown in Figure 5.2, and establish a mathematical model of stochastic groundwater dynamics. In addition to the stochastic equation (5.67), the boundary conditions are as follows:
Groundwater motion equation
Where h0 is a constant that does not change with time. The boundary conditions given in Equation (5.68) are definite. However, if h0 changes randomly with time, it will also affect the groundwater dynamics of the aquifer, which is not studied in this book. Because it is an unstable flow model, an initial condition is needed. However, for such a stochastic dynamic system, the initial conditions will only affect the initial water level dynamics, and the dynamics after a long time have little to do with the initial conditions.
Fig. 5.2 Schematic diagram of groundwater aquifer with random recharge
If the source-sink term is a stationary stochastic process, the expected value of w(t)
Groundwater motion equation
Is a constant, where f(w) is the probability density function of w. Another statistical feature of stationary stochastic process w(t) is the autocorrelation function, that is
Groundwater motion equation
It is only related to the time interval τ and has nothing to do with the specific time t, and μw(0) is the variance of this random process. As a result of the above-mentioned stationary random process W, the groundwater level H is also a stationary random process, but its expected value is a function of spatial coordinates, which is denoted as hm(x). Calculate the expected value of Equation (5.67), i.e.
Groundwater motion equation
Considering that the expected value of stationary stochastic processes does not change with time, there are
Groundwater motion equation
According to equation (5.68), the boundary condition of the expected value is
Groundwater motion equation
Equations (5.72) and (5.73) constitute the mathematical model of the expected water level. According to formula (2. 14), there are
Groundwater motion equation
Next, random variables are expressed as perturbations based on expected values, that is
Groundwater motion equation
Where the expected values of the disturbance terms H δ and W δ are zero. Substituting formula (5.75) into formula (5.67) and formula (5.68), and considering formula (5.72) and formula (5.73) at the same time, there are
Groundwater motion equation
Fourier transform is introduced into water level disturbance and source-sink disturbance;
Groundwater motion equation
Where Ω is the actual frequency, the result of Fourier transform of formula (5.76) is
Groundwater motion equation
The characteristic equation of equation (5.79) is
Groundwater motion equation
Where a = kha/sy, two characteristic roots can be obtained as follows.
Groundwater motion equation
Therefore, the general solution of formula (5.79) is
Groundwater motion equation
Where: C 1 and C2 are integer constants. Using the boundary conditions, we can get
Groundwater motion equation
Therefore,
Groundwater motion equation
In ...
Groundwater motion equation
Equation (5.85) actually reflects the relationship between water level and disturbance power spectra of source and sink terms (Gelher et al., 1974). For example, when x = l, that is, at the right boundary position, the power spectrum amplitude has the following ratio.
Groundwater motion equation
The variation of the amplitude ratio of the power spectrum obtained from Equation (5.87) with frequency is shown in Figure 5.3. As the frequency increases (the period decreases), the amplitude ratio decreases, indicating that the groundwater level has a strong response to the source-sink disturbance in the long period (low frequency) and is insensitive to the source-sink disturbance in the short period (high frequency).
Figure 5.3 Variation of Power Spectrum Amplitude Ratio with Frequency