Theis formula will be derived from the groundwater flow control equation and its definite solution conditions.
The flow equation of single well with constant flow in infinite aquifer is established based on the following assumptions: ① The aquifer is homogeneous, isotropic, of equal thickness and horizontally distributed, and the aquifer is assumed to be elastic; (2) There is no vertical supply and discharge, that is, w = 0；; ③ Seepage satisfies Darcy's law; (4) Completion, assuming that the flow rate is uniform along the borehole wall; ⑤ The groundwater release caused by head loss is instantaneous; ⑥ Before pumping, the water head is horizontal; ⑦ Infinitely small hole diameter and constant flow pumping; (8) The aquifer extends laterally and infinitely.
According to conditions ① to ⑤, the basic differential equation of axisymmetric flow (2-3- 17) or (2-3- 19) can be applied, and ⑤ is the initial condition; ⑦ and ⑧ are internal and external boundary conditions respectively. Therefore, the definite solution problem can be written as
Groundwater dynamics (fifth edition)
Where: a is the head diffusion coefficient/pressure conductivity coefficient of the aquifer; H is the water head, which is a function of r and t; H0 is the original person in charge; R is the distance from any point to the pumping well; T is the time from pumping; Q is the flow (constant) of the pumping well; T is the permeability coefficient of aquifer. In groundwater dynamics, it is customary to stipulate that the pumping flow is positive, so the formula (5- 1-4) is missing a symbol.
The definite solution problem can be solved by integral transformation, separation of variables or L.Boltzmann transformation. Boltzmann method was put forward in 1894. This method is characterized by introducing a binary function or other similar forms, transforming partial differential equations into ordinary differential equations, and then solving the latter. This method is simpler than the classical variable separation method and involves simpler mathematical knowledge than the integral transformation method, so this method is adopted in this textbook (Matthews et al., 1967).
Introduce the variable u
Groundwater dynamics (fifth edition)
rule
Groundwater dynamics (fifth edition)
H is a function of U, and U is a function of R and T. According to the derivative rule of composite function, there are
Groundwater dynamics (fifth edition)
Or write it as
Groundwater dynamics (fifth edition)
therefore
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and
Groundwater dynamics (fifth edition)
Substituting the above three relationships into the formula (5- 1- 1), we get
Groundwater dynamics (fifth edition)
In this way, the partial differential equation (5-1-kloc-0/) becomes an ordinary differential equation (5- 1-6), and the initial condition (5- 1-2) and the boundary condition (5- 1-3) can be merged. Note that the formula (5- 1-5) and the internal boundary condition (5- 1-4) can be written as the formula (5- 1-8), that is
Groundwater dynamics (fifth edition)
In this way, the problem [I] becomes the problem [I]'.
Equation (5- 1-6) is solved as follows:
manufacture
Groundwater dynamics (fifth edition)
Then (5- 1-6) can be written as
Groundwater dynamics (fifth edition)
that is
Groundwater dynamics (fifth edition)
Variable separation
Groundwater dynamics (fifth edition)
comprehensive
Groundwater dynamics (fifth edition)
that is
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In ...
Groundwater dynamics (fifth edition)
or
Groundwater dynamics (fifth edition)
According to the boundary condition (5- 1-8), the integral constant C 1 is determined.
Groundwater dynamics (fifth edition)
get
Groundwater dynamics (fifth edition)
Substituting the obtained C 1 value into the formula (5- 1- 10), we get
Groundwater dynamics (fifth edition)
Again, separate the variables.
Groundwater dynamics (fifth edition)
Integral, and pay attention to the condition (5- 1-7), so
Groundwater dynamics (fifth edition)
Get the Theis formula.
Groundwater dynamics (fifth edition)
In ...
Groundwater dynamics (fifth edition)
Groundwater dynamic habit
Groundwater dynamics (fifth edition)
W(u) is called the well function of Theis well flow.
Define the depth of the drop
Groundwater dynamics (fifth edition)
Then, from the above, we can get three basic equations of artesian well flow, namely
Groundwater dynamics (fifth edition)
and
Groundwater dynamics (fifth edition)
Where W- 1 is the anti-well function. If w (u) = a, then w-1(a) = u.
The problem of complete well flow in single-layer diving is obviously much more complicated than that in confined well flow. A comprehensive analysis of it is left in chapter 9, and only one problem is paid attention to here, that is, the thickness m of confined well flow is constant, while the saturation thickness h of submerged well flow is variable. If other differences are ignored, as long as the corresponding relationship between phreatic well flow H and confined well flow M is found (water level H replaces water head H and gravity specific yield μd replaces elastic specific yield μe), the solution of confined well flow can be applied to phreatic well flow.
If the phreatic well flow meets the above eight assumptions of confined well flow, and the first point is changed to "the aquifer is homogeneous, isotropic and of equal thickness, and the aquifer floor is horizontal", and the first condition is added, the depth reduction is much less than the thickness of phreatic aquifer, and the flow rate meets the Qiubuyi hypothesis, then the phreatic well flow can correspond to confined well flow.
The differential equation of phreatic well flow satisfying the above nine assumptions is (2-5- 14), namely
Groundwater dynamics (fifth edition)
At present, it is known that the solution to the problem of definite solution of confined well flow [I] which is composed of formulas (5-1-1) ~ (5-1-4) is formula (5-1). Therefore, in the basic differential equation of restricted flow (5- 1- 1).
Groundwater dynamics (fifth edition)
Multiplied by m at both ends, the definition of potential function φ of restricted flow is introduced.
Groundwater dynamics (fifth edition)
Then the differential equation becomes
Groundwater dynamics (fifth edition)
It can be seen that after the definition of potential function φ is introduced into confined well flow, the form of definite solution problem [Ⅲ] of confined intact well is exactly the same as that of definite solution problem [Ⅱ] of submerged intact well, so the form of its solution φ should be the same, just pay attention to the relationship between the two variables:
Groundwater dynamics (fifth edition)
Therefore, the formula (5-1-1) of complete well flow solution under pressure is rewritten as
Groundwater dynamics (fifth edition)
In ...
Groundwater dynamics (fifth edition)
By using three corresponding conversion relations between submersible well flow and confined well flow, the constant current pumping formula of submersible well can be directly obtained.
Groundwater dynamics (fifth edition)
If the water level drops by S = H0-h, that is
Groundwater dynamics (fifth edition)
Therefore, (5- 1- 19) can be written as follows.
Groundwater dynamics (fifth edition)
Compared with the equation of confined well flow (5- 1- 17), it can be seen that the average thickness hm of phreatic well flow can be approximately calculated as follows.
Groundwater dynamics (fifth edition)
In other words, we can introduce the definition of average aquifer thickness hm (5- 1-23) when pumping water, and correspond or convert the discharge equation of diving well with that of confined well. In some literatures, Jacob's improved depth reduction sc is used to transform the above two well flow equations, namely
Groundwater dynamics (fifth edition)
commemorate
Groundwater dynamics (fifth edition)
Where: sc is the corrected depth reduction. Therefore, the aquifer thickness h0 remains unchanged, and the depth of water level decline changes accordingly.
Therefore, corresponding to the three basic equations of complete well flow under pressure (5-1-14) ~ (5-1-16), the three basic equations of complete well flow under diving are as follows
Groundwater dynamics (fifth edition)
and
Groundwater dynamics (fifth edition)
The above six formulas (5-1-14) ~ (5-1-kloc-0/6) and (5-1-24) ~ (5-1-26).
Well function can also be expressed in series form, that is
Groundwater dynamics (fifth edition)
Fig. 5- 1- 1 W(u)-u curve
The function curve of Tesi Well is shown in Figure 5- 1- 1. For the convenience of calculation, a function table (table 1) is made.
When it is small enough, the Tess well function can be approximately expressed by the first two terms of equation (5- 1-27), namely
Groundwater dynamics (fifth edition)
When u≤0.0 1, that is, the error is within 0.25%, or when u≤0.05, that is, the error is within 2%, the Theis well function can be replaced by logarithmic approximation. Therefore, equations (5-1-14) ~ (5-1-kloc-0/6) and (5-1-24) ~ (5-1-26).
For restricted well flow
Groundwater dynamics (fifth edition)
For phreatic well flow
Groundwater dynamics (fifth edition)
The above six formulas can be called the approximate formula of Theis formula-Jacob formula.
Theis formula is the basic formula of steady flow and unstable complete well flow in aquifer without vertical recharge/discharge (w = 0). The formula describes the relationship between the time and spatial distribution of head/water level decline and aquifer parameters and borehole flow. Theis formula is the basis for determining parameters in aquifer pumping/water injection test and the theoretical basis for dynamic prediction of groundwater exploitation. Theis formula plays an extremely important role in groundwater dynamics and should be deeply understood.