In the prototype, we know that the differential equation of water head at any point in the seepage field is
Groundwater dynamics (second edition)
Through comprehensive comparison, it is not difficult to find that the differential equations describing physical quantities in various models have a form similar to (8-2).
In the sandbox model, except for different geometric dimensions, medium parameters and prototypes, both systems are water flowing through porous media, so their differential equations must be in the same form.
In the electric simulation model, when the current is conducted in the conductive medium of the charging and discharging element, the forms of the potential distribution equation and (8-2) are also the same. At this time, parameters μs and k represent capacitance and conductivity, gradH represents potential gradient, and t represents time.
The conditions for the definite solution are similar, including:
(1) Geometric similarity, that is, in the limited space of prototype and model, the coordinates or corresponding lengths of corresponding points should meet a fixed ratio, that is:
Groundwater dynamics (second edition)
Subscript m is the coordinate or length of the model; Prototype without subscript m coordinates or length, it is not difficult to infer that the area and volume of the two systems should also meet a fixed ratio:
Groundwater dynamics (second edition)
However, when converting anisotropic media into isotropic models, deformation models are often used. Strict geometric similarity cannot be maintained in the deformation model, and αx≠αy≠αz or α x = α y ≠ α z is often taken. These two systems, which are not similar in geometry, can still remain similar according to the changing law of physical quantities.
(2) The time is similar, and the prototype and model can run synchronously; But it is rarely used in seepage simulation. It is commonly used that the model process needs to be accelerated, so the time ratio between the prototype and the model should also be kept fixed, and αt should remain unchanged during the whole operation.
From the relationship between geometric similarity and time similarity, it can be seen that the velocity must be similar in the sand trough model and the fluid model, that is,
Groundwater dynamics (second edition)
From this, we can observe the real (undeformed) streamline or trace. However, in the deformation model, because α 1 is related to the same direction, αu will change with the direction of flow velocity, that is:
Groundwater dynamics (second edition)
This shows that the motion similarity cannot be maintained in the deformation model.
(3) The parameters are similar, and the corresponding physical parameters in the two systems must maintain a linear relationship.
(4) The initial values are similar. In these two systems, the initial values of the corresponding physical quantities should meet a fixed ratio.
(5) The boundary values are similar. In the two systems, the boundary values of the corresponding physical quantities and their derivatives on the boundary should also meet the fixed ratio. When the boundary values change with time, it is necessary to keep the time similarity of the boundary values.
In short, in the case of the same differential equation form, all the corresponding physical quantities should maintain a fixed proportion, which is a necessary and sufficient condition for the prototype to be similar to the model. These conditions become the similar basis for simulating seepage law.