First, Darcy's law and penetration rate

For the seepage of a homogeneous fluid in a porous medium, it is described by Darcy's law and proved to be suitable in many cases. In addition to the extremely high flow rate of the fluid, and the very low flow rate of the gas are two exceptions.

Figure 1 Darcy percolation experiment

The classical seepage experiment of Darcy is shown in Figure 1. h represents the height of the percolation layer, the cross-sectional area of ​​the percolation column layer is S, the diameter is d, and the pressures at the upper and lower ends of the percolation layer are P ₂ and P _{1 , respectively} . In terms of the top-down flow of a homogeneous fluid, the flow rate 'represents a volume flow rate Q, it is diafiltered _retentate velocity v

v _seepage = or (1)

Expressed by the pressure difference, since the flow direction is opposite to the pressure direction, then

Q'=-KeS (2)

If ρ and g are both constant, a new constant K'e is introduced, then

Q'= (3)

Equations (1) and (3) can be called Darcy's law.

The constant K'e characterizes the ability of a porous medium to penetrate a particular fluid, called the permeability coefficient, and the magnitude of its value is determined by the properties of both the medium and the fluid. Nutin defines this constant as K'c=K _permeation /μ, then

Q'=- (4)

Where the viscosity of the μ-fluid;

K _permeability - (specific) permeability of porous media.

Permeability K _seepage and permeability coefficient K'e will often be involved in heap leaching operations. The former characterizes the intrinsic permeability of porous media (or heap leaching materials) regardless of the fluid. The latter represents the permeability of a porous medium to a particular fluid, the magnitude of which is determined by the nature of both the porous medium and the fluid.

The unit of penetration rate, different countries, different industries have different representation methods. According to Darcy's law expressed by equation (3), the dimension of permeability is L ² . According to the provisions of China's measurement law, the unit of permeability is cm/s. However, many literatures, especially in many Western literatures, use Darcy and millidarcy to express permeability. For water, their conversion relationship is 1 cm/s = 1033.4 Darcy.

Second, the general expression of single saturated seepage

In porous media, the application of Darcy's law is limited if the height h of the percolating layer is large. AE Schrödinger applies the Darcy law expressed by equation (3) to the general single-saturated fluid percolation (isotropic porous medium), the percolation layer thickness h to a minimum value, and the pressure gradient instead of pressure. Poor, and use the state equation ρ = ρ (P) and the continuity condition is -e =div(ρV seepage) is associated with equation (3). The general expression for seeping a single saturated flow in a porous medium is as follows:

-e =div (5)

Where the porosity of the e-porous medium,

Ρ-fluid density;

Viscosity of the μ-fluid;

K _permeability - permeability of porous media;

gradP-fluid pressure gradient;

G - A vector of gravity directions whose value is equal to gravity.

Third, the expression of the unsaturated stream

The flow rate of the percolation in equation (5) is related to the fluid pressure, so it is only suitable for a flow pattern in which only one fluid flows through the porous medium in a saturated state, or when the porous medium is always filled with the same fluid flow. This formula can be applied. If the formula is extended to multiphase fluids while flowing in a porous medium, greater correction must be made. For the unsaturated flow, the rate of seepage is not related to the pressure of the applied fluid, but has a function relationship with the capillary pressure Pe, the saturation Se and the like. According to the conventional and actual situation, AE Schrödinger regards the air pressure in the unsaturated flow as 0, then the fluid pressure P _L =Pe(S), but in the opposite direction. It is also assumed that the density of water is constant. So he gave the following formula:

-e =div (6)

The characteristic parameter saturation Se in the unsaturated flow, the capillary pressure Pe, the porosity e, the relative permeability Kw, and K'e are the permeability coefficients, and the physical meanings of ρ and g are as in the formula (5). Relative permeability and permeability K Kw aforementioned different _{permeability,} K is the _permeability properties of porous media, it means a 100% saturation state, the permeability of porous media; and Kw is a flow state is not saturated, The permeability of porous media at different degrees of saturation, which is a function of saturation. The Kw is clearly different depending on the saturation of the porous medium. It will be appreciated reader, for a particular porous media, 100% of saturation and the relative permeability Kw _retentate permeability K are equal.

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