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Darcy's Law in Three Dimensions

  The one dimensional form of Darcy's Law can be generalized to three dimensions such that the head potential, is expanded to be a function of the three space coordinates, x, y, and z. Consequently the velocity v is a vector with components , and such that the potential head is dependent on the position.

The three dimensional generalization of Darcy's Law, can now be written as:

Each component takes the partial derivative of h with respect to the corresponding direction, since h is dependent on x, y and z.

  
Figure 2: Elementary Cube

Groundwater flow is a function of time and is essentially 3-dimensional in space. The velocity vector at any point has components along three mutually perpendicular axes x, y, and z of a Cartesian coordinate system, thus the velocity is a function of x, y, z and t. Consider the flow through a small elementary cube whose sides are of length , and (Figure 2). The law of conservation of matter holds for fluids. Specifically

The net excess of mass flux, per unit of time, into or out of any infinitesimal volume element in the fluid system is exactly equal to the change per unit of time of the fluid density in the element multiplied by the free volume of the element. [8]

or for each of the specific directions:
mass inflow =

where

  ¯ = fluid velocity (Darcian) in the x, y, z directions respectively [LT]

 		 = fluid density

giving the total net inward flux as the sum of differences between the inflow and outflow in each of the three directions.

where

 V ¯ =   the volume of the cube [L]

where

 M      = mass in the elementary cube

		=

      = effective porosity

giving us the Equation of Continuity:

where

          = fluid density [ML]

    = fluid velocity (Darcian) in the x, y, z directions respectively [LT]

          = effective porosity

with the specific directions x, y, z in Equation 1 we obtain