Saturated Flow, continued

 

The Laplace equation:

            : the second derivative of the hydraulic gradient is zero.  In other words, the hydraulic gradient is a linear function (in steady linear flow in a uniform medium).  This is similar to Darcy’s Law: Laplace says that, for constant flow, the gradient from one place to another is linear, and Darcy says the flow is a linear function of the hydraulic gradient.  You can use this to solve some odd problems.  If you know that flow is steady and the soil is uniform, you know that the total head (gravitational + pressure) decreases linearly across your sample, however odd the setup may look.

 

Limitations of Darcy’s law:

            At high flow velocities, flow becomes turbulent.  When flow is turbulent, energy is drained by “extra” viscosity – unnecessary fluid movement – so flux is less.  Bottom line: when flow velocity is too high, flux will be less than that predicted by Darcy’s law from measurements at lower velocities.

 

Clarification of units:

            Hydraulic conductivity has units of velocity: cm s-1.  But this is actually a flux density: cm3 cm-2 s-1.  The real velocity inside the medium (pore water velocity) is less because:

            Porosity < 1, so velocity inside the pores must be greater

            Tortuosity > 1, so the actual flow path is longer than delta-L

            We will see a similar adjustment (by a factor ) later when we look at diffusion.

 

The hydraulic conductivity has a medium and a fluid component.  Hillel calls the medium component “intrinsic permeability” (k, units of length squared), and the fluid component fluidity ().  Consider the analogy with Poiseuille’s law:

            Poiseuille:        

            Darcy:             

In Poiseuille’s law, the permeability part – what is specific to the medium – is given by , while the fluid part is given by the viscosity.  There is no gravity or density term.  Also missing from this approach is a fluid-solid interaction term, as described by the Klinkenberg effect.  The concept of separating the fluid and medium effects is a good one, but not well worked out.

 

Review the sections in the book on measuring the hydraulic conductivity of a soil using a simple, steady-state setup, a composite (2-layer) setup, and a falling head setup.