9/15/04

Preliminaries: further comments on significant digits, solutions to problems posed in the last lecture

I. Permeability, qualitative definition: ease with which water flows through a porous medium. Depends on effective porosity (greater open area, more flow) and pore size (smaller pores, more frictional resistance at water solid contact)

II. Aquifers and confining units (defined in Ch. 1 of S&Z) - aquifers have sufficient porosity to store water and sufficient permeabilty to supply water to a well. All other "hydrostratigraphic units" are confining units. Other terms for confining units include aquitard, aquiclude and aquifuge.

III Darcy's experiment

Henry Darcy conducted experiments in Dijon, France in the 1850s to aid in the design of sand filters to improve the quality of river water being used for the city's water supply. (A short biography of Darcy can be found in the January 1994 issue of the journal Ground Water.) He demonstrated that the rate of water flow through his sand columns was proportional to the change in "head" rather than the change in pressure across the column. This result is now known as Darcy's Law.

Darcy demonstrated two main things with his experiment:

1) the driving force for flow is proportional not to a gradient in pressure, but to the gradient in "head" (h) where

h = z + p/rwg

z is the elevation of the base of the manometer relative to some datum where zo=0

p/rwg = the height of water in the manometer above the measuring point at elevation z

rw is the density of water

g is the gravitational acceleration constant

2) a proportionality constant K exists for a given porous medium

IV. Physical Significance of Head

The physical significance of head was not clearly explained to the geologic community until almost a century after Darcy's experiment. This explanation was provided by M. King Hubbert, who was concerned that petroleum engineers and others had forgotten Darcy's result that flow is not proportional to a pressure gradient alone. Hubbert's analysis began by considering the energy require to move a mass of water from a reference state of z=0 (elevation is the datum), p = po (atmospheric pressure) and zero velocity to a state in the aquifer at elevation z1, pressure p1 and velocity v1.

Hubbert showed that head is actually a measure of mechanical energy per unit weight of water.

Hubbert reasoned that there were three components of mechanical energy:

1) potential energy = work to move water from reference datum to elevation z1

= force x distance

= mass x acceleration x distance = mgz1

2) kinetic energy =

3) pressure energy = work to raise pressure from atmospheric pressure (0 gage) to p1

= "PV work"

             = 

If we can assume that water is essentially incompressible (rw is constant)

               

which for pressures expressed as gage pressures relative to atmospheric (po = 0)

=

So the total mechanical energy =

Mechanical energy per unit mass = Hubbert's potential = F =

Mechanical energy per unit weight = Total head =

This is almost the same as Darcy's head except for the kinetic energy term. We can ignore the kinetic energy term primarily because the change in velocity from one point to another in an aquifer is very small. (This is not necessarily the case near a pumping well.) The reason that we are interested in the change in velocity and not velocity itself is that Darcy's law employs a gradient in head (change in head over distance) rather than the actual magnitude of head. A secondary reason (and the one mentioned in the text above equation 3.9) is that groundwater velocities are generally small. However, note that pressure head, p/rwg, and elevation head, z, can also be small at some places in an aquifer, so this second reason is not really sufficient to allow us to ignore velocity head.

While reviewing the fact that the gradient is a slope of a plot of head versus difference, it was noted that a negative gradient gives rise to flow in the positive direction, hence the minus sign in Darcy's law. Sometimes hydrogeologists get a bit sloppy in writing Darcy's law and leave out the minus sign. S&Z are clever in defining i as -dh/dl so they can avoid having to include the minus sigh in equations, but strictly speaking what they have defined as i is the vector that points in the opposite direction of the gradient.

You should review Example 3.1 in S&Z to confirm your understanding of the concepts of pressure head, pressure, and elevation head.

V. Defintion of a piezometer - see Figure 3.4 in text, a well open to a "point" in an aquifer with which you can measure head at that point. You will see piezometers in the sand box demonstration in lab this week.

VI. Hydraulic gradients in 2 and 3D

A. Graphical method for identifying an equipotential and directions of the gradient and flow

Example 3.2 from S&Z illustrates a graphical solution to the "3 point problem". Note that this is identical to the process of determining strike and dip of beds from three points. Limitations of the graphical method are that it requires careful drawing and measurement and is not very precise.