I. Hydraulic gradients in 2 and 3D (continued from last Wednesday)

Trigonometric method

This is a more precise method that is less subject to the types of errors associated with the graphical method. It requires knowing the x,y coordinates and heads at three locations A, B and C that are in the x,y plane and that are not along a single line (i.e. they form a triangle). Two additional locations D and E are then identified along one or more of the lines forming the sides of the triangle ABC. One of these must have the same x coordinate as one of the three original locations and the other must have the same y coordinate as one of the original locations. It is then possible to determine the remaining coordinates and the heads at these two locations using an understanding of the principles of similar right triangles.

The example below is equivalent to that demonstrated in class, although the figure is slightly different. Consider the case where heads and coordinates (x,y) are known at three points A, B, and C (shown on the figure below).

We first identify two additional points, D and E, such that xD = xA and yE = yB. These points are shown on the figure below.

We can then determine the y coordinate for D using the relationship

(yD - yC)/(yB - yC) = (xD - xC)/(xB - xC)

Similarly, we can determine the x coordinate for E using the relationship

(xE - xC)/(xA - xC) = (yE - yC)/(yA - yC)

We can then determine the heads at D and E from the relationships

(hD - hC)/(hB - hC) = (xD - xC)/(xB - xC)

(hE - hC)/(hA - hC) = (yE - yC)/(yA - yC)

The hydraulic gradient in the x direction is then computed as

= (hB - hE)/(xB - xE)

and that in the y direction as

= (hA - hD)/(yA - yD)

These two hydraulic gradient vectors are then added together to get a resultant gradient vector with magnitude

and direction

where a is an angle measured counterclockwise from the positive x-axis (if both gradients are in the positive x and y directions).

II. Definition of the water table

The definitions of hydraulic head, elevation head and pressure head allow us to define the water table as the location at which the following relations are true

a. Hydraulic head is equal to the elevation: h = z

b. The pressure is zero gage: p = po (atmospheric pressure)

c. The pressure head is zero: p/rwg = 0

III. Darcy's K

The proportionality constant in Darcy's Law, K, is known as the hydraulic conductivity and has units of L/T. The hydraulic conductivity is a function of both the properties of the porous medium that contribute to resistance to flow (i.e. the surface area of solids per unit volume of water), but also of the properties of the fluid itself, since internal friction in the fluid also causes mechanical energy to be converted to heat during flow. The properties of the porous medium alone are quantified by the "intrinsic permeability" k. Hydraulic conductivity is directly proportional to k. The relevant property of the fluid is its viscosity, mw, and K is inversely proportional to viscosity. Finally, since head is mechanical energy per unit weight of fluid, we need to put the unit weight of the fluid, gw = rwg, back into K so that we actually have mechanical energy in Darcy's Law. The result is the following definition

There are also a number of empirical correlations that have been proposed to allow estimation of K for granual porous media using measured values of porosity or grain size. Some of these are listed in Table 3.5 of S&Z. Useful parameters of a grain size distribution, as illustrated in Figure 3.7 of S&Z, include the median diameter, d50, the "effective diameter", d10 (10% of the grains are finer than this diameter), and the log standard deviation of the grain size distribution.

IV. Ranges of K for geologic materials

Table 3.4 of S&Z lists ranges of K for common geologic materials (both sediments and rocks). Note the very large range of values (at least 12 orders of magnitude).

V. Direct measures of K via laboratory permeameters

A. Constant Head Permeameters

Figure 3.8a in S&Z is a diagram of a constant head permeameter,
similar to those that will be used in lab. Note that the length*
h *indicated on this diagram actually corresponds to the *change
in head* (D*h*) across the
column. This change in head corresponds to hin
- hout. We have assumed that there is no
loss of head through the porous plates that enclose the sediment
filled portion of the column, and no change in head in the water
that fills the reservoirs and tubes outside of the sediment filled
column. Thus, the head at the inlet (bottom) end of the column
is equal to the head at the water surface in the reservoir and
the head at the column outlet is equal to the head at the water
surface above the column. In the case of the columns we will use
in lab, the entire reservoir above the column fills with water,
so that the free surface where water and air are in contact (and
at which the pressure head p/rwg is equal to 0) occurs at the opening
of the downward pointing spout.

Results of the constant head permeameter experiment can be analyzed directly by rearranging Darcy's law

Q = -KA(hin-hout)/(zin-zout) = +(KA)D h/L

to solve for K as

K = (Q/A)L/D h

Note that L is positive (by definition) while (zin-zout) is negative for the column set-up shown in the text. Hence the change in sign in Darcy's Law above.

B. Falling head permeameter

A diagram of a falling head permeameter is shown in Figure
3.8b of S&Z. As in Figure 3.8a, the quantities labeled as
*h*o and *h1* should actually be shown as D*ho* and D*h1* since these represent changes in head
across the column at times to and t1.

The text provides an equation for the falling head permeameters. This equation was derived by combining Darcy's law for flow through the column with an expression for the instantaneous flow rate out of the burette. At any instant in time, the flow rate through the column is assumed to be steady so that

Qin = Qout (or eqivalently DStorage = 0)

The flow rate out of the column is quantified using Darcy's law as in the case of a constant head permeameter, but now Dh is a function of time

Qout = (KA)(Dh/L)

We can derive an equation for water flowing into the column by recognizing that flow through the burette that serves as the inlet reservoir can be quantified by

Qin = - a (dhin/dt)

where hin is the head at the inlet boundary
of the column (equal to the water level in the burette) and *a*
is the cross sectional area of the burette. Note that by this
definition, a decrease in head results in a positive Q, which
is what we want to compare to the positive (upward) Q through
the column. Also, since the head at the column outlet does not
change over time

dhin/dt = dDh/dt

The "heads" ho and h1 shown
on the figure for the falling head permeameter in the text are
actually *changes in head*, Dho and Dh1,
at times to and t1
respectively.

Setting the flow rate through the column equal to the flow rate out of the burette

-a (dDh/dt) = KA (Dh/L)

This is an ordinary differential equation that can be solved by separation of variables

Integrating, we obtain

ln (Dh1/Dho) = - (KA/aL) (t1 - to)

which can be rearranged to give