Principles of Flow 3

3.1 IntroductionIn almost any investigation involving groundwater, questions arise about how much wateris moving and how fast it is owing. Typical questions in contamination remediationstudies are: What should the well discharge be to capture the entire plume of contam-inated water? or How long will it take for the contaminated groundwater to reach anearby stream? Regarding water supply issues, you might hear questions like If wepump 5 million gallons/day from this well eld, will it dry up a nearby wetland andnearby domestic wells? or How much discharge can we hope to get from a well 100 ftdeep in this aquifer at this location? The answers to such questions are based on ground-water ow analyses, which in turn are based on some straightforward physical principlesthat govern subsurface ow. An empirical relationship called Darcys law and conserva-tion of mass form the basis for many useful hand calculations and computer simulationsthat can be made to analyze groundwater ow. Principles of ow are also covered inbooks by Freeze and Cherry (1979), Todd (1980), Domenico and Schwartz (1998), andFetter (2001).

3.2 Darcys Law and Hydraulic ConductivityIn 1856, French engineer Henry Darcy was working for the city of Dijon, France ona project involving the use of sand to lter the water supply. He performed laboratoryexperiments to examine the factors that govern the rate of water ow through sand (Darcy,1856; Freeze, 1994). The results of his experiments dened basic empirical principles ofgroundwater ow that are embodied in an equation now known as Darcys law.

Darcys apparatus consisted of a sand-lled column with an inlet and an outlet similarto that illustrated in Figure 3.1. Two manometers (essentially very small piezometers)measure the hydraulic head at two points within the column (h1 and h2). The sample issaturated, and a steady ow of water is forced through at a discharge rate Q [L3/T].

Darcy found through repeated experiments with a specic sand thatQwas proportionalto the head dierence h between the two manometers and inversely proportional to ()the distance between manometers s:

40 Principles of Flow

Figure 3.1 Schematicillustrating steady owthrough a sand sample.The manometersmeasure heads h1 and h2at locations 1 and 2within the column. The scoordinate direction runsparallel to the column.

Q h, Q 1s

(3.1)

Obviously, Q is also proportional to the cross-sectional area of the column A.Combining these observations and writing an equation in dierential form gives

Darcys law for one-dimensional ow:

Qs = Ksdh

dsA (3.2)

where Qs is discharge in the s direction. The constant of proportionality Ks is thehydraulic conductivity in the s direction, a property of the geologic medium. Hydraulicconductivity is a measure of the ease with which a medium transmits water; higher Ksmaterials transmit water more easily than low Ks materials. The term hydraulic conduc-tivity is sometimes abbreviated to just conductivity. The minus sign on the right side ofthis equation is necessary because head decreases in the direction of ow. If there is owin the positive s direction, Qs is positive and dh/ds is negative. Conversely, when ow isin the negative s direction, Qs is negative and dh/ds is positive.

Consider the units of the entities in Darcys law. Head h and the coordinate s both havelength units, so dh/ds is dimensionless. The dimensionless quantity dh/ds represents therate that head changes in the s direction, and is known as the hydraulic gradient. Thedimensions ofQs are [L3/T] and of A are [L2], so the hydraulic conductivityKs is [L/T].

Rocks or soils with small pores allow only slow migration of water while materialswith larger, less constricted pores permit more rapid migration. Water traveling throughsmall, constricted pores must shear itself more in the process of traveling a given distancethan water traveling through larger pores. More shearing in the water causes more viscousresistance and slower ow. Other factors being equal, the average velocity of groundwatermigration is proportional to K. Hydraulic conductivity is an empirical constant measuredin laboratory or eld experiments.

Another term, permeability, has historically been synonymous with hydraulic conduc-tivity, but now its usage is associated with intrinsic permeability, a related property thatis not specic to the uid water (see Section 3.3).

Table 3.1 lists some typical ranges of hydraulic conductivity values for common rocksand soils. These ranges are mostly based on data compiled by Davis (1969) and

Darcys Law and Hydraulic Conductivity 41

Table 3.1 Typical Values of Hydraulic Conductivity

Material K(cm/sec) Source

Gravel 101 to 100 1Clean sand 104 to 1 1Silty sand 105 to 101 1Silt 107 to 103 1Glacial till 1010 to 104 1Clay 1010 to 106 1,2Limestone and dolomite 107 to 1 1Fractured basalt 105 to 1 1Sandstone 108 to 103 1Igneous and metamorphic rock 1011 to 102 1Shale 1014 to 108 2

Sources: (1) Freeze and Cherry (1979); (2) Neuzil (1994).

summarized by Freeze and Cherry (1979). Hydraulic conductivity varies over a tremen-dous range, 12 orders of magnitude, in common geologic materials. The wide varia-tions of fracture width and frequency in crystalline rocks account for the huge ranges inobserved hydraulic conductivities in such rock. Where carbonate rocks have been erodedby dissolution, fractures widen to form large openings and talk of underground riversis not just mythology. Some basalts are also very conductive due to open columnar jointsand voids at the bases and tops of successive lava ows. Groundwater ow velocitiesin basalts and limestones can be extremely high compared to velocities in more typicalgeologic materials where the pore sizes are on the order of millimeters or smaller.

The most common units for hydraulic conductivity are meters/day and feet/day foreld studies, and cm/sec for laboratory studies. Inconsistent units such as gallons/day/foot2 have been used in older irrigation and water supply studies. For conversion factors,see Appendix A.

Example 3.1 A sample of silty sand is tested in a laboratory experimentjust like that illustrated in Figure 3.1. The column has an inside diameter of10 cm and the length between manometers is s = 25 cm. With a steadyow of Q = 1.7 cm3/min, the head dierence between the manometers ish = 15 cm. Calculate the hydraulic conductivity Ks.

This is a direct application of Eq. 3.2, with a little rearrangement at the begin-ning to isolate Ks:

Ks = Qs1A

ds

dh

= 1.7 cm3/min(

1(5 cm)2

)(25 cm15 cm

)= 0.036 cm/min

= 6.0 104 cm/sec

The sign of ds/dh is negative because as s increases, h decreases (see Fig-ure 3.1).