The following discussions are really applicable to line frequencies. For higher frequencies, dielectric loss must be considered, as well as skin effect. For lower frequencies (i.e. DC) consideration should be given to electrolytic effects such as polarization.
The dominant effect for a ground is the current distribution within the earth. The ground rod or wire itself typically has negligible resistance, as does the interface between the rod and soil. As such, the soil conductivity has a very large influence on the ground resistance.
Resistivity of different soils and resistance of a single rod. The rod, in the table below, is a standard rod 5/8 inch in diameter and 10 feet long (16 mm diam by 3m long) Data taken from IEEE Std 1421991.
Soil Description  Group Symbol  Avg Resistivity (kohm cm) 
Resistance of rod 

Well graded gravel, gravelsand mixtures, little or no fines  GW  60100  180300 
Poorly graded gravels, gravel sand mixtures, little or no fines  GP  100250  300750 
Clayey gravel, poorly graded gravel, sandclay mixtures  GC  2040  60120 
Silty sands, porly graded sandsilts mixtures 
SM  1050  30150 
Clayey sands, poorly graded sandclay mixtures  SC  520  1560 
Silty or clyey fine sands with slight plasticity  ML  38  924 
Fine sandy or silty soils, elastic silts  MH  830  2490 
Gravelly clays, sandy clays, silty clays, lean clays  CL  2.56  1718 
Inorganic clays of high plasticity  CH  15.5  316 
Note that for the last two, clays, the resistivity is highly dependent on soil moisture.
The following table gives resistivity (kohm cm) for three types of soil, for moisture contents from 2% to 24% by weight. Data taken from IEEE Std 1421991.
Moisture content (% by weight) 
Top Soil

Sandy Loam

Red Clay


2 
185


4 
60


6 
135

38


8 
90

28


10 
60

22


12 
35

17

180

14 
25

14

55

16 
20

12

20

18 
15

10

14

20 
12

9

10

22 
10

8

9

24 
10

7

8

The following table gives variation in soil resistivity with temperature. The significant transition is at the freezing point, and above that the resitivity drops fairly linearly with temperature. I'm not sure about the double entry for 0 degrees.. perhaps it represents the variability at the transition temperature? Data taken from IEEE Std 1421991.
Temperature (deg C)

Resistivity (kohm cm)


5

70

0

30

0 (?)

10

10

8

20

7

30

6

40

5

50

4

An expression, accurate to 15%, for a single 10ft (3m) rod 5/8" (16mm) in diameter is:
Rground(rod) = rho/335 ohms
where rho is in ohmcm (note the tables above are in kohm/cm).
Multiple rods
Multiple ground rods are often used particularly where high currents may be involved. The ground resistance is not simply the resistance of one rod, divided by the number of rods, unless the rods are very far apart. However, even though the ground resistance of the combination may not be all that much better than a single rod, the current is shared among the rods, reducing the current per rod. A guideline from IEEE Std 1521991 to avoid "smoking rods" is:
Max Current(amps) = 34.8E3 * d * L / sqrt(rho * t)
where:
d is the rod diameter in meters
L is the rod length in meters
rho is the soil resistivity in ohm meters
t is the duration of the current in seconds (and is valid for short times only)
The text of IEEE Std 1421991 claims that for 1 ft 5/8" rod (32 cm x 16 mm) this expression yields 116A in 2500 ohmcm soil and 58A for 10,000 ohmcm soil, however I don't get these numbers.
Here's a table of some formulae to calculate ground resistance for combinations of rods. This table is taken from IEEE Std 1421991, but it's cited there as coming from: Dwight, H.B.,"Calculation of resistance to ground", AIEE Transactions, vol 55, Dec 1936, pp 13191328.
Picture  Description  expression for R 
Hemisphere, radius a  =rho/(2*pi*a)  
One rod, Length L, radius a  =rho/(2*pi*L)*(ln(4*L/a)1)  
Two ground rods spacing s, s>L  =rho/(4*pi*L)*(ln(4*L/a)1) + rho/(4*pi*s)* (1  1/3 * (L/s)^2 + 1/5 * (L/s)^4) ...) 

Two rods, spacing s, s<L  =rho/(2*pi*L)* (ln(4*L/a)  ln(4*L/s)  2 + 1/2 * s/L  1/16 * (s/L)^2 + 1/512 * (s/L)^4 ...) 

Buried horizontal wire, length 2L, depth s/2  =rho/(4*pi*L)* (ln(4*L/a) +ln(4*L/s)  2 + 1/2 * s/L  1/16 * (s/L)^2 + 1/512 * (s/L)^4 ...) 

right angle turn of wire, Arm length L, depth s/2 (essentially 2 arms at right angles) 
=rho/(4*pi*L)* (ln(2*L/a) +ln(2*L/s)  0.2373 + 0.2146 * s/L + 0.1035 * (s/L)^2  0.0424 * (s/L)^4 ...) * I'm not sure of this one, the sign pattern is very different from the others! 

3 point star  =rho/(6*pi*L)* (ln(2*L/a) +ln(2*L/s) + 1.071  0.209 * s/L + 0.238 * s^2/L^2  0.054* s^4/L^4  ...) 

4 point star (cross)  =rho/(8*pi*L)* (ln(2*L/a) +ln(2*L/s) + 2.912  1.071 * s/L + 0.645 * s^2/L^2  0.145 * s^4/L^4  ...) 

6 point star  =rho/(12*pi*L)* (ln(2*L/a) +ln(2*L/s) + 6.851  3.128 * s/L + 1.758 * s^2/L^2  0.490 * s^4/L^4  ...) 

8 point star  =rho/(16*pi*L)* (ln(2*L/a) +ln(2*L/s) + 10.98  5.51 * s/L + 3.26 * s^2/L^2  1.17 * s^4/L^4  ...) 

ring of wire, diameter of ring D, diameter of wire d, depth s/2  =rho/(2*pi^2*D)* (ln(8*D/s)+ln(4*D/s)) 

buried horizontal strip length 2L, section a x b, depth s/2, b<a/8 
=rho/(4*pi*L)* 

buried horizontal round plate, radius a, depth s/2 
= rho/(8*a) + 

buried vertical round plate, radius a, depth s/2  = rho/(8*a) + rho/(4*pi*s) * ( 1  7/24 * (a/s)^2 + 99/320 * (a/s)^4  ...) 
hv/grounds.htm  18 March 2003  Jim
Lux
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