The governing equation for groundwater flow can be written usingtotal head (h) or pressure (p).

Why do we typically use head (h) as the dependent variable?

= density of water

h = z + p/gTotal head (h) = elevation head + pressure head

K = k g /

K = hydraulic conductivity (L/T)k = permeability (L) = density = viscosityg = acceleration

Hydraulic conductivity is dependent on density and viscosity of water.

Density and viscosity are dependent on temperature of water.

Mean temperature of groundwater (C) at a depth of 10m.Annual variation is 5 - 10 C.

Mean Air Temperature for Wisconsin

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Month

Tem

p (

*C)

Rule of thumb: The average groundwater temperature at around10 m below surface is 1 to 2 C warmer than the average air temperature.

For constant density,K is larger at warmer temperatures.

3.98 C

K = k g /

K = hydraulic conductivity (L/T)k = permeability (L) = density = viscosityg = acceleration

Isotherms showing a plume of “hot” water in the aquifer(Winslow 1962)

Head

Temperature

(with advectionof groundwater)

Conduction only

If temperature is a variable, we needto couple the groundwater flow modelto a heat transport model.

groundwaterflow

Heat flow

Specified head; specified temperature

No flow;No heat flux

No flow; specified heat flux

No flow;No heat flux

Cross section through a groundwater basinBoundary conditions for a coupled

groundwater flow and heat flow model

Heat

Head - Shallow BasinResults from SHEMAT

Head – Deep Basin

Results from SHEMAT

Temperature - Shallow Basin10 – 18 C

Results from SHEMAT

Temperature - Deep Basin 10 – 28 C

Results from SHEMAT

In applications of groundwater models to shallow problemsin freshwater aquifers, we typically assume groundwater hasa constant temperature and density and viscosity of waterare constants.

Therefore, we can use total head, h, as the dependentvariable and hydraulic conductivity, K.

Furthermore, density is not constant when brines are presentor in coastal aquifers where sea water is present. A governing equationthat allows for variation in density is used in these applications as well.

In applications of groundwater models to geological problems(e.g., earthquakes, fluid movement along deep-seated faults,plate tectonics), temperature is an important variable and thegoverning equation is written in terms of pore pressure, p, allowingdensity and viscosity to vary. Permeability, k, is used instead of K.The flow model is coupled to a heat transport model that calculatestemperatures.