# 3 Theis Solution

The concept of storage is most frequently encountered in hydrogeology as the “*S*” parameter in Theis’ (1935) solution. Theis conceptualized the well-drawdown problem in heat conduction terms, which he expressed in correspondence to his former college classmate, Clarence Lubin, who had become a mathematics professor at the University of Cincinnati (Freeze, 1985).

“The flow of ground water has many analogies to the flow of heat by conduction. We have exact analogies in ground water theory for thermal gradient, thermal conductivity, and specific heat. I think a close approach to the solution of some of our problems are [sic] probably already worked out in the theory of heat conduction. Is this problem in radial flow worked out?: Given a plate of given constant thickness and with constant thermal characteristics at a uniform initial temperature to compute the temperatures thruout [sic] the plate at any time after the introduction of a sink kept at 0 temperature? And a more valuable one from our standpoint: Given the same plate under the same conditions to compute the temperatures after the introduction of a sink into which heat flows at a uniform rate? I forgot to say that the plate may be considered to have infinite areal extent.”

Lubin provided Theis with the solution from Carslaw[5] (1921) according to Banks (2015) which Theis duly noted in his paper. Theis prominently stated the heat flow analogy, as he did in his earlier letter to Lubin.

“Darcy’s law is analogous to the law of the flow of heat by conduction, hydraulic pressure being analogous to temperature, pressure-gradient to thermal gradient, permeability to thermal conductivity, and

specific yield to specific heat. Therefore, the mathematical theory of heat-conduction developed by Fourier and subsequent writers is largely applicable to hydraulic theory. This analogy has been recognized, at least since the work of Slichter[6], but apparently no attempt has been made to introduce the function of time into the mathematics of ground-water hydrology.”“In heat-conduction a specific amount of heat is lost concomitantly and instantaneously with fall in temperature. It appears probable, analogously, that

in elastic artesian aquifers a specific amount of water is discharged instantaneously from storage as the pressure falls.”

For groundwater storage, Theis used the variable *S*, which he originally called the “specific yield” in the quote above and not to be confused with “specific yield” of an unconfined aquifer. Tellingly, the analogy to specific heat[7] is the only attribute Theis provided for *S*, in contrast to a short description of the physical meaning of the coefficient of transmissibility *T*. Theis in his 1935 paper provided no mechanistic insight to groundwater storage; it was a property inferred from the heat flow analogy. However, Theis later elaborated on *S* in an Author’s Note added to a 1952 United States Geological Society (USGS) reprint of his 1935 paper.

“The factor S in the equations given is called ‘specific yield’ in the text of the paper. Later consideration has shown it advisable to call this term the “coefficient of storage” of the aquifer and to define it as the quantity of water in cubic feet that is discharged from each vertical prism of the aquifer with basal area equal to 1 square foot and height equal to that of the aquifer when the water level falls 1 foot.”

Here, Theis backs away from using the term “specific yield” for a confined aquifer. Today, “storativity” is used synonymously with “coefficient of storage,” and Theis’ verbal definition is the one commonly provided in textbooks and illustrated for a confined aquifer in Figure 1.

Theis’ definition of coefficient of storage is for two-dimensional radial flow. In three-dimensions, the analog of specific heat is specific storage, in which the amount of water removed from a representative elementary volume REV per unit head decline is normalized by the volume of the REV. For precision, the verbal definition will be translated into an equation, although most hydrogeology textbooks follow Theis and omit doing so. A key quantity to define is the increment of fluid content, *ζ* (Equation 1), which has its origin in soil mechanics and the theory of poroelasticity (Biot, 1941; Wang, 2000).

(1) |

where:

ΔV_{w} |
= | volume of water added to or removed from storage in an REV (L^{3}) |

V |
= | volume of representative elemental volume (L^{3}) |

The quantity Δ*V*_{w} is positive when water is added to the aquifer and negative when water is removed from the aquifer. The quantity Δ*V*_{w} represents a volume of water transported to or from an external source at a reference pressure, usually atmospheric pressure, because pressure gages typically measure the difference between the absolute pressure and atmospheric pressure (“gage pressure”). Thus, Δ*V*_{w} is an increment of water volume added to or removed from the aquifer, much as money might be added to or withdrawn from a bank account. The amount of water added to or removed from the aquifer is normalized by the volume *V* of the REV. Then translating Theis’ words into an equation gives a mathematical definition for specific storage as shown in Equation 2a and 2b.

(2a) |

Specific storage *S*_{s} has units of inverse height, such as inverse meters, 1/m. The definition can also be expressed for a change in fluid pressure because *Δp* = *ρ*_{w}*g*Δ*h*. Head (*h*) is defined and discussed in the GW-Project book by Woessner and Poeter (2020).

(2b) |

where:

ρ_{w} |
= | water density (M/L^{3}) |

g |
= | acceleration due to gravity (L/T^{2}) |

In two-dimensional radial flow, the storativity, *S*, for a confined aquifer of thickness *b*, is specific storage times aquifer thickness, *S=S*_{s}*b*, because storativity per Theis’ definition is the amount of water “discharged from each vertical prism of the aquifer” of unit area per unit decline in head. Storativity is dimensionless as the dimension of *S*_{s} is inverse length.

This recounting of Theis’ 1935 solution for drawdown due to a pumping well in a horizontal aquifer demonstrates the important role that the heat conduction analogy played in developing the concept of hydrogeologic storage. This analogy is both physical and mathematical (Table 1). Head/temperature are the driving potentials for flow of water/flow of heat according to Darcy’s/Fourier’s law. Specific storage (volume of water discharged from storage per unit aquifer volume per unit decline in head) is the analog of specific heat per unit-volume (addition of heat required to raise the temperature of a unit volume of material one degree). An interesting contrast between this analog pair is that the definition of specific storage is usually expressed as a double negative (*discharge per decline*) given the importance of extracting water from an aquifer, whereas the definition of specific heat is expressed as a double positive (*addition to raise*). Table 1 is not yet complete as the paired governing equations will be discussed in Section 5 on the diffusion equation.

**Table 1 –**Analogous quantities in groundwater flow and heat flow (Wang and Anderson, 1982; Wang, 2000; Anderson, 2007).

Groundwater Flow |
Heat Flow |

Fluid head, h = p/(ρ_{w} g) + z [m] |
Temperature, T [°K] |

Water added to storage, ΔV_{w} [m^{3}] |
Heat, ΔQ [J] |

Groundwater flux, [m^{3}/m^{2}/s = m/s] |
Heat flux, q [J/(m^{2}s) = W/m^{2}] |

Hydraulic Conductivity, K [m/s] |
Thermal Conductivity, K [J/(m °K s) = W/(m °K)] |

Specific Storage, S_{s} = S/b [1/m] |
Specific heat capacity (per unit volume), ρc [J/(°K m^{3})] |

Hydraulic diffusivity, K/S_{s} [m^{2}/s] |
Thermal diffusivity, K/(ρc) [m^{2}/s](here K is thermal conductivity, see above) |

Darcy’s law, q = –K (dh/dx) |
Fourier’s law, q = –K (dT/dx) |

*in*History of Geophysics, volume 3, The History of Hydrology, edited by Edward R. Landa and Simon Ince, pages 103112.