Theis Equation

  C.V. Theis first published in 1935 ``The Relation Between the Lowering of the Piezometric surface and the Rate and Duration of Discharge of a Well Using Groundwater Storage''. He developed an analytic solution for the drawdown for a non-steady flow in a confined aquifer. Theis found the non-steady flow of groundwater to be analagous to the unsteady flow of heat in a homogeneous solid. The Theis equation has become the most widely used equation in transient groundwater hydraulics and the solution in terms of drawdown is

where

      = drawdown at distance(r) at time (t) after the start of pumping [L]

Q         = discharge rate [LT]

      = well function of Theis

Specifically

where
Euler`s constant = .577215664901532860606512
Conventional Units

where

t       = time since pumping started(days)

    = drawdown [ft]

              at distance r and time t after pumping began

T       = aquifer transmissivity [gpd/ft]

S       = aquifer storativity

Q       = discharge or well-pumping rate [gpm]

The exponential integral is easily calculated with u defined as above and is known as the Well Function, W(u). Storativity is the addition or release of water to the storage space due to the increase or decrease of hydraulic head, while transmissivity is a function of the hydraulic conductivity and the thickness of the aquifer and descibes how easily the aquifer moves groundwater through its pore spaces. If T, S, and the pumping rate Q are known for the aquifer, the drawdown can be easily calculated. A plot of the cone of depression can be calculated using values of or the drawdown at various values of r for a given time t. (See Appendix E) Examples ``For a given aquifer the cone of depression increases in depth and extent with increasing time. Drawdown at any point at a given time is directly proportional to the pumping rate and inversely proportional to aquifer transmissivity and aquifer storativity.''[2] Figure 5 illustrates these relationships between high and low transmissivity and storativity. Low transmissivity produces a tight ``v'' shape, while high transmissivity pulls the cone out into a wider, more shallow shape. High and low storativity both create a wide cone with the low storativity having a deeper v-shape.

  
Figure 5: Effects of Storativity and Transmissivity on the Cone of Depression

The calculation of the cone of depression for different pumping rates for various periods of time is just one small part of a many faceted and detailed evaluation of a potential well site. To ensure the performance and efficiency of the well and the protection of the aquifer being pumped from requires many tests and a careful study of all the factors invloved. Assumptions underlying all of the derivations must be taken into consideration also. Due to the nature of the assumptions in the model for drawdown using the Theis equation, it is most commonly used for single well analysis.