6.3 General Assumptions Used to Develop Analytical Well Hydraulic Equations
Several simplifying assumptions are required to generate workable analytical solutions from linear differential equations. Such equations are used to forecast the effect of pumping on production and observation well water levels. The resulting analytical solutions are powerful, but their use has limitations as they are simplified to represent ideal conditions that are not usually satisfied in the real world.
Many textbooks list the general assumptions used to formulate standard analytical hydraulic test equations (e.g., Freeze & Cherry, 1979; Fetter, 2001; Kasenow, 2001). In this book we use a modified version of those listed in Fetter (2001) including twelve general assumptions.
- A confining layer underlies the geologic unit being pumped. This assumption states no water moves vertically upward from the underlying formations.
- Geologic units being pumped are essentially horizontal. This assumption eliminates the need to account for flow that is not horizontal.
- The geologic unit being pumped has infinite lateral extent. This assumption eliminates the presence of boundary conditions that could limit the expansion of the cone of depression (e.g., impermeable or recharge boundaries).
- The potentiometric surface and water table are stable (do not change with time) prior to pumping, and all subsequent changes in the potentiometric surface and water table are a result of pumping. This assumption eliminates the complications of having the background water level fluctuate prior to the start of pumping or during pumping. Under these conditions the potentiometric surface and water table are initially static, and all drawdown is caused by the pumping well. This removes those variations in drawdown that may be created by nearby pumping wells or natural water level variations (e.g., recharge, discharge, barometric, or tidal effects).
- The geologic unit being pumped is isotropic and homogeneous. This assumption means that hydrogeologic properties such as K, T, S, and Sy as uniform and constant. Under these conditions the cone of depression will be symmetrical.
- Flow in the pumped formation is radial. This assumption allows radial flow theory and mathematics to be applied. It also says observation wells located at different radial distances from the pumping well do not have to be in a straight line to describe how pumping effects water levels at various distances from the pumping well.
- Flow in the pumped formation is essentially horizontal. This factor eliminates the need to account for vertical flow and the resulting more complex mathematics.
- Darcy’s law governs groundwater flow. This assumption allows governing equations based on the validity of Darcy’s Law to be used to formulate the required mathematics (i.e., groundwater flow is laminar, as discussed by Woessner and Poeter (2020)).
- The viscosity and density of the groundwater is constant. This assumption eliminates the need to account for variations in viscosity and density which cause K to vary and produce gradients in addition to the hydraulic gradients caused by pumping the well. This assumption requires that temperature and dissolved solute concentration are reasonably uniform throughout the aquifer over the time period when the equation is applied. Again, formulating a solution that includes these factors requires additional mathematical rigor.
- The observation and pumping wells are screened over the entire geologic unit being tested and are 100 percent efficient. The fully screened assumption assures that the flow of the water in the geologic unit is horizontal. The 100 percent efficient requirement means there are no frictional nor turbulent head losses as water enters the pumped well. This is important because head loss causes water levels to be lower in the pumped well than in the aquifer immediately adjacent to the well, whereas the equations assume the head in the pumping well is equal to the head in the aquifer at the locations of the well. The 100 percent efficient requirement also means there is no skin effect. A skin is a zone in the formation around the borehole that has been altered as a result of drilling, often due to drilling fluid permeating the pores of the formation and decreasing hydraulic conductivity in the zone. Observation wells are not pumped so they do not have losses due to friction or turbulence although they may have a skin that may limit the rate of water level response to pumping.
- The location of the pumping well can be represented as a single point that has an infinitesimal diameter. This assumption simplifies the mathematics of calculating the drawdown at a distance, r, from the pumping well and eliminates the need to consider the effect of water stored in the well bore, a condition that would impact the drawdown observed in the pumping well (referred to as casing/wellbore storage effects).
- The well discharge is constant. This factor eliminates the need to define the well discharge as a variable entity in the mathematical solution.
Analytical well hydraulic equations can be thought of as tools to describe how any selected well discharge affects the head distribution in the hydrogeologic unit being pumped. Analytical solutions provide continuous solutions. This means under steady‑state conditions (Section 7) they can be used to generate an estimate of drawdown at any location in the problem domain as defined by the radial distance from the pumping well. When equations are transient (Sections 8‑12), they describe conditions at any location and time since pumping started.
Sections 7 through 12 develop analytical equations that describe how wells effect water levels in confined and unconfined units under steady‑state and transient conditions. These equations can be applied to estimate drawdown resulting from pumping of a well at a specified rate when hydrogeologic properties are known. These equations are also used to analyze pumping test results and compute hydrogeologic property values from the time/drawdown data for the hydrogeologic units pumped.