Step-Drawdown Tests (Step Tests)
by Glenn M. Duffield, President, HydroSOLVE, Inc.
What Is A
A step-drawdown test (or step test) is a single-well pumping test designed to investigate the performance of a pumping well under controlled variable discharge conditions. In a step-drawdown test, the discharge rate in the pumping well is increased from an initially low constant rate through a sequence of pumping intervals (steps) of progressively higher constant rates. Each step is typically of equal duration, lasting from approximately 30 minutes to 2 hours (Kruseman and de Ridder 1994). Each step should be of sufficient duration to allow dissipation of wellbore storage effects.
In addition to estimating hydraulic properties of an aquifer system such as transmissivity and hydraulic conductivity, the goal of a step-drawdown test is to evaluate well performance criteria such as well loss and well efficiency.
Typically, aquifer properties and well-loss coeffivients are estimated from a pumping test by fitting mathematical models (type curves) to drawdown data through a procedure known as curve matching (Figure 1).
Pumping rates used in a step-drawdown test should encompass the maximum design pumping rate for the well. The following table shows one possible scheme for the pumping rates in a step-drawdown test consisting of four steps:
Jacob (1947) proposed the following drawdown equation to account for linear and nonlinear head losses in the pumping well at time t:
Δhw(t) = B(rw*,t)Q + CQ²
where Δhw(t) is drawdown in the pumped well at time t [L], B is a time-dependent linear (laminar) head-loss coefficient [T/L²], rw* is effective radius of the pumped well [L], Q is pumping rate [L³/T], and C is a nonlinear (turbulent) well-loss coefficient.
The effective well radius, rw*, is defined as the radial distance from the center of the well at which the theoretical drawdown in the aquifer (aquifer loss) is equal to the total linear head loss in the well (total drawdown in the well neglecting turbulent loss).
Linear Well Loss
The linear head-loss coefficient, B, consists of two components, aquifer loss and linear well loss, which Kruseman and de Ridder (1994) define as B1 and B2, respectively, as follows:
B(rw*,t) = B1(rw,t) + B2
where rw is the nominal radius of the well [L]. B2 is assumed independent of time. B2 > 0 implies rw* < rw; B2 < 0 implies rw* > rw.
For a well experiencing no well loss, Jacob's drawdown equation reduces to:
Δhw(t) = B1(rw,t)Q
Ramey (1982) defines the linear well-loss coefficient in terms of a dimensionless wellbore skin factor, Sw, as follows:
Sw = B22πT
where T is transmissivity [L²/T].
Assuming nonlinear (turbulent) well loss is negligible, the skin effect is the difference between the total drawdown in the well and the theoretical drawdown (aquifer loss) at the well screen. A positive skin (B2 > 0) indicates permeability reduction at the wellbore (e.g., clogging from drilling debris or biofouling). Negative skin (B2 < 0) suggests permeability enhancement (e.g., stimulation).
Nonlinear Well Loss
Δhw(t) = B(rw*,t)Q + CQP
where P is the order of nonlinear well losses. According to Rorabaugh (1953), the value of P can assume values ranging from 1.5 to 3.5 depending on the value of Q, but many researchers accept the value of P=2 as proposed by Jacob (1947).
When P = 2, C has the peculiar units of T2/L5. Walton (1962) reported the following interpretation of nonlinear well loss coefficients:
|Nonlinear Well Loss Coefficient, C|
|mild||5 - 10||1900 - 3800|
When P ≠ 2, the units for C are TP/L3P-1.
Wwell efficiency, which expresses the ratio of aquifer loss (theoretical drawdown) to total (measured) drawdown in the pumped well, is computed from a step-drawdown test as follows (Kruseman and de Ridder 1994):
Ew = B1(rw,t)Q/Δhw(t) x 100%
When the linear and nonlinear well loss terms are zero, Ew = 100%.
It's a straightforward matter to adapt existing pumping test solutions to the analysis of step-drawdown tests by simply adding the nonlinear well loss term CQP to the drawdown calculated in the pumped well (Bear 1979).
Figure 2 illustrates the application of the Hantush and Jacob (1955) solution for the analysis of a step-drawdown test in a leaky confined aquifer.