Derivative Analysis
by Glenn M. Duffield, President, HydroSOLVE, Inc.
What Is Derivative Analysis?
Derivative analysis is an important diagnostic tool for the interpretation of pumping tests. This powerful technique was introduced in the petroleum industry literature (Bourdet et al. 1983; Bourdet et al. 1989) and has been growing in popularity among groundwater hydrologists for a number of years (Spane and Wurstner 1993).
The derivative used for pumping test interpretation is given by the slope of drawdown data plotted on a graph with semilog axes (linear drawdown and logarithmic time) as shown on the figure below.
Plot of drawdown (squares) and derivative (crosses) from a constant-rate pumping test in a nonleaky confined aquifer (Walton 1962).
Note that the derivative reaches a plateau when the Cooper-Jacob straight line method would be appropriate for matching drawdown data to estimate aquifer properties in a nonleaky confined aquifer of infinite extent (infinite-acting radial flow).
Derivative Calculation
The Bourdet derivative (Bourdet et al. 1989) uses the following simple three-point formula to compute derivatives from drawdown data by numerical differentiation:
where T is an appropriate time function (e.g., elapsed time or Agarwal equivalent time). Essentially, this formula is a weighted average of slopes computed from data points on either side of data point i. In the above formula, the two slopes are
and
These slopes are also known as the left and right derivative, respectively.
An important aspect of performing derivative analysis is the selection of an appropriate calculation method. Bourdet (2002) recommends using a nearest neighbor method (adjacent points) for preliminary derivative analysis; however, this method often results in noisy derivative data. To remove noise from calculated derivatives, the Bourdet method uses data points separated by a fixed distance measured in logarithmic time. Typically, the separation or differentiation interval required to remove noise ranges between 0.1 and 0.5 log cycle. In selecting the differentiation interval, one must exercise care to avoid overly smoothing the data.
Spane and Wurstner (1993) present an alternate method for computing derivatives. Like the Bourdet method, the Spane method uses a logarithmic differentiation interval; however, instead of using three points in the derivative computation, the Spane method computes the left and right derivatives by applying linear regression to all of the points falling within the differentiation interval. In some cases, one finds that the Spane method produces a smoother derivative than the Bourdet method.
Catalog of Derivative Plots
Derivative plots combine drawdown and derivative data on a single plot. The typical derivative plot used for diagnostic purposes is displayed on log-log axes.
On the derivative plots presented below, drawdown and derivative responses are displayed as solid blue and red curves, respectively. For reference, the Theis solution is shown on selected plots by a dashed black curve. The following table provides well and aquifer parameters assumed for the plots (unless otherwise noted):
| Pumping (Control) Well | |
| discharge rate = | 0.002 m3/min |
| casing radius = | 0.1 m |
| well radius = | 0.1 m |
| depth to top of screen = | 5 m |
| screen length = | 5 m |
| Piezometer | |
| radial distance = | 3.16 m |
| depth = | 0.75 m |
| Aquifer | |
| thickness = | 10 m |
| vertical-to-horizontal anisotropy = | 0.5 |
Nonleaky Confined Aquifer
» Finite-Diameter Source with Wellbore Storage
Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming a partially penetrating, finite-diameter pumping well with wellbore storage. Drawdown and derivative curves attain 1:1 slopes at early time. Derivative curve attains plateau at late time (infinite-acting radial flow).
» Observation Well, Line Source
Derivative plot for a piezometer in an infinite nonleaky confined aquifer assuming a partially penetrating, line-source pumping well. Derivative curve attains plateau at late time (infinite-acting radial flow).
» Observation Well, Finite-Diameter Source with Wellbore Storage
Derivative plot for a piezometer in an infinite nonleaky confined aquifer assuming a partially penetrating, finite-diameter pumping well with wellbore storage. Derivative curve attains plateau at late time (infinite-acting radial flow).
» Recharge Boundary
Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partially penetrating, line-source pumping well and a constant-head (recharge) boundary. Derivative plateau at intermediate time indicates infinite-acting radial flow. Recharge boundary produces constant drawdown (plateau) at late time.
» Barrier Boundary
Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partially penetrating, line-source pumping well and a no-flow (barrier) boundary. First derivative plateau indicates infinite-acting radial flow. Barrier boundary produces second derivative plateau (with twice the slope of infinite-acting period).
» Channel Aquifer
Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partially penetrating, line-source pumping well and a channel (strip) aquifer with impermeable walls. Derivative plateau at intermediate time indicates infinite-acting radial flow. Drawdown and derivative curves attain 1:2 slope at late time.
» Closed Aquifer
Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partially penetrating, line-source pumping well and a closed aquifer with impermeable walls. Derivative plateau at intermediate time indicates infinite-acting radial flow. Drawdown and derivative curves attain 1:1 slope at late time.
Leaky Confined Aquifer
» Partial Penetration, Incompressible Aquitard, Case 1
Derivative plot for a piezometer in an infinite leaky confined aquifer assuming a partially penetrating, line-source pumping well, an incompressible aquitard and a constant-head source aquifer (Hantush's Case 1). Derivative plateau at intermediate time indicates infinite-acting radial flow before drawdown departs from the Theis solution for a nonleaky confined aquifer.
» Full Penetration, Incompressible Aquitard, Case 1
Derivative plot for a piezometer in an infinite leaky confined aquifer assuming a fully penetrating, line-source pumping well, an incompressible aquitard and a constant-head source aquifer (Hantush's Case 1). Derivative plateau at intermediate time indicates infinite-acting radial flow before drawdown departs from the Theis solution for a nonleaky confined aquifer.
» Full Penetration, Compressible Aquitard, Case 1
Derivative plot for a piezometer in an infinite leaky confined aquifer assuming a fully penetrating, line-source pumping well, a compressible aquitard and a constant-head source aquifer (Hantush's Case 1). Release of water from storage in the aquitard results in early departure of drawdown from the Theis solution for a nonleaky confined aquifer.
» Full Penetration, Compressible Aquitard, Case 2
Derivative plot for a piezometer in an infinite leaky confined aquifer assuming a fully penetrating, line-source pumping well, a compressible aquitard and no source aquifer (Hantush's Case 2). Release of water from storage in the aquitard results in early departure of drawdown from the Theis solution for a nonleaky confined aquifer. Derivative plateau at late time is equivalent to infinite-acting radial flow in nonleaky confined aquifer.
Unconfined Aquifer
» Instantaneous Drainage at Water Table
Derivative plot for a piezometer in an infinite unconfined aquifer assuming a partially penetrating, line-source pumping well and delayed yield (delayed gravity response) with instantaneous drainage at water table.
» Noninstantaneous Drainage at Water Table
Derivative plot for a piezometer in an infinite unconfined aquifer assuming a partially penetrating, line-source pumping well and delayed yield (delayed gravity response) with noninstantaneous drainage at water table.
Double-Porosity Aquifer with Fracture Skin
» Line Source
Derivative plot for a piezometer in an infinite confined double-porosity aquifer assuming a partially penetrating, line-source pumping well and fracture skin.
» Finite-Diameter Source
Derivative plot for a piezometer in an infinite confined double-porosity aquifer assuming a partially penetrating, finite-diameter pumping well with wellbore storage and fracture skin.
Vertical Fracture
Derivative plot for a piezometer in an infinite confined fractured aquifer assuming a single fully penetrating, vertical-plane fracture intersecting both the pumping well and the piezometer. Drawdown and derivative curves attain 1:2 slope at early time. Derivative plateau at late time indicates infinite-acting radial flow.