Derivative Analysis

by Glenn M. Duffield, President, HydroSOLVE, Inc.

What Is
Derivative Analysis?

Derivative analysis is powerful diagnostic tool that enhances the interpretation of data from pumping tests. Features difficult to discern in drawdown data alone are often readily apparent through the application of the derivative procedure. Groundwater hydrologists use derivative analysis to identify important flow regimes encountered during a pumping test, detect aquifer boundaries and select appropriate aquifer models.

Derivative analysis has its origins in the petroleum industry literature (Bourdet et al. 1983; Bourdet et al. 1989) and has been gaining attention among groundwater hydrologists for a number of years (Spane and Wurstner 1993; Renard et al. 2009).

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.

Derivative plot
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:

s ln T i = Δ s i-1 / Δ ln T i-1 Δ ln T i+1 + Δ s i+1 / Δ ln T i+1 Δ ln T i-1 Δ ln T i-1 + Δ ln T i+1

where s is drawdown and 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

Δ s i-1 / Δ ln T i-1

and

Δ s i+1 / Δ ln T i+1

These slopes are also known as the left and right derivatives, 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 logarithmic separation or differentiation interval (L) required to remove noise ranges between 0.1 and 0.5 (Horne 1995); however, L values as large as 1.0 may be necessary for infrequently sampled data. 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.

End Effects

End effects occur when computing derivatives near the beginning or end of a set of drawdown data. For example, fewer data points are available for computing the right derivative near the end of a test. Bourdet et al. (1989) provide procedures for overcoming such computational limitations, but one often finds in practice that derivatives calculated near the end of a data set are less reliable (Horne 1995).

Application of Derivative Smoothing

Successful application of derivative analysis nearly always requires smoothing to remove noise from the calculated derivatives. The benefit of derivative smoothing is illustrated by the following example from a constant-rate pumping test in an unconfined aquifer (Kruseman and de Ridder 1994). Without smoothing, the derivative is noisy and yields little useful information. Application of smoothing produces a cleaner derivative signal that suggests delayed yield in an unconfined aquifer.

Derivative Plot Without Smoothing

Derivative plot, no smoothing
Plot of drawdown (squares) and derivative (crosses) from a piezometer monitored during a constant-rate pumping test in an unconfined aquifer (Kruseman and de Ridder 1994). The derivatives calculated without smoothing (nearest neighbor method) yield no important information.

Derivative Plot With Smoothing

Derivative smoothing, Bourdet method

Plot of drawdown (squares) and derivative (crosses) from a piezometer monitored during a constant-rate pumping test in an unconfined aquifer (Kruseman and de Ridder 1994). The smoothed derivatives calculated with the Bourdet method suggest delayed yield. For this example, smoothing with the Spane method produces a similar derivative plot.

Flow
Regimes

Derivative analysis is an invaluable tool for diagnosing of a number of distinct flow regimes. Examples of flow regimes that one may discern with derivative analysis include infinite-acting radial flow, wellbore storage, linear flow, bilinear flow, inter-porosity flow and boundaries.

To help identify flow regimes, it is convenient to classify them, in a broad sense, according to their time of occurrence during a constant-rate pumping test (early, intermediate or late). Of course, this classification is idealized and some of the features noted may not become apparent in a pumping test of short duration. Well locations and aquifer geometries also play a role in the chronology of flow regimes. For example, wells located near a river may not exhibit the derivative plateau associated with infinite-acting radial flow before a recharge boundary effect is observed.

Flow Regimes Classified by Time of Occurrence

Early Time Flow Regimes

Intermediate Time Flow Regimes

Late Time Flow Regimes

Summary of Flow Regime Characteristics

Flow Regime Characteristic
infinite-acting radial flow derivative plateau
wellbore storage 1:1 slope on log s vs log t
linear flow (1, 2, 3, 4) 1:2 slope on log s vs log t
bilinear flow (1, 2) 1:4 slope on log s vs log t
recharge boundary drawdown plateau
barrier boundary derivative plateaus separated by factor of two
pseudo-steady state flow 1:1 slope on log s vs log t

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. A catalog of derivative plots is invaluable to the practicing hydrogeologist by providing models (signatures) of drawdown and derivative responses for specific flow regimes and boundary conditions. The following catalog includes aquifer models and flow regimes not available in compilations by Spane and Wurstner (1993) and Renard et al. (2009).

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
constant 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

To identify wellbore storage in the control (pumped) well, look for coincident drawdown and derivative curves having a unit (1:1) slope at early time.

Derivative plot for pumped well with wellbore storage in confined aquifer
Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming a fully penetrating, finite-diameter pumping well with wellbore storage and no wellbore skin. Drawdown and derivative curves attain 1:1 slopes at early time. Derivative curve attains plateau at late time (infinite-acting radial flow) approximately 1.5 log cycles after peak.
Derivative plot for pumped well with wellbore storage and wellbore skin in confined aquifer
Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming a fully penetrating, finite-diameter pumping well with wellbore storage and wellbore skin. Drawdown and derivative curves attain 1:1 slopes at early time. Derivative curve attains plateau at late time (infinite-acting radial flow). Wellbore skin increases separation between the drawdown and derivative curves compared to no skin case (above).
Derivative plot for partially penetrating pumped well with wellbore storage in confined aquifer
Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming a partially penetrating, finite-diameter pumping well with wellbore storage and no wellbore skin. Drawdown and derivative curves attain 1:1 slopes at early time. Derivative curve attains plateau at late time (infinite-acting radial flow). Like wellbore skin, partial penetration (pseudoskin) increases separation between the drawdown and derivative curves compared to fully penetrating, no skin case (above).

Observation Well, Line Source

Derivative plot for observation well assuming partially penetrating 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 observation well assuming partially penetrating finite-diameter pumping well
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

To identify a single infinite recharge (constant-head) boundary, look for a drawdown plateau and derivative curve plunging toward zero at late time. This behavior is similar to a leaky confined aquifer with an incompressible aquitard and constant-head source aquifer.

Derivative plot for observation piezometer assumng partially penetrating line source and 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

To identify a single infinite barrier (no-flow) boundary, look for two derivative plateaus separated by a factor of two. On semi-log axes, the drawdown slope doubles.

Derivative plot for observation piezometer assuming partially penetrating line source and 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

To identify linear flow in a channel (strip) aquifer, look for drawdown and derivative curves having a 1:2 slope and factor of two separation at late time.

Derivative plot for an observation well assuming partially penetrating line source and 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 when flow is linear in the channel aquifer.

Channel Aquifer with Permeable Boundaries

To identify linear flow in a channel (strip) aquifer with permeable boundaries, look for drawdown and derivative curves having a 1:2 slope and factor of two separation at intermediate time. To identify bilinear flow, look for drawdown and derivative curves having a 1:4 slope and factor of four separation at late time.

Derivative plot for observation piezometer assuming fully penetrating line source and channel aquifer with permeable walls
Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a fully penetrating, line-source pumping well and a channel (strip) aquifer with permeable walls. Derivative plateau at early time indicates infinite-acting radial flow. Drawdown and derivative curves attain 1:2 slope at intermediate time when flow is linear in the channel aquifer; drawdown and derivative curves attain 1:4 slope at late time due to leakage from boundaries.

Closed Aquifer

To identify pseudo-steady state flow in a closed aquifer, look for coincident drawdown and derivative curves having a unit (1:1) slope at late time.

Derivative plot for observation piezometer assuming partially penetrating line source and 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 during pseudo-steady-state flow regime.

Leaky Confined Aquifer

Partial Penetration, Incompressible Aquitard, Case 1

Derivative plot for observation piezometer assuming partially penetrating line source in leaky confined aquifer with incompressible aquitard
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 observation piezometer assuming fully penetrating line source in a leaky confined aquifer with incompressible aquitard
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 observation piezometer assuming fully penetrating line source in leaky confined aquifer with 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 observation piezometer assuming fully penetrating line source in leaky confined aquifer with 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.

Full Penetration, Compressible Aquitard, Channel Aquifer

Derivative plot for observation piezometer assuming fully penetrating line source in a leaky confined channel aquifer
Derivative plot for a piezometer in a leaky confined channel aquifer assuming a fully penetrating, line-source pumping well, a compressible aquitard and source aquifer with drawdown. Linear horizontal flow in channel aquifer combined with linear vertical flow across aquitard produces bilinear flow (1:4 slope) at intermediate time. Depletion of aquitard storage culminates in linear flow (1:2 slope) at late time.

Unconfined Aquifer

Instantaneous Drainage at Water Table

Derivative plot for observation piezometer assuming a partially penetrating line source in unconfined aquifer with 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 observation piezometer assuming partially penetrating line source in unconfined aquifer with 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 observation piezometer assuming partially penetrating line source in a double-porosity aquifer with fracture skin
Derivative plot for a piezometer in an infinite nonleaky confined double-porosity aquifer assuming a partially penetrating, line-source pumping well and fracture skin.

Finite-Diameter Source

Derivative plot for observation piezometer assuming partially penetrating finite-diameter pumping well in a double-porosity aquifer with fracture skin
Derivative plot for a piezometer in an infinite nonleaky confined double-porosity aquifer assuming a partially penetrating, finite-diameter pumping well with wellbore storage and fracture skin.

Vertical Fracture

To identify linear flow to a well located along an infinite-conductivity vertical plane fracture, look for drawdown and derivative curves having a 1:2 slope and factor of two separation at early time.

Derivative plot for observation piezometer assuming a fully penetrating line source with vertical fracture
Derivative plot for a piezometer in an infinite nonleaky confined fractured aquifer assuming a single fully penetrating, infinite-conductivity, vertical-plane fracture intersecting both the pumping well and the piezometer. Drawdown and derivative curves attain 1:2 slope at early time. Flow to fracture becomes pseudo-radial at late time as indicated by derivative plateau.