Defination Derivative Analisys

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The Bourdet derivative (Bourdet et al. 1989) uses the following simple three-point formula to

compute derivatives from drawdown data by numerical differentiation:

where s is drawdown and Tis 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 anearest 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 (Horne1995); 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 leftand 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 drawdowndata. 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 requiressmoothing 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 unconfinedaquifer.

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Derivative Plot Without 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). Thederivatives calculated without smoothing (nearest neighbor method) yield no importantinformation.

Derivative Plot With Smoothing

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Plot of drawdown (squares) and derivative (crosses) from a piezometer monitored during aconstant-rate pumping test in an unconfined aquifer (Kruseman and de Ridder 1994). The

smoothed derivatives calculated with the Bourdet method suggestdelayed yield. For thisexample, 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). Ofcourse, 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

wellbore storage (confined aquifer)

linear flow (vertical fracture)

Intermediate Time Flow Regimes

infinite-acting radial flow (see most examples in catalog)

delayed yield (delayed gravity response)

inter-porosity flow (double porosity)

leakage (incompressible aquitard, Case 1; compressible aquitard, Case 1;compressible aquitard,

Case 2)

bilinear flow (compressible aquitard, channel aquifer)

Late Time Flow Regimes

infinite linear boundary (recharge boundary, barrier boundary)

linear flow (channel aquifer)

bilinear flow (channel aquifer with permeable boundaries)

pseudo-steady-state flow (closed aquifer)

Summary of Flow Regime Characteristics

Flow Regime Characteristic

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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 catalogincludes 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 derivativeresponses are displayed as

solid blue and redcurves, respectively. For reference, the Theis solution is shown on selectedplots by a dashed blackcurve. The following table provides well and aquifer parameters assumedfor the plots (unless otherwise noted):

Pumping (Control) Well

constant discharge rate = 0.002 m3/min

casing radius = 0.1 mwell 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

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

Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming afullypenetrating, 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 plateauat late time (infinite-acting radial flow) approximately 1.5 log cycles after peak.

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Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming afullypenetrating, 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 latetime (infinite-acting radial flow). Wellbore skin increases separation between the drawdown and

derivative curves compared to no skin case (above).

Derivative plot for pumped well in an infinite nonleaky confined aquifer assuming apartially

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 plateauat 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

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Derivative plot for a piezometer in an infinite nonleaky confined aquifer assuming a partiallypenetrating, 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 partiallypenetrating, finite-diameter pumping well with wellbore storage. Derivative curve attains plateau

at late time (infinite-acting radial flow).

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Recharge Boundary

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

Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partiallypenetrating, line-source pumping well and a constant-head (recharge) boundary. Derivativeplateau 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.

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Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partiallypenetrating, line-source pumping well and a no-flow (barrier) boundary. First derivative plateau

indicates infinite-acting radial flow. Barrier boundary produces second derivative plateau (withtwice 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.

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Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partiallypenetrating, line-source pumping well and a channel (strip) aquifer with impermeable walls.

Derivative plateau at intermediate time indicates infinite-acting radial flow. Drawdown andderivative 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 bilinearflow, look for drawdown and derivative curves having a 1:4 slope and factor of four separation at late time.

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Derivative plot for a piezometer in a bounded nonleaky confined aquifer assuming a partiallypenetrating, line-source pumping well and a closed aquifer with impermeable walls. Derivative

plateau at intermediate time indicates infinite-acting radial flow. Drawdown and derivative curvesattain 1:1 slope at late time during pseudo-steady-state flow regime.

Leaky Confined Aquifer

Partial Penetration, Incompressible Aquitard, Case 1

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Derivative plot for a piezometer in an infinite leaky confined aquifer assuming a partiallypenetrating, 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 radialflow 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

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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 ofdrawdown 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).

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Release of water from storage in the aquitard results in early departure of drawdown from theTheis 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 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 producesbilinear 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

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Derivative plot for a piezometer in an infinite unconfined aquifer assuming a partiallypenetrating, 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 partiallypenetrating, line-source pumping well and delayed yield (delayed gravity response) with

noninstantaneous drainage at water table.

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Double-Porosity Aquifer with Fracture Skin

Line Source

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

Finite-Diameter Source

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Derivative plot for a piezometer in an infinite nonleaky confined double-porosity aquifer assuminga 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 a piezometer in an infinite nonleaky confined fractured aquifer assuming a

single fully penetrating, infinite-conductivity, vertical-plane fracture intersecting both thepumping 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.

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