Pumping Test and Analysis [8 Hours]

5.1 INTRODUCTION — WHY PUMPING TESTS?

Laboratory tests on soil samples (e.g., constant-head or falling-head permeameter) provide values of hydraulic conductivity at a point, but they cannot capture the variability, layering, and effective connectivity of a real aquifer. A pumping test stresses the aquifer over a large volume and yields integrated (field-scale) parameters that directly govern well yield, drawdown prediction, and groundwater resource planning.

Objectives of a Pumping Test

1. Determine transmissivity (T = Kb) — the rate at which water is transmitted through a unit width of the aquifer under a unit hydraulic gradient. This controls well yield and drawdown.
2. Determine storage coefficient (S) — the volume of water released from (or taken into) storage per unit area per unit change in head. This controls how quickly the cone of depression expands.
3. Evaluate well performance — measure well efficiency, specific capacity, and well losses.
4. Assess aquifer boundaries — detect impermeable barriers or recharge boundaries via deviations in the drawdown curve.
5. Estimate radius of influence (R) — the distance beyond which pumping has negligible effect.
6. Design production wells — determine sustainable pumping rate, well spacing, and predict long-term drawdown.

Specific Capacity

A quick field measure of well productivity is the specific capacity, defined as the discharge per unit drawdown:

Higher specific capacity indicates a more productive well-aquifer system. It decreases with time (as drawdown increases) and with increasing Q (due to well losses).

Test Components

5.2 TYPES OF PUMPING TESTS

Pumping tests are classified according to the flow regime achieved during the test and the method of analysis used:

A. Constant-Rate Pumping Test (Aquifer Test)

The well is pumped at a constant discharge Q and drawdown is measured in one or more observation wells at increasing time intervals. This is the standard test for determining T and S. Two analysis approaches:

B. Step-Drawdown Test

The pumping rate is increased in steps (e.g., Q, 2Q, 3Q), each maintained until drawdown stabilizes. Determines:

• Well efficiency — ratio of theoretical to actual drawdown
• Well loss coefficient (C) — determines the magnitude of turbulent well losses
• Optimum pumping rate for the well

The total drawdown in the pumping well consists of aquifer loss (laminar, Darcian) and well loss (turbulent, non-Darcian):

where B = aquifer loss coefficient (depends on T, S, r, t), and C = well loss coefficient (depends on well construction). The well efficiency is:

A well efficiency > 70% is considered acceptable. Low efficiency indicates excessive well losses due to poor screen design, clogging, or insufficient development.

C. Recovery Test

After the pump is shut off, the water level recovers (rises) toward its original position. Substituting “residual drawdown” into the Theis/Cooper-Jacob equations yields T independently. This is a valuable check on T obtained during the pumping phase.

D. Slug Test (Bail Test)

A known volume of water is instantaneously added to or removed from a well, and the subsequent recovery is monitored. No pumping is required. Used in low-permeability formations or when observation wells are not available. Yields K at the well screen, but not T or S at field scale.

5.3 THEIS EQUATION — UNSTEADY RADIAL FLOW TO A WELL

Assumptions

1. The aquifer is confined, homogeneous, isotropic, and of uniform thickness b.
2. The aquifer extends infinitely in all horizontal directions (no boundaries).
3. The well is fully penetrating and pumps at a constant rate Q.
4. Flow is horizontal and radially symmetric toward the well.
5. Water is released from storage instantaneously with decline in head.
6. The well has infinitesimal diameter (no well storage effects).
7. No recharge occurs during the test.

Governing Equation

Starting from the radial form of the confined flow equation (combining Darcy’s law with continuity):

With boundary conditions: \(h(r, 0) = h_0\) (initial), \(h(\infty, t) = h_0\) (infinite boundary), and \(\lim_{r \to 0} r\,\partial h/\partial r = -Q/(2\pi T)\) (well discharge).

Theis Solution

Theis solved this by recognising an analogy with heat conduction in solids. The drawdown at distance r from the well at time t is:

where \(W(u)\) is the Theis well function:

and the dimensionless time parameter u is:

Tabulated Values of W(u)

Physical Meaning of Parameters

5.4 THEIS TYPE-CURVE METHOD FOR T AND S

Step-by-Step Procedure

1. Prepare the type curve: Plot W(u) vs 1/u on log-log paper (standard Theis curve). This is a fixed reference curve.
2. Plot field data: On a separate sheet of log-log paper at the same scale, plot drawdown s (y-axis) vs time t (x-axis) for each observation well.
3. Overlay and match: Place the field data sheet on top of the type curve. Keep axes parallel. Slide the field data sheet until the data points align with the type curve.
4. Pick a match point: Choose any convenient point on the overlapping sheets. Read four values:
   • From the type curve sheet: \(W(u)\) and \(1/u\)
   • From the field data sheet: \(s\) and \(t\)
5. Calculate T:
   $$ \boxed{T = \frac{Q \cdot W(u)}{4\pi\,s}} $$
6. Calculate S:
   $$ \boxed{S = \frac{4Tu}{r^2} = \frac{4Tt}{r^2 \cdot (1/u)}} $$

5.5 THIEM’S EQUILIBRIUM (STEADY-STATE) METHOD FOR T

Derivation

At steady state, for radial flow in a confined aquifer, by Darcy’s law at distance r from the well:

Separating variables and integrating between two observation wells at distances \(r_1\) and \(r_2\) (with heads \(h_1\) and \(h_2\)):

Since drawdown \(s = h_0 - h\), we have \(h_2 - h_1 = s_1 - s_2\). Therefore:

Step-by-Step Procedure (Exam Answer for 2078 Q5a)

Thiem’s Equation for Unconfined Aquifer

For an unconfined aquifer (where the saturated thickness varies with drawdown), the Dupuit-Thiem equation uses head h instead of drawdown:

where \(h_1\) and \(h_2\) are the saturated thicknesses at distances \(r_1\) and \(r_2\) from the well. For the case with one observation well and the well itself (radius \(r_w\), head \(h_w\)) to the radius of influence \(R\) (head \(H\) = original saturated thickness):

Jacob Correction for Unconfined Aquifer

When applying confined-aquifer equations (Theis, Cooper-Jacob) to unconfined aquifers, Jacob’s correction adjusts the observed drawdown \(s\) to an equivalent confined drawdown \(s_c\):

where b is the initial saturated thickness. Use \(s_c\) in place of s in all confined-aquifer formulas. This correction is significant when s > 0.25b.

Limitations of Thiem’s Method

• Requires true steady state — often takes hours to days to achieve (may never be reached in practice).
• Cannot determine S (storage coefficient), only T.
• Requires at least two observation wells (or the pumping well data with one observation well, although this introduces well losses).
• Assumes confined aquifer with no leakage, no boundaries.

5.6 COOPER-JACOB STRAIGHT-LINE APPROXIMATION

Mathematical Basis

Recall the series expansion of W(u):

When \(u < 0.01\), the terms beyond \(-\ln u\) become negligible (\(u < 0.01\) contributes less than 1%), so:

Substituting \(u = r^2S/(4Tt)\) into the Theis equation:

Converting to log₁₀ (multiply by 2.303):

Why This Is a Straight Line

For a fixed observation well at distance r:

This is of the form \(s = m \log_{10}(t) + c\), which is a straight line when s is plotted against log₁₀(t). The slope m per log cycle equals:

Validity Condition

5.7 COOPER-JACOB TIME-DRAWDOWN METHOD

This is the most widely used pumping test analysis method and the most frequently examined in this course.

Step-by-Step Procedure

Physical Interpretation

• Δs is small → the line is flat → high T (aquifer transmits water easily, so drawdown increases slowly).
• Δs is large → steep line → low T (aquifer resists flow, drawdown grows rapidly).
• t₀ is small → small S (little storage, cone expands quickly, intersecting s = 0 at a very early time).
• t₀ is large → large S (ample storage, takes more time for drawdown to manifest).

5.8 COOPER-JACOB DISTANCE-DRAWDOWN METHOD

When data from multiple observation wells at different distances is available (typically at steady or quasi-steady state), the same Cooper-Jacob logic can be applied spatially.

Procedure

1. Plot s (linear y-axis) vs r (logarithmic x-axis). Or equivalently, s vs log₁₀(r).
2. Fit a straight line through the data points.
3. Read Δs = drawdown change per log cycle of distance.
4. Calculate T:
   $$ \boxed{T = \frac{2.303\,Q}{2\pi\,\Delta s}} $$

   (Note: \(2\pi\) in denominator, not \(4\pi\), because this is the steady-state Thiem equation.)

5. Extend the line to s = 0 → the r-intercept is the radius of influence R₀ (the distance at which the aquifer is unaffected by pumping).

Time-Drawdown vs Distance-Drawdown: Key Differences

5.9 RECOVERY TEST

After the pump is stopped (at time t_p since pumping started), the water level recovers. By the principle of superposition, the recovery can be modelled as continuing the original pumping AND injecting at rate Q from time t_p. The residual drawdown s’ at time t’ after pump shutdown is:

where \(t = t_p + t'\) = total time since pumping started, and \(t'\) = time since pumping stopped.

Procedure

1. Plot residual drawdown s’ (y-axis, linear) vs \(t/t'\) (x-axis, log scale).
2. Fit a straight line through the data.
3. Determine Δs’ per log cycle of \(t/t'\).
4. Compute:
   $$ \boxed{T = \frac{2.303\,Q}{4\pi\,\Delta s'}} $$

Advantages of the Recovery Test

• Independent check on T obtained during the pumping phase.
• No flow meter needed during recovery (Q is known from the pumping phase).
• Not affected by well losses (there is no discharge during recovery, so the non-Darcian turbulence zone disappears).
• Simpler data collection — only one well (even the pumping well) is needed.

5.10 IMAGE WELL THEORY AND BOUNDARY EFFECTS

Real aquifers are not infinite — they encounter boundaries such as impermeable barriers (faults, clay layers) and recharge boundaries (rivers, lakes). The method of images handles these by placing imaginary wells symmetrically across the boundary:

Drawdown with Image Wells

Using superposition, the total drawdown at any point is the algebraic sum of drawdowns from the real and image wells. If the real well is at distance \(r\) and the image well at distance \(r_i\) from the observation point:

Impermeable (barrier) boundary — image well pumps at +Q:

Recharge boundary — image well recharges at −Q:

where \(u_r = r^2S/(4Tt)\) and \(u_i = r_i^2 S/(4Tt)\). At the boundary itself (barrier), the no-flow condition is automatically satisfied.

Detecting Boundaries from Pumping Test Data

• On a Cooper-Jacob time-drawdown plot, a barrier causes the late-time data to steepen (upward break in the line, slope doubles).
• A recharge boundary causes the data to flatten below the original line (downward break, slope approaches zero).
• Distance to the boundary can be estimated from the time at which the break occurs:

where \(t_b\) is the time at which the break in the semi-log line occurs, and \(r_i\) is the distance from the pumping well to the image well (= twice the distance to the boundary).

5.11 PAST QUESTION BANK FOR CHAPTER 5

Theory Questions

Numerical Questions (Solved in Numerical → Chapter 5)

CHAPTER SUMMARY AND EXAM FORMULA SHEET

Core Equations — Must Memorise

Key Relationships

Exam Tips for Chapter 5

Quick Reference: Which Method to Use?