← GW Home
Chapter 4

Well Hydraulics [10 Hours]

Well Hydraulics ~20+ Past Questions

MOST IMPORTANT TOPICS & MARKS DISTRIBUTION

High-Yield Topics (by exam frequency):

  1. Steady radial flow derivation (confined & unconfined) — Thiem/Dupuit equation — Asked in 2073, 2072 Magh [7–8 marks]
  2. Cone of depression, radius of influence, well terminology — Asked in 2078, 2072 Magh [6 marks]
  3. Numericals: discharge, drawdown calculations — Asked in 2078, 2073 [4–6 marks]

Marks Each Time Asked:

ExamQuestionsTotal
2078 ChaitraQ4a [6], Q4b [6 num]12
2073 BhadraQ4a [7], Q4b [4 num]11
2072 MaghQ4 [6], Q5b [8]14

CHAPTER ROADMAP

  1. Introduction to wells and well terminology
  2. Cone of depression, radius of influence, and area of influence
  3. Well yield and specific capacity
  4. Assumptions for steady-state well flow
  5. Derivation: Steady radial flow to a well in a confined aquifer (Thiem equation)
  6. Derivation: Steady radial flow to a well in an unconfined aquifer (Dupuit equation)
  7. Unsteady (transient) radial flow — Theis equation (overview)
  8. Partially penetrating wells
  9. Multiple well systems and interference
  10. Past question bank for Chapter 4
  11. Chapter summary and exam formula sheet

4.1 INTRODUCTION TO WELLS AND WELL TERMINOLOGY

A well is a hydraulic structure that penetrates an aquifer and allows extraction (or injection) of groundwater. Wells are the most common means of groundwater exploitation worldwide. Understanding the hydraulics of flow toward a pumping well is central to groundwater engineering.

Types of Wells

  • Fully penetrating well: A well whose screen extends through the entire thickness of the aquifer, so that water enters along the full saturated depth.
  • Partially penetrating well: A well that taps only a portion of the aquifer thickness, causing three-dimensional flow near the well.
  • Artesian (flowing) well: A well drilled into a confined aquifer where the piezometric surface is above the ground level, causing water to flow naturally without pumping.
  • Observation well (piezometer): A non-pumping well used to measure the water level or piezometric head at a given location.
Ground Surface Confining layer Confined Aquifer (thickness b) Unconfined Aquifer (thickness H) Impervious bedrock Confined Well (fully penetrating) Piezometric surface Unconfined Well Water table P P b H
Figure 4.1: Types of wells — confined (fully penetrating) and unconfined well with water levels.

Key Terminology

Term Definition
Static water level (SWL) The water level in a well before pumping begins; represents the natural piezometric surface or water table.
Pumping water level (PWL) The water level in the well during pumping; always lower than SWL.
Drawdown (\(s\) or \(s_w\)) The difference between SWL and PWL: \(s = \text{SWL} - \text{PWL}\). Represents the depression of the water surface caused by pumping.
Well radius (\(r_w\)) The radius of the well casing/screen.
Radius of influence (\(R\)) The radial distance from the well beyond which the drawdown is negligible; the boundary of the cone of depression.
Well discharge / yield (\(Q\)) The volume of water pumped per unit time (m³/s or L/s).
Specific capacity The discharge per unit drawdown: \(Q/s_w\). Indicates well efficiency.

4.2 CONE OF DEPRESSION, RADIUS OF INFLUENCE, AND AREA OF INFLUENCE

PAST EXAM QUESTIONS ADDRESSED IN THIS SECTION:

  1. Define cone of depression, radius of influence, well yield, and specific capacity of a well. Write down the assumptions used for the governing equation. [6] (2078 Chaitra, Q4a)
  2. Define cone of depression, radius of influence, and area of influence with a neat sketch in well hydraulics. [6] (2072 Magh, Q4)
DEFINITION: When a well is pumped, the water level drops at and around the well, forming a funnel-shaped depression in the piezometric surface (confined) or water table (unconfined). This depression is called the cone of depression (or drawdown cone).
Ground surface Original piezometric surface (SWL) Confined Aquifer (thickness b) Confining layer Q sw (drawdown) R (radius of influence) R rw Cone of depression The area of influence is the circular area (in plan) enclosed by the radius of influence R.
Figure 4.2: Cone of depression around a pumping well in a confined aquifer showing drawdown sw and radius of influence R.

Cone of Depression

When pumping starts, water is first extracted from the immediate vicinity of the well. As pumping continues, the drawdown propagates outward, forming a three-dimensional conical surface. Key features:

  • The drawdown is maximum at the well face and decreases with radial distance.
  • The cone expands with time until a steady state is reached (when recharge equals discharge).
  • The steepness of the cone depends on the aquifer transmissivity: low-T aquifers produce steep, narrow cones; high-T aquifers produce flat, wide cones.
  • In unconfined aquifers, the cone represents the actual lowering of the water table, reducing the saturated thickness near the well.
TYPICAL EXTENT: In unconfined aquifers, the cone usually extends less than 100 m but may reach several hundred metres. In confined aquifers, the cone may extend hundreds to thousands of metres because drawdown represents pressure-head reduction rather than physical water-table lowering.

Radius of Influence (R)

The radius of influence is the maximum distance from the pumping well at which measurable drawdown occurs. Beyond this distance, the aquifer is essentially undisturbed. It depends on:

Radius of Influence & Area of Influence Area of influence (plan view) Pumping well R (radius of influence) OW₁ (s > 0) OW₂ (s > 0) No effect Inside R: drawdown s > 0 R depends on: pumping rate Q, aquifer T & S, and time t
Figure 4.2b: Plan view — radius of influence (R) defines the boundary beyond which pumping has no measurable effect on the water table.
  • Pumping rate (\(Q\))
  • Aquifer transmissivity (\(T\))
  • Storage coefficient (\(S\))
  • Duration of pumping (\(t\))

Empirical estimates include:

$$ R = 3000 \cdot s_w \cdot \sqrt{K} \quad \text{(Sichardt formula, R in m, K in m/s)} $$

Area of Influence

The area of influence is the circular area (in plan view) within which the drawdown due to pumping is measurable. It equals \(\pi R^2\), where \(R\) is the radius of influence.

4.3 WELL YIELD AND SPECIFIC CAPACITY

Well Yield

The well yield (or discharge, \(Q\)) is the volume of water that can be pumped from a well per unit time under specified conditions. It is typically expressed in liters per second (L/s), cubic meters per day (m³/day), or gallons per minute (gpm).

Specific Capacity

The specific capacity of a well is defined as the discharge per unit drawdown:

Specific Capacity — Q/s Relationship Drawdown, s (m) Discharge, Q (m³/s) Linear region Nonlinear (well losses) Slope = Sc Specific Capacity: Sc = Q/s [m²/s or m³/day/m] Higher Sc = more efficient Sc decreases with time & Q
Figure 4.3b: Specific capacity (Q/s) measures well productivity — it decreases at higher pumping rates due to non-linear well losses.
$$ \text{Specific capacity} = \frac{Q}{s_w} \quad \text{(m}^3\text{/s per m, or L/s per m)} $$

Significance of specific capacity:

  • A higher specific capacity indicates a more productive well (or higher transmissivity aquifer).
  • It provides a quick field estimate of aquifer transmissivity: \(T \approx 1.22 \times (Q/s_w)\) (empirical, for confined aquifers).
  • Specific capacity decreases with increasing drawdown in unconfined aquifers because the saturated thickness diminishes.
  • It is also affected by well losses (well efficiency), partial penetration, and turbulent flow near the well.

EXAM TIP

The specific capacity \(Q/s_w\) is NOT constant in unconfined aquifers because as drawdown increases, the effective saturated thickness decreases, reducing the available transmissivity. In confined aquifers (where \(b\) stays constant), the specific capacity remains approximately constant for small drawdowns.

4.4 ASSUMPTIONS FOR STEADY-STATE WELL FLOW

PAST EXAM QUESTIONS ADDRESSED IN THIS SECTION:

  1. Write down the assumptions used for the governing equation. [Part of Q4a] (2078 Chaitra)

The derivation of steady-state radial flow equations (Thiem / Dupuit) relies on the following assumptions:

  1. The aquifer is underlain by an impermeable (confining) layer.
  2. Geological formations are horizontal and of infinite lateral extent (no boundary effects).
  3. The potentiometric surface is horizontal before pumping begins (steady initial conditions).
  4. All changes in hydraulic head are caused only by the pumping well.
  5. The aquifer is homogeneous and isotropic with uniform hydraulic conductivity \(K\).
  6. Groundwater flow is radial toward the well.
  7. Groundwater flow is horizontal (two-dimensional in plan).
  8. Flow obeys Darcy's law (laminar flow conditions).
  9. Groundwater has constant density and viscosity.
  10. Pumping and observation wells are fully penetrating — they receive water from the entire aquifer thickness.
  11. The pumping well has negligible diameter (well storage is negligible) and is 100% efficient (no well losses).
  12. For unconfined aquifers: the Dupuit assumptions apply — flow is horizontal and the hydraulic gradient equals the slope of the water table.
DUPUIT ASSUMPTIONS 1. Flow lines are horizontal 2. Hydraulic gradient = slope of the free surface (dh/dr) 3. Velocity is uniform over the entire depth STEADY-STATE CONDITION Pumping rate Q = Recharge ∂h/∂t = 0 Water levels are constant Cone of depression is stable DARCY'S LAW VALID Laminar flow conditions Re < 1 to 10 (pore scale) No turbulent well losses v = -K(dh/dr)
Figure 4.3: Key groups of assumptions underlying steady-state well equations.

4.5 DERIVATION: STEADY RADIAL FLOW TO A WELL IN A CONFINED AQUIFER (THIEM EQUATION)

PAST EXAM QUESTIONS ADDRESSED IN THIS SECTION:

  1. Derive an expression for steady radial flow into a well in a confined aquifer with a neat sketch. [7] (2073 Bhadra, Q4a)
  2. A well penetrates an artesian aquifer. Determine the maximum yield for a given drawdown and the percentage change in discharge if the well diameter is reduced. [6] (2078 Chaitra, Q4b)
Impervious base Confining layer Original piezometric surface Q b H hw sw R (radius of influence) rw r Cylindrical surface Confined Aquifer
Figure 4.4: Steady radial flow to a fully penetrating well in a confined aquifer — definition sketch for Thiem equation derivation.

Derivation

STEADY STATE → STORAGE IGNORED

Under steady-state conditions, water levels are constant and no new water is released from storage. Therefore, the storage coefficient (S) does not appear in steady-state equations — only transmissivity (T) and hydraulic conductivity (K) matter.

Consider a fully penetrating well of radius \(r_w\) pumping at a constant rate \(Q\) from a confined aquifer of thickness \(b\) and hydraulic conductivity \(K\). At steady state, the flow toward the well is purely radial and horizontal.

At any radial distance \(r\) from the well, consider a cylindrical surface of height \(b\) and circumference \(2\pi r\). By continuity, all the water flowing through this surface must equal the pumping rate:

$$ Q = A \cdot v = (2\pi r \cdot b) \cdot v $$

By Darcy's law, the velocity at distance \(r\) is:

$$ v = K \frac{dh}{dr} $$

Substituting:

$$ Q = 2\pi r b K \frac{dh}{dr} $$

Separating variables:

$$ \frac{Q}{2\pi K b} \frac{dr}{r} = dh $$

Integrating between the well face (\(r = r_w\), \(h = h_w\)) and the radius of influence (\(r = R\), \(h = H\)):

$$ \frac{Q}{2\pi K b} \int_{r_w}^{R} \frac{dr}{r} = \int_{h_w}^{H} dh $$
$$ \frac{Q}{2\pi K b} \ln\left(\frac{R}{r_w}\right) = H - h_w $$

Since the drawdown at the well face \(s_w = H - h_w\) and the transmissivity \(T = Kb\):

$$ \boxed{Q = \frac{2\pi T s_w}{\ln(R/r_w)} = \frac{2\pi K b \, s_w}{\ln(R/r_w)}} $$

This is the Thiem equation for steady radial flow in a confined aquifer.

General form between two observation wells

If head measurements \(h_1\) and \(h_2\) are available at distances \(r_1\) and \(r_2\) from the pumping well:

Thiem Equation — Two Observation Wells (Confined) Confined aquifer (T = Kb) Pumping Well Q Piezometric surface OW₁ (r₁, h₁) OW₂ (r₂, h₂) r₁ r₂ T = Q·ln(r₂/r₁) / [2π(h₂−h₁)]
Figure 4.5b: Thiem equation with two observation wells — T is determined from the measured heads h₁ and h₂ at distances r₁ and r₂.
$$ Q = \frac{2\pi T (h_2 - h_1)}{\ln(r_2/r_1)} $$

THIEM EQUATION — CONFINED

\(Q = \dfrac{2\pi K b \, s_w}{\ln(R/r_w)}\)

or equivalently: \(Q = \dfrac{2\pi T \, s_w}{\ln(R/r_w)}\)

KEY INSIGHT

Discharge \(Q\) depends on \(\ln(R/r_w)\), not on \(R/r_w\) directly. Since the logarithm varies slowly, doubling the well diameter changes \(Q\) by only a few percent.

Symbol Meaning Typical Units
\(Q\) Well discharge (pumping rate) m³/s or m³/day
\(K\) Hydraulic conductivity m/s or m/day
\(b\) Confined aquifer thickness m
\(T\) Transmissivity = \(Kb\) m²/s or m²/day
\(s_w\) Drawdown at the well face m
\(R\) Radius of influence m
\(r_w\) Well radius m

4.6 DERIVATION: STEADY RADIAL FLOW TO A WELL IN AN UNCONFINED AQUIFER (DUPUIT EQUATION)

PAST EXAM QUESTIONS ADDRESSED IN THIS SECTION:

  1. Derive the equation for steady radial flow in an unconfined aquifer. [8] (2072 Magh, Q5b)
  2. A fully penetrating well in an unconfined aquifer is pumped under steady conditions. Compute the steady-state discharge. [4] (2073 Bhadra, Q4b)
Impervious base Original water table Ground surface Q H hw sw h r R Unconfined Aquifer
Figure 4.5: Steady radial flow to a fully penetrating well in an unconfined aquifer — definition sketch for Dupuit equation derivation.

Derivation

Consider a fully penetrating well of radius \(r_w\) pumping at a constant rate \(Q\) from an unconfined aquifer resting on an impervious base. The original (pre-pumping) saturated thickness is \(H\). Under the Dupuit assumptions, flow is horizontal and the hydraulic gradient equals the slope of the free surface.

At radial distance \(r\), the saturated thickness is \(h\) (which varies with \(r\)). The area of the cylindrical surface through which flow passes is \(A = 2\pi r \cdot h\). By continuity and Darcy's law:

$$ Q = 2\pi r \cdot h \cdot K \frac{dh}{dr} $$

Separating variables:

$$ \frac{Q}{2\pi K} \frac{dr}{r} = h \, dh $$

Integrating between the well face (\(r = r_w\), \(h = h_w\)) and the radius of influence (\(r = R\), \(h = H\)):

$$ \frac{Q}{2\pi K} \int_{r_w}^{R} \frac{dr}{r} = \int_{h_w}^{H} h \, dh $$
$$ \frac{Q}{2\pi K} \ln\left(\frac{R}{r_w}\right) = \frac{H^2 - h_w^2}{2} $$

Solving for \(Q\):

$$ \boxed{Q = \frac{\pi K (H^2 - h_w^2)}{\ln(R/r_w)}} $$

This is the Dupuit equation for steady radial flow in an unconfined aquifer.

General form between two observation wells

If heads \(h_1\) and \(h_2\) are measured at distances \(r_1\) and \(r_2\):

$$ Q = \frac{\pi K (h_2^2 - h_1^2)}{\ln(r_2/r_1)} $$

DUPUIT EQUATION — UNCONFINED

\(Q = \dfrac{\pi K (H^2 - h_w^2)}{\ln(R/r_w)}\)

DUPUIT VALIDITY

The Dupuit equation is most accurate when the drawdown is small relative to the saturated thickness: \(s_w / H < 0.5\). For large drawdowns, vertical flow components become significant and the equation overestimates discharge.

Comparison: Confined vs. Unconfined Well Equations

Feature Confined (Thiem) Unconfined (Dupuit)
Aquifer thickness Constant (\(b\)) Variable (\(h\), decreases near well)
Flow cross-section area \(2\pi r b\) \(2\pi r h\)
Equation form \(Q = \dfrac{2\pi Kb \, s_w}{\ln(R/r_w)}\) \(Q = \dfrac{\pi K(H^2 - h_w^2)}{\ln(R/r_w)}\)
Head term Linear in \(h\) (or \(s_w\)) Difference of squares (\(H^2 - h_w^2\))
Specific capacity Approximately constant Decreases with increasing drawdown
Thiem vs Dupuit — Steady-State Well Formulas CONFINED (Thiem) Constant thickness b T = Kb (constant) Q = 2πT(h₂−h₁)/ln(r₂/r₁) h varies linearly with ln(r) Piezometric surface is lowered but aquifer stays fully saturated UNCONFINED (Dupuit) Variable thickness h(r) T = Kh (varies with r) Q = πK(h₂²−h₁²)/ln(r₂/r₁) h² varies linearly with ln(r) Water table drops → saturated thickness reduces near well
Figure 4.6b: Comparison of steady-state well equations — Thiem (confined, h-based) vs Dupuit (unconfined, h²-based).

4.7 UNSTEADY (TRANSIENT) RADIAL FLOW — THEIS EQUATION (OVERVIEW)

In reality, when a well starts pumping, the cone of depression expands with time and steady state may not be achieved quickly (especially in confined aquifers). The Theis (1935) solution addresses this transient behavior.

Governing Equation

The radial form of the groundwater flow equation in a confined aquifer is:

$$ \frac{\partial^2 h}{\partial r^2} + \frac{1}{r}\frac{\partial h}{\partial r} = \frac{S}{T}\frac{\partial h}{\partial t} $$

This equation is analogous to the heat conduction (diffusion) equation, showing that groundwater flow under transient conditions behaves as a diffusion process. Theis exploited this analogy to develop his solution.

Theis Solution

Theis expressed the drawdown at distance \(r\) and time \(t\) as:

$$ s = \frac{Q}{4\pi T} W(u) $$

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

$$ W(u) = \int_u^{\infty} \frac{e^{-\eta}}{\eta} d\eta = -0.5772 - \ln u + u - \frac{u^2}{2 \cdot 2!} + \frac{u^3}{3 \cdot 3!} - \cdots $$

and the dimensionless parameter \(u\) is:

$$ u = \frac{r^2 S}{4Tt} $$
Symbol Meaning
\(s\) Drawdown at distance \(r\) and time \(t\)
\(Q\) Constant pumping rate
\(T\) Transmissivity of the aquifer
\(S\) Storage coefficient (storativity)
\(W(u)\) Theis well function
\(u\) Dimensionless time parameter

Cooper–Jacob Approximation

For large time or small distance (i.e., \(u < 0.01\)), the well function simplifies:

$$ s \approx \frac{Q}{4\pi T} \left[ -0.5772 - \ln\left(\frac{r^2 S}{4Tt}\right) \right] = \frac{2.303\, Q}{4\pi T} \log_{10}\left(\frac{2.25\, Tt}{r^2 S}\right) $$

This is the basis for the Cooper–Jacob straight-line method used in pumping test analysis (covered in detail in Chapter 5).

Theis Type-Curve Method

The Theis method uses graphical curve-matching to determine aquifer parameters:

Theis Type-Curve Matching 1/u or r²/t (log scale) W(u) or s (log scale) Type curve W(u) Field data (s vs r²/t) Match point At match: T = QW(u)/(4πs), S = 4Tu/r²×t
Figure 4.7b: Theis type-curve matching — overlay field data (s vs r²/t) on the standard W(u) vs 1/u curve to find T and S at the match point.
  1. Plot \(W(u)\) versus \(u\) on log–log paper — this is the type curve.
  2. Plot observed drawdown \(s\) versus \(r^2/t\) (or \(t\)) on a separate log–log sheet of the same scale.
  3. Superimpose the field data on the type curve and shift (keeping axes parallel) until the curves match.
  4. Select a match point and read off \(s\), \(W(u)\), \(r^2/t\), and \(u\).
  5. Calculate \(T\) from \(T = \dfrac{Q \cdot W(u)}{4\pi s}\) and \(S\) from \(S = \dfrac{4Tu}{r^2/t}\).

Cooper–Jacob Straight-Line Method

For small \(u\) (i.e., small \(r\) and/or large \(t\)), drawdown varies linearly with the logarithm of time:

Cooper–Jacob Straight-Line Method log(t) Drawdown s (↓) Best-fit line Δs per log cycle t₀ (s=0) T = 2.3Q/(4πΔs) S = 2.25Tt₀/r² Valid when u < 0.01 (late-time data)
Figure 4.7c: Cooper–Jacob method — plot s vs log(t), fit a straight line, and compute T from the slope Δs and S from the t₀ intercept.
  • Plot \(s\) versus \(\log t\) on semi-log paper — the data form a straight line at later times.
  • The slope \(\Delta s\) (drawdown per log cycle of time) gives transmissivity: \(T = \dfrac{2.303\,Q}{4\pi\,\Delta s}\).
  • The time intercept \(t_0\) (where the straight line meets \(s = 0\)) gives storativity: \(S = \dfrac{2.25\,T\,t_0}{r^2}\).

Chow Method

An alternative graphical method that avoids type-curve fitting:

  1. Plot \(s\) versus \(\log t\) on semi-log paper (same as Cooper–Jacob).
  2. Select any point on the fitted curve.
  3. Draw a tangent to the curve at that point.
  4. Measure \(\Delta s\) — the drawdown difference per log cycle along the tangent.
  5. Compute \(F(u) = s / \Delta s\) at the selected point.
  6. Use a standard chart of \(F(u)\) versus \(W(u)\) and \(u\) to read off \(W(u)\) and \(u\).
  7. Calculate \(T\) and \(S\) using the Theis equations.

Recovery Test

After pumping stops, water levels in the well and aquifer rise (recover). The remaining drawdown is called the residual drawdown (\(s'\)).

Recovery Test — Residual Drawdown Method Time Drawdown (↓) Pumping phase Pump OFF (t'=0) Recovery phase s' (residual) Plot s' vs log(t/t') → T = 2.3Q/(4πΔs') Recovery data gives T independent of S — useful for verification
Figure 4.7d: Recovery test — after the pump is shut off, residual drawdown s' is plotted vs log(t/t') to determine T from the straight-line slope.
  • Plot \(s'\) versus \(\log(t/t')\) on semi-log paper, where \(t\) is total time since pumping started and \(t'\) is time since pumping stopped.
  • At late recovery times, the data plot as a straight line.
  • Transmissivity \(T\) is obtained from the slope of this line: \(T = \dfrac{2.303\,Q}{4\pi\,\Delta s'}\).
  • Storage coefficient \(S\) cannot be determined from the recovery test.

EXAM NOTE

The Theis equation and Cooper–Jacob method are more commonly examined in Chapter 5 (Pumping Tests). In Chapter 4, the focus is primarily on steady-state equations (Thiem for confined, Dupuit for unconfined).

4.8 PARTIALLY PENETRATING WELLS

DEFINITION: A partially penetrating well is one whose length of water entry (screen length) is less than the full thickness of the aquifer it penetrates. Many artesian wells and wells with partial screens fall into this category.

Effect on Flow and Drawdown

In a partially penetrating well, there is a vertical convergence of streamlines near the well. The average flow-line length into a partially penetrating well exceeds that into a fully penetrating well, causing greater resistance to flow. This results in:

Partially vs Fully Penetrating Well Full b Horizontal flow only Partial l 3D flow (vertical comp.) Partially penetrating well: sp > sf (extra head loss) Convergence of flow lines → 3D effects near the well screen At distance r > 1.5b, effect of partial penetration vanishes
Figure 4.8b: Fully penetrating well has horizontal flow only; partially penetrating well develops vertical flow components, increasing drawdown near the well.
  • If the discharge is the same (\(Q_p = Q\)), then the drawdown in the partially penetrating well is greater: \(s_p > s\).
  • If the drawdown is the same (\(s_p = s\)), then the discharge from the partially penetrating well is less: \(Q_p < Q\).

The effect of partial penetration on the flow pattern and drawdown is negligible beyond a radial distance of about \(0.5b\) to \(2b\) (where \(b\) is the saturated thickness), depending on the degree of penetration.

Wein Hsiung Li (1954) Empirical Formula

Based on electric analog model tests, Wein Hsiung Li proposed empirical formulae for drawdown and discharge ratios in a partially penetrating well. Let \(\alpha = L/b\) be the penetration ratio (screen length \(L\) / aquifer thickness \(b\)):

$$ \frac{s_{wp}}{s_w} = \frac{1}{\alpha} \left[1 + 7\sqrt{\frac{r_w}{2b}} \cos\left(\frac{\pi\alpha}{2}\right)\right] $$
$$ \frac{Q_p}{Q} = \frac{\alpha}{1 + 7\sqrt{\frac{r_w}{2b}} \cos\left(\frac{\pi\alpha}{2}\right)} $$

where \(r_w\) = well radius, \(R\) = radius of influence, and these formulae are applicable for \(\alpha > 0.1\).

Worked Example

Q) A 40 cm diameter well penetrates an aquifer of 30 m thickness and the length of the screened portion is 10 m. The well yields 2000 litres per minute with a drawdown of 3 m. If the screen length is increased to 20 m, determine: (a) the drawdown in the well, and (b) the percentage increase in specific capacity of the well.

Solution:

Given: \(d = 0.40\) m, \(r_w = 0.20\) m, \(b = 30\) m, \(Q = 2000\) lpm, \(s_{w1} = 3\) m.

Penetration ratios: \(\alpha_1 = 10/30 = 1/3\), \(\alpha_2 = 20/30 = 2/3\).

For same \(Q\), using the drawdown ratio formula:

$$ \frac{s_{w2}}{s_{w1}} = \frac{\alpha_1}{\alpha_2} \times \frac{1 + 7\sqrt{\frac{r_w}{2b}} \cos\left(\frac{\pi\alpha_2}{2}\right)}{1 + 7\sqrt{\frac{r_w}{2b}} \cos\left(\frac{\pi\alpha_1}{2}\right)} $$

Computing: \(\sqrt{r_w / 2b} = \sqrt{0.20/60} = 0.0577\)

\(s_{w2} \approx 1.92\) m

(a) Drawdown with 20 m screen = 1.92 m.

(b) Specific capacities:

  • \(S_{c1} = Q/s_{w1} = 2000/3 = 666.7\) lpm/m
  • \(S_{c2} = Q/s_{w2} = 2000/1.92 = 1041.7\) lpm/m
  • Percentage increase = \(\dfrac{1041.7 - 666.7}{666.7} \times 100 = \textbf{56.3\%}\)

Partially Penetrating Well in a Confined Aquifer

The drawdown in a partially penetrating well in a confined aquifer is:

$$ s_{wp} = s_w + \Delta s_w $$

where \(\Delta s_w\) is the additional drawdown due to convergence of flow lines caused by partial penetration.

Partially Penetrating Well in an Unconfined Aquifer

For a fully penetrating well in an unconfined aquifer under steady state:

$$ Q = \frac{\pi K(H^2 - h_w^2)}{\ln(R/r_w)} $$

For a partially penetrating well (usually bottom-half to one-third of the aquifer screened), the penetration ratio is \(\alpha = l/h_w\), where \(l\) is the screen length and \(h_w = H - s_w\). The discharge is reduced by the partial penetration factor.

Worked Example (Unconfined)

Q) A 30 cm well fully penetrates a homogeneous aquifer having a permeability of 40 m/day. The saturated thickness is 15 m. What is the drawdown when pumped at 1000 lpm? What will be the drawdown if only the bottom half of the aquifer is screened? Assume \(R = 300\) m.

Solution:

Given: \(d = 0.30\) m, \(r_w = 0.15\) m, \(K = 40\) m/day, \(H = 15\) m, \(Q = 1000\) lpm = 1440 m³/day, \(R = 300\) m.

For the fully penetrating well: \(Q = \dfrac{\pi K(H^2 - h_w^2)}{\ln(R/r_w)}\)

$$ 1440 = \frac{\pi \times 40 \times (15^2 - h_w^2)}{\ln(300/0.15)} $$

Solving: \(h_w \approx 14.12\) m, so \(s_w \approx 0.88\) m for the fully penetrating well.

For the bottom-half screened well (\(\alpha = 0.5\)), the partial penetration factor increases the drawdown. Using the empirical equations, the drawdown will be greater than 0.88 m due to the convergence effects.

4.9 MULTIPLE WELL SYSTEMS AND INTERFERENCE

DEFINITION: When several wells pump from the same aquifer, their individual cones of depression may overlap, creating well interference. The combined drawdown at any point is the sum of drawdowns from all pumping wells.

Well Interference

When the cones of depression of two or more wells overlap, the groundwater table of both wells combines to form a new composite drawdown surface. This phenomenon is called well interference.

Effects of Well Interference

  • The total groundwater output is less than the sum of the discharge capacities of the individual wells operating alone.
  • The efficiency of each well decreases.
  • The drawdown is increased, resulting in a higher pumping lift.
  • The pumping cost increases due to decreased efficiency and increased pumping lift.

Principle of Superposition (Confined Aquifers)

In confined aquifer systems, the principle of superposition applies because the governing Laplace equation (steady state) is linear. The total drawdown at any point is the algebraic sum of the drawdowns produced by each well independently:

Superposition Principle — Multiple Wells Ground surface Static WT Individual cones Combined drawdown Well A Well B stotal = sA + sB (superposition) Valid for linear (confined) aquifers; approximate correction needed for unconfined
Figure 4.9b: Superposition principle — total drawdown at any point equals the sum of individual drawdowns from each well.
$$ s_{total} = s_1 + s_2 + s_3 + \cdots = \sum_{i=1}^{n} s_i $$

This principle is used to determine composite drawdowns in well fields, essential for designing efficient water supply systems.

Correction for Unconfined Aquifers

For unconfined aquifers, the linear superposition principle is not directly applicable because significant drawdown reduces the saturated thickness and thus the aquifer transmissivity. A correction procedure is required:

  1. First, calculate the theoretical confined drawdown for each well using superposition.
  2. Then apply a correction using the initial saturated thickness \(H\):
$$ s_{unconfined} = s_{confined} - \frac{s_{confined}^2}{2H} $$

Uses and Well Configurations

Multiple well systems are used for:

Common Multi-Well Configurations Line of Wells Dewatering along excavation / trench Interference increases total drawdown Circular Array Dewatering for circular excavation Well + Recharge −Q +Q Image well for boundary effects In all configurations: stotal at any point = ∑ individual drawdowns (superposition)
Figure 4.9c: Common multi-well configurations — line of wells for dewatering, circular arrays, and image wells for boundary simulation.
  • Water supply: wells spaced to minimise interference and reduce drawdown.
  • Subsurface drainage and dewatering for excavation/mining: wells placed closely to maximise interference and lower the water table.
  • Seawater intrusion management in coastal aquifers.

Common configurations include linear, triangular, and square arrangements, analysed for their discharge characteristics under steady-state conditions.

4.10 PAST QUESTION BANK FOR CHAPTER 4

A. Theory Questions

2078 Chaitra Q4(a) 6 marks Theory

Define cone of depression, radius of influence, well yield, and specific capacity of a well. Write down the assumptions used for the governing equation.

2073 Bhadra Q4(a) 7 marks Theory

Derive an expression for steady radial flow into a well in a confined aquifer with a neat sketch.

2072 Magh Q4 6 marks Theory

Define cone of depression, radius of influence, and area of influence with a neat sketch in well hydraulics.

2072 Magh Q5(b) 8 marks Theory

Derive the equation for steady radial flow in an unconfined aquifer.

B. Numerical Questions

2078 Chaitra Q4(b) 6 marks Numerical

A well penetrates an artesian aquifer. Determine the maximum yield for a given drawdown and the percentage change in discharge if the well diameter is reduced.

Well diameter D = 30 cm
Aquifer thickness b = 23 m
Drawdown s_w = 3.5 m
Hydraulic conductivity K = 2.8 × 10−4 m/s
Radius of influence R = 450 m
Modified well diameter D' = 20 cm
2073 Bhadra Q4(b) 4 marks Numerical

A fully penetrating well in an unconfined aquifer is pumped under steady conditions. Compute the steady-state discharge.

Well radius r_w = 0.5 m
Saturated thickness H = 50 m
Hydraulic conductivity K = 30 m/day
Water level at well h_w = 40 m
Radius of influence R = 500 m

C. Quick Revision Matrix

Theme Years asked What to prepare
Definitions: cone of depression, radius/area of influence, specific capacity 2078, 2072 Magh Sketch of cone, definitions of all terms, formula for specific capacity
Assumptions for well flow equations 2078 List all 12 assumptions; distinguish Dupuit assumptions for unconfined
Derivation: Thiem equation (confined) 2073 Full derivation with neat figure, arrive at Q = 2πKb sw / ln(R/rw)
Derivation: Dupuit equation (unconfined) 2072 Magh Full derivation with figure, arrive at Q = πK(H² − hw²) / ln(R/rw)
Numerical: confined well discharge 2078 Thiem formula application; effect of well diameter on Q (logarithmic insensitivity)
Numerical: unconfined well discharge 2073 Dupuit formula application; Dupuit validity check (sw/H < 0.5)

CHAPTER SUMMARY AND EXAM FORMULA SHEET

Concept Key relation / statement
Drawdown \( s_w = H - h_w \) (confined: measured from piezometric surface; unconfined: from water table)
Specific capacity \( Q / s_w \) — discharge per unit drawdown
Thiem equation (confined) \( Q = \dfrac{2\pi K b \, s_w}{\ln(R/r_w)} = \dfrac{2\pi T \, s_w}{\ln(R/r_w)} \)
Dupuit equation (unconfined) \( Q = \dfrac{\pi K (H^2 - h_w^2)}{\ln(R/r_w)} \)
Between two observation wells (confined) \( Q = \dfrac{2\pi T (h_2 - h_1)}{\ln(r_2/r_1)} \)
Between two observation wells (unconfined) \( Q = \dfrac{\pi K (h_2^2 - h_1^2)}{\ln(r_2/r_1)} \)
Transmissivity \( T = K \cdot b \)
Sichardt radius of influence \( R = 3000 \, s_w \sqrt{K} \) (K in m/s, R in m)
Theis equation (transient) \( s = \dfrac{Q}{4\pi T} W(u) \), where \( u = \dfrac{r^2 S}{4Tt} \)
Cooper–Jacob approximation \( s \approx \dfrac{2.303\, Q}{4\pi T} \log_{10}\!\left(\dfrac{2.25\, Tt}{r^2 S}\right) \) for \(u < 0.01\)
Cooper–Jacob: \(T\) from slope \( T = \dfrac{2.303\, Q}{4\pi\, \Delta s} \), where \(\Delta s\) = drawdown per log cycle
Cooper–Jacob: \(S\) from intercept \( S = \dfrac{2.25\, T\, t_0}{r^2} \), where \(t_0\) = time at \(s = 0\)
Recovery test Plot \(s'\) vs \(\log(t/t')\); \( T = \dfrac{2.303\, Q}{4\pi\, \Delta s'} \); \(S\) cannot be found
Partial penetration: drawdown ratio \( s_{wp}/s_w = (1/\alpha)\left[1 + 7\sqrt{r_w/2b}\,\cos(\pi\alpha/2)\right] \), \(\alpha = L/b\)
Well interference (confined) \( s_{total} = \sum s_i \) — superposition of individual drawdowns
Unconfined correction \( s_{unconf} = s_{conf} - s_{conf}^2/(2H) \)
Dupuit assumptions Horizontal flow lines, gradient = water-table slope, uniform velocity over depth

LAST-MINUTE EXAM FOCUS

  1. Be able to sketch and label the cone of depression for both confined and unconfined wells, marking \(s_w\), \(R\), \(r_w\), \(b\) (or \(H\)), \(h_w\).
  2. Know all definitions: cone of depression, radius of influence, area of influence, specific capacity, well yield.
  3. List all 12 assumptions for steady-state well equations; highlight the Dupuit assumptions for unconfined aquifers.
  4. Derive from first principles both the Thiem (confined) and Dupuit (unconfined) equations — start from Darcy's law + continuity on a cylindrical surface.
  5. For numericals, remember: confined uses \(s_w\), unconfined uses \((H^2 - h_w^2)\). Always check units and convert diameters to radii.
  6. Understand the logarithmic insensitivity of discharge to well diameter — a common exam insight question.
  7. Know the effect of partial penetration: same Q → greater drawdown; same drawdown → less discharge.
  8. For multiple wells: superposition works directly in confined aquifers; apply unconfined correction \(s - s^2/2H\) when needed.