Permeability, Seepage and Groundwater Flow

Introduction

Groundwater flow governs the performance of foundations, excavations, earth dams, retaining systems, tunnels and groundwater-control works. Excessive seepage may cause uplift, loss of effective stress, piping, internal erosion or unacceptable settlement. Conversely, inadequate drainage can prolong consolidation and increase pore-water pressures.

The governing principles are conceptually simple: water flows in response to hydraulic-head gradients, while the soil fabric controls resistance. Their professional application is more complex because soils are heterogeneous, anisotropic, layered and sometimes unsaturated. Hydraulic conductivity may vary by several orders of magnitude within one site, making geological interpretation and uncertainty management as important as calculation.

Learning Objectives

After studying this chapter, the reader should be able to:

• apply Darcy’s law and distinguish discharge velocity from seepage velocity;
• determine hydraulic conductivity using laboratory and field methods;
• analyse flow through anisotropic and layered soils;
• construct and interpret flow nets;
• calculate seepage forces, uplift pressure and critical hydraulic gradients;
• evaluate piping, boiling, suffusion and other internal-erosion mechanisms;
• assess granular-filter compatibility;
• formulate steady and transient groundwater-flow equations; and
• select appropriate groundwater-control and monitoring measures.

1. Hydraulic Head and Darcy Flow

1.1 Components of hydraulic head

For incompressible groundwater, the Bernoulli head is

\[

h=z+\frac{u}{\gamma_w}+\frac{v^2}{2g},

\]

where \(h\) is total head, \(z\) is elevation head, \(u\) is pore-water pressure, \(\gamma_w\) is the unit weight of water, \(v\) is water velocity and \(g\) is gravitational acceleration. Velocity head is usually negligible in soils, giving

\[

h\approx z+\frac{u}{\gamma_w}.

\]

A piezometer measures the sum of elevation and pressure heads at its intake. Groundwater flows from greater total head toward lower total head, not necessarily from higher pore pressure to lower pore pressure.

The one-dimensional hydraulic gradient is

\[

i=\frac{\Delta h}{L},

\]

where \(\Delta h\) is the head loss over flow length \(L\). It is dimensionless.

1.2 Darcy’s law

For one-dimensional saturated flow,

\[

Q=KiA,

\]

where \(Q\) is volumetric discharge, \(K\) is hydraulic conductivity, and \(A\) is the gross cross-sectional area normal to flow. The Darcy velocity or specific discharge is

\[

q=\frac{Q}{A}=Ki.

\]

In vector form, with the sign convention that the gradient points toward increasing head,

\[

\mathbf q=-\mathbf K\nabla h.

\]

The negative sign indicates flow toward decreasing head. \(\mathbf K\) is a scalar for isotropic soil and a second-order tensor for anisotropic soil.

Darcy velocity is not the actual average velocity in pore channels. The average linear seepage velocity is approximated by

\[

v_s=\frac{q}{n_e},

\]

where \(n_e\) is effective porosity. Effective porosity excludes pores that do not participate significantly in through-flow.

Darcy’s law is applicable to laminar flow through a representative soil volume under conditions where hydraulic conductivity is reasonably constant. Deviations may occur in very coarse media at high Reynolds number, in low-permeability clays with threshold-like behaviour, and in unsaturated soils where conductivity varies strongly with water content.

1.3 Hydraulic conductivity and intrinsic permeability

Hydraulic conductivity depends on both the porous medium and the permeating fluid:

\[

K=\frac{k\rho_w g}{\mu},

\]

where \(k\) is intrinsic permeability with units of area, \(\rho_w\) is fluid density, and \(\mu\) is dynamic viscosity. Thus, a reported conductivity should be associated with fluid type and temperature. Increasing temperature generally reduces water viscosity and increases measured \(K\).

Soil factors controlling conductivity include particle size, void ratio, pore connectivity, fabric, fissuring, stratification, saturation and stress history. Empirical grain-size correlations are useful only for soils similar to those from which the correlations were derived.

Worked Example 1: Darcy discharge and seepage velocity

A saturated stratum has \(K=3.0\times10^{-5}\) m/s. The head loss is 1.8 m over 12 m, and the flow area is 25 m². Effective porosity is 0.30.

\[

i=\frac{1.8}{12}=0.15

\]

\[

Q=(3.0\times10^{-5})(0.15)(25)=1.125\times10^{-4}\ \text{m}^3/\text{s}.

\]

This equals approximately 9.72 m³/day. The Darcy velocity is

\[

q=Ki=4.50\times10^{-6}\ \text{m/s},

\]

and the seepage velocity is

\[

v_s=\frac{4.50\times10^{-6}}{0.30}=1.50\times10^{-5}\ \text{m/s}.

\]

2. Measurement of Hydraulic Conductivity

2.1 Constant-head test

The constant-head test is generally suitable for sands and gravels. For a specimen of length \(L\), area \(A\), constant head difference \(\Delta h\), collected volume \(V\), and elapsed time \(t\),

\[

K=\frac{VL}{A\Delta h\,t}.

\]

The specimen should be saturated, sidewall leakage prevented, and hydraulic gradients selected to avoid particle migration or non-laminar flow.

2.2 Falling-head test

For silts and clays, a standpipe of area \(a\) may be connected to a specimen of area \(A\) and length \(L\). If the standpipe head falls from \(h_1\) to \(h_2\) in time \(t\), integration of Darcy’s law gives

\[

K=\frac{aL}{At}\ln\left(\frac{h_1}{h_2}\right).

\]

The derivation assumes saturated one-dimensional flow, negligible storage within the apparatus and a constant specimen volume during the test.

2.3 Field methods

Field estimates include pumping tests, slug tests, packer tests, borehole permeability tests and infiltration tests. Pumping tests investigate a much larger soil volume than laboratory tests and may capture fissures and stratification. Their interpretation nevertheless depends on assumptions concerning aquifer geometry, well penetration, boundaries, storage and pumping history.

Laboratory samples may lose fissures or undergo density changes during sampling. Professional assessment should therefore integrate laboratory measurements, field tests, geological logging and inverse calibration against monitored groundwater response.

3. Anisotropic and Layered Media

3.1 Conductivity tensor

For two-dimensional flow aligned with the principal conductivity directions,

\[

q_x=-K_x\frac{\partial h}{\partial x},\qquad q_z=-K_z\frac{\partial h}{\partial z}.

\]

Under steady, incompressible flow without internal sources,

\[

K_x\frac{\partial^2h}{\partial x^2}+K_z\frac{\partial^2h}{\partial z^2}=0,

\]

provided \(K_x\) and \(K_z\) are spatially constant. In anisotropic media, flow need not be parallel to the hydraulic gradient.

The coordinate transformation

\[

X=x\sqrt{\frac{K_z}{K_x}},\qquad Z=z

\]

converts the preceding equation into Laplace’s equation in transformed coordinates. A flow net may then be constructed in the transformed section. The equivalent conductivity used for discharge per unit width is

\[

K_{eq}=\sqrt{K_xK_z}.

\]

This result applies to homogeneous orthotropic media whose principal conductivity axes coincide with the selected coordinates.

3.2 Equivalent conductivity of horizontal layers

For flow parallel to layers, each layer experiences the same hydraulic gradient. The equivalent horizontal conductivity is the thickness-weighted arithmetic mean:

\[

K_{\parallel}=\frac{\sum K_jH_j}{\sum H_j}.

\]

For flow perpendicular to layers, the discharge is continuous and head losses add. The equivalent vertical conductivity is the thickness-weighted harmonic mean:

\[

K_{\perp}=\frac{\sum H_j}{\sum(H_j/K_j)}.

\]

These expressions assume laterally extensive layers and one-dimensional flow respectively parallel or normal to bedding.

Worked Example 2: Layered deposit

A deposit comprises 2 m of sand with \(K_1=1.0\times10^{-4}\) m/s above 3 m of silt with \(K_2=1.0\times10^{-7}\) m/s.

For horizontal flow,

\[

K_{\parallel}=\frac{2(10^{-4})+3(10^{-7})}{5}

=4.006\times10^{-5}\ \text{m/s}.

\]

For vertical flow,

\[

K_{\perp}=\frac{5}{2/(10^{-4})+3/(10^{-7})}

=1.67\times10^{-7}\ \text{m/s}.

\]

The conductivity ratio is approximately 240. The thin, low-conductivity silt has limited influence on horizontal flow but dominates vertical resistance.

4. Flow Nets

A flow net is a graphical solution to two-dimensional, steady-state seepage governed by Laplace’s equation. It consists of:

• flow lines, tangent to the direction of seepage; and
• equipotential lines, along which total head is constant.

For homogeneous isotropic media, the two families intersect at right angles and form approximate curvilinear squares.

4.1 Boundary conditions

A constant-head boundary, such as an upstream reservoir, is an equipotential boundary. An impermeable boundary is a flow line because normal discharge is zero. A free surface is generally a flow line at atmospheric pressure, but its position is initially unknown. A seepage face is also at atmospheric pressure, although flow may emerge across it.

4.2 Discharge from a flow net

If the total head loss is \(H\), the number of flow channels is \(N_f\), and the number of equipotential drops is \(N_d\), discharge per unit width for homogeneous isotropic soil is

\[

q=KH\frac{N_f}{N_d}.

\]

For a transformed homogeneous anisotropic section, \(K\) is replaced by \(\sqrt{K_xK_z}\). This formula is not directly applicable to arbitrary heterogeneous sections without appropriate numerical or piecewise analysis.

The head loss per potential drop is

\[

\Delta h=\frac{H}{N_d}.

\]

The local gradient may be estimated as \(i\approx\Delta h/\Delta s\), where \(\Delta s\) is the local distance between adjacent equipotential lines measured along a flow line. The exit gradient is particularly important because high upward gradients near the downstream toe can initiate boiling or erosion.

Worked Example 3: Flow-net discharge

For a foundation with \(K=2.5\times10^{-5}\) m/s, \(H=6\) m, \(N_f=4\), and \(N_d=12\),

\[

q=(2.5\times10^{-5})(6)\frac{4}{12}

=5.0\times10^{-5}\ \text{m}^2/\text{s}

\]

per metre width.

5. Seepage Force, Effective Stress and Uplift

5.1 Seepage force

Water flowing through soil exerts drag on the skeleton. The seepage force per unit bulk volume has magnitude

\[

f_s=i\gamma_w

\]

and acts in the direction of flow. Its effects must be interpreted together with gravity and effective stress.

For saturated soil under upward one-dimensional seepage, the effective vertical stress at depth \(z\), measured downward from the top surface, is

\[

\sigma'_v=(\gamma'-i\gamma_w)z,

\]

where

\[

\gamma'=\gamma_{sat}-\gamma_w

\]

is the submerged unit weight. Therefore, the effective unit weight under upward seepage is

\[

\boxed{\gamma'_{eff}=\gamma'-i\gamma_w}.

\]

It is not \(\gamma_{sat}-i\gamma_w\), because buoyancy is already present before the additional seepage force is considered. For downward seepage, the effective unit weight becomes \(\gamma'+i\gamma_w\).

5.2 Critical hydraulic gradient

The quick or boiling condition begins when upward seepage reduces effective stress to zero:

\[

\gamma'-i_c\gamma_w=0.

\]

Thus,

\[

i_c=\frac{\gamma'}{\gamma_w}.

\]

For a saturated soil,

\[

\gamma'=\frac{G_s-1}{1+e}\gamma_w,

\]

so

\[

\boxed{i_c=\frac{G_s-1}{1+e}},

\]

where \(G_s\) is specific gravity of solids and \(e\) is void ratio. A basic exit-gradient safety factor is

\[

FS=\frac{i_c}{i_{exit}}.

\]

Required values depend on uncertainty, geometry, consequences and applicable standards. Local gradients can be much higher than average gradients, particularly at corners or defects.

The quick condition is not identical to earthquake-induced liquefaction. Both involve low effective stress, but their loading mechanisms and assessments differ.

Worked Example 4: Upward seepage

For sand with \(G_s=2.68\) and \(e=0.72\),

\[

i_c=\frac{2.68-1}{1+0.72}=0.977.

\]

Taking \(\gamma_w=9.81\) kN/m³,

\[

\gamma'=i_c\gamma_w=9.58\ \text{kN/m}^3.

\]

At an upward gradient of 0.75,

\[

\gamma'_{eff}=9.58-0.75(9.81)=2.22\ \text{kN/m}^3.

\]

The remaining effective weight is small, and the gradient safety factor is \(0.977/0.75=1.30\).

5.3 Uplift pressure

Pore pressure at an elevation \(z\) is

\[

u=\gamma_w(h-z).

\]

The uplift force beneath a slab is the integral of pore pressure over its base. If the pressure distribution is linear, the resultant equals the average pressure multiplied by area and acts through the centroid of the pressure diagram. Stability checks should include uplift, sliding, bearing resistance and structural capacity. Drainage systems should not be credited without allowances for clogging, maintenance and loss of power where pumping is required.

6. Piping and Internal Erosion

Internal erosion is the detachment and transport of soil particles by seepage. Important mechanisms include:

1. Backward erosion piping: erosion begins at an unfiltered exit and progresses upstream in a pipe-like channel.

2. Suffusion: fine particles migrate through the pore network formed by coarser particles while the coarse skeleton remains largely intact.

3. Contact erosion: fines are removed where a fine soil contacts a much coarser layer under tangential flow.

4. Concentrated-leak erosion: soil is eroded along cracks, poorly compacted interfaces, conduits or animal burrows.

5. Heave or boiling: upward seepage reduces effective stress and destabilises a soil mass.

The critical-gradient equation alone is not a complete piping assessment. Backward erosion depends on exit conditions, hydraulic loading, soil erodibility and the capacity of a pipe to progress. Suffusion depends on grading, constriction sizes, density and stress state. Defects and cracks may dominate the behaviour of otherwise low-permeability cores.

Risk reduction measures include cut-off walls, grout curtains, upstream blankets, drainage blankets, relief wells, staged excavation, controlled pumping and properly designed filters. Monitoring should include piezometers, weirs, turbidity observations, settlement points and inspections for sand boils or sinkholes.

7. Filtration and Drainage Criteria

A granular filter must simultaneously:

• retain the protected base soil;
• provide sufficient drainage capacity;
• remain internally stable; and
• resist segregation and construction contamination.

A widely used preliminary retention criterion is

\[

\frac{D_{15,F}}{d_{85,B}}\leq4\text{ to }5,

\]

where \(D_{15,F}\) is the particle size for which 15% of the filter is finer and \(d_{85,B}\) is the size for which 85% of the base soil is finer. A common preliminary permeability criterion is

\[

\frac{D_{15,F}}{d_{15,B}}\geq4.

\]

These are not universal design laws. Applicable limits depend on whether the base soil is broadly graded, dispersive, gap graded or susceptible to suffusion, and on the governing authority’s standard. For gap-graded soils, the base fraction requiring protection may need to be redefined. Filters must also satisfy limits on maximum particle size, segregation, internal stability and constructability.

Geotextile filters are assessed using characteristic opening size, permittivity, clogging resistance, survivability and compatibility with the soil. Long-term biological or chemical clogging can govern performance.

8. Transient Groundwater Flow

8.1 Governing equation

Transient flow occurs when groundwater storage changes with time. For a saturated, heterogeneous anisotropic medium,

\[

\nabla\cdot(\mathbf K\nabla h)+R=S_s\frac{\partial h}{\partial t},

\]

where \(R\) is a volumetric source term per unit bulk volume and \(S_s\) is specific storage with units m⁻¹. The sign assigned to extraction or recharge must be consistent with the selected convention.

For a homogeneous confined aquifer of thickness \(b\), transmissivity and storativity are

\[

T=Kb,\qquad S=S_sb.

\]

Specific storage can be expressed approximately as

\[

S_s=\rho_wg(\alpha+n\beta_w),

\]

where \(\alpha\) is compressibility of the porous skeleton, \(n\) is porosity and \(\beta_w\) is water compressibility. Definitions of skeleton compressibility must be consistent with the derivation used.

For unconfined aquifers, delayed drainage from pores is usually represented by specific yield \(S_y\), although elastic storage can matter at early time. Linearisation may be inappropriate where drawdown is a substantial fraction of saturated thickness.

Hydraulic diffusivity is \(K/S_s\) for a three-dimensional confined medium and \(T/S\) for an aquifer formulation. High diffusivity produces rapid propagation of hydraulic-head changes.

8.2 Theis solution

For constant-rate pumping from a fully penetrating well in an ideal confined aquifer,

\[

s(r,t)=\frac{Q}{4\pi T}W(u),

\qquad

u=\frac{r^2S}{4Tt},

\]

where \(s\) is drawdown, \(Q\) is pumping rate, \(r\) is radial distance, \(t\) is elapsed time, and

\[

W(u)=\int_u^\infty\frac{e^{-y}}{y}\,dy.

\]

The ideal assumptions include a homogeneous, isotropic, laterally infinite confined aquifer; negligible well radius; instantaneous release from storage; and a fully penetrating well. Real boundaries, partial penetration, leakage, wellbore storage and variable pumping require modified solutions or numerical modelling.

For small \(u\), the Cooper–Jacob approximation is

\[

s\approx\frac{2.3Q}{4\pi T}

\log_{10}\left(\frac{2.25Tt}{r^2S}\right).

\]

Worked Example 5: Pumping-induced drawdown

A confined aquifer has \(T=1.0\times10^{-3}\) m²/s and \(S=5.0\times10^{-4}\). A well pumps at 0.020 m³/s for 86,400 s. Estimate drawdown 50 m away.

\[

u=\frac{50^2(5.0\times10^{-4})}{4(1.0\times10^{-3})(86400)}=0.00362,

\]

which is sufficiently small for the Cooper–Jacob approximation.

\[

s=\frac{2.3(0.020)}{4\pi(0.001)}

\log_{10}\left[\frac{2.25(0.001)(86400)}{50^2(5.0\times10^{-4})}\right].

\]

The logarithm argument is 155.52, giving

\[

s\approx3.661(2.192)=8.02\ \text{m}.

\]

This substantial drawdown could induce settlement in compressible strata and should be assessed beyond the excavation itself.

8.3 Unsaturated flow

Above the water table, hydraulic conductivity depends strongly on water content or matric suction. Unsaturated flow is commonly represented by Richards’ equation, combining Darcy–Buckingham flow with moisture conservation. Because the equation is nonlinear and constitutive relationships exhibit hysteresis, predictions are sensitive to soil-water characteristic curves, initial conditions and boundary fluxes.

9. Illustrative Case: Cofferdam Excavation in Layered Alluvium

Background

This is an illustrative case. A 30 m wide cofferdam is required for a river structure. The excavation base lies in saturated sand overlying a lower-permeability stratum. The external-to-internal head difference is 7 m. Preliminary transformed-section analysis gives \(K_{eq}=1.8\times10^{-5}\) m/s.

Problem

The initial flow net has \(N_f=4\) and \(N_d=14\). The final potential drop occurs over approximately 0.35 m near the excavation toe. The soil has \(G_s=2.68\) and \(e=0.72\). The design must address seepage inflow, boiling, uplift and effects on neighbouring assets.

Analysis

Discharge per metre width is

\[

q=(1.8\times10^{-5})(7)\frac{4}{14}=3.60\times10^{-5}\ \text{m}^2/\text{s}.

\]

Across 30 m,

\[

Q=1.08\times10^{-3}\ \text{m}^3/\text{s}=93.3\ \text{m}^3/\text{day}.

\]

The final head drop is \(7/14=0.50\) m, so

\[

i_{exit}\approx\frac{0.50}{0.35}=1.43.

\]

With \(i_c=0.977\), the exit-gradient safety factor is only

\[

FS=\frac{0.977}{1.43}=0.68.

\]

The preliminary arrangement is therefore unacceptable even though the pumping rate appears manageable.

Approach and solution

A deeper cut-off wall and downstream filter-drain are introduced. A revised flow net gives \(N_d=24\), with a final-field length of approximately 0.60 m:

\[

i_{exit}\approx\frac{7/24}{0.60}=0.486,

\qquad FS\approx\frac{0.977}{0.486}=2.01.

\]

The revised inflow estimate is

\[

Q=(1.8\times10^{-5})(7)\frac{4}{24}(30)

=6.30\times10^{-4}\ \text{m}^3/\text{s},

\]

or approximately 54.4 m³/day.

The design also includes piezometers inside and outside the cofferdam, standby pumps, filtered relief points, turbidity monitoring and trigger levels for evacuation or flooding. A numerical seepage model is used to assess stratigraphic variability and wall defects. Drawdown limits are imposed near adjacent foundations.

Outcome

The deeper cut-off reduces both exit gradient and inflow. The filter-drain provides a controlled exit that prevents loss of base soil. Monitoring verifies that measured heads remain within the observational-method trigger envelope.

Learning points

• Pump capacity alone does not demonstrate seepage safety.
• Local exit gradients may govern design.
• Cut-offs lengthen the seepage path but must be combined with controlled drainage.
• Uncertainty in stratigraphy and wall continuity requires monitoring and contingency planning.
• Groundwater lowering can transfer risk to neighbouring structures and water users.

10. Modelling, Technology and Professional Practice

Finite-difference and finite-element programs can represent anisotropy, complex boundaries, staged excavation, unsaturated flow and soil–structure interaction. Model sophistication does not eliminate uncertainty. Mesh refinement, mass-balance error, boundary placement and parameter sensitivity should be documented. Calibration should use independent head and flow observations where possible.

Automated piezometers, distributed fibre-optic sensing, remote turbidity measurements and real-time pumping records increasingly support observational design. Machine-learning tools may detect anomalous trends, but they should not replace physical seepage models or engineering judgement. Training data may not represent rare failure conditions, and automated alarms require validated thresholds and human oversight.

Good professional practice includes:

• developing a defensible geological and hydrogeological conceptual model;
• characterising spatial variability rather than relying on one conductivity value;
• checking both steady and transient conditions;
• examining credible defects and blocked-drain scenarios;
• using parameter sensitivity and consequence-based safety margins;
• validating numerical models against limiting calculations and field observations; and
• maintaining drainage systems throughout their design life.

Groundwater abstraction, discharge quality, wetland impacts, contaminated-water migration and effects on third-party wells may be regulated. Engineers must obtain required permits, manage contaminated effluent, preserve monitoring records and communicate foreseeable risks. Deliberately optimistic conductivity selection or unreported model uncertainty is both technically unsound and ethically unacceptable.

Chapter Summary

Darcy’s law relates specific discharge to hydraulic gradient through hydraulic conductivity. Conductivity depends on soil fabric and fluid properties, while actual seepage velocity also depends on effective porosity. Layering produces arithmetic-mean conductivity parallel to bedding and harmonic-mean conductivity normal to bedding. Anisotropic flow is governed by a conductivity tensor and may be examined through coordinate transformation or numerical analysis.

Flow nets provide discharge, head and gradient estimates for two-dimensional steady seepage. Seepage force has magnitude \(i\gamma_w\). Under upward flow, the effective unit weight is \(\gamma'-i\gamma_w\), leading to the critical gradient \(i_c=\gamma'/\gamma_w=(G_s-1)/(1+e)\). Local uplift and exit-gradient checks are essential.

Internal erosion includes backward erosion, suffusion, contact erosion and concentrated-leak erosion. Filters must retain the base soil while remaining permeable and internally stable. Transient groundwater analysis incorporates storage, and pumping responses may be estimated using analytical solutions where their assumptions are satisfied. Reliable engineering combines calculation, site characterisation, numerical modelling, monitoring, maintenance and responsible groundwater governance.

Key Terms

Hydraulic head: Mechanical energy of groundwater per unit weight, expressed as a water-column height.

Hydraulic gradient: Head loss per unit flow length.

Hydraulic conductivity: Coefficient relating Darcy discharge to hydraulic gradient.

Intrinsic permeability: Fluid-independent measure of a porous medium’s ability to transmit fluid.

Darcy velocity: Discharge divided by gross cross-sectional area.

Seepage velocity: Average water velocity through hydraulically effective pore space.

Effective porosity: Fraction of bulk volume contributing to connected through-flow.

Anisotropy: Directional dependence of hydraulic conductivity.

Flow net: Network of flow lines and equipotential lines representing two-dimensional seepage.

Equipotential line: Line along which total hydraulic head is constant.

Seepage force: Drag force exerted by flowing water on soil per unit bulk volume.

Critical hydraulic gradient: Upward gradient at which effective stress becomes zero under ideal one-dimensional conditions.

Uplift pressure: Upward-acting pore-water pressure beneath a structure or soil plug.

Piping: Progressive erosion that forms a subsurface channel.

Suffusion: Selective migration of fines through a coarser load-bearing skeleton.

Granular filter: Graded material designed to retain soil while permitting drainage.

Specific storage: Volume of water released from a unit volume of aquifer per unit decline in head.

Storativity: Water released from a vertical aquifer column of unit area per unit decline in head.

Transmissivity: Product of aquifer hydraulic conductivity and saturated thickness.

Hydraulic diffusivity: Ratio controlling the rate at which head disturbances propagate through an aquifer.

Review Questions

Short Answer Questions

1. Distinguish hydraulic conductivity from intrinsic permeability.

2. Why is Darcy velocity smaller than average seepage velocity in a saturated soil?

3. State the boundary conditions represented by flow lines and equipotential lines.

4. Define the critical hydraulic gradient and state its physical significance.

5. Distinguish backward erosion piping from suffusion.

Long Answer Questions

1. Derive the falling-head permeability equation and state its assumptions.

2. Explain how anisotropy affects groundwater-flow direction, flow-net construction and discharge calculation.

3. Derive the equivalent conductivities for flow parallel and perpendicular to horizontal soil layers.

4. Discuss the mechanisms, detection and mitigation of internal erosion in earth structures.

5. Develop the confined transient groundwater-flow equation and discuss the limitations of the Theis solution.

Analytical / Application Questions

1. A 6 m thick deposit consists of three horizontal layers. Determine equivalent horizontal and vertical conductivities for specified layer thicknesses and conductivities, and explain which layer controls each result.

2. Using a supplied cofferdam flow net, calculate seepage discharge, pore pressures, exit gradient and factor of safety against boiling. Recommend modifications if safety is inadequate.

3. Design a preliminary granular filter from given base-soil and filter gradations. Check retention, permeability and internal-stability requirements, and identify additional laboratory or field verification needed.

Multiple Choice Questions with Answers

Question 1

A saturated soil has K = 2.0 × 10^-5 m/s, hydraulic gradient 0.25 and flow area 20 m². What is the discharge?

A. 1.0 × 10^-4 m³/s

B. 4.0 × 10^-4 m³/s

C. 2.5 × 10^-5 m³/s

D. 1.6 × 10^-3 m³/s

Correct Answer: A. 1.0 × 10^-4 m³/s

Question 2

For one-dimensional upward seepage through saturated soil, the effective unit weight is:

A. γsat − iγw

B. γ′ − iγw

C. γ′ + iγw

D. γsat − γ′

Correct Answer: B. γ′ − iγw

Question 3

For homogeneous anisotropic soil with principal conductivities Kx and Kz, the conductivity used in the transformed flow-net discharge equation is:

A. Kx + Kz

B. (Kx + Kz)/2

C. √(KxKz)

D. KxKz

Correct Answer: C. √(KxKz)

Question 4

Which average gives the equivalent conductivity for one-dimensional flow perpendicular to horizontal layers?

A. Thickness-weighted arithmetic mean

B. Thickness-weighted harmonic mean

C. Unweighted geometric mean

D. Largest layer conductivity

Correct Answer: B. Thickness-weighted harmonic mean

Question 5

A flow net has K = 3 × 10^-5 m/s, H = 4 m, Nf = 3 and Nd = 12. What is the discharge per metre width?

A. 1.0 × 10^-5 m²/s

B. 3.0 × 10^-5 m²/s

C. 9.0 × 10^-5 m²/s

D. 1.2 × 10^-4 m²/s

Correct Answer: B. 3.0 × 10^-5 m²/s

Question 6

For soil with Gs = 2.65 and void ratio e = 0.65, the critical upward hydraulic gradient is:

A. 0.65

B. 1.00

C. 1.65

D. 2.54

Correct Answer: B. 1.00

Question 7

A preliminary filter criterion requires D15,F/d85,B ≤ 5. If d85,B = 0.20 mm, which filter value fails the criterion?

A. D15,F = 0.60 mm

B. D15,F = 0.80 mm

C. D15,F = 1.00 mm

D. D15,F = 1.20 mm

Correct Answer: D. D15,F = 1.20 mm

Question 8

In a confined aquifer, transmissivity is defined as:

A. Hydraulic conductivity divided by aquifer thickness

B. Hydraulic conductivity multiplied by aquifer thickness

C. Storativity multiplied by drawdown

D. Specific storage divided by hydraulic conductivity

Correct Answer: B. Hydraulic conductivity multiplied by aquifer thickness

Question 9

Which boundary is represented by a flow line in an ideal two-dimensional flow net?

A. A constant-head reservoir boundary

B. An impermeable foundation boundary

C. A fully screened constant-head well

D. An equipotential boundary

Correct Answer: B. An impermeable foundation boundary

Question 10

A soil has Darcy velocity 6 × 10^-6 m/s and effective porosity 0.30. What is its average seepage velocity?

A. 1.8 × 10^-6 m/s

B. 6.0 × 10^-6 m/s

C. 2.0 × 10^-5 m/s

D. 5.0 × 10^-2 m/s

Correct Answer: C. 2.0 × 10^-5 m/s