Effective Stress, Permeability and Seepage

Introduction and Context

Effective stress, hydraulic conductivity and seepage form the basis for analysing how water influences the strength, deformation and stability of soil. Almost every geotechnical structure—foundations, retaining walls, embankments, earth dams, cuttings, tunnels and drainage systems—is affected by groundwater flow and pore-water pressure.

A soil mass transmits load through a combination of its solid skeleton and pore fluid. The total stress measured at a point does not, by itself, determine soil deformation or shear strength. The stress carried by the soil skeleton, called effective stress, is the principal stress variable governing consolidation, frictional resistance and volume change in saturated soil. Seepage modifies pore-water pressure and therefore changes effective stress. Upward seepage may reduce effective stress to zero and cause boiling or piping, whereas downward seepage can increase effective stress.

This chapter develops the principles needed to calculate effective stress, hydraulic conductivity and seepage forces, and to apply flow nets, layered-soil methods, filter criteria and drainage concepts in professional practice. Introductory aspects of unsaturated-soil behaviour are also considered because many field problems involve partially saturated soils rather than ideal fully saturated conditions.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain total stress, pore-water pressure and effective stress.

2. Calculate effective stress under hydrostatic and seepage conditions.

3. Determine hydraulic conductivity using laboratory and field approaches.

4. Apply Darcy’s law and distinguish velocity, discharge velocity and seepage velocity.

5. Analyse one-dimensional flow through homogeneous, layered and anisotropic soils.

6. Construct and interpret flow nets for seepage and uplift assessment.

7. Evaluate piping, boiling, uplift and internal erosion risks.

8. Select appropriate drainage and filter measures.

9. Explain the principal differences between saturated and unsaturated flow.

10. Recognise assumptions, limitations and professional uncertainties in seepage analysis.

1. Stress in a Saturated Soil

1.1 Total stress

For a horizontal soil layer under a vertical overburden, the total vertical stress at depth z is the weight of material above that depth per unit area. For uniform unit weight γ:

σv = γz

For several layers:

σv = Σ(γi hi)

where γi is the total unit weight of layer i and hi is its thickness. The equation assumes a laterally extensive deposit and neglects surface surcharge. A surcharge q is added directly to the vertical stress where its distribution may reasonably be treated as uniform.

Total stress is an externally measurable or calculable quantity, but it does not indicate how the load is shared between water and soil particles.

1.2 Pore-water pressure

In static water, pore-water pressure is hydrostatic:

u = γw h

where u is gauge pore-water pressure, γw is the unit weight of water, and h is the vertical height of water above the point. With a water table at elevation zw and a point below it at depth z, u = γw(z − zw), provided the pressure is hydrostatic and the point is saturated.

Pore pressure is usually expressed as gauge pressure. Absolute pressure includes atmospheric pressure and is generally unnecessary in ordinary geotechnical stress calculations.

1.3 Effective stress principle

For saturated soil under conditions where water pressure acts equally in all directions, Terzaghi’s effective stress equation is

σ′ = σ − u

where σ′ is effective normal stress, σ is total normal stress and u is pore-water pressure. The corresponding incremental form is

dσ′ = dσ − du

The equation applies most directly to saturated soils with incompressible grains and water, and to stresses normal to the plane being considered. In three dimensions, effective stress is applied to each principal stress, provided the pore fluid is continuous and isotropic pressure is appropriate.

The soil skeleton carries effective stress. Consequently, shear strength and compressibility are governed primarily by effective stress rather than total stress. For a saturated frictional soil, the Mohr–Coulomb effective-stress expression is

τf = c′ + σ′ tan φ′

where τf is shear strength, c′ is effective cohesion intercept and φ′ is effective friction angle. In clean sands, c′ is often approximately zero, although apparent cohesion may arise from suction, cementation or testing conditions.

2. Effective Stress Under Different Hydraulic Conditions

2.1 Hydrostatic conditions

Below the water table, a soil with saturated unit weight γsat has total vertical stress γsat z and pore pressure γw z when z is measured downward from the water table. Thus:

σ′v = (γsat − γw)z = γ′z

where γ′ is the submerged or buoyant unit weight. The expression is valid for a homogeneous saturated layer under hydrostatic conditions.

Above the water table, the use of total unit weight depends on the degree of saturation and suction. In a simplified analysis of a dry or nearly dry soil, pore pressure is taken as zero gauge and σ′ approximately equals σ. This is not a complete unsaturated-soil treatment.

2.2 Seepage and hydraulic gradient

When water flows through soil, hydraulic head changes with position. The hydraulic gradient is

i = Δh/L

where Δh is the loss of total hydraulic head over flow length L. Hydraulic head is the sum of elevation head and pressure head:

h = z + u/γw

with z measured relative to a chosen datum. In a vertical soil column, downward flow increases effective stress, while upward flow decreases it.

For vertical upward seepage, the effective vertical stress gradient is approximately

dσ′/dz = γ′ − iγw

when z is measured in the downward direction and i is the upward hydraulic gradient magnitude. At the critical gradient, effective stress becomes zero:

ic = γ′/γw = (Gs − 1)/(1 + e)

where Gs is specific gravity of soil solids and e is void ratio. This ideal expression assumes saturated cohesionless soil, one-dimensional upward flow and no stabilising structural or capillary effects. For Gs = 2.65 and e = 0.65, ic = 1.65/1.65 = 1.00.

Boiling or quick condition occurs when upward seepage reduces interparticle contact stress to approximately zero. It does not mean that individual grains literally reach their melting or boiling temperature. Cohesive soils may not show visible boiling at the same gradient because apparent cohesion and low permeability alter the mechanism, but internal erosion may still occur.

Worked Example 1: Effective stress with upward seepage

A saturated sand has Gs = 2.65 and e = 0.65. Its saturated unit weight is 19.9 kN/m³. At a point 4 m below the ground surface, the water table is at the surface and an upward hydraulic gradient of 0.40 exists. Determine total stress, pore pressure and effective stress.

Take γw = 9.81 kN/m³.

Total stress:

σv = γsat z = 19.9 × 4 = 79.6 kPa

Hydrostatic pore pressure at 4 m:

u0 = γw z = 9.81 × 4 = 39.24 kPa

The upward seepage force per unit volume is iγw = 0.40 × 9.81 = 3.924 kN/m³. Over 4 m, the pressure increase associated with upward flow, relative to the hydrostatic reference, is 15.70 kPa. Hence:

u = 39.24 + 15.70 = 54.94 kPa

σ′v = 79.6 − 54.94 = 24.66 kPa

The hydrostatic effective stress would have been 40.36 kPa; upward seepage has therefore reduced it by 15.70 kPa. The critical gradient is approximately 1.00, so the current gradient is below the ideal critical value.

3. Hydraulic Conductivity and Darcy’s Law

3.1 Hydraulic conductivity

Hydraulic conductivity k expresses the ease with which a particular soil–fluid system transmits water. It depends on pore size, void ratio, soil fabric, degree of saturation, fluid viscosity and temperature. It is not solely a property of the soil.

Typical orders of magnitude are approximately:

• clean gravel: 10⁻² to 1 m/s;
• clean sand: 10⁻⁵ to 10⁻² m/s;
• silt: 10⁻⁹ to 10⁻⁵ m/s;
• clay: 10⁻¹² to 10⁻⁸ m/s.

These ranges overlap and should not replace testing.

3.2 Darcy’s law

For one-dimensional saturated flow through a representative soil volume:

q = kiA

where q is discharge, k is hydraulic conductivity, i is hydraulic gradient and A is gross cross-sectional area normal to flow. The Darcy flux or discharge velocity is

vD = q/A = ki

Darcy’s law is valid for laminar flow through soil pores, which is the usual condition in sands, silts and clays. It is a macroscopic law and does not represent the actual velocity in each pore.

The average seepage velocity through the voids is

vs = vD/n = ki/n

where n is porosity. Since n is less than one, seepage velocity exceeds Darcy flux. Confusing the two can produce significant errors in contaminant travel-time estimates.

3.3 Hydraulic conductivity tests

The constant-head test is suitable for relatively permeable soils. If a specimen of length L and cross-sectional area A carries a discharge Q under head difference Δh for time t:

k = QL/(AΔh t)

The falling-head test is commonly used for less permeable soils. If a standpipe of area a is connected to a specimen of length L and area A, and the head falls from h1 to h2 during time t:

k = (aL/At) ln(h1/h2)

These equations assume saturated, one-dimensional flow; constant specimen geometry; negligible leakage; and consistent units. Laboratory results can differ from field values because of sample disturbance, scale, fabric, fissures and anisotropy.

Field methods include pumping tests, slug tests, borehole permeameter tests, infiltration tests and interpretation of piezometric responses. For earth structures, field-scale tests are often more representative than small laboratory specimens.

Worked Example 2: Darcy discharge and seepage velocity

A saturated sand layer has k = 2 × 10⁻⁵ m/s, hydraulic gradient i = 0.30, cross-sectional area A = 12 m² and porosity n = 0.35. Determine discharge, Darcy flux and average seepage velocity.

q = kiA = (2 × 10⁻⁵)(0.30)(12) = 7.2 × 10⁻⁵ m³/s

vD = q/A = 7.2 × 10⁻⁵/12 = 6.0 × 10⁻⁶ m/s

vs = vD/n = (6.0 × 10⁻⁶)/0.35 = 1.71 × 10⁻⁵ m/s

The seepage velocity is approximately 2.86 times the Darcy flux because n = 0.35.

4. Flow Through Layered and Anisotropic Soils

4.1 Layers normal to flow

For layers of thickness Li arranged in series along the direction of flow, the same discharge passes through every layer. The equivalent hydraulic conductivity is the harmonic average:

kH = Ltotal / Σ(Li/ki)

where Ltotal = ΣLi. A thin, very low-permeability layer can therefore control the overall flow.

4.2 Layers parallel to flow

For layers arranged side by side with flow parallel to their boundaries, each layer experiences approximately the same hydraulic gradient. The equivalent conductivity is the arithmetic, thickness-weighted average:

kV or kparallel = Σ(ki Ai)/ΣAi

For layers of equal width and thickness representation, this becomes Σ(ki Li)/ΣLi. The notation kH and kV must be tied to the actual geometry; different textbooks use different subscripts.

4.3 Anisotropy

Many natural deposits have horizontal conductivity greater than vertical conductivity because of stratification. Let kx and kz denote principal conductivities in horizontal and vertical directions. Darcy’s law becomes a tensor relationship:

q = −K∇h

where q is the Darcy flux vector, K is the hydraulic-conductivity tensor and ∇h is the hydraulic-gradient vector. In two-dimensional orthogonal principal directions, coordinate transformation or an equivalent-section method may be used. For homogeneous anisotropic soil, a common transformation uses a scale factor related to √(kx/kz), but the exact construction depends on the chosen coordinates and boundary conditions. Numerical methods are preferable for complicated geometry.

Anisotropy affects pore-pressure dissipation, slope drainage, consolidation, contaminant transport and seepage exits. Assuming isotropy without evidence may be unconservative.

5. Flow Nets and Seepage Analysis

A flow net is a graphical representation of two orthogonal families: flow lines, which indicate paths followed by seepage, and equipotential lines, which join points of equal total head. In an ideal homogeneous, isotropic soil, the two families intersect approximately at right angles and form curvilinear squares.

For a two-dimensional flow net, the total discharge per unit width is

q = kH(Nf/Nd)

where H is total head loss, Nf is the number of flow channels and Nd is the number of potential drops. The formula assumes steady two-dimensional flow, homogeneous isotropic soil, negligible dimensional thickness effects and a correctly drawn net. If the potential drops are equal, the head loss per drop is H/Nd.

The pore pressure at a point can be estimated from its total-head position. The pressure head is total head minus elevation head, and u = γw times pressure head. This allows uplift pressures beneath floors, dams and weirs to be assessed.

Constructing a flow net

A practical procedure is:

1. Define upstream and downstream water levels and impermeable boundaries.

2. Identify constant-head boundaries and no-flow boundaries.

3. Sketch approximate flow lines and equipotential lines.

4. Adjust the network until intersections are approximately orthogonal and individual elements have similar proportions.

5. Count flow channels and potential drops.

6. Calculate discharge and pressure distribution.

7. Check exit gradients and the plausibility of the result.

Flow nets are powerful conceptual and design tools, but they are approximate. Complex stratigraphy, anisotropy, drains, cracks and transient conditions may require finite-element or finite-difference seepage analysis.

6. Uplift, Exit Gradient and Piping

6.1 Uplift pressure

Water pressure acting upward beneath a slab or hydraulic structure produces uplift. The net uplift force is obtained by integrating pressure over the relevant area:

U = ∫u dA

For a constant pressure u over area A, U = uA. In a flow-net analysis, pressures vary spatially and must be evaluated at several points or integrated numerically. Structural stability checks normally compare uplift with resisting weight, anchor forces, downstream water forces and friction, while also considering sliding and overturning.

6.2 Exit gradient

The exit gradient is the hydraulic gradient where seepage emerges at a downstream surface. It is often critical for assessing erosion and piping. A high exit gradient creates seepage forces that can detach particles, particularly at an unprotected soil–water interface.

A factor of safety against piping may be expressed as

FS = ic/iexit

where ic is the critical gradient and iexit is the computed or measured exit gradient. The selected allowable factor of safety is a design and regulatory matter and must reflect uncertainty, consequences and the reliability of drainage and filter measures. It should not be adopted automatically without reference to project standards.

6.3 Internal erosion and piping

Piping is a progressive form of internal erosion in which seepage removes soil particles through concentrated paths. It may occur through the foundation, embankment, cracks, poorly compacted zones, animal burrows or interfaces. It is distinct from a simple exceedance of the ideal critical gradient, although high gradients can initiate it.

Indicators include turbid seepage, increasing discharge, sinkholes, muddy boils, wet spots, settlement and unexplained changes in piezometric levels. Early detection through instrumentation and inspection is essential.

7. Filters and Drainage Systems

7.1 Filter function

A filter must retain the protected base soil while allowing water to pass. It also prevents migration of fines into drains and relieves pore pressure. A filter that is too fine may clog; one that is too coarse may permit internal erosion.

Filter design is commonly based on particle-size relationships between the base soil and filter. Several empirical criteria exist, often involving D15 of the filter and d85 or d15 of the base soil. Because criteria differ with soil gradation, uniformity, plasticity and flow direction, they should be applied using the governing standard and project-specific testing. No single ratio is universally valid for every soil.

A filter system may include:

• a drainage layer;
• graded transition zones;
• chimney and blanket drains in embankment dams;
• toe drains;
• geotextiles;
• relief wells or drainage galleries;
• cutoff trenches, slurry walls or grout curtains.

7.2 Drainage design principles

Good drainage design must provide adequate capacity, continuity, durability, cleanability and resistance to clogging. Geotextiles require separate checks for retention, permeability, clogging, survivability and installation damage. Drain outlets must remain accessible and protected from blockage.

Drainage reduces pore-water pressure and can improve slope and foundation stability, but it may also mobilise fines, alter groundwater chemistry, affect neighbouring properties or create uncontrolled discharge. Construction sequencing is important: drains should not be installed in a way that creates preferential erosion paths before filters and outlets are complete.

8. Unsaturated-Soil Behaviour: An Introduction

Above the water table, soil commonly contains both air and water. Negative pore-water pressure relative to air pressure is called matric suction. In unsaturated soils, stress behaviour cannot generally be represented by σ − u alone because air and water pressures differ and the degree of saturation changes.

A useful conceptual description uses two independent stress variables, such as net stress (σ − ua) and suction (ua − uw), where ua is pore-air pressure and uw is pore-water pressure. An extended shear-strength expression is often written in the form

τf = c′ + (σ − ua)tanφ′ + χ(ua − uw)tanφ′

where χ is a parameter between approximately zero and one that depends on saturation and soil characteristics. This is a conceptual or constitutive representation rather than a universal design equation; χ must be selected or measured appropriately.

Hydraulic conductivity in unsaturated soil decreases markedly as suction increases because water-filled pathways become discontinuous. Unsaturated flow is therefore nonlinear. Soil-water-retention curves relate water content or degree of saturation to suction, and hydraulic-property models such as van Genuchten-type relationships are often calibrated from laboratory data. Hysteresis may occur: drying and wetting paths do not coincide.

Unsaturated conditions influence rainfall-induced slope failure, expansive soils, infiltration, landfill covers, pavement subgrades and capillary barriers. Ignoring suction may be conservative or unconservative depending on the problem and the drainage or wetting scenario.

9. Illustrative Case: Seepage Control at an Earth Embankment

Background

An illustrative earth embankment retains water upstream. The foundation consists of a permeable sand layer over relatively low-permeability silt. Field observations show wetness and occasional cloudy seepage near the downstream toe after reservoir levels rise.

Problem

The owner must determine whether the seepage indicates normal drainage or progressive internal erosion. The principal risks are excessive exit gradient, migration of foundation sand into the silt interface, uplift beneath a downstream apron and deterioration during prolonged high water levels.

Analysis

Investigations include boreholes, grain-size testing, piezometers along the centreline and toe, discharge measurement at drains, and inspection during reservoir fluctuation. The piezometric data show a steep head drop near the downstream toe. A two-dimensional seepage model, calibrated against measured heads, predicts an elevated exit gradient at an unprotected interface. The measured discharge is modest, but turbidity increases during rapid reservoir rise.

A simple isotropic model underestimates the lateral flow observed in the sand layer. Incorporating horizontal-to-vertical anisotropy produces a better match to piezometric levels. This demonstrates the importance of field-scale hydraulic properties and model calibration.

Approach and solution

The remedial scheme includes a properly graded downstream filter and drainage blanket connected to a toe drain, together with erosion-resistant outlet protection. Piezometers are added upstream and downstream of the filter, and a turbidity and flow-monitoring programme is established. Construction quality control verifies filter gradation, continuity, compaction of adjacent zones and protected outlet discharge.

Outcome

Following installation, downstream pore pressures decrease, the exit gradient is reduced and seepage becomes clear. Continued monitoring confirms stable discharge during reservoir cycles. The intervention does not attempt to eliminate all seepage; instead, it controls pressure and prevents particle migration.

Learning points

1. Visible seepage is not automatically evidence of failure; its quantity, clarity, pressure and trend are important.

2. Filters must be designed for both retention and drainage.

3. Instrumentation is valuable for detecting changes before visible distress occurs.

4. Anisotropy and interfaces can dominate seepage behaviour.

5. Remediation should address the mechanism, not merely conceal the symptom.

10. Professional Practice, Limitations and Contemporary Tools

Seepage analysis is affected by uncertain stratigraphy, spatial variability, seasonal groundwater changes, boundary-condition errors, scale effects and imperfect hydraulic-conductivity data. Sensitivity analysis should examine plausible ranges of k, water levels, anisotropy, drain performance and filter properties.

Numerical groundwater models can represent complex geometry, nonlinear unsaturated flow and transient reservoir changes. However, a sophisticated model does not compensate for poor conceptualisation or inadequate data. Model verification, calibration, mesh-quality checks, mass-balance checks and independent engineering review are essential.

Remote sensing, automated piezometers, fibre-optic sensing, drones and data dashboards increasingly support embankment and slope surveillance. Artificial intelligence can help identify anomalous trends in pore pressure, seepage discharge or turbidity, but it should supplement—not replace—mechanistic interpretation, site inspection and professional judgement. Training data may be sparse precisely for rare failure events, and false alarms or missed alarms have safety implications.

Ethically and legally, engineers must communicate uncertainty, maintain auditable calculations, follow applicable dam, earthwork, groundwater and environmental regulations, and avoid presenting model outputs as measurements. Groundwater extraction, drainage discharge and filter materials may affect ecosystems, neighbouring landowners and water quality. Design decisions should therefore consider safety, environmental impact, maintainability and long-term monitoring.

Best-Practice Checklist

Before approving a seepage design, confirm that:

• groundwater levels and seasonal extremes have been established;
• total and effective stresses use consistent reference elevations and units;
• hydraulic conductivity data represent field fabric and anisotropy;
• Darcy flux has not been confused with pore-water velocity;
• layered-soil equivalent conductivity matches the flow geometry;
• boundary conditions and transient stages are documented;
• uplift and exit gradients have been checked;
• filter retention, permeability and clogging criteria are satisfied;
• drains have continuous, maintainable outlets;
• construction tolerances and quality-control tests are specified;
• instrumentation, trigger levels and response actions are defined;
• uncertainty and consequences of failure are explicitly considered.

Chapter Summary

Effective stress is the difference between total stress and pore-water pressure and governs the mechanical response of saturated soil. Hydrostatic water pressure produces submerged effective unit weight, while seepage changes effective stress through the hydraulic gradient. Upward seepage reduces effective stress and may lead to boiling or piping when the critical gradient is approached.

Darcy’s law relates discharge to hydraulic conductivity, hydraulic gradient and area. Darcy flux is not the same as average seepage velocity, which must be divided by porosity. Hydraulic conductivity depends on soil, fluid and fabric, and is commonly determined by constant-head, falling-head or field tests.

Layered soils require equivalent-conductivity methods suited to flow geometry. Anisotropy is common in stratified deposits and may strongly influence seepage. Flow nets provide graphical estimates of discharge, pressure distribution, uplift and exit gradient, although numerical models are often needed for complex or transient systems.

Filters and drainage systems must retain soil particles while safely conveying water and reducing pore pressure. Piping is a progressive internal-erosion mechanism requiring attention to gradients, interfaces, particle migration and monitoring. Unsaturated soils require consideration of suction, separate air and water pressures, nonlinear conductivity and changing saturation. Safe professional practice combines sound theory, representative investigation, calibrated analysis, constructible drainage, instrumentation and transparent treatment of uncertainty.

Key Terms

Effective stress: Stress carried by the soil skeleton, commonly σ − u in saturated soil.

Total stress: Stress caused by the total weight and applied loads above a plane.

Pore-water pressure: Pressure exerted by water within soil voids.

Hydraulic head: Elevation head plus pressure head.

Hydraulic gradient: Head loss per unit length in the direction of flow.

Hydraulic conductivity: Capacity of a soil–fluid system to transmit water.

Darcy flux: Discharge per unit gross cross-sectional area, ki.

Seepage velocity: Average water velocity through the void space, usually ki/n.

Porosity: Volume of voids divided by total soil volume.

Void ratio: Volume of voids divided by volume of solids.

Critical hydraulic gradient: Ideal gradient at which effective stress in saturated cohesionless soil becomes zero.

Flow line: A line tangent to the direction of seepage.

Equipotential line: A line joining points of equal total hydraulic head.

Flow net: Intersecting flow-line and equipotential-line network.

Uplift: Upward water force acting beneath a structure or soil mass.

Exit gradient: Hydraulic gradient at the point where seepage exits soil.

Piping: Progressive internal erosion caused by seepage-induced particle migration.

Filter: Graded soil or geosynthetic layer that retains base soil and permits drainage.

Matric suction: Difference between pore-air and pore-water pressure in unsaturated soil.

Anisotropy: Directional variation of a soil property, such as hydraulic conductivity.

Review Questions

Short Answer Questions

1. Define effective stress and state the saturated-soil equation commonly used to calculate it.

2. Distinguish Darcy flux from average seepage velocity.

3. What is meant by the critical hydraulic gradient?

4. Why is the harmonic mean used for conductivity of layers normal to flow?

5. State two functions of a properly designed filter.

Long Answer Questions

1. Explain how hydrostatic and seepage conditions affect vertical effective stress in a saturated soil deposit.

2. Derive and discuss the constant-head and falling-head permeability equations, including their assumptions.

3. Describe the construction, interpretation and limitations of a two-dimensional flow net.

4. Discuss the mechanisms of uplift, exit-gradient failure and piping in hydraulic earth structures.

5. Explain why anisotropy and unsaturated conditions complicate seepage analysis and design.

Analytical / Application Questions

1. A 5 m saturated sand layer has γsat = 20 kN/m³ and is subjected to an upward hydraulic gradient of 0.35. Taking γw = 9.81 kN/m³, calculate the total stress, hydrostatic pore pressure, seepage-induced pressure increment and effective stress at the base.

2. A soil profile contains 1 m of sand with k = 1 × 10⁻⁴ m/s over 3 m of clay with k = 2 × 10⁻⁸ m/s. Calculate the equivalent conductivity for vertical flow and explain which layer controls the discharge.

3. A flow net has k = 4 × 10⁻⁶ m/s, total head loss H = 6 m, Nf = 4 and Nd = 12. Calculate discharge per metre width and explain what additional checks are needed before accepting the design.

Multiple Choice Questions with Answers

Question 1

Under saturated hydrostatic conditions, the effective vertical stress in a homogeneous layer below the water table increases with depth at approximately which rate?

A. γsat

B. γw

C. γsat − γw

D. γsat + γw

Correct Answer: C. γsat − γw

Question 2

A soil has hydraulic conductivity 2 × 10⁻⁵ m/s and hydraulic gradient 0.30. What is its Darcy flux?

A. 6.0 × 10⁻⁶ m/s

B. 6.0 × 10⁻⁵ m/s

C. 1.5 × 10⁻⁵ m/s

D. 1.67 × 10⁻⁴ m/s

Correct Answer: A. 6.0 × 10⁻⁶ m/s

Question 3

Average seepage velocity is obtained from Darcy flux by dividing by:

A. void ratio

B. porosity

C. specific gravity

D. hydraulic gradient

Correct Answer: B. porosity

Question 4

For soil layers arranged in series normal to flow, the equivalent hydraulic conductivity is based on the:

A. arithmetic mean of conductivities

B. geometric mean of conductivities

C. harmonic, thickness-weighted mean

D. largest conductivity only

Correct Answer: C. harmonic, thickness-weighted mean

Question 5

In an ideal homogeneous isotropic flow net, flow lines and equipotential lines intersect approximately at:

A. 0°

B. 45°

C. 60°

D. 90°

Correct Answer: D. 90°

Question 6

The ideal critical hydraulic gradient for saturated cohesionless soil is given by:

A. γw/γ′

B. γ′/γw

C. γsat/γw

D. e/Gs

Correct Answer: B. γ′/γw

Question 7

Which observation is most directly suggestive of particle migration and possible internal erosion?

A. Clear, stable seepage

B. A constant piezometric level

C. Increasing cloudy or turbid seepage

D. A low hydraulic gradient

Correct Answer: C. Increasing cloudy or turbid seepage

Question 8

The primary combined function of a properly designed filter is to:

A. increase total stress and eliminate all groundwater

B. retain base soil while permitting drainage

C. reduce soil specific gravity

D. increase pore-water pressure

Correct Answer: B. retain base soil while permitting drainage

Question 9

In unsaturated soil, matric suction is commonly defined as:

A. uw − ua

B. ua − uw

C. σ − u

D. γw h only

Correct Answer: B. ua − uw

Question 10

For a two-dimensional flow net, discharge per unit width is commonly calculated as:

A. kH(Nd/Nf)

B. kH(Nf/Nd)

C. H(Nf/Nd)/k

D. kNfNd/H

Correct Answer: B. kH(Nf/Nd)