The Ultimate Guide to Your Soil Mechanics Past Year Question Bank

Importance of Soil Mechanics in Civil Engineering:

Soil mechanics is a fundamental branch of civil engineering that deals with the study of the physical properties and mechanical behavior of soil. Its importance is paramount as virtually every civil engineering structure is founded on or in the earth. A thorough understanding of soil mechanics is crucial for:

• Foundation Design: The stability and longevity of any structure depend entirely on its foundation. Soil mechanics provides principles to design foundations that can safely transmit structural loads to underlying soil without causing shear failure or excessive settlement.
• Pavement Design: Design of flexible and rigid pavements depends on the strength and properties of subgrade soil. Soil mechanics principles assess California Bearing Ratio (CBR) value and design appropriate pavement layer thickness.
• Slope Stability and Earth-Retaining Structures: Critical for preventing landslides in excavations, embankments, and hillsides. Used to analyze slope stability and design retaining structures.
• Underground Structures and Tunneling: Requires understanding of soil behavior, pressure distribution, and ground movement potential.
• Earth Dams and Embankments: Essential for selecting suitable soil types, determining compaction standards, and analyzing stability and seepage.

Soil Formation:

Soils are formed through continuous geological process involving weathering and decomposition of parent rocks:

• Physical Weathering (Mechanical Disintegration): Rocks break into smaller particles without chemical change. Agents include temperature changes, frost action, abrasion, and plant/animal action.
• Chemical Weathering (Chemical Decomposition): Alters mineralogical composition forming new minerals. Processes include hydration, carbonation, oxidation, and solution.

Weathered particles may remain in place (residual soils) or be transported (transported soils).

Clay minerals are primary constituents of fine-grained cohesive soils, crystalline in nature, formed from chemical weathering of rocks. Fundamental building blocks are Silica Tetrahedron and Alumina Octahedron.

Kaolinite Group:

• Structure: 1:1 crystal structure – one silica tetrahedral sheet bonded to one alumina octahedral sheet with strong hydrogen bonds.
• Characteristics: Strong bonding prevents water entry, low swelling potential, low plasticity, relatively stable, non-expanding lattice.
• Sketch Description: Tetrahedral sheet directly on octahedral sheet, stacked with hydrogen bonds between layers.

Illite Group:

• Structure: 2:1 crystal structure – alumina octahedral sheet sandwiched between two silica tetrahedral sheets, bonded by potassium ions (K+).
• Characteristics: Moderate swelling potential and plasticity, most common clay mineral in engineering soils.
• Sketch Description: Two tetrahedral sheets with octahedral sheet between, potassium ions between stacked layers.

Montmorillonite Group (Smectite):

• Structure: 2:1 structure similar to illite, but bonding by very weak van der Waals forces with significant isomorphous substitution.
• Characteristics: Weak bonds allow water entry, significant swelling upon wetting, very high plasticity and compressibility.
• Sketch Description: 2:1 layer structure with large variable space between layers containing water molecules and exchangeable cations.

Definition of Soil Mechanics:

Soil Mechanics is the branch of engineering science that applies principles of mechanics (including hydraulics and statics) and properties of engineering materials to predict mechanical behavior of soils in response to loads, water, and temperature changes. It provides theoretical basis for soil engineering.

Significance of Fluid Mechanics in Soil Mechanics:

Fluid mechanics is integral to soil mechanics due to presence and movement of water in soil’s void spaces:

• Effective Stress Principle: Governs strength and compressibility of soil, directly related to pore water pressure. Fluid mechanics principles essential to calculate pore pressures.
• Permeability and Seepage: Critical for estimating seepage under dams, analyzing slope stability under seepage pressure, and designing drainage systems. Governed by Darcy’s law from fluid mechanics.
• Consolidation: Settlement of saturated fine-grained soils under load is time-dependent process controlled by rate of water expulsion from soil pores. Theory based on fluid flow through porous medium.

Building Blocks of Clay Minerals:

Fundamental building blocks are microscopic, crystalline sheet-like structures:

• Silica Tetrahedron: Central silicon atom (Si⁴⁺) equidistant from four oxygen atoms at corners of tetrahedron. Link together to form silica sheet.
• Alumina Octahedron: Central aluminum atom (Al³⁺) surrounded by six hydroxyl (OH⁻) groups at corners of octahedron. Link together to form gibbsite sheet.

Common Groups of Clay Minerals:

Formed by stacking of basic sheets:

• Kaolinite: 1:1 mineral formed by alternating silica and gibbsite sheets, bonded strongly by hydrogen bonds. Stable with low swelling potential.
• Illite: 2:1 mineral with gibbsite sheet between two silica sheets. Layers bonded by potassium ions, giving moderate swelling potential.
• Montmorillonite: 2:1 mineral similar to illite, but with very weak van der Waals forces between layers. Allows water entry causing very high swelling and shrinking potential.

Definition of Soil:

From civil engineering perspective, soil is defined as uncemented or weakly cemented accumulation of mineral particles and decayed organic matter (solid particles) with liquid and gas in empty spaces between particles. It is loose surface material of Earth’s crust that can be excavated with common tools.

Various Soil Engineering Problems:

• Foundation Failure: Bearing capacity failure or excessive settlement of foundations supporting structures.
• Slope Instability: Landslides and failures of natural slopes, embankments, and cuts.
• Seepage: Uncontrolled flow of water through or under structures like dams and levees leading to instability.
• Low Soil Strength: Soils too weak to support pavements or structures.
• High Compressibility: Soils undergoing large volume changes (settlement) under load, damaging structures.
• Expansive Soils: Soils that swell significantly when wet and shrink when dry, exerting large forces on foundations and walls.

Basic Structural Units of Clay Minerals:

The two fundamental structural units are:

• Silica Tetrahedron: Unit with one silicon atom surrounded by four oxygen atoms.
• Alumina Octahedron: Unit with one aluminum atom surrounded by six hydroxyl groups.

These units combine to form silica sheets and alumina (gibbsite) sheets, stacked to create different clay minerals.

Difference between Kaolinite, Illite, and Montmorillonite:

Clay particles have net negative charge on their surface due to isomorphous substitution. To balance this charge, they attract positive ions (cations) from surrounding water (pore fluid). These cations form concentrated layer close to clay surface, known as adsorbed layer. Concentration decreases with distance forming more diffuse layer. This combination of dense adsorbed layer and subsequent diffuse layer of cations surrounding negatively charged clay particle is called Diffuse Double Layer. Thickness influences repulsive forces between clay particles, thus affecting soil’s engineering behavior.

Difference between Residual and Transported Soils:

Solutions to Soil Engineering Problems:

Solutions involve either improving soil properties or designing structures to accommodate them:

• Soil Improvement: Techniques like compaction, drainage, soil stabilization with admixtures (cement, lime), or ground reinforcement (geosynthetics).
• Foundation Design: For weak soils, deep foundations (piles, piers) transfer loads to stronger strata. Raft foundations distribute loads over larger area.
• Retaining Structures: Building retaining walls or soil nailing to support unstable slopes or excavations.

Basic Structural Units of Clay Minerals:

Fundamental structural units are:

• Silica Tetrahedron: Unit composed of central silicon ion bonded to four oxygen ions. Link to form tetrahedral sheet.
• Alumina Octahedron: Unit composed of central aluminum ion bonded to six hydroxyl ions. Link to form octahedral (gibbsite) sheet.

Structure of Kaolinite, Illite, and Montmorillonite:

• Kaolinite: 1:1 structure – one silica sheet stacked on one alumina sheet. Composite layers held by strong hydrogen bonds. Rigid structure prevents water penetration between layers – no swelling.
  Structural Representation: [Silica Sheet]–(H-Bond)–[Alumina Sheet] — [Silica Sheet]–(H-Bond)–[Alumina Sheet]
• Illite: 2:1 structure – alumina sheet sandwiched between two silica sheets. 2:1 layers bonded by potassium ions (K⁺). Bond weaker than Kaolinite.
  Structural Representation: [Silica Sheet]–[Alumina Sheet]–[Silica Sheet]–(K⁺ ions)–[Silica Sheet]–[Alumina Sheet]–[Silica Sheet]
• Montmorillonite: 2:1 structure similar to Illite. Bond between layers by very weak van der Waals forces. Weak bond allows water and exchangeable cations to enter space between layers, causing significant swelling.
  Structural Representation: [Silica Sheet]–[Alumina Sheet]–[Silica Sheet] ~ (Water + Cations) ~ [Silica Sheet]–[Alumina Sheet]–[Silica Sheet]

Soil structure refers to geometric arrangement of soil particles and interparticle forces acting between them. Main types:

• Single-grained Structure: Found in coarse-grained soils (sands, gravels). Particles in direct contact. Structure can be loose or dense depending on packing.
• Honeycomb Structure: Found in fine sands and silts. Particles form small arch-like structures trapping large voids. Unstable, can collapse under load or vibration.
• Flocculated Structure: Found in clays deposited in saltwater. Clay particles orient in edge-to-face arrangement due to electrical charges, forming loose but relatively strong “card-house” structure.
• Dispersed Structure: Found in clays deposited in freshwater or remolded clays. Particles orient in face-to-face or parallel arrangement, resulting in dense but weaker structure.

Adsorbed Water:

Adsorbed water is water held onto surface of soil particles, particularly clay particles, by powerful molecular forces of electrical attraction. Clay surfaces have net negative charge, attracting positive (hydrogen) side of dipolar water molecules. Creates thin, dense, highly viscous film of water physically bound to particle surface, not considered free or gravitational water.

Double Diffusive Layer Formation:

Formation of diffuse double layer is electrochemical phenomenon occurring around clay particles suspended in water:

1. Surface Charge: Clay minerals develop net negative electrical charge mainly due to isomorphous substitution (lower-valence cation replaces higher-valence cation), leaving unsatisfied negative charge.
2. Attraction of Cations: To maintain electrical neutrality, negatively charged surfaces attract positive ions (cations like Na⁺, K⁺, Ca²⁺) from surrounding pore water.
3. Layer Formation: Attracted cations arrange in two layers:
   • Adsorbed/Stern Layer: Fixed, dense layer of cations strongly held to clay surface.
   • Diffuse Layer: Second, more diffuse layer where cation concentration decreases with distance until reaching equilibrium concentration of pore water.

This combined two-part system of negative surface charge and distributed positive charge of cations is diffuse double layer.

a) Importance of Studying Soil Mechanics:

Studying soil mechanics is essential for civil engineers because soil is most common material they encounter. It provides knowledge to:

• Design safe and stable foundations for all types of structures (buildings, bridges)
• Analyze stability of slopes and design retaining structures to prevent landslides
• Design and construct earthworks such as dams, embankments, and roads
• Understand and predict how soil will behave under different loading and environmental conditions, preventing structural failures, excessive settlements, and other geotechnical problems

b) Types of Soil:

Based on Formation:

• Residual Soil: Soil that remains at location of its formation
• Transported Soil: Soil moved from place of origin by agents like water (alluvial), wind (aeolian), ice (glacial), or gravity (colluvial)

Based on Grain Size:

• Coarse-grained Soil: Gravels and sands
• Fine-grained Soil: Silts and clays

Based on Cohesiveness:

• Cohesionless Soil: Soils where particles do not stick together (sand, gravel). Strength derived from inter-particle friction
• Cohesive Soil: Soils where particles stick together due to intermolecular attraction (clay). Possess cohesion and plasticity

Soil Fabric and Soil Structure:

• Soil Fabric: Refers to spatial arrangement of soil particles and associated voids. Deals with orientation and distribution of particles on small scale.
• Soil Structure: Broader term including not only soil fabric but also interparticle forces (electrical forces) acting between them. Describes combined effect of fabric and bonding.

1:1 and 2:1 Types of Clay Minerals:

1:1 Type Clay Mineral (e.g., Kaolinite):

• Explanation: Consists of alternating layers of one silica tetrahedral sheet and one alumina octahedral sheet in 1:1 ratio. Layers held together by strong hydrogen bonds between hydroxyls of alumina sheet and oxygen of silica sheet. Strong bonding makes structure stable, prevents water entry between layers.
• Sketch Description: Single tetrahedral sheet linked to single octahedral sheet. 1:1 unit stacked on another 1:1 unit with hydrogen bonds indicated between them.

2:1 Type Clay Mineral (e.g., Illite, Montmorillonite):

• Explanation: Consists of one alumina octahedral sheet sandwiched between two silica tetrahedral sheets (2 silica sheets to 1 alumina sheet). Properties vary based on bonding between stacked layers:
  • In Illite: layers bonded by potassium ions
  • In Montmorillonite: layers held by very weak van der Waals forces
• Sketch Description: Octahedral sheet in middle with tetrahedral sheet above and below, forming single 2:1 layer. Bonding agent (potassium ions for illite, or water/cations for montmorillonite) depicted in space between stacked 2:1 layers.

Soil formation, or pedogenesis, is process of rock weathering. Begins with breakdown of solid parent rock at Earth’s surface. Breakdown occurs in two main ways:

• Physical Weathering: Mechanical disintegration of rock into smaller particles due to forces like temperature changes, freezing and thawing of water (frost action), and abrasion from wind and water. Does not change chemical makeup of rock.
• Chemical Weathering: Decomposition of rock through chemical reactions like oxidation, hydration, and carbonation. Changes mineral composition of rock, often resulting in formation of fine-grained particles like clay minerals.

Weathered particles can either stay in place to form residual soils or be transported by wind, water, or ice to new location to form transported soils.

Early Period (Pre-18th Century): Engineers relied on experience and empirical rules. Notable early works include Coulomb’s 1776 theory on earth pressure and Rankine’s 1857 theory. Early attempts to apply mechanics to soil problems but lacked understanding of soil properties like permeability and shear strength.

Classical Soil Mechanics: Modern era began in early 20th century. Karl von Terzaghi recognized as “Father of Soil Mechanics.” In 1925, published “Erdbaumechanik” laying out fundamental principles including crucial concept of effective stress. Transformed subject from empirical art into mature engineering science.

Modern Era: Since Terzaghi, field advanced significantly with contributions from researchers like A. Casagrande (soil classification, seepage) and A.W. Skempton (pore pressure parameters), and development of sophisticated laboratory testing, in-situ testing, and numerical modeling techniques.

a) Types of Clay Minerals:

Clay minerals formed by stacking sheets of basic units. Main sheets are Silica (tetrahedral) sheet and Gibbsite (octahedral) sheet (central ion Aluminum). If central ion in octahedral sheet is Magnesium, called Brucite sheet. Main types based on stacking:

• Kaolinite (1:1 type): One Silica sheet bonded to one Gibbsite sheet
• Illite (2:1 type): One Gibbsite sheet bonded between two Silica sheets
• Montmorillonite (2:1 type): Same sheet structure as Illite but with weaker bonds
• Vermiculite and Chlorite: Other complex minerals where Brucite sheets can be present between main 2:1 layers

b) Soil Structures and Double Diffuse Layer:

Soil Structures Based on Compaction:

• Compaction Dry of Optimum: When cohesive soil compacted at water content below Optimum Moisture Content (OMC), repulsive forces between particles high, leading to random, edge-to-face flocculated structure. Results in higher strength but higher permeability.
• Compaction Wet of Optimum: When compacted at water content above OMC, more water lubricates particles, allowing alignment in more parallel, face-to-face dispersed structure. Results in lower strength, lower permeability, higher compressibility.

Double Diffuse Layer:

Distribution of cations (positive ions) in pore water surrounding negatively charged clay particle. Consists of dense layer of cations strongly adsorbed to clay surface and more diffuse layer extending into pore fluid. Layer neutralizes particle’s surface charge, thickness influences repulsive forces between particles, controlling whether flocculated or dispersed structure forms.

Geotechnical Problems in Civil Engineering:

Common geotechnical problems include bearing capacity failure of foundations, excessive settlement of structures, instability of slopes and embankments, soil liquefaction during earthquakes, swelling and shrinkage of expansive soils, and issues related to seepage and drainage.

Solutions to Geotechnical Problems:

Solutions broadly categorized into:

• Ground Improvement: Improving engineering properties of existing soil on-site using techniques like compaction, preloading, soil stabilization with admixtures (lime, cement), reinforcement with geosynthetics.
• Appropriate Foundation Design: Bypassing problematic soil using deep foundations (piles) to transfer loads to deeper, stronger stratum, or designing foundations (rafts) accommodating expected movements.
• Construction of Retaining and Drainage Structures: Building retaining walls supporting slopes or drainage systems controlling pore water pressure and seepage.

Various Minerals in Clay Soil:

Most common clay minerals:

• Kaolinite: 1:1 mineral with strong hydrogen bonds between layers. Structurally stable, low plasticity, does not swell.
• Illite: 2:1 mineral with layers bonded by potassium ions. Medium plasticity, moderate swelling.
• Montmorillonite: 2:1 mineral with very weak van der Waals bonds between layers. Large surface area, high plasticity, very high potential for swelling and shrinkage.

Definitions:

• Specific Surface: Specific surface area (SSA) of soil defined as total surface area of particles per unit mass (or unit volume). Expressed in m²/g. Fine-grained soils like clays have very large specific surface compared to coarse-grained soils, why behavior dominated by surface forces.
• Diffuse Double Layer: Combination of negatively charged clay particle surface and cloud of positively charged cations (in two layers: one fixed, one diffuse) surrounding it in pore fluid to maintain electrical neutrality.

Difference between Residual and Transported Soils:

Solutions to Soil Engineering Problems:

Solutions involve adapting design to soil conditions or improving soil itself:

• For weak soils: use deep foundations (piles) or improve ground using compaction or chemical stabilization
• For slope instability: build retaining walls, flatten slope, install drainage
• For expansive soils: stabilize soil with lime or design foundations resisting or moving with swelling pressures

Basic Structural Units of Clay Minerals:

Fundamental building blocks are sheet-like structures formed from two basic units:

• Silica Tetrahedron: Pyramid-shaped unit with central silicon atom and four oxygen atoms at corners
• Alumina Octahedron: Diamond-shaped unit with central aluminum atom and six hydroxyl groups at corners

Difference between Silica, Gibbsite, and Brucite Sheets:

Understanding and Need for Soil Mechanics:

Soil mechanics is application of laws of mechanics and hydraulics to engineering problems dealing with sediments and other unconsolidated accumulations of solid particles. Essential to study because nearly all civil engineering structures—buildings, bridges, roads, dams—are founded on or built with soil. Proper understanding necessary to ensure structures safe, stable, economical, and prevent failures like building collapse, road cracking, or dam breaches.

Solutions to Soil Engineering Problems:

Solutions involve either:

• Modifying the Soil: Using techniques like compaction, soil stabilization (with cement/lime), installing drainage systems to improve soil’s strength and behavior
• Adapting the Design: If soil cannot be improved, structure’s design adapted. May involve using deep foundations (piles) reaching stronger soil layers or lightweight fill material reducing load on soil

Specific Surface Area (SSA):

Specific surface area is property of solids defined as total surface area of material per unit of mass (m²/g) or bulk volume (m²/m³). For soils, represents total surface area of all individual particles contained in given mass.

Effect on Fine-Grained Soil:

Effect profound in fine-grained soils (clays and silts), negligible in coarse-grained soils:

• Dominance of Surface Forces: As particle size decreases, specific surface area increases dramatically. For clay particles, SSA so large that forces acting on particle surfaces (electrochemical forces) become much more significant than gravitational forces (mass)
• Water Holding Capacity: Larger surface area allows soil to hold much greater amount of adsorbed water
• Plasticity: High SSA and associated adsorbed water give fine-grained soils characteristic property of plasticity (ability to be molded)
• Swelling and Shrinkage: Soils with minerals having high SSA (like montmorillonite) exhibit high swelling and shrinkage behavior because large area available for water adsorption

High specific surface area is primary reason for cohesive, plastic, and often problematic engineering behavior of fine-grained soils.

Civil Engineering Problems Related to Soils:

• Foundation Stability: Ensuring foundations can support structures without failing (bearing capacity) or settling excessively
• Slope Stability: Preventing landslides in natural slopes, cuts, and embankments
• Earth Pressure: Designing retaining structures (walls, bulkheads) to withstand lateral pressure from soil
• Seepage Control: Managing flow of water through soil under dams and levees to prevent erosion and instability
• Expansive Soils: Dealing with soils that swell on wetting and shrink on drying, causing damage to foundations and pavements

Solutions to Such Problems:

General approach to solving problems is to either improve soil (through compaction, stabilization with lime/cement, or drainage) or design structure accommodating poor soil conditions (using deep pile foundations bypassing weak soil). Specific solution depends on project, soil type, and economic considerations.

Double Diffuse Layer:

Diffuse double layer describes electrochemical interface between charged clay particle surface and surrounding pore fluid. Clay particles carry net negative surface charge. Charge attracts layer of positive ions (cations) from fluid. Layer of cations has two parts: inner, tightly bound “Stern layer” and outer, more mobile “diffuse layer.” Concentration of cations highest at particle surface, decreases with distance. Entire system of charged surface and neutralizing cation cloud is diffuse double layer. Governs repulsive forces between clay particles, influencing soil’s structure and behavior.

Swelling of Clay Minerals:

Among three minerals, Montmorillonite swells the most.

Reason:

Swelling potential of clay mineral determined by strength of bond between fundamental layers and ability to draw water into structure:

• Weak Interlayer Bonds: Montmorillonite is 2:1 clay mineral where individual layers held together by very weak van der Waals forces
• Water Ingress: Weak bonds easily broken, allowing water molecules and exchangeable cations to penetrate space between layers
• Hydration and Expansion: Cations attracted into interlayer space (due to isomorphous substitution) become hydrated, influx of water forces layers apart, causing entire mineral mass to swell significantly

In contrast, Kaolinite’s layers held by strong hydrogen bonds, Illite’s layers held by moderately strong potassium ion bonds, both largely preventing water entry, limiting swelling.

Fields of Application of Soil Mechanics:

Soil mechanics applied in nearly every aspect of civil engineering, including:

• Foundation engineering (shallow and deep foundations)
• Transportation engineering (design of highway and airfield pavements)
• Geotechnical engineering (slope stability, retaining walls, tunnels)
• Hydraulic engineering (design of dams, levees, canals)
• Environmental engineering (design of landfills and waste containment systems)

Factors Determining Characteristics of Residual Soil:

Characteristics of residual soil depend on:

• Parent Rock Composition: Mineralogy of original rock dictates minerals present in soil
• Climate: Temperature and rainfall influence rate and type (physical vs. chemical) of weathering. Warm, humid climates promote rapid chemical weathering
• Topography: Slope of ground affects drainage and rate of erosion, influencing thickness of soil profile
• Time: Duration of weathering process determines extent of decomposition and depth of soil layer

Isomorphous Substitution:

Isomorphous substitution is process during formation of clay minerals where one type of atom (ion) in crystal lattice replaced by another type of atom of similar size but typically of lower positive charge (valence), without changing overall crystalline structure of mineral. For example, replacement of Al³⁺ ion by Mg²⁺ ion in octahedral sheet. Process is primary source of net negative charge on surfaces of clay minerals like montmorillonite and illite.

Comparison between 1:1 and 2:1 Minerals:

The differences between index properties and engineering properties of soil are tabulated below:

Index Tests for Different Soil Types:

• Coarse-grained soils (e.g., sand, gravel):
  • Sieve Analysis (for grain size distribution)
  • Relative Density Test
• Fine-grained soils (e.g., silt, clay):
  • Atterberg Limits (Liquid Limit, Plastic Limit, Shrinkage Limit)
  • Hydrometer Analysis (for grain size distribution of fine particles)

The stress-strain behavior of soil varies significantly with its consistency, which is related to its water content. The general behavior at different states is shown below:

• Liquid State: The soil behaves like a viscous fluid and has no shear strength. It deforms continuously under a constant load.
• Plastic State: The soil deforms plastically without rupture. It shows some shear strength.
• Semi-Solid State: The soil becomes stiffer and exhibits brittle failure upon reaching its peak strength.
• Solid State: The soil is hard and brittle, showing a distinct peak strength followed by sudden failure.

Let’s denote the in-situ (borrow pit) conditions with subscript 1 and the compacted (embankment) conditions with subscript 2.

Given Data:

• In-situ (Borrow Pit): Bulk unit weight, $\gamma_{t1} = 18 \text{ kN/m³}$, Water content, $w_1 = 10\% = 0.10$
• Compacted (Embankment): Dry unit weight, $\gamma_{d2} = 19 \text{ kN/m³}$, Water content, $w_2 = 18\% = 0.18$, Volume of compacted soil, $V_2 = 1 \text{ m³}$

Step 1: Calculate the dry unit weight of the in-situ soil ($\gamma_{d1}$)

The dry unit weight is related to the bulk unit weight by: $\gamma_d = \frac{\gamma_t}{1 + w}$

$\gamma_{d1} = \frac{18}{1 + 0.10} = 16.36 \text{ kN/m³}$

Step 2: Calculate the weight of solids required for 1 m³ of compacted soil ($W_s$)

The weight of soil solids remains constant. $W_s = \gamma_{d2} \times V_2$

$W_s = 19 \text{ kN/m³} \times 1 \text{ m³} = 19 \text{ kN}$

Step 3: Calculate the volume of soil to be excavated from the borrow pit ($V_1$)

$V_1 = \frac{W_s}{\gamma_{d1}}$

$V_1 = \frac{19 \text{ kN}}{16.36 \text{ kN/m³}} = 1.16 \text{ m³}$

Therefore, 1.16 m³ of soil needs to be excavated for every 1 m³ of compacted embankment.

The Massachusetts Institute of Technology (MIT) classification system is one of the early systems for classifying soils based solely on particle size. It categorizes soil into four main fractions:

This system is straightforward and easy to use but has a significant limitation: it does not consider the plasticity and other physical characteristics of the soil, which are crucial for predicting its engineering behavior. Modern systems like the Unified Soil Classification System (USCS) incorporate plasticity characteristics for a more comprehensive classification.

Step 1: Determine if the soil is coarse-grained or fine-grained.

The percentage of fines (passing 75µm sieve) is 4%, which is less than 50%. Therefore, the soil is coarse-grained.

Step 2: Determine if the coarse fraction is primarily sand or gravel.

• Total coarse fraction = 100% – 4% = 96%
• Percentage of gravel (retained on 4.75 mm sieve) = 35%
• Percentage of sand = 96% – 35% = 61%

Since the percentage of sand (61%) is greater than the percentage of gravel (35%), the soil is a sand.

Step 3: Classify the sand based on the percentage of fines.

The percentage of fines is 4% (<5%), so it is a clean sand (SW or SP).

Step 4: Check the gradation criteria for sand.

For a well-graded sand (SW), $C_u \ge 6$ and $1 \le C_c \le 3$.

• Given $C_u = 5$. This condition ($C_u \ge 6$) is not met.
• Given $C_c = 2$. This condition ($1 \le C_c \le 3$) is met.

Since both conditions are not satisfied, the soil is poorly graded.

Final Classification: The soil is a Poorly graded sand (SP).

Index Properties: These are properties used for identification and classification of soil, determined by simple tests. Examples include particle size distribution, Atterberg limits, and specific gravity.

Engineering Properties: These properties assess the performance of soil as a construction material, used in engineering design. Examples include permeability, compressibility, and shear strength.

For identification and classification, index properties are more significant because:

1. Simplicity and Cost-Effectiveness: Index tests are simple, quick, and inexpensive compared to engineering property tests.
2. Correlation: There are well-established empirical correlations between index properties and engineering properties, allowing for reasonable estimates of behavior.
3. Fundamental Nature: Index properties describe the fundamental physical makeup and state of the soil, which govern its complex engineering behavior.

Relative Consistency (or Liquidity Index, $I_L$): It quantifies the in-situ consistency of a fine-grained soil. It is defined as: $I_L = \frac{w – w_P}{w_L – w_P}$

Numerical Problem:

Step 1: Determine the mass of soil solids ($M_s$)

$V_L – V_S = \frac{(w_L – w_S) \times M_s}{\rho_w}$

$40 – 23.5 = \frac{(0.60 – 0.20) \times M_s}{1}$

$M_s = \frac{16.5}{0.40} = 41.25 \text{ g}$

Step 2: Determine the specific gravity of solids ($G_s$)

$V_{\text{solids}} = V_S – \frac{w_S \times M_s}{\rho_w} = 23.5 – \frac{0.20 \times 41.25}{1} = 15.25 \text{ cc}$

$G_s = \frac{M_s}{V_{\text{solids}} \times \rho_w} = \frac{41.25}{15.25 \times 1} = 2.705$

Specific gravity of solids, $G_s \approx 2.71$

Step 3: Determine the Shrinkage Ratio (SR)

$SR = \frac{V_L – V_S}{V_S} \times \frac{100}{w_L – w_S} = \frac{40 – 23.5}{23.5} \times \frac{100}{60 – 20} = 1.755$

Shrinkage Ratio, $SR \approx 1.76$

Phase Diagram at Liquid Limit ($w_L = 60\%$):

Phase Diagram at Shrinkage Limit ($w_S = 20\%$):

Purpose of Soil Classification: To arrange soils into groups based on their properties to predict behavior, provide a common language for engineers, organize data, and guide design decisions.

Soil Classification Problem:

Step 1: Coarse-grained or fine-grained? Fines = 8% (< 50%), so it is coarse-grained.

Step 2: Sand or gravel?

• % Gravel (retained on 4.75 mm) = 100% – 42% = 58%
• % Coarse fraction = 100% – 8% = 92%
• % Sand = 92% – 58% = 34%

Since % Gravel > % Sand, the soil is a gravel.

Step 3: Fines classification. Fines = 8% (between 5% and 12%), so it’s a borderline case requiring a dual symbol.

Step 4: Gradation criteria for gravel. For well-graded gravel (GW), $C_u \ge 4$ and $1 \le C_c \le 3$.

• Given $C_u = 6$. This condition ($C_u \ge 4$) is met.
• Given $C_c = 4$. This condition ($1 \le C_c \le 3$) is not met.

The gravel is poorly graded (GP).

Step 5: Classify the fines. Given PI = 4. This falls in the silty (M) category for fines. The second symbol is GM.

Final Classification: The soil is a Poorly graded gravel with silt (GP-GM).

• Toughness Index ($I_t$): The ratio of the plasticity index to the flow index ($I_t = \frac{I_p}{I_f}$). It represents the shear strength of a soil at its plastic limit.
• Coefficient of Curvature ($C_c$): A measure of the shape of the particle-size distribution curve, indicating gradation. $C_c = \frac{(D_{30})^2}{D_{60} \times D_{10}}$. For well-graded soils, $1 \le C_c \le 3$.
• Activity of Soil (A): An index for the swelling potential of clay soils. $A = \frac{I_p}{\% \text{ clay fraction}}$.
• Air Content ($a_c$): The ratio of the volume of air to the total volume of voids ($a_c = \frac{V_a}{V_v}$). It is related to saturation by $a_c = 1 – S$.

1. Water Content (w):

$w = \frac{\text{Mass of water}}{\text{Mass of solids}} = \frac{20 – 16.5}{16.5} = 0.2121$. $w = 21.21\%$

2. Dry Density ($\rho_d$):

$\rho_d = \frac{\text{Mass of solids}}{\text{Total volume}} = \frac{16.5 \text{ kg}}{0.011 \text{ m³}} = 1500 \text{ kg/m³}$

3. Porosity (n):

First find void ratio (e): $\rho_d = \frac{G_s \rho_w}{1+e} \Rightarrow 1500 = \frac{2.70 \times 1000}{1+e} \Rightarrow e = 0.8$

$n = \frac{e}{1+e} = \frac{0.8}{1.8} = 0.4444$. $n = 44.44\%$

4. Degree of Saturation (S):

$S \times e = w \times G_s \Rightarrow S \times 0.8 = 0.2121 \times 2.70 \Rightarrow S = 0.7159$. $S = 71.59\%$

Importance of Soil Classification:

Soil classification is fundamentally important in geotechnical engineering because it provides a systematic way to organize soils into groups with similar engineering properties. Its importance lies in predicting behavior, providing a common language, structuring site data, and guiding further testing.

Classification of Soils A and B:

(Note: The properties for Soil A and Soil B were not provided in the question. The following is a general procedure that would be followed once the data is available.)

The general procedure for USCS classification involves determining if the soil is coarse or fine-grained based on the percentage passing the 75-micron sieve. If coarse-grained, it is further classified as sand or gravel, and then by its gradation (well or poorly graded) and the nature of its fines (silty or clayey). If fine-grained, it is classified based on its liquid limit and plasticity index using the plasticity chart.

1. Core Cutter Method: A steel cutter of known volume is driven into the ground. Best for soft to stiff cohesive soils.
2. Sand Replacement Method: The volume of an excavated hole is found by filling it with calibrated sand. Suitable for granular and gravelly soils.
3. Water Displacement Method: The volume of a cohesive soil lump is found by water displacement after coating it in wax. This is a lab method for field samples.
4. Nuclear Density Gauge: A non-destructive method using a radioactive source to measure density by backscattered radiation. It is fast and suitable for a wide range of soils.

Step 1: Calculate mass and volume of water.

Mass of water, $M_w = 38 \text{ gm} – 28 \text{ gm} = 10 \text{ gm}$. Volume of water, $V_w = \frac{10 \text{ gm}}{1 \text{ gm/cc}} = 10 \text{ cc}$.

Step 2: Calculate volume of voids and solids.

Since the soil is saturated, Volume of voids, $V_v = V_w = 10 \text{ cc}$.

Volume of solids, $V_s = \text{Total Volume} – V_v = 20 \text{ cc} – 10 \text{ cc} = 10 \text{ cc}$.

Step 3: Calculate the Void Ratio (e).

$e = \frac{V_v}{V_s} = \frac{10 \text{ cc}}{10 \text{ cc}} = 1.0$.

Step 4: Calculate the Specific Gravity ($G_s$).

$G_s = \frac{M_s}{V_s \times \rho_w} = \frac{28 \text{ gm}}{10 \text{ cc} \times 1 \text{ gm/cc}} = 2.8$.

Field Distinction between Clay and Silt:

• Dilatancy Test: Silt shows a reaction (water appears on shaking), clay does not.
• Dry Strength Test: Dry clay lumps are strong, dry silt lumps crumble easily.
• Toughness Test: Clay can be rolled into strong, thin threads; silt cannot.

Purpose of Identification and Classification: To group soils to predict engineering behavior, provide a common language, and guide design.

Three Important Classification Systems: USCS, AASHTO, ISSCS.

Description of USCS: It classifies soils into coarse-grained (Gravel-G, Sand-S), fine-grained (Silt-M, Clay-C, Organic-O), and highly organic (Peat-Pt). Coarse soils are further defined by gradation (W or P) and fines (M or C). Fine soils are defined by plasticity (Low-L or High-H).

Limitations of USCS: It is qualitative, not suitable for all applications (e.g., agriculture), and field classification requires experience.

Index Properties: Fundamental physical properties used for soil identification and classification (e.g., grain size distribution, Atterberg limits).

Engineering Properties: Properties that measure the soil’s mechanical behavior for design purposes (e.g., shear strength, permeability).

Necessity of Determining Index Properties:

1. Soil Classification: They form the basis of all classification systems.
2. Preliminary Assessment: They provide a quick and cost-effective way to estimate engineering behavior.
3. Empirical Correlations: They are used in established correlations to predict complex engineering properties.
4. Compaction Control: They are essential for quality control during the construction of earthworks.

To determine if the work meets the specification, we need to calculate the in-situ void ratio of the compacted soil and compare it to the required maximum value of 0.75.

Given Data:

• Bulk unit weight, $\gamma = 18 \, \text{kN/m}^3$
• Water content, $w = 15\% = 0.15$
• Specific gravity of soil solids, $G_s = 2.7$
• Unit weight of water, $\gamma_w = 9.81 \, \text{kN/m}^3$ (assumed)

Step 1: Calculate the dry unit weight ($\gamma_d$)

The relationship between bulk unit weight ($\gamma$), dry unit weight ($\gamma_d$), and water content ($w$) is:

Step 2: Calculate the void ratio (e)

The dry unit weight is also related to the specific gravity ($G_s$) and void ratio ($e$) by the formula:

Rearranging the formula to solve for $e$:

Step 3: Compare with Specification

The calculated void ratio is $e = 0.692$. The specification requires the void ratio to be less than 0.75.

Since $0.692 < 0.75$, the condition is satisfied.

Conclusion: Yes, I would pass the bill for the work because the void ratio of the compacted base course is within the specified limit.

Similarities:

• Both systems are based on similar soil properties: particle size distribution for coarse-grained soils and plasticity characteristics (Atterberg limits) for fine-grained soils.
• Both use the 0.075 mm (No. 200) sieve as the primary boundary between coarse-grained and fine-grained soils.
• Both use letter symbols to designate different soil groups.

Differences:

The basic requirements for a useful soil classification system are:

• Comprehensiveness: It should be applicable to all types of natural soils without ambiguity.
• Simplicity and Practicality: It should be based on a limited number of easily determinable properties through standard laboratory or field tests.
• Descriptive and Systematic: The names and symbols used should be systematic and convey a concise yet clear picture of the soil’s key characteristics.
• Engineering Relevance: The system should group soils with similar engineering properties (such as strength, permeability, and compressibility), allowing engineers to make preliminary judgments about the soil’s behavior for engineering purposes.

Step 1: Determine if the soil is coarse-grained or fine-grained.

• Percentage of fines (passing No. 200 sieve) = 40%.
• Since less than 50% of the material passes the No. 200 sieve, the soil is coarse-grained.

Step 2: Determine if the coarse fraction is sand or gravel.

• Total coarse fraction = 100% – Fines = 100% – 40% = 60%.
• Gravel fraction = % of total soil retained on No. 4 sieve = 55%.
• Sand fraction = Total coarse fraction – Gravel fraction = 60% – 55% = 5%.
• Since the gravel fraction (55%) is greater than half of the coarse fraction (60%/2 = 30%), the soil is primarily Gravel.

Step 3: Check the grading of the gravel.

We calculate the Coefficient of Uniformity ($C_u$) and the Coefficient of Curvature ($C_c$).

For a well-graded gravel (GW), the criteria are $C_u > 4$ and $1 < C_c < 3$.

• Our calculated $C_u = 3.17$, which does not satisfy the $C_u > 4$ condition.
• Therefore, the gravel is poorly graded (GP).

Step 4: Classify based on the percentage of fines.

• The percentage of fines is 40%, which is greater than 12%. This indicates a dual symbol is required, incorporating the properties of the fines.
• To fully classify the soil, the Atterberg limits (Liquid Limit and Plastic Limit) of the fines are needed to determine if they are silty (M) or clayey (C).

Step 5: Final Classification

Since the Atterberg limits are not provided, we cannot definitively classify the fines. The classification will be one of the following, depending on the plasticity of the fines:

• GP-GM (Poorly graded gravel with silt)
• GP-GC (Poorly graded gravel with clay)

a) Phase Diagram and Definitions

A three-phase diagram for a partially saturated soil shows the distribution of solids, water, and air by volume and weight.

• Volumes: $V_a$ (Air), $V_w$ (Water), $V_s$ (Solids). Total Volume $V = V_s + V_w + V_a$. Volume of Voids $V_v = V_w + V_a$.
• Weights: $W_a \approx 0$ (Air), $W_w$ (Water), $W_s$ (Solids). Total Weight $W = W_s + W_w$.

Definitions:

• Void Ratio (e): The ratio of the volume of voids ($V_v$) to the volume of soil solids ($V_s$). It indicates the amount of empty space in a soil mass. $$e = \frac{V_v}{V_s}$$
• Degree of Saturation (S): The ratio of the volume of water ($V_w$) to the total volume of voids ($V_v$), expressed as a percentage. It indicates the extent to which the voids are filled with water. $$S = \frac{V_w}{V_v} \times 100\%$$

b) Calculations

Given Data:

• Weight of empty core-cutter = 22.80 N
• Weight of soil and core-cutter = 50.05 N
• Diameter (d) = 90.0 mm = 0.09 m
• Height (h) = 180.0 mm = 0.18 m
• Weight of wet sample = 0.5405 N
• Weight of dry sample = 0.5112 N
• Specific gravity, $G_s = 2.72$

Step 1: Calculate basic parameters.

• Weight of soil, $W = 50.05 – 22.80 = 27.25 \, \text{N}$
• Volume of soil, $V = \frac{\pi}{4}d^2h = \frac{\pi}{4}(0.09)^2(0.18) = 1.145 \times 10^{-3} \, \text{m}^3$
• Bulk unit weight, $\gamma = \frac{W}{V} = \frac{27.25}{1.145 \times 10^{-3}} = 23799 \, \text{N/m}^3 = 23.8 \, \text{kN/m}^3$
• Water content, $w = \frac{W_{wet} – W_{dry}}{W_{dry}} = \frac{0.5405 – 0.5112}{0.5112} = 0.0573 \, \text{or} \, 5.73\%$

(i) Dry density (dry unit weight, $\gamma_d$)

(ii) Void-ratio (e)

(iii) Degree of saturation (S)

a) Field Identification Tests:

• For Sandy Soil: Visual inspection (visible grains), feel test (gritty), dry strength test (crumbles easily, no strength when dry).
• For Clayey Soil: Dry strength test (very hard to break when dry), dilatancy test (no reaction to shaking), toughness test (can be rolled into a thin thread of 3mm without crumbling), feel test (smooth and greasy).

b) Classification Systems:

Systems that use both particle size and plasticity characteristics include:

• Unified Soil Classification System (USCS)
• Indian Standard Soil Classification System (ISSCS)
• American Association of State Highway and Transportation Officials (AASHTO) System

c) Soil Classification (USCS):

Step 1: Determine if Coarse or Fine Grained.

• Percentage of fines (passing No. 200 sieve) = 30%.
• Since fines are less than 50%, the soil is Coarse-Grained.

Step 2: Determine if Sand or Gravel.

• Coarse fraction = 100% – 30% = 70%.
• Percentage retained on No. 4 sieve (Gravel fraction) = 100% – 70% = 30%.
• Percentage passing No. 4 but retained on No. 200 (Sand fraction) = 70% – 30% = 40%.
• Since the Sand fraction (40%) is greater than the Gravel fraction (30%), the soil is primarily Sand.

Step 3: Classify Fines.

• Liquid Limit (LL) = 33%
• Plastic Limit (PL) = 11%
• Plasticity Index (PI) = LL – PL = 33 – 11 = 22%.
• A-line equation: $PI_{A-line} = 0.73(LL – 20) = 0.73(33 – 20) = 9.49$.
• Since the soil’s PI (22) is above the A-line PI (9.49) and LL is less than 50, the fines are classified as Clay (C).

Step 4: Final Classification.

• The soil is a Sand with 30% fines. Since fines > 12%, a dual symbol is required.
• The fines are clayey (C).
• We need gradation data ($C_u$ and $C_c$) to determine if the sand is well-graded (SW) or poorly graded (SP). As this data is not provided, we must assume the sand is poorly graded.
• Final Classification: SP-SC (Poorly graded sand with clay) or SW-SC (Well-graded sand with clay), depending on gradation.

Given Data:

• Container diameter, d = 0.2 m
• Container height, h = 0.2 m
• Void ratio, $e = 0.54$
• Specific gravity, $G_s = 2.65$

Step 1: Calculate Volumes

• Total volume of container (and soil), $V = \frac{\pi}{4}d^2h = \frac{\pi}{4}(0.2)^2(0.2) = 0.006283 \, \text{m}^3$.
• From the relationship $V = V_s(1+e)$, we can find the volume of solids $V_s$: $$ V_s = \frac{V}{1+e} = \frac{0.006283}{1+0.54} = 0.00408 \, \text{m}^3 $$
• The volume of voids $V_v$ is: $$ V_v = e \times V_s = 0.54 \times 0.00408 = 0.002203 \, \text{m}^3 $$

Step 2: Calculate Weight of Solids ($W_s$)

(This calculated weight is very close to the 10kg provided. We will use the calculated Ws for consistency.)

(i) Water for Full Saturation (S = 100%)

• At full saturation, the volume of water ($V_w$) equals the volume of voids ($V_v$). $$ V_w = V_v = 0.002203 \, \text{m}^3 $$
• The amount (weight) of water needed is: $$ W_w = V_w \cdot \gamma_w = 0.002203 \times 9.81 \, \text{kN/m}^3 = 0.0216 \, \text{kN} $$ In mass, this is $2.203 \, \text{kg}$. Amount of water needed = 2.203 kg.
• The water content at full saturation ($w_{sat}$) is: $$ w_{sat} = \frac{W_w}{W_s} = \frac{0.0216 \, \text{kN}}{0.106 \, \text{kN}} = 0.2038 = \mathbf{20.38\%} $$

(ii) Water for 80% Saturation (S = 80%)

• The volume of water required is $V_w = S \times V_v$. $$ V_w = 0.80 \times 0.002203 \, \text{m}^3 = 0.001762 \, \text{m}^3 $$
• The amount (weight) of water to be added is: $$ W_w = V_w \cdot \gamma_w = 0.001762 \times 9.81 \, \text{kN/m}^3 = 0.01728 \, \text{kN} $$ In mass, this is $1.762 \, \text{kg}$. Amount of water to be added = 1.762 kg.

a) Cohesive vs. Cohesionless Soils

b) Field Identification of Silt vs. Clay

The following tests help distinguish between silt and clay in the field:

• Dry Strength Test: A dry lump of soil is crushed between the fingers. Clay has high dry strength and is difficult to break, while silt has low dry strength and powders easily.
• Dilatancy (Shaking) Test: A pat of moist soil is shaken in the palm of the hand. Silt will show a glossy, wet surface quickly, which disappears when the pat is squeezed. Clay shows no or very slow reaction to shaking.
• Toughness (Plasticity) Test: The soil is rolled into a thread of about 3mm diameter. Clay is plastic and can be rolled into a thin, strong thread without crumbling. Silt has low plasticity and the thread will break easily.

c) IS Plasticity Chart

The IS Plasticity Chart is a graphical plot of Plasticity Index (PI) versus Liquid Limit (LL) used to classify fine-grained soils. The chart is divided into different regions by the “A-line” and vertical lines at LL=35% and LL=50%.

• A-Line: An empirical boundary separating more clay-like soils (above the line) from more silt-like soils (below the line). Its equation is $PI = 0.73(LL – 20)$.
• Regions:
  • CL: Clays of Low Plasticity (LL < 35%)
  • CI: Clays of Intermediate Plasticity (35% < LL < 50%)
  • CH: Clays of High Plasticity (LL > 50%)
  • ML / OL: Silts and Organic soils of Low Plasticity (LL < 35%)
  • MI / OI: Silts and Organic soils of Intermediate Plasticity (35% < LL < 50%)
  • MH / OH: Silts and Organic soils of High Plasticity (LL > 50%)
  • A special zone (hatched area) for CL-ML (silty clay) is also present.

Given Data:

• Specific gravity, $G_s = 2.65$
• Dry unit weight, $\gamma_d = 15 \, \text{kN/m}^3$
• Bulk unit weight, $\gamma = 18 \, \text{kN/m}^3$ (since 1m³ weighs 18 kN)

(i) Water content (w)

Using the relationship $\gamma_d = \gamma / (1+w)$:

(ii) Degree of saturation (S)

First, we need to find the void ratio ($e$).

Now, use the relationship $S \cdot e = w \cdot G_s$:

(iii) Submerged unit weight ($\gamma’$)

The submerged unit weight is defined for a saturated soil ($\gamma’ = \gamma_{sat} – \gamma_w$). First, we calculate the saturated unit weight ($\gamma_{sat}$).

Now, we can find the submerged unit weight:

Step 1: Determine if the soil is granular or silt-clay material.

• Percent passing No. 200 sieve = 58%.
• Since this is greater than 35%, the soil is a silt-clay material (Groups A-4, A-5, A-6, or A-7).

Step 2: Check against group classification criteria.

• Liquid Limit (LL) = 30%
• Plasticity Index (PI) = 10%

We check the criteria for the silt-clay groups:

• Group A-4: LL ≤ 40 and PI ≤ 10. (This fits: 30% ≤ 40 and 10% ≤ 10).
• Group A-5: LL > 40 and PI ≤ 10. (Does not fit).
• Group A-6: LL ≤ 40 and PI > 10. (Does not fit).
• Group A-7: LL > 40 and PI > 10. (Does not fit).

The soil falls into group A-4.

Step 3: Calculate the Group Index (GI).

The formula for GI is:

Where:

• F = Percent passing No. 200 sieve = 58
• LL = Liquid Limit = 30
• PI = Plasticity Index = 10

The GI is rounded to the nearest whole number, so GI = 3.

Final Classification: The soil is classified as A-4(3).

a) Phase Diagram:

A phase diagram, also known as a block diagram, is a schematic representation used in soil mechanics to visualize the different phases of a soil mass. It separates the total volume or mass of the soil into its constituent components: solid particles (soil grains), water, and air. This helps in establishing the relationships between weights and volumes of these components.

Phase Diagrams for Different Soil Conditions:

• Saturated Soil (Two-Phase Diagram): The void spaces are completely filled with water. There is no air phase.
• Partially Saturated Soil (Three-Phase Diagram): The void spaces contain both water and air. This is the most common state for natural soils.
• Dry Soil (Two-Phase Diagram): The void spaces are completely filled with air. There is no water phase.

b) Index and Engineering Properties of Soil:

Index Properties: These are properties used for the identification, description, and classification of soils. They provide a general indication of the soil’s behavior without conducting complex tests.

• Water Content
• Specific Gravity of soil solids
• Particle Size Distribution (Sieve and Hydrometer analysis)
• In-situ Density (Unit Weight)
• Consistency Limits (Liquid Limit, Plastic Limit, Shrinkage Limit)
• Relative Density (for cohesionless soils)

Engineering Properties: These are properties that quantify the mechanical behavior of soil and are essential for the design and analysis of geotechnical structures. They are determined through laboratory or in-situ tests that simulate field conditions.

• Permeability
• Compressibility (Consolidation parameters)
• Shear Strength (Cohesion and Angle of Internal Friction)
• Bearing Capacity
• Collapsibility and Swelling Potential

c) Stress-Strain Curve for Different Consistency States:

The consistency of a fine-grained soil changes with its water content. The stress-strain behavior varies significantly across these states:

• Liquid State: The soil behaves like a viscous fluid and has negligible shear strength. It deforms continuously under a small shear stress.
• Plastic State: The soil behaves like a plastic material. It undergoes permanent deformation (yields) without rupture after a certain yield stress is reached.
• Semi-Solid State: The soil behaves as a brittle material. It shows some resistance to stress but fails suddenly with little plastic deformation.
• Solid State: The soil is hard and brittle. It behaves like a brittle solid, exhibiting high strength and failing at a very low strain.

a) Soil Classification Systems and Plasticity Chart:

Systems based on particle size and/or plasticity include:

• Unified Soil Classification System (USCS)
• Indian Standard Soil Classification System (ISSCS)
• American Association of State Highway and Transportation Officials (AASHTO) System
• MIT Classification System (based primarily on particle size)
• British Standard Soil Classification System

Plasticity Chart (ISSCS/USCS):

The plasticity chart is a graphical plot of the Plasticity Index ($I_P$) versus the Liquid Limit ($W_L$) for fine-grained soils. It is the primary tool for classifying silts, clays, and organic soils. The chart features two important lines:

• A-Line: An empirically derived boundary line that separates clays from silts. The equation for the A-line is $I_P = 0.73(W_L – 20)$. Soils plotting above the A-line are generally clays (C), and those below are silts (M) or organic soils (O).
• U-Line: An upper-boundary line representing the likely upper limit for natural soils. The equation is $I_P = 0.9(W_L – 8)$. Test results plotting above the U-line are considered suspect.

The chart is also divided by vertical lines at $W_L = 35\%$ and $W_L = 50\%$ to distinguish between low (L), intermediate (I), and high (H) plasticity.

b) Classification of Soil A and Soil B (USCS):

Assuming both soils are fine-grained (more than 50% passing the No. 200 sieve).

For Soil A:

• Liquid Limit, $W_L = 45\%$
• Plastic Limit, $W_P = 15\%$
• Plasticity Index, $I_P = W_L – W_P = 45 – 15 = 30\%$

Classification:

1. Since $35\% < W_L = 45\% < 50\%$, the soil has intermediate plasticity (I).
2. Calculate the A-line’s $I_P$ value at $W_L = 45\%$:
   $I_{P(A-line)} = 0.73(45 – 20) = 0.73 \times 25 = 18.25\%$
3. Compare the soil’s $I_P$ with the A-line value: $30\% > 18.25\%$. The point plots above the A-line.
4. Therefore, the soil is an inorganic clay of intermediate plasticity.

Group Symbol: CI

For Soil B:

• Liquid Limit, $W_L = 25\%$
• Plastic Limit, $W_P = 10\%$
• Plasticity Index, $I_P = W_L – W_P = 25 – 10 = 15\%$

Classification:

1. Since $W_L = 25\% < 35\%$, the soil has low plasticity (L).
2. Calculate the A-line’s $I_P$ value at $W_L = 25\%$:
   $I_{P(A-line)} = 0.73(25 – 20) = 0.73 \times 5 = 3.65\%$
3. Compare the soil’s $I_P$ with the A-line value: $15\% > 3.65\%$. The point plots above the A-line.
4. Therefore, the soil is an inorganic clay of low plasticity.

Group Symbol: CL

a) Difference between Index Property and Engineering Property:

b) Index Tests:

• Tests for individual soil grains: Specific Gravity Test.
• Tests for soil mass as a whole: Sieve Analysis, Hydrometer Analysis, Atterberg Limits Tests (Liquid Limit, Plastic Limit, Shrinkage Limit), Field Density Tests, Relative Density Test.

c) Stress-Strain Behaviour of Consistency States:

This is a representation of how a soil’s resistance to deformation (stress) changes as it is deformed (strain) at different water contents. The four states (liquid, plastic, semi-solid, solid) show distinct behaviors from fluid-like to brittle-solid. [Refer to answer Q.2.26(c) for the curve and description].

d) Embankment Volume Calculation:

The volume of solid particles ($V_s$) excavated from the borrow pit is the same as the volume of solids in the compacted embankment.

Given:

• Volume of soil from borrow pit, $V_{pit} = 1,00,000 \text{ m}^3$
• Void ratio of borrow pit soil, $e_{pit} = 0.8$
• Void ratio of embankment soil, $e_{emb} = 0.6$

Step 1: Calculate the volume of soil solids ($V_s$).

The total volume is related to the volume of solids by $V = V_s(1+e)$.

$V_s = \frac{V_{pit}}{1 + e_{pit}} = \frac{1,00,000}{1 + 0.8} = \frac{1,00,000}{1.8} = 55,555.56 \text{ m}^3$

Step 2: Calculate the volume of the embankment ($V_{emb}$).

Using the same volume of solids, we find the new total volume with the new void ratio.

$V_{emb} = V_s(1 + e_{emb}) = 55,555.56 \times (1 + 0.6) = 55,555.56 \times 1.6 = 88,888.89 \text{ m}^3$

The volume of the embankment will be 88,888.89 m³.

Usefulness of Plasticity Chart:

The plasticity chart is an indispensable tool for classifying fine-grained soils. Its primary uses are:

• Differentiation of Soil Types: The A-line on the chart effectively separates inorganic clays (plotting above) from inorganic silts and organic soils (plotting below).
• Assessment of Plasticity: By plotting a soil’s liquid limit on the x-axis, the chart categorizes it into low ($W_L < 35\%$), intermediate ($35\% \leq W_L \leq 50\%$), or high ($W_L > 50\%$) plasticity. This gives a direct indication of the soil’s potential for swelling, shrinkage, and compressibility.
• Unified Language: It provides a standardized and widely accepted method for soil classification, ensuring engineers have a common basis for describing and comparing fine-grained soils.

Classification of Given Soil (ISSCS):

Given Data:

• Percentage passing No. 200 sieve (75µ) = 55%
• Percentage of coarse fraction passing No.4 sieve (4.75mm) = 60%
• Liquid limit, $W_L = 68\%$
• Plastic limit, $W_P = 22\%$

Step-by-Step Classification:

1. Determine if Coarse or Fine-Grained:
   • Percent of fines (passing 75µ) = 55%.
   • Since this is more than 50%, the soil is classified as a Fine-Grained Soil.
2. Classify the Fine-Grained Soil:
   • $W_L = 68\%$. Since $W_L > 50\%$, the soil has high plasticity (H).
   • Calculate the Plasticity Index: $I_P = W_L – W_P = 68 – 22 = 46\%$.
   • Determine its position relative to the A-line: $I_{P(A-line)} = 0.73(W_L – 20) = 0.73(68 – 20) = 35.04\%$.
   • Since the soil’s $I_P (46\%) > I_{P(A-line)} (35.04\%)$, it plots above the A-line.
   • The group symbol for the fine fraction is CH (Clay of high plasticity).
3. Check Coarse Fraction for Description:
   • Percent of coarse fraction = $100\% – 55\% = 45\%$.
   • Percent of sand in coarse fraction = 60%. Percent of gravel in coarse fraction = 40%.
   • Percent of sand in total sample = $60\%$ of $45\% = 0.60 \times 45 = 27\%$.
   • Percent of gravel in total sample = $40\%$ of $45\% = 0.40 \times 45 = 18\%$.
   • Since the coarse fraction is greater than 30% and the percentage of sand (27%) is greater than the percentage of gravel (18%), the soil can be described as sandy.

Final Classification (ISSCS): The group symbol is CH. The descriptive name is Sandy Clay of High Plasticity.

Given Data:

• $e_{max} = 1.25$
• $e_{min} = 0.45$
• Initial Relative Density, $D_{r1} = 40\% = 0.4$
• Final Relative Density, $D_{r2} = 60\% = 0.6$
• Specific Gravity, $G_s = 2.65$
• Initial thickness, $H_1 = 3 \text{ m}$
• Unit weight of water, $\gamma_w = 9.81 \text{ kN/m}^3$

1. Initial Dry Density ($\gamma_{d1}$):

First, find the initial void ratio ($e_1$) using the relative density formula: $D_r = \frac{e_{max} – e}{e_{max} – e_{min}}$

$0.4 = \frac{1.25 – e_1}{1.25 – 0.45} \implies 0.4 = \frac{1.25 – e_1}{0.80}$

$e_1 = 1.25 – (0.4 \times 0.80) = 1.25 – 0.32 = 0.93$

Now, calculate the initial dry density:

$\gamma_{d1} = \frac{G_s \gamma_w}{1+e_1} = \frac{2.65 \times 9.81}{1 + 0.93} = \frac{26.0}{1.93} = 13.47 \text{ kN/m}^3$

2. Reduction in Thickness ($\Delta H$):

First, find the final void ratio ($e_2$) at $D_{r2} = 60\%$:

$0.6 = \frac{1.25 – e_2}{1.25 – 0.45} \implies 0.6 = \frac{1.25 – e_2}{0.80}$

$e_2 = 1.25 – (0.6 \times 0.80) = 1.25 – 0.48 = 0.77$

The reduction in thickness (settlement) is given by:

$\Delta H = H_1 \frac{e_1 – e_2}{1 + e_1} = 3 \times \frac{0.93 – 0.77}{1 + 0.93} = 3 \times \frac{0.16}{1.93} = 0.249 \text{ m}$

3. New Dry and Saturated Densities:

New Dry Density ($\gamma_{d2}$):

$\gamma_{d2} = \frac{G_s \gamma_w}{1+e_2} = \frac{2.65 \times 9.81}{1 + 0.77} = \frac{26.0}{1.77} = 14.69 \text{ kN/m}^3$

New Saturated Density ($\gamma_{sat2}$):

$\gamma_{sat2} = \frac{(G_s + e_2) \gamma_w}{1+e_2} = \frac{(2.65 + 0.77) \times 9.81}{1 + 0.77} = \frac{3.42 \times 9.81}{1.77} = 18.96 \text{ kN/m}^3$

a) Field Identification of Fine-Grained Soils:

Simple manual tests can be used in the field to distinguish between silts and clays:

• Dilatancy (Shaking) Test: A small pat of moist soil is placed in the palm and shaken horizontally. If water appears on the surface giving a glossy look, the soil is likely a silt or very fine sand. If there is no change or a slow reaction, it is likely a clay.
• Toughness (Plasticity) Test: A sample of moist soil is rolled by hand on a smooth surface into a thread about 3 mm in diameter. If the thread can be formed at a plastic consistency and has high tensile strength, it is clay. Silt threads are weak and crumble easily.
• Dry Strength Test: A small pat of soil is allowed to dry. Its crushing strength is tested by hand. A high dry strength indicates clay, while a low dry strength (easily powdered) indicates silt.

b) Types of Soil Classifications:

• Particle Size Classification (e.g., MIT system)
• Textural Classification (e.g., USDA chart)
• AASHTO Soil Classification System
• Unified Soil Classification System (USCS)
• Indian Standard Soil Classification System (ISSCS)

c) Soil Classification for Highways:

The AASHTO Soil Classification System is primarily used for the suitability of soils as subgrade for highways.

• Groups: Soils are divided into 8 major groups, A-1 through A-7 (for mineral soils) and A-8 (for organic soils).
• Subgroups: Groups are further divided into subgroups like A-1-a, A-2-4, A-7-6 etc.
• General Rating as Subgrade:
  • Excellent to Good: A-1, A-3, A-2-4, A-2-5
  • Fair to Poor: A-2-6, A-2-7, A-4, A-5, A-6, A-7

d) Plasticity Chart (USCS):

The USCS plasticity chart is a graph of Plasticity Index versus Liquid Limit. It is used to classify fine-grained soils. Key features are the A-line ($I_P = 0.73(W_L – 20)$) and U-line, which help separate clays (C), silts (M), and organic soils (O) into categories of low (L) or high (H) plasticity based on the liquid limit ($W_L=50\%$ is the boundary). [Refer to answer Q.2.27(a) for image and detailed description].

a) Graph of Stress-Strain Behavior vs. Consistency:

The stress-strain behavior of fine-grained soil is highly dependent on its consistency (water content). A schematic graph shows that as the soil goes from liquid to solid state, its strength increases and its failure behavior changes from viscous flow to brittle fracture. [Refer to answer Q.2.26(c) for the curve and description].

b) Embankment Calculation:

The principle is that the weight of soil solids ($W_s$) required for the embankment must be excavated from the borrow pit.

Given Data:

• Embankment Volume, $V_{emb} = 1,00,000 \text{ m}^3$
• Embankment Dry Unit Weight, $\gamma_{d,emb} = 16 \text{ kN/m}^3$
• Borrow Pit Bulk Unit Weight, $\gamma_{bulk,pit} = 12 \text{ kN/m}^3$
• Borrow Pit Water Content, $w_{pit} = 15\% = 0.15$

Step 1: Calculate the total weight of solids ($W_s$) needed for the embankment.

$W_s = V_{emb} \times \gamma_{d,emb} = 1,00,000 \text{ m}^3 \times 16 \text{ kN/m}^3 = 1,600,000 \text{ kN}$

Step 2: Calculate the dry unit weight of the soil in the borrow pit ($\gamma_{d,pit}$).

$\gamma_{d,pit} = \frac{\gamma_{bulk,pit}}{1 + w_{pit}} = \frac{12}{1 + 0.15} = \frac{12}{1.15} = 10.435 \text{ kN/m}^3$

Step 3: Calculate the volume of soil to be excavated from the pit ($V_{pit}$).

This volume must contain the required weight of solids ($W_s$).

$V_{pit} = \frac{W_s}{\gamma_{d,pit}} = \frac{1,600,000 \text{ kN}}{10.435 \text{ kN/m}^3} = 153,330.14 \text{ m}^3$

Step 4: Calculate the total weight of soil to be excavated from the pit ($W_{pit}$).

This is the bulk weight of the excavated volume.

$W_{pit} = V_{pit} \times \gamma_{bulk,pit} = 153,330.14 \text{ m}^3 \times 12 \text{ kN/m}^3 = 1,839,961.7 \text{ kN}$

Alternatively, $W_{pit} = W_s (1 + w_{pit}) = 1,600,000 \times (1.15) = 1,840,000 \text{ kN}$.

Answer: Volume to be excavated is 153,330 m³ and weight to be excavated is 1,840,000 kN.

MIT Grain Size Ranges:

The Massachusetts Institute of Technology (MIT) soil classification system defines soil particle sizes as follows:

Field Identification Methods for Fine-Grained Soils:

These methods are used for a quick, qualitative assessment of fine-grained soils (silts and clays) in the field without laboratory equipment.

1. Dilatancy Test (Shaking Test):
   • A small amount of soil is mixed with water to make a soft, putty-like paste.
   • The paste is held in the open palm of one hand and shaken horizontally, striking vigorously against the other hand.
   • A positive reaction (dilatancy) occurs when water appears on the surface, giving it a shiny or “livery” appearance. Squeezing the sample causes the water to disappear. This rapid reaction is characteristic of silts and very fine sands.
   • Clays show a slow or no reaction.
2. Toughness Test (Plasticity Test):
   • A sample is moistened to its plastic limit consistency.
   • It is then rolled by hand on a smooth surface or between the palms into a thread of about 3 mm in diameter.
   • The resistance to crumbling and the tensile strength of the thread are observed. High-plasticity clays (CH) form strong threads that can be reformed and re-rolled multiple times. Low-plasticity clays (CL) are weaker. Silts (ML) have very little plasticity, and their threads are very weak and crumble easily.
3. Dry Strength Test:
   • A small pat of soil is molded and allowed to dry completely.
   • The effort required to break the dry pat with one’s fingers is assessed.
   • High dry strength is characteristic of high-plasticity clays. Low dry strength is characteristic of silts. A medium dry strength suggests a low-plasticity clay.

Definitions:

• Thixotropy: It is the property of certain fine-grained soils, particularly clays, to regain a portion of their shear strength over time when left undisturbed at a constant water content after being remolded. Remolding destroys the soil structure, causing a loss of strength. Thixotropy is this time-dependent, reversible stiffening process.
• Flow Index ($I_f$ or $F_I$): It is the slope of the flow curve, which is obtained by plotting water content against the logarithm of the number of blows (N) in a Casagrande liquid limit test. It indicates the rate at which a soil loses shear strength with an increase in water content. A steeper slope (higher flow index) means a more rapid loss of strength. It is given by the formula: $I_f = \frac{w_1 – w_2}{\log_{10}(N_2/N_1)}$.

Calculations:

Given Data:

• Initial saturated volume, $V_1 = 97 \text{ cm}^3$
• Initial saturated mass, $M_1 = 202 \text{ gm}$
• Dry volume, $V_d = 87 \text{ cm}^3$
• Dry mass (mass of solids), $M_s = 167 \text{ gm}$
• Density of water, $\rho_w = 1 \text{ gm/cm}^3$

i) Initial Water Content ($w$):

Mass of water, $M_w = M_1 – M_s = 202 – 167 = 35 \text{ gm}$

$w = \frac{M_w}{M_s} \times 100\% = \frac{35}{167} \times 100\% = 20.96\%$

ii) Specific Gravity of Soil Solids ($G_s$):

Volume of water in saturated sample, $V_w = \frac{M_w}{\rho_w} = \frac{35 \text{ gm}}{1 \text{ gm/cm}^3} = 35 \text{ cm}^3$

Volume of solids, $V_s = V_1 – V_w = 97 – 35 = 62 \text{ cm}^3$

$G_s = \frac{\text{Density of solids}}{\text{Density of water}} = \frac{M_s/V_s}{\rho_w} = \frac{167 \text{ gm} / 62 \text{ cm}^3}{1 \text{ gm/cm}^3} = 2.69$

iii) Shrinkage Limit ($W_s$):

The shrinkage limit is the water content at which the volume of the soil becomes constant upon further drying. At this point, the volume of water is equal to the volume of voids in the dry pat ($V_d – V_s$).

Volume of water at shrinkage limit, $V_{w,sl} = V_d – V_s = 87 – 62 = 25 \text{ cm}^3$

Mass of water at shrinkage limit, $M_{w,sl} = V_{w,sl} \times \rho_w = 25 \times 1 = 25 \text{ gm}$

$W_s = \frac{M_{w,sl}}{M_s} \times 100\% = \frac{25}{167} \times 100\% = 14.97\%$

i) Soil a:

• % passing 75µ sieve (Fines) = 4%
• % of coarse fraction passing 4.75mm sieve = 62%
• $C_u = 5$, $C_c = 2.6$

Classification:

1. Coarse vs Fine: Fines = 4%, which is < 5%. It's a clean Coarse-Grained Soil.
2. Sand vs Gravel: Coarse fraction = 96%. Sand portion = $0.62 \times 96\% = 59.5\%$. Gravel portion = $0.38 \times 96\% = 36.5\%$. Since Sand > Gravel, it’s a Sand.
3. Well vs Poorly Graded: For well-graded sand (SW), conditions are $C_u \ge 6$ and $1 \le C_c \le 3$. Here, $C_u = 5$, which does not meet the criterion.

Group Symbol: SP (Poorly graded Sand)

ii) Soil b:

• % passing 75µ sieve (Fines) = 62%
• Liquid limit, $W_L = 54\%$
• Plastic limit, $W_P = 23\%$

Classification:

1. Coarse vs Fine: Fines = 62%, which is > 50%. It’s a Fine-Grained Soil.
2. Clay vs Silt & Plasticity:
   • $W_L = 54\%$, which is > 50%. So it has high plasticity (H).
   • $I_P = W_L – W_P = 54 – 23 = 31\%$.
   • A-line value: $I_{P(A-line)} = 0.73(54 – 20) = 24.8\%$.
   • Since the soil’s $I_P (31\%) > I_{P(A-line)}$, it plots above the A-line.

Group Symbol: CH (Clay of high plasticity)

iii) Soil c:

• % passing 75µ sieve (Fines) = 39%
• Liquid limit, $W_L = 33\%$
• Plastic limit, $W_P = 18\%$

Classification:

1. Coarse vs Fine: Fines = 39%. Since $5\% < \text{Fines} < 50\%$, it's a Coarse-Grained Soil with fines (borderline case).
2. Sand vs Gravel: Assuming more than half of the coarse fraction is sand, the soil is a Sand. (This information is missing but required for full classification).
3. Nature of Fines: Use plasticity chart for the fines.
   • $I_P = W_L – W_P = 33 – 18 = 15\%$.
   • A-line value: $I_{P(A-line)} = 0.73(33 – 20) = 9.5\%$.
   • Since the soil’s $I_P (15\%) > I_{P(A-line)}$, the fines are clayey (C).

Group Symbol: SC (Clayey Sand)

Definition

A flow net is a graphical representation of seepage (groundwater flow) through a porous medium, like soil. It consists of two families of curves:

• Flow Lines: The paths that individual particles of water follow as they flow through the soil.
• Equipotential Lines: Lines connecting points of equal total head (pressure head + elevation head).

These two sets of lines intersect to form a grid of “curvilinear squares.”

Properties of a Flow Net

1. Orthogonality: Flow lines and equipotential lines intersect at right angles ($90^\circ$).
2. Curvilinear Squares: The figures formed by adjacent flow lines and equipotential lines are “curvilinear squares,” meaning their average width and length are approximately equal.
3. Boundary Conditions (Flow): Flow lines run parallel to and cannot cross impermeable boundaries (e.g., concrete dams, sheet piles, impervious clay layers).
4. Boundary Conditions (Head): Equipotential lines run parallel to and are perpendicular to impermeable boundaries. They coincide with permeable boundaries of constant head (e.g., the ground surface of a reservoir or the tailwater surface).
5. Constant Flow: The rate of flow ($\Delta q$) is constant between any two adjacent flow lines (which form a “flow channel”).
6. Constant Head Drop: The head drop ($\Delta h$) is constant between any two adjacent equipotential lines.

Uses of a Flow Net

1. Estimate Seepage Quantity (Discharge): To calculate the total volume of water seeping under a dam or into an excavation.
2. Determine Seepage Pressure: To find the uplift pressure at any point beneath a hydraulic structure, which is crucial for stability analysis.
3. Calculate Hydraulic Gradient: To find the hydraulic gradient ($i$) at any point, especially the “exit gradient” where seepage emerges.
4. Assess Piping Risk: To check if the exit gradient is high enough to cause “piping” or “quick sand condition,” which can lead to structural failure.

Proof of Discharge Equation

Let:

• $q$ = Total discharge per unit length of the dam
• $k$ = Coefficient of permeability of the soil
• $h$ = Total head loss across the flow regime
• $N_f$ = Total number of flow channels in the flow net
• $N_d$ = Total number of equipotential drops in the flow net
• $\Delta q$ = Discharge through a single flow channel
• $\Delta h$ = Head loss across a single equipotential drop

From the definitions of $N_f$ and $N_d$:

Total discharge $q = N_f \times \Delta q$

Total head loss $h = N_d \times \Delta h$, so $\Delta h = h / N_d$

Now, consider a single curvilinear square from the flow net, with average length $l$ (in the direction of flow) and average width $b$ (perpendicular to flow).

From Darcy’s Law, the flow through this single square is:

$\Delta q = k \cdot i \cdot A$

Where:

• $i$ = hydraulic gradient $= \Delta h / l = (h/N_d) / l$
• $A$ = cross-sectional area $= b \times 1$ (for unit length)

Substituting these into Darcy’s Law:

$\Delta q = k \cdot \left( \frac{h}{N_d \cdot l} \right) \cdot (b \times 1)$

$\Delta q = k \cdot \frac{h}{N_d} \cdot \left( \frac{b}{l} \right)$

A key property of a flow net is that it is drawn to consist of curvilinear squares, for which the aspect ratio $(b/l)$ is approximately 1.
Therefore, for a single flow channel square, $\frac{b}{l} \approx 1$.

This simplifies the equation for $\Delta q$ to:

$\Delta q = k \cdot \frac{h}{N_d}$

This is the flow through one flow channel. To find the total flow $q$, we multiply the flow per channel by the total number of flow channels, $N_f$:

$q = N_f \times \Delta q = N_f \times \left( k \cdot \frac{h}{N_d} \right)$

Rearranging gives the final equation:

Flow Net

A flow net is a graphical representation of seepage (groundwater flow) through a porous medium. It consists of two families of curves: flow lines (paths of water particles) and equipotential lines (lines of equal total head). These lines intersect at right angles, forming a grid of “curvilinear squares.” It is used to visualize flow patterns and to calculate seepage quantity, pressure, and hydraulic gradients.

Mechanism of Piping

Piping is a form of internal erosion in soil that can lead to the failure of hydraulic structures like earthen dams or levees. The mechanism is as follows:

1. Seepage Flow: Water from the upstream (high head) side seeps through the foundation soil to the downstream (low head) side.
2. Upward Force: As the water flows upwards towards the downstream ground surface (the “toe” of the dam), it exerts an upward drag force, known as seepage pressure, on the soil particles.
3. Critical Condition: This upward seepage force counteracts the downward gravitational force (the submerged weight) of the soil particles.
4. Particle Detachment: If the upward seepage force becomes equal to or greater than the submerged weight of the soil, the effective stress in the soil becomes zero. The soil particles are lifted and become suspended in the flowing water. This is the “quick sand condition.”
5. Erosion: These detached particles are then washed away by the flowing water. This creates a small channel or “pipe” in the soil.
6. Progressive Failure: This pipe provides an easier path for water, increasing the flow velocity and erosion rate. The pipe progressively enlarges and erodes backward (upstream) towards the reservoir.
7. Structural Collapse: Once the pipe becomes large enough, it can undermine the foundation of the dam, leading to a rapid, catastrophic failure and collapse of the structure.

Piping risk is highest at the downstream toe, where the upward hydraulic gradient (the “exit gradient,” $i_{exit}$) is at its maximum. Failure occurs if $i_{exit} \geq i_c$ (the critical hydraulic gradient).

Given Data:

• Discharge, $q = 200 \text{ cc/day} = 200 \text{ cm}^3/\text{day}$
• Number of flow channels, $N_f = 5$
• Number of equipotential drops, $N_d = 10$
• Net head, $h = 2.5 \text{ m} = 250 \text{ cm}$

Required:

• Coefficient of permeability, $k$

Unit Conversion:

We must use consistent units. Let’s convert the discharge $q$ to $\text{cm}^3/\text{s}$.
1 day = 24 hours $\times$ 60 min/hr $\times$ 60 s/min = 86,400 s

Calculation:

We use the flow net discharge equation:

Rearranging the formula to solve for $k$:

Plugging in the values:

Answer:

The permeability of the soil is $1.852 \times 10^{-5} \text{ cm/s}$.

1. Coefficient of Transmissibility (T):
   It is a measure of how much water can be transmitted horizontally through the entire saturated thickness of an aquifer. It is defined as the rate of flow of water (per unit width) through a vertical strip of the aquifer of unit width and full saturated height, under a hydraulic gradient of 1. It is the product of the coefficient of permeability ($k$) and the saturated thickness of the aquifer ($B$). $$ T = k \times B $$ The SI unit is $\text{m}^2/\text{s}$.
2. Seepage Pressure ($p_s$):
   It is the pressure or drag force exerted by flowing water on the solid particles of a soil mass. This force is transferred to the soil skeleton through friction. The seepage pressure per unit volume of soil is given by $j = i \cdot \gamma_w$, where $i$ is the hydraulic gradient and $\gamma_w$ is the unit weight of water. The total seepage force over a length $L$ is $P_s = i \cdot L \cdot A \cdot \gamma_w = h \cdot A \cdot \gamma_w$, where $h$ is the head loss over length $L$.
3. Quick Sand Condition:
   This is a condition, not a type of soil. It occurs in cohesionless (sandy or silty) soils when the upward seepage pressure becomes so high that it equals the buoyant (submerged) unit weight of the soil. When this happens, the effective stress in the soil drops to zero, and the soil loses all its shear strength. The soil behaves like a liquid and cannot support any load. This condition occurs when the upward hydraulic gradient $i$ reaches the “critical hydraulic gradient” ($i_c$). $$ i_c = \frac{\gamma’}{\gamma_w} = \frac{G_s – 1}{1 + e} $$
4. Held Water:
   Held water is the water found in a soil mass that is held in place and does not move under the force of gravity. It is the portion of soil water that remains after gravitational water has drained away. It is broadly classified into:
   • Structural Water: Water chemically bonded within the crystalline structure of the soil minerals.
   • Adsorbed (Hygroscopic) Water: Water held tightly to the surface of soil particles by strong electrochemical forces.
   • Capillary Water: Water held in the small pores (capillaries) between soil particles by surface tension, existing above the water table.

Given Data:

• Diameter, $D = 7.5 \text{ cm}$
• Initial porosity, $n_1 = 44\% = 0.44$
• Volume of water, $Q = 626 \text{ ml} = 626 \text{ cm}^3$
• Time, $t = 60 \text{ s}$
• Sample length, $L = 18 \text{ cm}$
• Head loss, $h = 24.7 \text{ cm}$
• New porosity, $n_2 = 39\% = 0.39$

Calculations (Part 1):

1. Cross-sectional Area ($A$): $$ A = \frac{\pi}{4} D^2 = \frac{\pi}{4} (7.5 \text{ cm})^2 \approx 44.1786 \text{ cm}^2 $$
2. Hydraulic Gradient ($i$): $$ i = \frac{h}{L} = \frac{24.7 \text{ cm}}{18 \text{ cm}} \approx 1.3722 $$
3. Coefficient of Permeability ($k$):
   Using Darcy’s Law for a constant head test: $Q = k \cdot i \cdot A \cdot t$ $$ k = \frac{Q}{i \cdot A \cdot t} $$ $$ k = \frac{626 \text{ cm}^3}{1.3722 \cdot 44.1786 \text{ cm}^2 \cdot 60 \text{ s}} $$ $$ k = \frac{626}{3638.1} \approx 0.17206 \text{ cm/s} $$
4. Flow Velocity ($v$): (also called discharge velocity)
   This is the apparent velocity through the total cross-section. $$ v = \frac{Q}{A \cdot t} = \frac{626 \text{ cm}^3}{44.1786 \text{ cm}^2 \cdot 60 \text{ s}} \approx 0.2361 \text{ cm/s} $$

   Alternatively, $v = k \cdot i = 0.17206 \cdot 1.3722 \approx 0.2361 \text{ cm/s}$. The values match.

5. Seepage Velocity ($v_s$):
   This is the actual velocity of water flowing through the soil pores. $$ v_s = \frac{v}{n_1} = \frac{0.2361 \text{ cm/s}}{0.44} \approx 0.5366 \text{ cm/s} $$

Calculations (Part 2): Permeability at $n_2 = 39\%$

We use the Kozeny-Carman equation, which relates permeability $k$ to void ratio $e$:

Therefore, $\frac{k_2}{k_1} = \left( \frac{e_2^3}{1+e_2} \right) / \left( \frac{e_1^3}{1+e_1} \right)$

First, find the void ratios $e_1$ and $e_2$ from porosities $n_1$ and $n_2$.

Now, substitute these into the ratio:

Finally, calculate $k_2$:

Answers:

• Permeability ($k$): $0.172 \text{ cm/s}$
• Flow Velocity ($v$): $0.236 \text{ cm/s}$
• Seepage Velocity ($v_s$): $0.537 \text{ cm/s}$
• Permeability at 39\% porosity ($k_2$): $0.101 \text{ cm/s}$

Flow Net Construction for Sheet Pile

1. Draw the Domain: Draw the cross-section of the sheet pile wall, the soil layer, and the impervious boundary to scale. Define the upstream (US) and downstream (DS) water levels.
2. Identify Boundaries:
   • Equipotential Lines (Known Head): The US soil surface is the highest equipotential line ($h = h_{max}$). The DS soil surface is the lowest equipotential line ($h = h_{min}$).
   • Flow Lines (No Flow): The surface of the impervious stratum (e.g., bedrock) is a flow line. The sheet pile itself is also an impermeable boundary and thus a flow line.
3. Sketch Initial Lines: Sketch 3 to 5 (a small number, $N_f$) flow lines that follow the general path of water. They start at the US surface, flow down, loop around the bottom tip of the sheet pile, and flow up to the DS surface. They must be parallel to the impervious boundary and the pile.
4. Sketch Orthogonal Lines: Start drawing equipotential lines. Begin at the US side, crossing the flow lines at 90-degree angles. These lines must be parallel to the US and DS ground surfaces where they meet them.
5. Iterate and Adjust: This is a trial-and-error process. Adjust the flow lines and equipotential lines iteratively until:
   • All intersections are orthogonal (at $90^\circ$).
   • All figures are “curvilinear squares” (average length $\approx$ average width).
   • All boundary conditions are met.
6. Count $N_f$ and $N_d$: Once the net is satisfactory, count the number of flow channels ($N_f$) and the number of equipotential drops ($N_d$) to use in calculations.

Graded Filter Design

A graded filter is a layer of granular material (like sand or gravel) placed between the base soil (the soil being protected) and a drainage exit (like a pipe or open air).

Purpose:

1. Prevent Piping: The filter’s pores must be small enough to prevent the fine particles of the base soil from being washed through it.
2. Allow Drainage: The filter must be significantly more permeable than the base soil to allow water to flow freely, preventing the buildup of hydrostatic pressure.

Sketch Description (as text):

Imagine a diagram with three layers from left to right:
1. Base Soil: The original, fine-grained soil that needs protection.
2. Filter Layer: A layer of coarser material (e.g., sand) next to the base soil.
3. Drain: A layer of very coarse material (e.g., gravel) or a perforated pipe that collects and removes the water.
Arrows show water flowing from the Base Soil, through the Filter, and into the Drain.

Design Criteria (Terzaghi’s Criteria):

The design relates the particle size distribution of the filter material to the base material. The criteria are based on the $D_{15}$, $D_{50}$, and $D_{85}$ sizes (the particle diameter for which 15\%, 50\%, and 85\% of the material is finer).

1. Piping/Clogging Criterion (Retention): To stop base soil particles from entering the filter: $$ \frac{D_{15} \text{ (filter)}}{D_{85} \text{ (base)}} \leq 5 $$ This ensures the 15\%-finest pores of the filter are small enough to hold back the 85\%-coarsest particles of the base.
2. Permeability Criterion (Drainage): To ensure the filter is permeable enough to drain the base soil: $$ \frac{D_{15} \text{ (filter)}}{D_{15} \text{ (base)}} \geq 5 $$ This ensures the filter is at least 5 times more permeable than the base.

Additional criteria for uniformly graded soils and to prevent segregation are also used:

• $D_{50} \text{ (filter)} / D_{50} \text{ (base)} \leq 25$
• The grain size curves of the filter and base should be roughly parallel.

This problem treats the soil clog inside the pipe as a soil sample in a permeameter.

Given Data:

• Permeability, $k = 10 \text{ m/day}$
• Head loss, $h = 20 \text{ m}$ (This is the head driving the flow through the clog)
• Discharge, $q = 0.15 \text{ m}^3/\text{day}$
• Area, $A = 200 \text{ cm}^2$

Required:

• Volume of soil, $V$

Unit Conversion:

All units should be consistent. Let’s use meters (m) and days.

• $k = 10 \text{ m/day}$ (OK)
• $h = 20 \text{ m}$ (OK)
• $q = 0.15 \text{ m}^3/\text{day}$ (OK)
• $A = 200 \text{ cm}^2 = 200 / (100 \times 100) \text{ m}^2 = 0.02 \text{ m}^2$

Calculation:

The unknown is the length of the soil clog, $L$.
We use Darcy’s Law: $q = k \cdot i \cdot A$
The hydraulic gradient, $i$, is $h / L$.

We rearrange the formula to solve for $L$:

Plugging in the values:

Now, calculate the volume of the soil clog:

Answer:

The volume of soil in the pipe is $0.533 \text{ m}^3$.

The top flow line in an earthen dam is called the phreatic line. Below this line, the soil is saturated; above it, the soil is unsaturated (or moist). The phreatic line for a homogeneous dam with a horizontal filter takes the shape of a parabola.

Method (Casagrande’s Graphical Method):

Sketch Description (as text):

Draw a cross-section of an earthen dam.

• Label the upstream (US) slope, the crest (top), and the downstream (DS) slope.
• Show the water in the reservoir, intersecting the US slope at a point A.
• Show the horizontal filter (drainage blanket) at the base of the dam on the DS side.
• Label the upstream-most point of the horizontal filter as F. This is the Focus of the parabola.

Construction Procedure:

1. Draw the Dam Section: Draw the dam cross-section to scale.
2. Locate Starting Point (B): The basic parabola does not start at point A (water surface), but at a point B on the water surface. Point B is located a horizontal distance of $0.3 \times L’$ from A, where $L’$ is the horizontal projection of the wetted upstream slope.
3. Locate the Focus (F): Identify the focus F of the parabola, which is the upstream starting point of the horizontal filter.
4. Find the Directrix: The directrix of a parabola is a vertical line at a horizontal distance $p$ from the focus, where every point on the parabola is equidistant from the focus and the directrix. The vertex of the parabola is at $p/2$ from the focus. The basic parabola passes through B, so its equation can be used to find $p$.
5. Draw the Basic Parabola: Using the focus F and starting point B (or by plotting points), sketch the basic parabola. It extends from B, through the dam, and is tangent to the horizontal filter (meaning it becomes horizontal as it approaches the filter).
6. Apply Upstream Correction: The actual phreatic line must start at A, not B. It must also be perpendicular to the upstream slope (which is an equipotential line). The sketched parabola is manually corrected (re-drawn) at the upstream end to smoothly connect from point A (perpendicular to the slope) to the main body of the basic parabola.
7. Downstream Section: The phreatic line simply follows the basic parabola until it intersects the horizontal filter. The top of the horizontal filter itself then becomes the flow line.

Note: The figure mentioned in the question (“figure below”) was not provided. A numerical solution is impossible without the geometric data from the figure (specifically, the upstream head $h$ and the base length $d$).

However, the method to solve the problem is as follows:

Formula:

The quantity of seepage (discharge) $q$ per unit length through a homogeneous earthen dam with a horizontal filter is given by the formula:

where:

• $k$ = coefficient of permeability
• $S_0$ = focal distance of the phreatic line parabola.

Given:

Procedure:

1. Find $S_0$ from the Figure: The focal distance $S_0$ (which is $2p$ in some notations) is found from the dam’s geometry.
2. Method: Using Casagrande’s method, the focus F is at the toe of the filter. The directrix is a vertical line. The basic parabola’s equation is $z^2 = 2S_0 (x + S_0/2)$ (if origin is at focus).
3. Direct Calculation: $S_0$ can be calculated if the horizontal distance $d$ (from the focus F to the point on the upstream slope at water level) and the head $h$ (upstream water depth) are known. The formula is: $$ S_0 = \sqrt{d^2 + h^2} – d $$
4. Calculate $q$: Once $S_0$ is determined from the figure (in meters), the seepage $q$ is calculated: $$ q = (3 \times 10^{-6} \text{ m/s}) \times S_0 \text{ (m)} $$ The resulting unit will be $\text{m}^3/\text{s}$ per meter length of the dam.

Conclusion:

Please provide the figure or the values of $h$ (upstream head) and $d$ (base length for the parabola) to complete the calculation.

Given Data:

• Permeability, $k = 4 \times 10^{-2} \text{ cm/s} = 4 \times 10^{-4} \text{ m/s}$
• Aquifer depth, $H_{total} = 10 \text{ m}$
• Pile depth, $D = 7 \text{ m}$
• Net head, $h = 3 \text{ m}$ (The difference between upstream and downstream water levels)

Flow Net Sketch (Description):

1. Boundaries: We draw a 10 m deep soil layer on top of an impervious ledge. A 7 m deep sheet pile is drawn in the center.
2. Equipotential Lines: The ground surface on the upstream (US) side is the $h=3$ m line. The ground surface on the downstream (DS) side is the $h=0$ m line.
3. Flow Lines: The impervious ledge at 10 m depth is a flow line. The sheet pile (from 0 m to 7 m) is also a flow line.
4. Sketching: We draw flow lines starting from the US surface, arcing down under the 7m pile, and arcing back up to the DS surface. We draw equipotential lines starting from the US surface and dropping in value as they move towards the DS surface, crossing the flow lines at 90-degree angles.
5. Estimating $N_f$ and $N_d$: A flow net must be sketched to find the ratio $N_f/N_d$. For this standard geometry (a pile driven part-way into a stratum), a properly sketched flow net yields approximately:
   • $N_f \approx 4$ (Number of flow channels)
   • $N_d \approx 8$ (Number of equipotential drops)
   *Note: These values are estimates from a standard sketch. The exact values depend on the precise drawing.

Seepage Calculation:

We use the flow net discharge equation:

Plugging in the given values and our flow net estimates:

If we want the answer in $\text{m}^3/\text{day}$:

Answer:

Based on an estimated flow net ($N_f=4, N_d=8$), the seepage quantity is $6.0 \times 10^{-4} \text{ m}^3/\text{s}$ per meter length of the wall.

Note: This question repeats material from Q.1 and Q.6.

Filter Requirements for Controlling Piping

To control piping, a filter (a layer of granular material) is designed to allow water to pass through freely while holding the base soil particles in place. The two main requirements are:

1. Piping/Clogging Criterion (Retention): The filter’s pores must be small enough to prevent the particles of the base soil from being washed into and through the filter. $$ \frac{D_{15} \text{ (filter)}}{D_{85} \text{ (base)}} \leq 5 $$
2. Permeability Criterion (Drainage): The filter must be significantly more permeable than the base soil to avoid the buildup of seepage pressure and to ensure free drainage. $$ \frac{D_{15} \text{ (filter)}}{D_{15} \text{ (base)}} \geq 5 $$

Where $D_{15}$ and $D_{85}$ are the particle sizes from the grain size distribution curve.

Properties and Application of Flow Net

Properties:

• A flow net is a grid of flow lines and equipotential lines.
• Flow lines and equipotential lines are orthogonal (intersect at $90^\circ$).
• The spaces between them are “curvilinear squares” (average width $\approx$ average length).
• The flow $\Delta q$ between any two adjacent flow lines is constant.
• The head drop $\Delta h$ between any two adjacent equipotential lines is constant.
• Flow lines are parallel to impermeable boundaries.
• Equipotential lines are parallel to permeable boundaries (constant head).

Applications (Uses):

• Estimate Seepage Quantity ($q$): Using the formula $q = k \cdot h \cdot (N_f/N_d)$.
• Determine Seepage/Uplift Pressure: The pressure at any point can be found by determining its total head from the equipotential lines.
• Calculate Hydraulic Gradient ($i$): The gradient at any point is $\Delta h / \Delta l$, which is used to check for piping risk (especially the exit gradient).

Darcy’s Law and Discharge Velocity

Darcy’s Law (1856) states that for laminar flow of water through a saturated soil mass, the discharge per unit time ($Q$) is directly proportional to the hydraulic gradient ($i$) and the total cross-sectional area ($A$) of the soil through which the flow occurs.

Mathematically:

$$ Q \propto i \cdot A $$

$$ Q = k \cdot i \cdot A $$

Where:

• $Q$ = discharge (volume of water per unit time, e.g., $m^3/s$).
• $k$ = coefficient of permeability (or hydraulic conductivity), a property of the soil (e.g., $m/s$).
• $i$ = hydraulic gradient, which is the head loss per unit length of flow ($i = \frac{\Delta h}{L}$).
• $A$ = total cross-sectional area of the soil mass, perpendicular to the direction of flow.

The discharge velocity ($v$), also known as the superficial velocity, is the fictitious average velocity calculated as if the water were flowing through the entire cross-sectional area $A$ (including both solids and voids).

It is defined as:

$$ v = \frac{Q}{A} $$

By substituting this into Darcy’s Law, we get the law in terms of discharge velocity:

$$ \frac{Q}{A} = k \cdot i $$

$$ \mathbf{v = k \cdot i} $$

This equation is the most common form of Darcy’s Law. It shows a linear relationship between the discharge velocity and the hydraulic gradient.

Tests for Coefficient of Permeability

1. Laboratory Tests:
   • Constant Head Permeability Test: Suitable for coarse-grained soils (e.g., sands, gravels) with high permeability.
   • Falling Head (or Variable Head) Permeability Test: Suitable for fine-grained soils (e.g., silts, clays) with low permeability.
   • Capillary Permeability Test
2. Field Tests (In-situ Tests):
   • Pumping Out Tests: Used for large-scale determination of permeability, typically for aquifers.
     • Pumping Test in Unconfined Aquifer
     • Pumping Test in Confined Aquifer
   • Pumping In Tests: Also known as bore-hole tests.
     • Open End Test
     • Packer Test

Flow Net

A flow net is a graphical representation of two-dimensional seepage or fluid flow through a porous medium (like soil). It consists of two families of curves that intersect orthogonally (at right angles):

1. Flow Lines (or Streamlines): These represent the paths along which individual particles of water flow through the soil. No flow can cross a flow line.
2. Equipotential Lines: These are lines connecting points of equal total head ($h$).

The space between two adjacent flow lines is called a flow channel, and the space between two adjacent equipotential lines is called an equipotential drop.

Properties of a Flow Net

• Flow lines and equipotential lines intersect each other at right angles ($90^\circ$).
• The fields (elements) formed by the intersection are “curvilinear squares.” This means the average width and length of each element are approximately equal.
• The quantity of seepage ($ \Delta q $) is the same in each flow channel.
• The head drop ($ \Delta h_d $) is the same between any two adjacent equipotential lines.
• The size of the curvilinear squares varies with the hydraulic conductivity; they are smaller in regions of high conductivity (or high gradient) and larger in regions of low conductivity.
• A flow line cannot intersect another flow line, and an equipotential line cannot intersect another equipotential line.

Applications of a Flow Net

• Estimation of Seepage Discharge (Quantity of Seepage): To calculate the total flow $Q$ under or through a structure like a dam.

  $$ Q = k \cdot H \cdot \frac{N_f}{N_d} $$

  where $H$ = total head loss, $N_f$ = number of flow channels, $N_d$ = number of equipotential drops.
• Determination of Pore Water Pressure (Uplift Pressure): To find the pressure at any point within the soil, which is crucial for stability analysis (e.g., uplift pressure on the base of a dam).
• Calculation of Hydraulic Gradient: To find the gradient at any point, especially the exit gradient, which is used to check for safety against piping or quicksand conditions.
• Stability Analysis: To assess the stability of earth dams and levees against seepage-induced failures.

This proof relates to the A. Casagrande solution for the phreatic line (top flow line) in an earthen dam, which is approximated by a parabola.

Proof:

1. Basic Parabola: The Casagrande solution models the phreatic line as a parabola with its focus at the start of the horizontal filter (toe drain). Let this point be the origin (0,0).
2. The equation of this base parabola is:

   $$ x = \frac{y^2 – s^2}{2s} \quad \text{or} \quad y^2 = 2sx + s^2 $$

   where $s$ is the focal length (the distance from the focus to the directrix).
3. Darcy’s Law: The discharge per unit width ($q$) is given by $q = k \cdot i \cdot A$.
4. Assumptions (Dupuit): We assume that for small inclinations of the phreatic line, the hydraulic gradient ($i$) at any point $(x, y)$ is equal to the slope of the phreatic line at that point.

   $$ i = \frac{dy}{dx} $$

5. The area of flow ($A$) at that section, per unit width, is $A = y \times 1 = y$.
6. Substituting these into Darcy’s Law:

   $$ q = k \cdot \left(\frac{dy}{dx}\right) \cdot y $$

7. Finding the Slope ($dy/dx$): We differentiate the parabola’s equation ($y^2 = 2sx + s^2$) with respect to $x$:

   $$ 2y \frac{dy}{dx} = 2s $$

   $$ \frac{dy}{dx} = \frac{s}{y} $$

8. Substitution: Now, substitute this expression for the slope back into the discharge equation:

   $$ q = k \cdot \left(\frac{s}{y}\right) \cdot y $$

9. Result: The $y$ terms cancel out, leaving:

   $$ \mathbf{q = k \cdot s} $$

Thus, it is proven that the discharge per unit width ($q$) through an earthen dam with a horizontal filter at its toe is equal to the coefficient of permeability ($k$) times the focal length ($s$) of the base parabola. (Q.E.D.)

This is a constant head permeability test.

Given:

• Time of collection, $t = 15 \text{ min} = 15 \times 60 = 900 \text{ s}$
• Volume of water, $Q = 500 \text{ ml} = 500 \text{ cm}^3$
• Diameter of permeameter, $d = 5 \text{ cm}$
• Length of sample (distance between gauging points), $L = 15 \text{ cm}$
• Head difference (head loss), $\Delta h = 40 \text{ cm}$
• Dry weight of sample, $W_d = 4.86 \text{ N}$
• Specific gravity of solids, $G_s = 2.65$
• Unit weight of water, $\gamma_w = 9.81 \text{ kN/m}^3 = 9810 \text{ N/m}^3$

To Find:

1. Coefficient of permeability, $k$
2. Seepage velocity, $v_s$

Solution:

Part 1: Coefficient of Permeability ($k$)

1. Cross-sectional Area ($A$):

   $$ A = \frac{\pi d^2}{4} = \frac{\pi (5 \text{ cm})^2}{4} = 19.635 \text{ cm}^2 $$

2. Hydraulic Gradient ($i$):

   $$ i = \frac{\Delta h}{L} = \frac{40 \text{ cm}}{15 \text{ cm}} = 2.667 $$

3. Darcy’s Law (Constant Head):

   $$ Q = k \cdot i \cdot A \cdot t $$

4. Solve for $k$:

   $$ k = \frac{Q}{i \cdot A \cdot t} $$

   $$ k = \frac{500 \text{ cm}^3}{(2.667) \cdot (19.635 \text{ cm}^2) \cdot (900 \text{ s})} $$

   $$ k = \frac{500}{47125.8} $$

   $$ \mathbf{k = 0.01061 \text{ cm/s}} $$

Part 2: Seepage Velocity ($v_s$)

1. Relationship: Seepage velocity ($v_s$) is related to discharge velocity ($v$) by porosity ($n$):

   $$ v_s = \frac{v}{n} $$

2. Discharge Velocity ($v$):

   $$ v = k \cdot i = (0.01061 \text{ cm/s}) \cdot (2.667) = 0.02829 \text{ cm/s} $$

3. Find Porosity ($n$): We need to find $n$ from the given weight and volume.
   • Total Volume of Sample ($V$):

     $$ V = A \times L = (19.635 \text{ cm}^2) \times (15 \text{ cm}) = 294.52 \text{ cm}^3 $$

     $$ V = 294.52 \times 10^{-6} \text{ m}^3 $$

   • Dry Unit Weight ($\gamma_d$):

     $$ \gamma_d = \frac{W_d}{V} = \frac{4.86 \text{ N}}{294.52 \times 10^{-6} \text{ m}^3} = 16500 \text{ N/m}^3 = 16.5 \text{ kN/m}^3 $$

   • Void Ratio ($e$): We use the relationship $\gamma_d = \frac{G_s \gamma_w}{1+e}$.

     $$ 16.5 \text{ kN/m}^3 = \frac{(2.65) \cdot (9.81 \text{ kN/m}^3)}{1+e} $$

     $$ 1+e = \frac{26.0}{16.5} = 1.5758 $$

     $$ e = 0.5758 $$

   • Porosity ($n$):

     $$ n = \frac{e}{1+e} = \frac{0.5758}{1.5758} = 0.3654 $$

4. Calculate Seepage Velocity ($v_s$):

   $$ v_s = \frac{v}{n} = \frac{0.02829 \text{ cm/s}}{0.3654} $$

   $$ \mathbf{v_s = 0.0774 \text{ cm/s}} $$

1. Drawing the Flow Net

Drawing a precise flow net is a graphical procedure done to scale. It involves:

1. Drawing the cross-section of the earthen dam to scale with all given dimensions.
2. Height $H_{dam} = 30\text{m}$. Freeboard = 5m.
3. Height of water $H = 30 – 5 = 25\text{m}$.
4. Slopes = 2H:1V. Top Width = 15m. Horizontal drain length = 30m.
5. Sketching the phreatic line (top flow line), starting from the upstream water level, intersecting the upstream slope orthogonally. It follows a parabolic path and ends at the toe of the horizontal drain.
6. Sketching a few more flow lines (e.g., 3-4 total, $N_f = 3$ or $4$) below the phreatic line, following its general shape.
7. Sketching equipotential lines, starting from the upstream bed (equipotential 1) and ending at the horizontal drain (final equipotential). They must intersect all flow lines at right angles ($90^\circ$).

Since this is a text-based format, the graphical flow net cannot be drawn here. A placeholder is shown.

Let’s assume after drawing the flow net, we counted:

• Number of flow channels, $N_f = 4$
• Number of equipotential drops, $N_d = 10$

(Note: These are assumed values for demonstration.)

The discharge from the flow net would be:

$$ q = k H \frac{N_f}{N_d} = (4 \text{ m/day}) \cdot (25 \text{ m}) \cdot \frac{4}{10} = 100 \times 0.4 = \mathbf{40 \text{ m}^3/\text{day/m}} $$

2. Theoretical Calculation (Casagrande’s Method)

The theoretical discharge $q$ is given by $q = k \cdot s$, where $s$ is the focal length of the base parabola. The focal length $s$ can be calculated from the geometry.

$$ s = \sqrt{d^2 + H^2} – d $$

Where:

• $H$ = height of water = $25 \text{ m}$.
• $d$ = horizontal distance from the focus $F$ (start of the filter) to the point $P$ where the upstream water level ($y=H$) intersects the upstream slope.

Let’s find $d$:

• Place the origin (0,0) at the focus $F$, which is the start of the 30m drain.
• The downstream toe is at $x = 30 \text{ m}$.
• The D/S slope (2:1) goes up 30m, so it covers $2 \times 30 = 60 \text{ m}$ horizontally. It goes from $x=30$ to $x=30+60=90$.
• The crest is 15m wide, from $x=90$ to $x=105$.
• The U/S slope (2:1) goes from $x=105, y=30$ down to $x=105+60=165, y=0$.
• The upstream water ($y=H=25\text{m}$) hits this slope. The horizontal distance from the crest edge ($x=105$) is $(30-25) \times 2 = 10 \text{ m}$.
• So, the point $P$ is at $x = 105 + 10 = 115 \text{ m}$.
• $d$ is the horizontal distance from $P(115, 25)$ to $F(0, 0)$.
• $d = 115 \text{ m}$.

Now, calculate $s$:

$$ s = \sqrt{115^2 + 25^2} – 115 = \sqrt{13225 + 625} – 115 = \sqrt{13850} – 115 $$

$$ s = 117.69 – 115 = 2.69 \text{ m} $$

Theoretical Discharge ($q_{th}$):

$$ q_{th} = k \cdot s = (4 \text{ m/day}) \cdot (2.69 \text{ m}) $$

$$ \mathbf{q_{th} = 10.76 \text{ m}^3/\text{day/m}} $$

3. Comparison

The discharge from the (hypothetical) flow net was $40 \text{ m}^3/\text{day/m}$. The theoretical discharge is $10.76 \text{ m}^3/\text{day/m}$.

The large discrepancy is due to the $N_f$ and $N_d$ values being assumed. A properly drawn flow net would yield $N_f/N_d$ ratio that gives a discharge value very close to the theoretical $10.76 \text{ m}^3/\text{day/m}$. For example, if $N_f=3$ and $N_d=28$, $q = 4 \times 25 \times (3/28) \approx 10.71 \text{ m}^3/\text{day/m}$, which is a close match.

Darcy’s Law for Inclined Flow

Darcy’s Law remains the same regardless of the orientation of the soil sample, as it is based on total head, not just pressure head.

The Law:

$$ Q = k \cdot i \cdot A $$

Explanation of Terms:

• $Q$: The discharge, or the volume of water flowing through the soil per unit time (e.g., $m^3/s$).
• $A$: The total cross-sectional area of the soil sample, measured perpendicular to the direction of flow (e.g., $m^2$).
• $k$: The coefficient of permeability (or hydraulic conductivity). This is a property of the soil that measures the ease with which water can flow through it (e.g., $m/s$).
• $i$: The hydraulic gradient. This is the driving force for the flow. It is the total head loss per unit length of flow.

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

• $L$: The length of the soil sample along the path of flow (from entry to exit).
• $\Delta h$: The total hydraulic head difference (or head loss) between the entry point (1) and the exit point (2). This is the ‘$h$’ mentioned in the question.

Total head at any point is the sum of its elevation head ($z$) and pressure head ($p/\gamma_w$).

• Total Head at Entry (point 1): $h_1 = z_1 + \frac{p_1}{\gamma_w}$
• Total Head at Exit (point 2): $h_2 = z_2 + \frac{p_2}{\gamma_w}$
• Head Loss: $\Delta h = h_1 – h_2$

The flow always occurs from a point of higher total head to a point of lower total head.

Consider a soil mass composed of $N$ horizontal layers, each with thickness $H_i$ and permeability $k_i$. The total thickness is $H = \sum H_i$.

1. Horizontal Flow (Parallel to Bedding Plane)

In this case, the flow is parallel to the layers.

• The hydraulic gradient ($i$) is the same for all layers.
• The total discharge ($Q$) is the sum of the discharges through each layer ($Q = Q_1 + Q_2 + …$).

$$ Q = Q_1 + Q_2 + … + Q_N $$

$$ (k_H \cdot i \cdot A) = (k_1 \cdot i \cdot A_1) + (k_2 \cdot i \cdot A_2) + … + (k_N \cdot i \cdot A_N) $$

Where $A = H \times 1$ and $A_i = H_i \times 1$ (for unit width).

$$ k_H \cdot i \cdot (H \cdot 1) = (k_1 \cdot i \cdot H_1 \cdot 1) + (k_2 \cdot i \cdot H_2 \cdot 1) + … $$

Dividing by $i$:

$$ k_H \cdot H = k_1 H_1 + k_2 H_2 + … + k_N H_N $$

$$ \mathbf{k_H = \frac{k_1 H_1 + k_2 H_2 + … + k_N H_N}{H_1 + H_2 + … + H_N} = \frac{\sum k_i H_i}{\sum H_i}} $$

2. Vertical Flow (Perpendicular to Bedding Plane)

In this case, the flow is normal to the layers.

• The discharge ($q$) is the same through all layers (by continuity).
• The total head loss ($h$) is the sum of the head losses in each layer ($h = h_1 + h_2 + …$).

$$ h = h_1 + h_2 + … + h_N $$

Since $h = i \cdot L$ and $q = k \cdot i \cdot A$, we have $h = \frac{qL}{kA}$.

$$ \frac{q H}{k_V A} = \frac{q H_1}{k_1 A} + \frac{q H_2}{k_2 A} + … + \frac{q H_N}{k_N A} $$

Dividing by $q$ and $A$:

$$ \frac{H}{k_V} = \frac{H_1}{k_1} + \frac{H_2}{k_2} + … + \frac{H_N}{k_N} $$

$$ \mathbf{k_V = \frac{H}{\frac{H_1}{k_1} + \frac{H_2}{k_2} + … + \frac{H_N}{k_N}} = \frac{\sum H_i}{\sum (H_i / k_i)}} $$

• Unconfined Aquifer (or Water Table Aquifer): An aquifer in which the upper boundary is the free water table (phreatic surface). The water level in a well penetrating this aquifer will be at the water table, which is at atmospheric pressure.
• Confined Aquifer (or Artesian Aquifer): An aquifer that is bounded above and below by impermeable or semi-permeable layers (aquitards or aquicludes). The water in this aquifer is under pressure, and the water level in a well will rise to the piezometric surface, which is above the top of the aquifer.

Equations for Permeability ($k$) (from Field Pumping Tests)

These equations are used to find $k$ by pumping water from a central well at a constant rate $Q$ and observing the drawdown in two observation wells at radii $r_1$ and $r_2$.

1. Confined Aquifer (Thiem’s Equation):

   $$ k = \frac{Q \cdot \ln(r_2 / r_1)}{2 \pi B (h_2 – h_1)} $$

   Where:

   • $Q$ = steady state discharge from the pumping well.
   • $r_1, r_2$ = radii of the two observation wells.
   • $h_1, h_2$ = steady state heights of the piezometric surface (heads) at $r_1, r_2$.
   • $B$ = thickness of the confined aquifer.
2. Unconfined Aquifer (Dupuit’s Equation):

   $$ k = \frac{Q \cdot \ln(r_2 / r_1)}{\pi (h_2^2 – h_1^2)} $$

   Where:

   • $Q$ = steady state discharge from the pumping well.
   • $r_1, r_2$ = radii of the two observation wells.
   • $h_1, h_2$ = steady state heights of the water table (heads) at $r_1, r_2$ (measured from the impermeable base).

Confined and Unconfined Seepage Flow

• Confined Flow: This is seepage flow where the flow domain is confined between two impermeable (or practically impermeable) boundaries. The flow lines are known at all boundaries. A common example is the seepage flow underneath a concrete gravity dam.
• Unconfined Flow: This is seepage flow where one boundary, the upper boundary, is a free water surface (phreatic line). The position of this free surface is not known in advance and must be determined as part of the solution. A common example is the seepage through an earthen dam.

Purpose of a Filter on the Downstream of an Earth Dam

A filter, typically a multi-layered graded sand and gravel drain (or a geotextile), is used on the downstream side (e.g., as a toe drain or chimney drain) for two primary reasons:

1. To Prevent Piping: The filter is designed to be fine enough to prevent the fine soil particles of the dam core from being washed away by the seeping water (this erosion is called piping), but coarse enough to be much more permeable than the dam core.
2. To Control the Phreatic Line: By providing a high-permeability path, the filter freely drains the seepage, keeping the phreatic line low. This prevents the downstream slope from becoming saturated, which is crucial for maintaining its shear strength and stability.

Proof of Orthogonality of Flow and Equipotential Lines

1. Definitions:
   • An equipotential line is a line of constant total head, $\Phi(x, y) = C_1$.
   • A flow line is defined by a flow function, $\Psi(x, y) = C_2$.
2. Velocity Components: The seepage velocity components ($v_x, v_y$) are related to the potential function ($\Phi$) by Darcy’s law:

   $$ v_x = -k \frac{\partial \Phi}{\partial x} \quad ; \quad v_y = -k \frac{\partial \Phi}{\partial y} $$

   They are related to the flow function ($\Psi$) by the continuity equation:

   $$ v_x = -\frac{\partial \Psi}{\partial y} \quad ; \quad v_y = \frac{\partial \Psi}{\partial x} $$

   (Assuming $k=1$ or absorbing it into the functions for simplicity of proof).
3. Slope of Equipotential Line ($m_e$):

   Since $\Phi(x, y) = C_1$ (constant), its total differential $d\Phi$ is zero.

   $$ d\Phi = \frac{\partial \Phi}{\partial x} dx + \frac{\partial \Phi}{\partial y} dy = 0 $$

   $$ \frac{\partial \Phi}{\partial y} dy = – \frac{\partial \Phi}{\partial x} dx $$

   $$ m_e = \frac{dy}{dx} = – \frac{\partial \Phi / \partial x}{\partial \Phi / \partial y} = – \frac{(-v_x / k)}{(-v_y / k)} = – \frac{v_x}{v_y} $$

4. Slope of Flow Line ($m_f$):

   Since $\Psi(x, y) = C_2$ (constant), its total differential $d\Psi$ is zero.

   $$ d\Psi = \frac{\partial \Psi}{\partial x} dx + \frac{\partial \Psi}{\partial y} dy = 0 $$

   $$ \frac{\partial \Psi}{\partial y} dy = – \frac{\partial \Psi}{\partial x} dx $$

   $$ m_f = \frac{dy}{dx} = – \frac{\partial \Psi / \partial x}{\partial \Psi / \partial y} = – \frac{v_y}{-v_x} = \frac{v_y}{v_x} $$

5. Product of Slopes:

   $$ m_e \times m_f = \left( – \frac{v_x}{v_y} \right) \times \left( \frac{v_y}{v_x} \right) = -1 $$

Since the product of their slopes is -1, the flow lines and equipotential lines are orthogonal (intersect at right angles). (Q.E.D.)

Effective Stress Variation with Flow

The effective stress ($\sigma’$) at any point in a soil mass is the total stress ($\sigma$) minus the pore water pressure ($u$).

$$ \sigma’ = \sigma – u $$

Under a no-flow (static) condition at a depth $z$ below the surface of a saturated soil:

• $\sigma = z \cdot \gamma_{sat}$
• $u = z \cdot \gamma_w$
• $\sigma’ = z \cdot \gamma_{sat} – z \cdot \gamma_w = z (\gamma_{sat} – \gamma_w) = z \cdot \gamma’$

Where $\gamma’$ is the submerged (or buoyant) unit weight.

1. Downward Flow

• When water flows downward, it exerts a drag force (seepage force) on the soil particles in the direction of flow (downward).
• This seepage force adds to the gravitational weight of the particles, increasing the inter-granular pressure.
• The effective stress at depth $z$ increases.
• The head loss over the depth $z$ is $h_L = i \cdot z$. The pore pressure at depth $z$ is $u = z \cdot \gamma_w – h_L \cdot \gamma_w = z \cdot \gamma_w – i z \gamma_w$.
• $\sigma’ = \sigma – u = (z \cdot \gamma_{sat}) – (z \cdot \gamma_w – i z \gamma_w)$
• $\sigma’ = z (\gamma_{sat} – \gamma_w) + i z \gamma_w = \mathbf{z \cdot \gamma’ + i z \gamma_w}$
• This increased effective stress leads to greater soil stability.

2. Upward Flow

• When water flows upward, it exerts a drag force (seepage force) on the soil particles in the opposite direction of gravity (upward).
• This seepage force reduces the inter-granular pressure.
• The effective stress at depth $z$ decreases.
• The head gain over the depth $z$ (relative to the exit) is $h_L = i \cdot z$. The pore pressure at depth $z$ is $u = z \cdot \gamma_w + h_L \cdot \gamma_w = z \cdot \gamma_w + i z \gamma_w$.
• $\sigma’ = \sigma – u = (z \cdot \gamma_{sat}) – (z \cdot \gamma_w + i z \gamma_w)$
• $\sigma’ = z (\gamma_{sat} – \gamma_w) – i z \gamma_w = \mathbf{z \cdot \gamma’ – i z \gamma_w}$
• Critical Condition: If the upward gradient ($i$) is large enough, the effective stress can become zero ($\sigma’ = 0$). At this point, the soil loses all its shear strength and behaves like a fluid. This is called the quick condition or piping. The gradient that causes this is the critical hydraulic gradient ($i_c$).

  $$ z \cdot \gamma’ – i_c z \gamma_w = 0 \implies i_c = \frac{\gamma’}{\gamma_w} $$

Discharge Velocity ($v$):

(This is the same as in Q.16b). It is a fictitious or superficial velocity, calculated as if flow occurs through the entire cross-section of the soil (solids + voids). It is given by $v = Q/A$ or $v = ki$.

Given:

• Standpipe area, $a = 130 \text{ mm}^2$
• Permeameter (sample) area, $A = 4560 \text{ mm}^2$
• Initial head, $h_1 = 900 \text{ mm}$
• Final head, $h_2 = 135 \text{ mm}$

Case 1: Soil 1 only

• Sample length = $L_1$
• Time, $t_1 = 5 \text{ s}$
• Permeability = $k_1$

The formula for a variable head test is:

$$ k = \frac{a L}{A t} \ln\left(\frac{h_1}{h_2}\right) $$

$$ k_1 = \frac{a L_1}{A t_1} \ln\left(\frac{h_1}{h_2}\right) $$

Let’s find the value of the constant $\frac{k_1}{L_1}$:

$$ \frac{k_1}{L_1} = \frac{a}{A t_1} \ln\left(\frac{h_1}{h_2}\right) = \frac{130}{4560 \times 5} \ln\left(\frac{900}{135}\right) $$

$$ \frac{k_1}{L_1} = \frac{130}{22800} \ln(6.667) = 0.0057017 \times 1.8971 $$

$$ \frac{k_1}{L_1} = 0.010817 \text{ s}^{-1} \quad \text{— (Equation 1)} $$

Case 2: Soil 1 and Soil 2 (Layered)

• Soil 1 length = $L_1$, permeability = $k_1$
• Soil 2 length = $L_2 = 60 \text{ mm}$, permeability = $k_2$
• Total length, $L_{total} = L_1 + L_2 = L_1 + 60$
• Time, $t_2 = 150 \text{ s}$

This is a composite sample with vertical flow. The equivalent permeability ($k_{eq}$) is:

$$ k_{eq} = \frac{L_{total}}{\frac{L_1}{k_1} + \frac{L_2}{k_2}} = \frac{L_1 + 60}{\frac{L_1}{k_1} + \frac{60}{k_2}} $$

Using the variable head formula for the composite sample:

$$ k_{eq} = \frac{a L_{total}}{A t_2} \ln\left(\frac{h_1}{h_2}\right) $$

$$ k_{eq} = \frac{130 \cdot (L_1 + 60)}{4560 \times 150} \ln(6.667) $$

$$ k_{eq} = \frac{130 \cdot (L_1 + 60)}{684000} \times 1.8971 $$

$$ k_{eq} = 0.0003611 \cdot (L_1 + 60) \quad \text{— (Equation 2)} $$

Solve for $k_2$:

Now we have two expressions for $k_{eq}$.

From (Equation 2):

$$ \frac{k_{eq}}{L_1 + 60} = 0.0003611 $$

From the definition of $k_{eq}$:

$$ \frac{k_{eq}}{L_1 + 60} = \frac{1}{\frac{L_1}{k_1} + \frac{60}{k_2}} $$

Set them equal:

$$ 0.0003611 = \frac{1}{\frac{L_1}{k_1} + \frac{60}{k_2}} $$

$$ \frac{L_1}{k_1} + \frac{60}{k_2} = \frac{1}{0.0003611} = 2769.3 $$

From Equation 1, we know $\frac{L_1}{k_1} = \frac{1}{0.010817} = 92.446$.

Substitute this value:

$$ 92.446 + \frac{60}{k_2} = 2769.3 $$

$$ \frac{60}{k_2} = 2769.3 – 92.446 = 2676.85 $$

$$ k_2 = \frac{60}{2676.85} $$

$$ \mathbf{k_2 = 0.02241 \text{ mm/s}} $$

This is the same as Q.13a.

A flow net is a graphical representation of two-dimensional seepage flow through a porous soil. It consists of two families of curves:

1. Flow Lines: The paths that water particles follow.
2. Equipotential Lines: Lines connecting points of equal total head.

These two sets of lines intersect orthogonally (at 90 degrees) to form a grid of curvilinear squares. A flow net is used to estimate seepage quantity, pore water pressures, and hydraulic gradients.

Derivation of Laplace Equation

The derivation is based on two principles: Darcy’s Law and the Continuity Equation for an incompressible fluid in a non-deforming porous medium.

1. Continuity Equation: For steady-state flow, the flow rate into any element must equal the flow rate out. Consider a 2D element with dimensions $dx$, $dy$, and unit thickness $dz=1$.

   Flow in ($x$-dir): $v_x \cdot dy \cdot 1$

   Flow out ($x$-dir): $(v_x + \frac{\partial v_x}{\partial x} dx) \cdot dy \cdot 1$

   Flow in ($y$-dir): $v_y \cdot dx \cdot 1$

   Flow out ($y$-dir): $(v_y + \frac{\partial v_y}{\partial y} dy) \cdot dx \cdot 1$

   (Flow In) – (Flow Out) = 0

   $$ (v_x dy) – (v_x + \frac{\partial v_x}{\partial x} dx) dy + (v_y dx) – (v_y + \frac{\partial v_y}{\partial y} dy) dx = 0 $$

   $$ – \frac{\partial v_x}{\partial x} dx dy – \frac{\partial v_y}{\partial y} dy dx = 0 $$

   Dividing by $dx \cdot dy$:

   $$ \mathbf{\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} = 0} \quad \text{(Continuity Equation)} $$

2. Darcy’s Law: This relates velocity to the total head ($h$).

   $$ v_x = k_x i_x = -k_x \frac{\partial h}{\partial x} $$

   $$ v_y = k_y i_y = -k_y \frac{\partial h}{\partial y} $$

3. Combine: Substitute Darcy’s Law into the Continuity Equation:

   $$ \frac{\partial}{\partial x} \left( -k_x \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( -k_y \frac{\partial h}{\partial y} \right) = 0 $$

   $$ k_x \frac{\partial^2 h}{\partial x^2} + k_y \frac{\partial^2 h}{\partial y^2} = 0 $$

   This is the general equation for 2D seepage in an anisotropic soil.

4. Isotropic Condition: If the soil is isotropic, its permeability is the same in all directions ($k_x = k_y = k$).

   $$ k \frac{\partial^2 h}{\partial x^2} + k \frac{\partial^2 h}{\partial y^2} = 0 $$

   Dividing by $k$ (which is non-zero):

   $$ \mathbf{\frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} = 0} \quad \text{(Laplace Equation)} $$

(Assuming C is the tailwater/exit surface, A is the top of the soil, and B is the bottom of the soil layer).

Let’s assume:

• Datum is at point A (top of soil).
• Point C (tailwater) is at the same level as A.
• Point A is at depth $z = 0$.
• Point B is at depth $z = L$.
• Head loss between B and A is $h$.

1. Total Stress ($\sigma$)

Total stress is the weight of everything above a point.

At $z=0$ (Point A): $\sigma_A = 0$

At $z=L$ (Point B): $\sigma_B = L \cdot \gamma_{sat}$

The plot is a straight line, increasing with depth.

2. Pore Water Pressure ($u$)

We need to find the total head ($h_T$) at each point. $h_T = \text{elevation head } (h_e) + \text{pressure head } (h_p)$.

• At Point A ($z=0$):

  $h_{eA} = 0$ (at datum).

  Water exits at A (or C), so the total head here is the tailwater level. The head loss $h$ occurs *before* this point.

  $h_{TA} = 0$.

  $h_p = h_{TA} – h_{eA} = 0 – 0 = 0$.

  $u_A = h_p \cdot \gamma_w = 0$.

• At Point B ($z=L$):

  $h_{eB} = -L$ (below datum).

  Total head at B is $h$ higher than at A. $h_{TB} = h_{TA} + h = 0 + h = h$.

  $h_p = h_{TB} – h_{eB} = h – (-L) = h + L$.

  $u_B = (h+L) \cdot \gamma_w$.

The plot is a straight line from $0$ at A to $(h+L)\gamma_w$ at B.

3. Effective Stress ($\sigma’$)

$\sigma’ = \sigma – u$

• At Point A ($z=0$):

  $\sigma’_A = \sigma_A – u_A = 0 – 0 = 0$.

• At Point B ($z=L$):

  $\sigma’_B = \sigma_B – u_B = (L \cdot \gamma_{sat}) – (h+L)\gamma_w$

  $\sigma’_B = L \cdot \gamma_{sat} – h \cdot \gamma_w – L \cdot \gamma_w$

  $\sigma’_B = L (\gamma_{sat} – \gamma_w) – h \cdot \gamma_w$

  $\sigma’_B = L \cdot \gamma’ – h \cdot \gamma_w$

Since $i = h/L$, this can be written $\sigma’_B = L \cdot \gamma’ – i L \gamma_w$, which matches the formula from Q.18a.

The plot is a straight line from $0$ at A to $(L\gamma’ – h\gamma_w)$ at B.

Diagrams:

(Stress vs. Depth $z$, with $z=0$ at A and $z=L$ at B)

Capillarity

Capillarity (or capillary action) is the phenomenon where water is drawn up into the small pores (capillaries) of a soil mass above the free water table. This rise is caused by the surface tension of the water and its adhesion to the soil particles.

In the capillary fringe (the zone of soil above the water table that is saturated or partially saturated by capillary water), the pore water pressure is negative ($u < 0$). This negative pressure (or tension) pulls the soil particles together, increasing the effective stress.

$$ \sigma’ = \sigma – u = \sigma – (-\left|u\right|) = \sigma + \left|u\right| $$

Effect of Water Table (WT) Variation on Effective Stress

The position of the water table is a primary factor controlling effective stress.

1. WT Rises:
   • If the WT rises, the pore water pressure ($u$) at any point below the new WT increases (becomes less negative or more positive).
   • Since $\sigma’ = \sigma – u$, an increase in $u$ causes a decrease in effective stress ($\sigma’$).
   • This reduces the soil’s shear strength and bearing capacity.
2. WT Falls (Lowers):
   • If the WT falls, the pore water pressure ($u$) at any point above the new WT (but below the old WT) decreases (becomes more negative or less positive).
   • This decrease in $u$ causes an increase in effective stress ($\sigma’$).
   • This increases the soil’s shear strength. This is why temporary lowering of the WT (dewatering) is often used to stabilize excavations.

(Note: Without the figure, we must make standard assumptions.)

Assumptions:

1. The flow is uniform and parallel to the inclined layer.
2. The flow is driven by gravity only (i.e., the water table is at the surface of the permeable layer).

Given:

• $k = 4.8 \times 10^{-5} \text{ m/s}$
• Thickness of layer, $H = 3 \text{ m}$
• Angle of inclination, $\alpha = 5^\circ$

(i) Hydraulic Gradient ($i$)

• The hydraulic gradient ($i$) is the head loss ($\Delta h$) per unit length of flow ($L$).
• $i = \frac{\Delta h}{L}$
• The total head at any point is $h_T = z + p/\gamma_w$.
• Take two points, 1 and 2, a distance $L$ apart along the flow path.
• Since the water table is at the surface, the pressure head ($p/\gamma_w$) is zero at the surface.
• Head at 1: $h_1 = z_1 + 0 = z_1$
• Head at 2: $h_2 = z_2 + 0 = z_2$
• The head loss is $\Delta h = h_1 – h_2 = z_1 – z_2$.
• From trigonometry, the change in elevation ($z_1 – z_2$) over a slope length $L$ is $L \sin(\alpha)$.
• $\Delta h = L \sin(\alpha)$
• Therefore, the hydraulic gradient $i$ is:

  $$ i = \frac{\Delta h}{L} = \frac{L \sin(\alpha)}{L} = \sin(\alpha) $$

  $$ i = \sin(5^\circ) = \mathbf{0.08716} $$

(ii) Rate of Water Flow ($q$)

We will calculate the flow rate per unit width (perpendicular to the diagram).

• Discharge, $q = k \cdot i \cdot A$
• The cross-sectional area ($A$) for a unit width (1 m) is:

  $$ A = \text{Thickness} \times 1 \text{ m} = 3 \text{ m} \times 1 \text{ m} = 3 \text{ m}^2 $$

• $q = (4.8 \times 10^{-5} \text{ m/s}) \cdot (0.08716) \cdot (3 \text{ m}^2) $
• $q = 1.255 \times 10^{-5} \text{ m}^3/\text{s}$

The rate of water flow is $1.255 \times 10^{-5} \text{ m}^3/\text{s per meter width}$.

This is the same as Q.16b. See Q.16b for the detailed table.

• Properties of Flow Net: See Q.13a.
• Proof of Orthogonality: See Q.17.
• Testing Methods for Permeability: See Q.12a.

(Note: The solution requires the figure, which is not provided. The following is a general procedure.)

General Procedure:

1. Establish a Datum: Choose a convenient horizontal line as the datum (e.g., the level of point 3, or the tailwater level). Let’s assume the datum is at $z=0$.
2. Find Elevation Heads ($z$):
   • Find the vertical distance from the datum to point 1 ($z_1$).
   • Find the vertical distance from the datum to point 2 ($z_2$).
   • Find the vertical distance from the datum to point 3 ($z_3$).
3. Find Pressure Heads ($h_p$):
   • The piezometer at point 1 shows the height of water $h_1$ relative to point 1. So, $h_{p1} = h_1$.
   • The piezometer at point 2 shows $h_2$. So, $h_{p2} = h_2$.
   • The piezometer at point 3 shows $h_3$. So, $h_{p3} = h_3$.
4. Calculate Total Heads ($H = z + h_p$):
   • Total Head at 1: $\mathbf{H_1 = z_1 + h_{p1}}$
   • Total Head at 2: $\mathbf{H_2 = z_2 + h_{p2}}$
   • Total Head at 3: $\mathbf{H_3 = z_3 + h_{p3}}$
5. Calculate Hydraulic Gradient ($i$):
   • The flow is from (1) to (3). The total head loss is $\Delta H = H_1 – H_3$.
   • The length of the soil sample between points 1 and 3 is $L_{13}$.
   • The hydraulic gradient is $\mathbf{i = \frac{\Delta H}{L_{13}} = \frac{H_1 – H_3}{L_{13}}}$.

Derivation of Critical Hydraulic Gradient ($i_c$)

1. Quick Condition: This is a condition of zero effective stress ($\sigma’ = 0$) in a soil mass, caused by a sufficiently high upward hydraulic gradient.
2. Effective Stress (Upward Flow): From Q.18a, the effective stress at a depth $z$ with upward seepage is:

   $$ \sigma’_z = z \cdot \gamma’ – i z \gamma_w $$

3. Set $\sigma’ = 0$: The quick condition occurs when $\sigma’ = 0$. The gradient at this point is the critical hydraulic gradient, $i_c$.

   $$ 0 = z \cdot \gamma’ – i_c z \gamma_w $$

4. Solve for $i_c$:

   $$ i_c z \gamma_w = z \cdot \gamma’ $$

   $$ \mathbf{i_c = \frac{\gamma’}{\gamma_w}} $$

   where $\gamma’$ is the submerged unit weight ($\gamma’ = \gamma_{sat} – \gamma_w$).

5. Express in terms of $G_s$ and $e$:

   We know $\gamma’ = \left(\frac{G_s – 1}{1+e}\right) \gamma_w$.

   $$ i_c = \frac{\left(\frac{G_s – 1}{1+e}\right) \gamma_w}{\gamma_w} $$

   $$ \mathbf{i_c = \frac{G_s – 1}{1+e}} $$

Likelihood of Quick Condition (Sand vs. Clay)

Quick condition is much more likely in sands (or cohesionless silts) than in clays.

• Sands (Cohesionless): The shear strength of sand is purely frictional, given by $\tau = \sigma’ \tan(\phi’)$. When the effective stress $\sigma’$ becomes zero, the shear strength $\tau$ also becomes zero. The particles are cohesionless and easily separated by the upward flow, leading to the “boiling” or “quick” state. The $i_c$ for typical sand is $\approx 1.0$.
• Clays (Cohesive): The shear strength of clay has two components: $\tau = c’ + \sigma’ \tan(\phi’)$. Even if the effective stress $\sigma’$ becomes zero, the clay still retains significant shear strength due to its cohesion ($c’$). The electrochemical bonds between clay particles hold them together. A much higher hydraulic gradient would be needed to physically lift and erode the entire clay mass, which is a different failure mechanism (heave) rather than quick condition.
• Permeability: Furthermore, clays have extremely low permeability ($k$). To achieve a high gradient ($i = v/k$), an impossibly large discharge velocity ($v$) would be required. Sands have high permeability, allowing large flows to develop easily, which can lead to gradients approaching $i_c$.

Requirements for Protective Filter Design

A protective filter must satisfy two main criteria (Terzaghi’s criteria) to be effective:

1. Piping Requirement (Retention Criterion): The filter pores must be small enough to prevent the particles of the protected soil (base soil) from being washed into or through the filter.

   $$ \frac{D_{15} \text{ (of Filter)}}{D_{85} \text{ (of Soil)}} \leq 5 $$

2. Permeability Requirement (Drainage Criterion): The filter must be significantly more permeable than the base soil to allow water to drain freely without building up hydrostatic pressure.

   $$ \frac{D_{15} \text{ (of Filter)}}{D_{15} \text{ (of Soil)}} \geq 4 $$

(Where $D_{15}$ is the grain size for which 15% of the material is finer).

Flow Through an Earth Dam

The flow through the body of an earthen dam is unconfined flow, as its upper boundary is the phreatic line (a free water surface).

(The flow under the foundation of a dam could be confined).

Proof of Orthogonality

This is the same as Q.17. See Q.17 for the full derivation.

Factors Influencing Capillary Rise ($h_c$)

• Grain Size: $h_c$ is inversely proportional to the pore diameter, which is related to the effective grain size ($D_{10}$). $h_c \propto 1/D_{10}$. Finer soils (clays, silts) have much higher capillary rise than coarse soils (sands).
• Void Ratio ($e$): Denser soils (lower $e$) have smaller, more effective pores, leading to a higher capillary rise. $h_c \propto 1/e$.
• Surface Tension ($T_s$): Capillary rise is directly proportional to the surface tension of the fluid (water).
• Unit Weight of Fluid ($\gamma_w$): $h_c$ is inversely proportional to $\gamma_w$.
• Temperature: Affects $T_s$ and $\gamma_w$.

Relationship: Seepage vs. Superficial Velocity

This is the same as Q.16b.

• Superficial (Discharge) Velocity, $v = Q/A$
• Seepage Velocity, $v_s = Q/A_v$
• Area of Voids, $A_v = n \cdot A$ (where $n$ = porosity)
• Substitute: $v_s = \frac{Q}{n \cdot A} = \frac{1}{n} \left( \frac{Q}{A} \right)$
• $\mathbf{v_s = v / n}$

Numerical Problem

Given:

• Soil thickness, $H_s = 1.15 \text{ m}$
• Porosity, $n = 30\% = 0.30$
• Specific gravity, $G_s = 2.7$
• Upward seepage head (head loss), $h = 1.95 \text{ m}$
• Required Factor of Safety, $F.S. = 2.0$
• Coarse material has same $n$ and $G_s$.

To Find:

• Thickness of coarse material, $H_c$.

Solution:

1. Find properties ($e$ and $i_c$):
   • Void ratio, $e = \frac{n}{1-n} = \frac{0.30}{1 – 0.30} = \frac{0.30}{0.70} = 0.4286$
   • Critical hydraulic gradient, $i_c = \frac{G_s – 1}{1+e} = \frac{2.7 – 1}{1 + 0.4286} = \frac{1.7}{1.4286} = 1.1896$
2. Find actual hydraulic gradient ($i$):
   • The head loss $h=1.95\text{m}$ occurs over the thickness of the soil stratum $H_s = 1.15\text{m}$.
   • $i = \frac{h}{H_s} = \frac{1.95 \text{ m}}{1.15 \text{ m}} = 1.6956$
   • Note: $i > i_c$ ($1.696 > 1.19$), so the soil is unstable without the surcharge.
3. Factor of Safety ($F.S.$):

   The F.S. against piping is the ratio of resisting forces to driving forces.

   • Resisting Force: Submerged weight of the soil AND the surcharge. $W’_{total} = (H_s \cdot \gamma’) + (H_c \cdot \gamma’_c)$
   • Driving Force: Seepage force due to upward flow. $F_{seepage} = i \cdot H_s \cdot \gamma_w$

   This is complex. A simpler method is $F.S. = \frac{i_c}{i}$, but this doesn’t account for the surcharge.

   Let’s use the $F.S.$ definition as the ratio of total downward forces (submerged weight) to total upward forces (seepage force) on the soil mass.

   $$ F.S. = \frac{\text{Total Submerged Weight}}{\text{Total Seepage Uplift Force}} $$

   $$ \text{Total Submerged Weight} = (H_s + H_c) \cdot \gamma’ $$

   (Assuming $\gamma’_c = \gamma’$ since $G_s, n$ are same)

   $$ \text{Total Seepage Uplift Force} = h \cdot \gamma_w $$

   (The total head loss $h$ creates an uplift pressure of $h \gamma_w$ at the bottom).

   $$ F.S. = \frac{(H_s + H_c) \gamma’}{h \cdot \gamma_w} $$

4. Solve for $H_c$:

   We know $\frac{\gamma’}{\gamma_w} = i_c = 1.1896$.

   $$ 2.0 = \frac{(1.15 + H_c) \cdot (\gamma’ / \gamma_w)}{1.95} $$

   $$ 2.0 = \frac{(1.15 + H_c) \cdot 1.1896}{1.95} $$

   $$ 2.0 \times 1.95 = (1.15 + H_c) \cdot 1.1896 $$

   $$ 3.9 = (1.15 + H_c) \cdot 1.1896 $$

   $$ \frac{3.9}{1.1896} = 1.15 + H_c $$

   $$ 3.2784 = 1.15 + H_c $$

   $$ H_c = 3.2784 – 1.15 $$

   $$ \mathbf{H_c = 2.128 \text{ m}} $$

For anisotropic soil ($k_x \neq k_y$), the 2D seepage equation (from Q.19b) is:

$$ k_x \frac{\partial^2 h}{\partial x^2} + k_y \frac{\partial^2 h}{\partial y^2} = 0 $$

To solve this, we use a transformed section by scaling one coordinate. Let’s create a new coordinate $x_t$:

$$ x_t = x \sqrt{\frac{k_y}{k_x}} \quad ; \quad y_t = y $$

Now we find the partial derivatives in terms of the new coordinates ($x_t, y_t$):

$$ \frac{\partial h}{\partial x} = \frac{\partial h}{\partial x_t} \cdot \frac{\partial x_t}{\partial x} = \frac{\partial h}{\partial x_t} \sqrt{\frac{k_y}{k_x}} $$

$$ \frac{\partial^2 h}{\partial x^2} = \frac{\partial^2 h}{\partial x_t^2} \cdot \left(\sqrt{\frac{k_y}{k_x}}\right)^2 = \frac{\partial^2 h}{\partial x_t^2} \left(\frac{k_y}{k_x}\right) $$

And since $y_t = y$:

$$ \frac{\partial^2 h}{\partial y^2} = \frac{\partial^2 h}{\partial y_t^2} $$

Substitute these back into the original anisotropic equation:

$$ k_x \left[ \frac{\partial^2 h}{\partial x_t^2} \left(\frac{k_y}{k_x}\right) \right] + k_y \left[ \frac{\partial^2 h}{\partial y_t^2} \right] = 0 $$

$$ k_y \frac{\partial^2 h}{\partial x_t^2} + k_y \frac{\partial^2 h}{\partial y_t^2} = 0 $$

$$ \frac{\partial^2 h}{\partial x_t^2} + \frac{\partial^2 h}{\partial y_t^2} = 0 $$

This is the Laplace equation, which describes flow in an isotropic medium.

This means we can solve the seepage problem by:

1. Transforming the geometry (scaling the $x$-dimension by $\sqrt{k_y/k_x}$).
2. Drawing a standard flow net (with curvilinear squares) on this transformed geometry.
3. Counting $N_f$ and $N_d$ from this transformed flow net.

The discharge $q$ in the transformed section is given by $q = k’ H \frac{N_f}{N_d}$.

The equivalent isotropic permeability ($k’$) for this transformed section is:

$$ k’ = \sqrt{k_x k_y} $$

Therefore, the relationship for seepage discharge per unit width ($q$) through an anisotropic soil is:

$$ \mathbf{q = \sqrt{k_x k_y} \cdot H \cdot \frac{N_f}{N_d}} $$

Where $N_f$ and $N_d$ are counted from the flow net drawn on the section transformed by scaling $x \to x \sqrt{k_y/k_x}$.

Given:

• Upstream head, $H_{us} = 8 \text{ m}$
• Downstream head, $H_{ds} = 1 \text{ m}$
• Number of flow channels, $N_f = 9$
• Number of equipotential drops, $N_d = 12$
• Coefficient of permeability, $k = 3 \times 10^{-4} \text{ cm/s}$

To Find:

• Seepage discharge, $q$ (per unit width/meter run)

Solution:

1. Total Head Loss ($H$):

   $$ H = H_{us} – H_{ds} = 8 \text{ m} – 1 \text{ m} = 7 \text{ m} $$

2. Convert Permeability ($k$):

   We need $k$ in m/s to be consistent with $H$ in meters.

   $$ k = 3 \times 10^{-4} \text{ cm/s} = (3 \times 10^{-4}) / 100 \text{ m/s} = 3 \times 10^{-6} \text{ m/s} $$

3. Flow Net Discharge Formula (Isotropic):

   The discharge per unit width ($q$) is given by:

   $$ q = k \cdot H \cdot \frac{N_f}{N_d} $$

4. Calculate $q$:

   $$ q = (3 \times 10^{-6} \text{ m/s}) \cdot (7 \text{ m}) \cdot \left(\frac{9}{12}\right) $$

   $$ q = (21 \times 10^{-6}) \cdot (0.75) $$

   $$ q = 15.75 \times 10^{-6} \text{ m}^3/\text{s per meter} $$

The seepage discharge is $1.575 \times 10^{-5} \text{ m}^3/\text{s}$ per meter run of the dam.