14.333 GEOTECHNICAL LABORATORY BERNOULLI S EQUATION h u w v 2 2g Z h = Total Head u = Pressure v = Velocity g = Acceleration due to Gravity w = Unit Weight of Water Slide 1 of 14
h 14.333 GEOTECHNICAL LABORATORY BERNOULLI S EQUATION IN SOIL u w v 2 2g Z v 0 v 0 (i.e. velocity of water in soil is negligible). Therefore: u h Z w Slide 2 of 14
h h A h B CHANGE IN HEAD FROM POINTS A & B (H) h h h A B h h A h B Figure 5.1. Das FGE (2005). h u A w i Z A u h L B w Z h can be expressed in non-dimensional form i = Hydraulic Gradient L = Length of Flow between Points A & B B Slide 3 of 14
VELOCITY (v) VS. HYDRAULIC GRADIENT (i) General relationship shown in Figure 5.2 Three Zones: 1. Laminar Flow (I) 2. Transition Flow (II) 3. Turbulent Flow (III) For most soils, flow is laminar. Therefore: v i Figure 5.2. Das FGE (2005). Slide 4 of 14
DARCY S LAW (1856) v = Discharge Velocity (i.e. quantity of water in unit time through unit cross-sectional area at right angles to the direction of flow) k = Hydraulic Conductivity (i.e. coefficient of permeability) i = Hydraulic Gradient * Based on observations of flow of water through clean sands Slide 5 of 14
SOIL PERMEABILITY AND DRAINAGE 14.333 GEOTECHNICAL LABORATORY after Casagrande and Fadum (1940) and Terzagi et al. (1996). Slide 6 of 14
SOIL PERMEABILITY AND DRAINAGE COEFFICIENT OF PERMEABILITY CM/S (LOG SCALE) 10 2 10 1 1.0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 Drainage property Good drainage Poor drainage Practically impervious Application in earth dams and dikes Pervious sections of dams and dikes Impervious sections of earth dams and dikes Type of soil Clean gravel Clean sands, Clean sand and gravel mixtures Very fine sands, organic and inorganic silts, mixtures of sand, silt, and clay glacial till, stratified clay deposits, etc. Impervious soils e.g., homogeneous clays below zone of weathering Impervious soils which are modified by the effect of vegetation and weathering; fissured, weathered clays; fractured OC clays Direct determination of coefficient of permeability Direct testing of soil in its original position (e.g., well points). If properly conducted, reliable; considerable experience required. Constant Head Permeameter; little experience required. (Note: Considerable experience also required in this range.) Constant head test in triaxial cell; reliable w ith experience and no leaks. Reliable; Little experience required Falling Head Per meameter; Range of unstable permeability;* much experience necessary to correct interpretation Fairly reliable; considerable experience necessary (do in triaxial cell) Indirect determination of coefficient of permeability Computat ion: From the grain size distribution (e.g., Hazen s formula). Only applicable to clean, cohesionless sands and gravels Horizontal Capillarity Test: Very little experience necessary; especially useful for rapid testing of a large number of samples in the field w ithout laboratory facilities. Computations: from consolidation tests; expensive laboratory equipment and considerable experience required. 10 2 10 1 1.0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 *Due to migration of fines, channels, and air in voids. From FHWA IF-02-034 Evaluation of Soil and Rock Properties. Slide 7 of 14
FACTORS AFFECTING PERMEABILITY is not a fundamental soil property but depends upon a number of factors: Particle size distribution Particle shape and texture Mineralogical composite Invariable for a given soil Void ratio Degree of saturation Soil fabric Dependent upon placing and treatment of the soil Nature of fluid Type of Flow Temperature Relate to the permeability Temp. Correction: Slide 8 of 14
HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Constant Head (ASTM D2434) Falling Head (no ASTM) Figure 5.4. Das FGE (2005). Figure 5.5. Das FGE (2005). Slide 9 of 14
LABORATORY TESTING: CONSTANT HEAD Constant Head (ASTM D2434) Q Avt A( ki) t Figure 5.4. Das FGE (2005). Q = Quantity of water collected over time t t = Duration of water collection Q k A k QL Aht h t L Slide 10 of 14
LABORATORY TESTING: CONSTANT HEAD Slide 11 of 14
14.333 GEOTECHNICAL LABORATORY LABORATORY TESTING: FALLING HEAD Falling Head (No ASTM) h dh q k A a L dt A = Cross-sectional area of Soil a = Cross-sectional area of Standpipe Integrate from limits 0 to t t al Ak dt Figure 5.5. Das FGE (2005). 2 after rearranging above equation h log e 1 h al Ak k dh h after integration or 2.303 al At Integrate from limits h 1 to h 2 Log 10 h h 1 2 Slide 12 of 14
LABORATORY TESTING: FALLING HEAD Slide 13 of 14
HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS Uniform Sands - Hazen Formula (Hazen, 1930): k( cm / sec) cd 2 10 c = Constant between 1 to 1.5 D 10 = Effective Size (in mm) Sands Kozeny-Carman (Loudon 1952 and Perloff and Baron 1976): k C 1 e 1 C = Constant (to be determined) e = Void Ratio 3 e Sands Casagrande (Unpublished): k 2 1.4e k0.85 e = Void Ratio k 0.85 = Hydraulic Conductivity @ e = 0.85 Normally Consolidated Clays (Samarasinghe, Huang, and Drnevich, 1982): k C 2 n e 1 e C 2 = Constant to be determined experimentally n = Constant to be determined experimentally e = Void Ratio Slide 14 of 14