Chapter 2 Permeability of Soil 2 1 Introduction

第二章 土的渗透性 Chapter 2 Permeability of Soil 第一节 简介 2. 1 Introduction In this lecture, we’ll learn • Darcy’s Law (达西定律) • Determination of Coefficient of Permeability (渗透系数的测定) • Seepage and Flow Net (渗流和流网) • Effective stress (有效应力)

Arch dam (Courtesy of H. Chanson) Aix-en-Provence, France (1854) Drac river, France (1962) Height = 155 m

第二节 总水头 2. 2 Total head (i) Hydrostatic Groundwater Condition 地下水 静水条件 Pore-water pressure (孔隙水压力 ) at a depth z z below the water table u where w (k. N/m 3) is unit weight of the water (水的重度) z (m) is depth (深度) of the water below water table

(ii) Total head 总水头 地下水 u. A/ w where h is total head, u/ w is pressure head and z is elevation head Two piezometers (测 压 计 ) are installed at A and B. At hydrostatic groundwater condition, levels in both piezometers will be the same, i. e. h. A = h. B. u. B/ w A z. A B z. B Datum

Water will flow from A to B if there is a difference between the water levels in the piezometers (in this case h. A > h. B). u. A/ w A z. A Datum B u. B/ w z. B

第三节 达西定律 2. 2 Darcy’s Law (i) Darcy’s Law 达西定律 h where q = flow rate (m 3/s) 渗流量 h = total head difference (m) 水头差 L = length of flow (m) 渗径长度 i = hydraulic gradient (-) 水力梯度 k = coefficient of permeability (m/s) 渗透系数 A = cross-sectional area of the specimen (m 2) 横截面积 L

(ii) Validity of Darcy's law 达西定律的有效性 P 63 Darcy’s law is valid for laminar flow (层 流 ) condition where Reynolds number is smaller than or equal to 1. Reynolds number (雷诺数) is defined as follows: where is density of water (水的密度), v is velocity of water (流速), is viscosity of water (水的粘滞系数) and d is average diameter of soil grains (土粒子平均粒径).

第四节 渗透系数的测定 Determination of Coefficient of Permeability (i) Constant head test 常水头测试 where Q is volume of water collected, L is length of the specimen, A is cross-sectional area of the specimen, h is total head difference and t is duration of the test.

(ii) Falling head test 变水头测试 2 -19 -P 68 where L is length of the specimen, A 1 is cross-sectional area of the specimen, A 2 is crosssectional area of the standpipe, h 1 is total head at t = 0 and h 2 is total head at t = T

第五节 二维渗流和流网 2 -D Seepage and Flow Net (i) Laplace Equation 拉普拉斯方程 § Considering a two-dimensional element of soil of sizes dx and dz in the x and z directions, respectively. § It is assumed that the soil is homogeneous and isotropic with respect to permeability.

§ The governing differential equation for groundwater flow is obtained by equating the flow rates into and out of the element. ---- § For most practical geotechnical problems, the Laplace’s equation for 2 -D seepage is solved graphically by drawing flow nets.

(ii-a) Flow Net 流网 § A flow net consists of two sets of curves – equipotential lines (等势线) and flow lines (流线) – that intersect each other at 90°. § Along an equipotential, the total head is constant.

§ A pair of adjacent flow lines define a flow channel through which the rate of flow of pore fluid is constant. § The loss of head between two successive equipotentials is called the equipotential drop.

(ii-b) Properties of Flow Net – sketching rules 流网特性 § A flow lines cross the equipotentials at right angles. § A flow line cannot cross other flow lines. § An equipotential line cannot cross other equipotential lines. § The flow net must be constructed so that each element is a curvilinear square such that a circle may be inscribed within it that touches all four of its sides as shown in the figure on the right.

§ Impermeable boundaries and lines of symmetry are flow lines, e. g. lines EF and FG in the figure on the top right are flow lines. § Bodies of water, such as reservoirs behind a dam, are equipotentials, e. g. line AB in the figure on the bottom right is an equipotential line.

(ii-c) Typical Flow Nets 流网类型

(ii-d) Flow Net - Flow Rate Calculations 流网计算 § Consider water flows through the flow element shown in the figure on the right. § The flow rate through this element is given by: b=l

§ If NF is number of flow channels, Nd is number of equipotential drops and the total head difference is H, the total flow rate is

第六节 有效应力 Effective Stress (i) Effective Stress Principle 有效应力原理 § The total stress ( ) carried by a saturated soil is the sum of effective stress ( ’) carried by the soil particles and the pressure carried by the pore water (u). § Deformation of soil is a function of the change in effective stress and not total stress.

§ Effective stress is not the contact stress between two soil particles but is the average stress on a plane through the soil mass as shown in the following figure. P Ns X X P

(ii) Effective Stress under Hydrostatic Condition 静水条件下的有效应力 § Consider a soil element at a depth z below ground surface with water table at the ground surface as shown in the figure on the right: 地下水 z

(iii-a) Effective Stress under Downward Seepage Condition 向下渗流时的有效应力 § h z Water flow Consider a soil element at a depth z below ground surface with water table at the ground surface and water flows downwards as shown in the figure on the right: 地下水

(iii-b) Effective Stress under Upward Seepage Condition 向上渗流时的有效应力 § 地下水 h z Water flow Consider a soil element at a depth z below ground surface with water table at the ground surface and water flows upwards as shown in the figure on the right:

第七节 渗透力和临界梯度 Seepage force and critical hydraulic gradient 7. 1 seepage force 渗透力 As water flows through soil it exerts a drag on the soil paricles resulting in head losses. if the head losses over a flow distance L, is h, the seepage force is In seepage condition the effective stress contain two parts seepage downwards seepage upwards

7. 2 Types of seepage failure 渗透破坏形式 When flow is upward, with the increase of gradient i，the vertical effective stress become zero: The soil loses its strength and behaves like a viscous fluid. when the upward seepage forces exceeds the downward force of the silt, s "boiling" occurs. when the seepage force push the bottom of an excavation upward, We call this "Heaving". If the upward seepage forces exceed the submerged weight, the particles may be carried upwards to be deposited at the ground sueface and a "pipe" is formed in the soil near the surface.

pipe boiling

7. 3 Critical Hydraulic Gradient 临界水力梯度 § There exist a critical head difference (hc) such that ’ = 0.