SOIL PERMEABILITY AND SEEPAGE

Transcription

1 CHAPTER 4 SOIL PERMEABILITY AND SEEPAGE 4.1 SOIL PERMEABILITY A material is permeable if it contains continuous voids. All materials such as rocks, concrete, soils etc. are permeable. The flow of water through all of them obeys approximately the same laws. Hence, the difference between the flow of water through rock or concrete is one of degree. The permeability of soils has a decisive effect on the stability of foundations, seepage loss through embankments of reservoirs, drainage of subgrades, excavation of open cuts in water bearing sand, rate of flow of water into wells and many others. Hydraulic Gradient When water flows through a saturated soil mass there is certain resistance for the flow because of the presence of solid matter. However, the laws of fluid mechanics which are applicable for the flow of fluids through pipes are also applicable to flow of water through soils. As per Bernoulli's equation, the total head at any point in water under steady flow condition may be expressed as Total head = pressure head + velocity head + elevation head This principle can be understood with regards to the flow of water through a sample of soil of length L and cross-sectional area A as shown in Fig. 4.1 (a). The heads of water at points A and B as the water flows from A to B are given as follows (with respect to a datum) Total head at A, H. = Z A + ^ + -^- Y 2g p V 2 Total head at B, H K =Z K -\ + 87

2 88 Chapter 4 Figure 4.1 (a) Flow of water through a sample of soil As the water flows from A to B, there is an energy loss which is represented by the difference in the total heads H, and H D or PA PR» H A -H B =\Z c o u i _, A where, p A and p B = pressure heads, V A and V B = velocity, g - acceleration due to gravity, y w = unit weight of water, h = loss of head. For all practical purposes the velocity head is a small quantity and may be neglected. The loss of head of h units is effected as the water flows from A to B. The loss of head per unit length of flow may be expressed as h i = (4.1) where / is called the hydraulic gradient. Laminar and Turbulent Flow Problems relating to the flow of fluids in general may be divided into two main classes: 1. Those in which the flow is laminar. 2. Those in which the flow is turbulent. There is a certain velocity, v c, below which for a given diameter of a straight tube and for a given fluid at a particular temperature, the flow will always remain laminar. Likewise there is a higher velocity, v r above which the flow will always be turbulent. The lower bound velocity, v p of turbulent flow is about 6.5 times the upper bound velocity v of laminar flow as shown in Fig. 4.1(b). The upper bound velocity of laminar flow is called the lower critical velocity. The fundamental laws that determine the state existing for any given case were determined by Reynolds (1883). He found the lower critical velocity is inversely proportional to the diameter of

3 Soil Permeability and Seepage 89 Flow always Flow always laminarlaminar turbulent Flow always turbulent log/ logv V T Figure 4.Kb) Relationship between velocity of flow and hydraulic gradient for flow of liquids in a pipe the pipe and gave the following general expression applicable for any fluid and for any system of units. = 2000 where, A^ = Reynolds Number taken as 2000 as the maximum value for the flow to remain always laminar, D = diameter of pipe, v c = critical velocity below which the flow always remains laminar, y 0 = unit weight of fluid at 4 C, fjl = viscosity of fluid, g = acceleration due to gravity. The principal difference between laminar flow and turbulent flow is that in the former case the velocity is proportional to the first power of the hydraulic gradient, /, whereas in the latter case it is 4/7 the power of /. According to Hagen-Poiseuille's' Law the flow through a capillary tube may be expressed as R 2 ai (4.2a) or (4.2b) where, R = radius of a capillary tube of sectional area a, q = discharge through the tube, v = average velocity through the tube, ^ = coefficient of viscosity. 4.2 DARCY'S LAW Darcy in 1856 derived an empirical formula for the behavior of flow through saturated soils. He found that the quantity of water q per sec flowing through a cross-sectional area of soil under hydraulic gradient / can be expressed by the formula q = kia or the velocity of flow can be written as (4.3)

4 90 Chapter \ \ \ \ \ \ ^^k^ n Temperature C 30 Figure 4.2 Relation between temperature and viscosity of water v = -j = & (4.4) where k is termed the hydraulic conductivity (or coefficient of permeability)with units of velocity. A in Eq. (4.4) is the cross-sectional area of soil normal to the direction of flow which includes the area of the solids and the voids, whereas the area a in Eq. (4.2) is the area of a capillary tube. The essential point in Eq. (4.3) is that the flow through the soils is also proportional to the first power of the hydraulic gradient i as propounded by Poiseuille's Law. From this, we are justified in concluding that the flow of water through the pores of a soil is laminar. It is found that, on the basis of extensive investigations made since Darcy introduced his law in 1856, this law is valid strictly for fine grained types of soils. The hydraulic conductivity is a measure of the ease with which water flows through permeable materials. It is inversely proportional to the viscosity of water which decreases with increasing temperature as shown in Fig Therefore, permeability measurements at laboratory temperatures should be corrected with the aid of Fig. 4.2 before application to field temperature conditions by means of the equation k ~ (4.5) where k f and k T are the hydraulic conductivity values corresponding to the field and test temperatures respectively and /^,and ^rare the corresponding viscosities. It is customary to report the values of k T at a standard temperature of 20 C. The equation is ^20 (4.6) 4.3 DISCHARGE AND SEEPAGE VELOCITIES Figure 4.3 shows a soil sample of length L and cross-sectional area A. The sample is placed in a cylindrical horizontal tube between screens. The tube is connected to two reservoirs R^ and R 2 in which the water levels are maintained constant. The difference in head between R { and R 2 is h. This difference in head is responsible for the flow of water. Since Darcy's law assumes no change in the

5 Soil Permeability and Seepage 91 Sample B \ Screen Screen Figure 4.3 Flow of water through a sample of soil volume of voids and the soil is saturated, the quantity of flow past sections AA, BB and CC should remain the same for steady flow conditions. We may express the equation of continuity as follows Qaa = <lbb = 3cc If the soil be represented as divided into solid matter and void space, then the area available for the passage of water is only A v. If v s is the velocity of flow in the voids, and v, the average velocity across the section then, we have A v = Av or v = v s A A 1 l + e Since, ~7~ = ~ = A.. n e + e (4.7) Since (1 + e)le is always greater than unity, v s is always greater than v. Here, v s is called the seepage velocity and v the discharge velocity. 4.4 METHODS OF DETERMINATION OF HYDRAULIC CONDUCTIVITY OF SOILS Methods that are in common use for determining the coefficient of permeability k can be classified under laboratory and field methods. Laboratory methods: 1. Constant head permeability method 2. Falling head permeability method Field methods: 1. Pumping tests 2. Bore hole tests Indirect Method: Empirical correlations The various types of apparatus which are used in soil laboratories for determining the permeability of soils are called permeameters. The apparatus used for the constant head permeability test is called a constant head permeameter and the one used for the falling head test is a falling headpermeameter. The soil samples used in laboratory methods are either undisturbed or disturbed. Since it is not

6 92 Chapter 4 possible to obtain undisturbed samples of cohesionless soils, laboratory tests on cohesionless materials are always conducted on samples which are reconstructed to the same density as they exist in nature. The results of tests on such reconstructed soils are often misleading since it is impracticable to obtain representative samples and place them in the test apparatus to give exactly the same density and structural arrangement of particles. Direct testing of soils in place is generally preferred in cases where it is not possible to procure undisturbed samples. Since this method is quite costly, it is generally carried out in connection with major projects such as foundation investigation for dams and large bridges or building foundation jobs where lowering of the water table is involved. In place of pumping tests, bore hole tests as proposed by the U.S. Bureau of Reclamation are quite inexpensive as these tests eliminate the use of observation wells. Empirical correlations have been developed relating grain size and void ratio to hydraulic conductivity and will be discussed later on. 4.5 CONSTANT HEAD PERMEABILITY TEST Figure 4.4(a) shows a constant head permeameter which consists of a vertical tube of lucite (or any other material) containing a soil sample which is reconstructed or undisturbed as the case may be. The diameter and height of the tube can be of any convenient dimensions. The head and tail water levels are kept constant by overflows. The sample of length L and cross-sectional area A is subjected to a head h which is constant during the progress of a test. A test is performed by allowing water to flow through the sample and measuring the quantity of discharge Q in time t. The value of k can be computed directly from Darcy's law expressed as follows Supply,c-.Soil sample Filter skin T h 1 Screen Graduated jar ~\ (a) Figure 4.4 (b) Constant head permeability test

7 Soil Permeability and Seepage 93 Q = k At (4.8) 01 = The constant head permeameter test is more suited for coarse grained soils such as gravelly sand and coarse and medium sand. Permeability tests in the laboratory are generally subjected to various types of experimental errors. One of the most important of these arises from the formation of a filter skin of fine materials on the surface of the sample. The constant head permeameter of the type shown in Fig. 4.4(b) can eliminate the effect of the surface skin. In this apparatus the loss of head is measured through a distance in the interior of the sample, and the drop in head across the filter skin has no effect on the results. < 4-9 > 4.6 FALLING HEAD PERMEABILITY TEST A falling head permeameter is shown in Fig. 4.5(a). The soil sample is kept in a vertical cylinder of cross-sectional area A. A transparent stand pipe of cross sectional area, a, is attached to the test cylinder. The test cylinder is kept in a container filled with water, the level of which is kept constant by overflows. Before the commencement of the test the soil sample is saturated by allowing the water to flow continuously through the sample from the stand pipe. After saturation is complete, the stand pipe is filled with water up to a height of h Q and a stop watch is started. Let the initial time be t Q. The time t l when the water level drops from h Q to h } is noted. The hydraulic conductivity k can be determined on the basis of the drop in head (h Q - hj and the elapsed time (t l -? 0 ) required for the drop as explained below. Let h be the head of water at any time t. Let the head drop by an amount dh in time dt. The quantity of water flowing through the sample in time dt from Darcy's law is h dq = kiadt = k Adt v(4.10) L ' where, i = h/l the hydraulic gradient. The quantity of discharge dq can be expressed as dq = -adh (4.11) Since the head decreases as time increases, dh is a negative quantity in Eq. (4.11). Eq. (4.10) can be equated to Eq. (4.11) h -adh = k Adt (4.12) The discharge Q in time (t^ - f Q ) can be obtained by integrating Eq. (4.10) or (4.11). Therefore, Eq. (4.12) can be rearranged and integrated as follows *i Cdh ka C hn ka -a \ = \ dt or '- The general expression for k is

8 94 Chapter 4 dh T Stand pipe E! 1 L I #Vc?** /> Scree "X'^;< V J? k'v's&s J Sample (a) (b) Figure 4.5 Falling head permeability test al k = A(t, or k = 23aL (4.13) The setup shown in Fig. 4.5(a) is generally used for comparatively fine materials such as fine sand and silt where the time required for the drop in head from h Q to h l is neither unduly too long nor too short for accurate recordings. If the time is too long evaporation of water from the surface of the water might take place and also temperature variations might affect the volume of the sample. These would introduce serious errors in the results. The set up is suitable for soils having permeabilities ranging from 10~ 3 to 10~ 6 cm per sec. Sometimes, falling head permeameters are used for coarse grained soils also. For such soils, the cross sectional area of the stand pipe is made the same as the test cylinder so that the drop in head can conveniently be measured. Fig. 4.5(b) shows the test set up for coarse grained soils. When a = A, Eq. (4.13) is reduced to 2.3L, h Q log, n (4.14) Example 4.1 A constant head permeability test was carried out on a cylindrical sample of sand 4 in. in diameter and 6 in. in height. 10 in 3 of water was collected in 1.75 min, under a head of 12 in. Compute the hydraulic conductivity in ft/year and the velocity of flow in ft/sec. Solution The formula for determining k is Ait

9 Soil Permeability and Seepage Q = 10 in 3, A = 3.14 x = in O i = = = 2, t = 105 sec. L 6 Therefore fc = = 3.79 x 10~ 3 in./sec = x 10~ 5 ft/sec = 9960 ft/year 12.56x2x105 Velocity of flow = Id = x 10~ 5 x2 = x 10~ 4 ft/sec Example 4.2 A sand sample of 35 cm 2 cross sectional area and 20 cm long was tested in a constant head permeameter. Under a head of 60 cm, the discharge was 120 ml in 6 min. The dry weight of sand used for the test was g, and G s = Determine (a) the hydraulic conductivity in cm/sec, (b) the discharge velocity, and (c) the seepage velocity. Solution Use Eq. (4.9), k = hat where Q = 120 ml, t = 6 min, A = 35 cm 2, L = 20 cm, and h = 60 cm. Substituting, we have k = - = x 10~ 3 cm/sec 60x35x6x60 Discharge velocity, v = ki = x 10~ 3 x = 9.52 x 10~ 3 cm/sec Seepage velocity v s W 1120 Y G G From Eq. (3.18a), Y f t ~ w s or e ~ ~~^ since y = 1 g/cm 3 l + e y d Substituting, e = -- 1 = = l + e v 952xlO~ 3 Now, v = = ' = 2.36 x 10" 2 J cm/sec n Example 4.3 Calculate the value of A: of a sample of 2.36 in. height and 7.75 in 2 cross-sectional area, if a quantity of water of in 3 flows down in 10 min under an effective constant head of in. On oven

10 96 Chapter 4 drying, the test specimen weighed 1.1 Ib. Assuming G s = 2.65, calculate the seepage velocity of water during the test. Solution From Eq. (4.9), k = ^- = x236 - = 0.8x 10~ 3 hat 15.75x7.75x10x60 in./sec Discharge velocity, v = ki = k = 0.8xlO~ 3 x : = 5.34xlO~ 3 in./sec L 2.36 W 1 1 Y = s- = -:- = lb/in 3 = lb/ft 3 d V 7.75x2.36 Y G FromEq. (3.18a), e = ^-^ \ Yd 62.4x2.65 or e = =0.372 l + e v 5.34 xlo~ 3 Seepage velocity, v = = - = x 10~ 3 in./sec 6 s n Example 4.4 The hydraulic conductivity of a soil sample was determined in a soil mechanics laboratory by making use of a falling head permeameter. The data used and the test results obtained were as follows: diameter of sample = 2.36 in, height of sample = 5.91 in, diameter of stand pipe = 0.79 in, initial head h Q = in. final head h l = in. Time elapsed = 1 min 45 sec. Determine the hydraulic conductivity in ft/day. Solution The formula for determining k is [Eq. (4.13)],.,,. k = - log, 0 - where t is the elapsed time. A 3.14x0.79x A 4, 2 Area of stand pipe, a = - = 34 x 10 4 ft^ 4x12x12 Area of sample, A = 3-14x2-36x236 = 304 x 10~ 4 ft 2 4x12x12 Height of sample, L = ( ~ 1L81 ) = ft Head, /z0 n = -^ = ft, h, ] = = ft 12 12

11 Soil Permeability and Seepage 97 Elapsed time, t = 105 sec = = x 10~ 4 days 60x60x24 2.3x34x10-4x0.4925, t f t f l j k = x log = 18 ft/day 304 x 10-4 x x ID" DIRECT DETERMINATION OF k OF SOILS IN PLACE BY PUMPING TEST The most reliable information concerning the permeability of a deposit of coarse grained material below the water table can usually be obtained by conducting pumping tests in the field. Although such tests have their most extensive application in connection with dam foundations, they may also prove advisable on large bridge or building foundation jobs where the water table must be lowered. The arrangement consists of a test well and a series of observation wells. The test well is sunk through the permeable stratum up to the impermeable layer. A well sunk into a water bearing stratum, termed an aquifer, and tapping free flowing ground water having a free ground water table under atmospheric pressure, is termed a gravity or unconfined well. A well sunk into an aquifer where the ground water flow is confined between two impermeable soil layers, and is under pressure greater than atmospheric, is termed as artesian or confined well. Observation wells are drilled at various distances from the test or pumping well along two straight lines, one oriented approximately in the direction of ground water flow and the other at right angles to it. A minimum of two observation wells and their distances from the test well are needed. These wells are to be provided on one side of the test well in the direction of the ground water flow. The test consists of pumping out water continuously at a uniform rate from the test well until the water levels in the test and observation wells remain stationary. When this condition is achieved the water pumped out of the well is equal to the inflow into the well from the surrounding strata. The water levels in the observation wells and the rate of water pumped out of the well would provide the necessary additional data for the determination of k. As the water from the test well is pumped out, a steady state will be attained when the water pumped out will be equal to the inflow into the well. At this stage the depth of water in the well will remain constant. The drawdown resulting due to pumping is called the cone of depression. The maximum drawdown D Q is in the test well. It decreases with the increase in the distance from the test well. The depression dies out gradually and forms theoretically, a circle around the test well called the circle of influence. The radius of this circle, /?., is called the radius of influence of the depression cone. Equation for k for an Unconfined Aquifer Figure 4.6 gives the arrangement of test and observation wells for an unconfined aquifer. Only two observation wells at radial distances of r { and r 2 from the test well are shown. When the inflow of water into the test well is steady, the depths of water in these observation wells are h { and h 2 respectively. Let h be the depth of water at radial distance r. The area of the vertical cylindrical surface of radius r and depth h through which water flows is A = Inrh The hydraulic gradient is i = dr

12 98 Chapter 4 Ground level Test well Observation wells Figure 4.6 Pumping test in an unconfined aquifer As per Darcy's law the rate of inflow into the well when the water levels in the wells remain stationary is q = kia Substituting for A and / the rate of inflow across the cylindrical surface is, dh^, q - k 2nrh dr Rearranging the terms, we have dr r Inkhdh q The integral of the equation within the boundary limits is dr Ink r q. hdh (4.15) The equation for k after integration and rearranging is k = - (4.16) Proceeding in the same way as before another equation for k in terms of r Q, h Q and R { can be established as (referring to Fig. 4.6)

13 Soil Permeability and Seepage <? /?,- * log- 1 - (4.17) If we write h Q = (H- D 0 ) in Eq. (4.17), where D Q is the depth of maximum drawdown in the test well, we have 2.3 q -log^- \ Y 0 Now from Eq. (4.18), the maximum yield from the well may be written as (4.18) _7rD 0 k(2h-d Q ) q ~ 23 I (4.19) Radius of Influence R^ Based on experience, Sichardt (1930) gave an equation for estimating the radius of influence for the stabilized flow condition as /?. = 3000D 0 V& meters (4.20) where D Q = maximum drawdown in meters k = hydraulic conductivity in m/sec Equation for k in a Confined Aquifer Figure 4.7 shows a confined aquifer with the test and observation wells. The water in the observation wells rises above the top of the aquifer due to artesian pressure. When pumping from such an artesian well two cases might arise. They are: Case 1. The water level in the test well might remain above the roof level of the aquifer at steady flow condition. Observation wells \ Piezometnc level during pumping Case 1 h 0 > H 0 Case 2h 0 <H 0 Impermeable Impermeable stratum Figure 4.7 Pumping test in confined aquifer

14 100 Chapter 4 Case 2. The water level in the test well might fall below the roof level of the aquifer at steady flow condition. If H Q is the thickness of the confined aquifer and h Q is the depth of water in the test well at the steady flow condition Case 1 and Case 2 may be stated as Casel. When/z 0 >// 0. Case 2. When/i Q <// 0. Case 1. When h 0 > H 0 In this case, the area of a vertical cylindrical surface of any radius r does not change, since the depth of the water bearing strata is limited to the thickness H Q. Therefore, the discharge surface area is A. dh. ~. _,, Again writing i -, the now equation as per Darcy s law is dr dh_ dr The integration of the equation after rearranging the terms yields (4.21) dr_ a r or (A 2 -/i 1 ) = Tr7^1og, (4-22) The equation for k is,., k = - log 2 -A,) r, Alternate Equations As before we can write the following equation for determining k 23q r, k = i u ti. TT log ~~ (4.24a) - ^ ',. or k- --- log 1 - OF,. t 27rH 2xH Q L> D 0 g ~r~ r 0 Case 2. When h 0 < H 0 Under the condition when h Q is less than H Q, the flow pattern close to the well is similar to that of an unconfmed aquifer whereas at distances farther from the well the flow is artesian. Muskat (1946) developed an equation to determine the hydraulic conductivity. The equation is *,- log L

15 Soil Permeability and Seepage BOREHOLE PERMEABILITY TESTS Two types of tests may be carried out in auger holes for determining k. They are (a) Falling water level method (b) Rising water level method Falling Water Level Method (cased hole and soil flush with bottom) In this test auger holes are made in the field that extend below the water table level. Casing is provided down to the bottom of the hole (Fig. 4.8(a)). The casing is filled with water which is then allowed to seep into the soil. The rate of drop of the water level in the casing is observed by measuring the depth of the water surface below the top of the casing at 1, 2 and 5 minutes after the start of the test and at 5 minutes intervals thereafter. These observations are made until the rate of drop becomes negligible or until sufficient readings have been obtained. The coefficient of permeability is computed as [Fig. 4.8(a)] k = 2-3 nr Q H { log - -f,) ff, where, H { = piezometric head ait = t l,h 2 = piezometric head at t - t 2 - (4.26) Rising Water Level Method (cased hole and soil flush with bottom) This method, most commonly referred to as the time-lag method, consists of bailing the water out of the casing and observing the rate of rise of the water level in the casing at intervals until the rise in water level becomes negligible. The rate is observed by measuring the elapsed time and the depth of the water surface below the top of the casing. The intervals at which the readings are required will vary somewhat with the permeability of the soil. Eq. (4.26) is applicable for this case, [Fig. 4.8(b)]. A rising water level test should always be followed by sounding the bottom of the holes to determine whether the test created a quick condition. HI at t = H at t = t (a) Falling water head method (b) Rising water head method Figure 4.8 Falling and rising water method of determining k

16 102 Chapter APPROXIMATE VALUES OF THE HYDRAULIC CONDUCTIVITY OF SOILS The coefficients of permeability of soils vary according to their type, textural composition, structure, void ratio and other factors. Therefore, no single value can be assigned to a soil purely on the basis of soil type. The possible coefficients of permeability of some soils are given in Table HYDRAULIC CONDUCTIVITY IN STRATIFIED LAYERS OF SOILS Hydraulic conductivity of a disturbed sample may be different from that of the undisturbed sample even though the void ratio is the same. This may be due to a change in the structure or due to the stratification of the undisturbed soil or a combination of both of these factors. In nature we may find fine grained soils having either flocculated or dispersed structures. Two fine-grained soils at the same void ratio, one dispersed and the other flocculated, will exhibit different permeabilities. Soils may be stratified by the deposition of different materials in layers which possess different permeability characteristics. In such stratified soils engineers desire to have the average permeability either in the horizontal or vertical directions. The average permeability can be computed if the permeabilities of each layer are determined in the laboratory. The procedure is as follows: k {, k 2,..., k n = hydraulic conductivities of individual strata of soil either in the vertical or horizontal direction. z r Z 2 z n = thickness of the corresponding strata. k h = average hydraulic conductivity parallel to the bedding planes (usually horizontal). k v - average hydraulic conductivity perpendicular to the bedding planes (usually vertical). Flow in the Horizontal Direction (Fig. 4.9) When the flow is in the horizontal direction the hydraulic gradient / remains the same for all the layers. Let Vj, v 2,..., v n be the discharge velocities in the corresponding strata. Then Therefore, Q = kiz = (v^j + v 2 z v n z n ) = (k [ iz l +k 2 iz 2 + +k n iz n ) '"+k n z n ) (4.27) k (cm/sec) 10 1 to ' to IO- 4 io- 5 io- 6 IO- 7 to IO- 9 Table 4.1 Hydraulic conductivity of some soils (after Casagrande and Fadum, 1939) Soils type Clean gravels Clean sand Clean sand and gravel mixtures Very fine sand Silt Clay soils Drainage conditions Good Good Good Poor Poor Practically impervious

17 Soil Permeability and Seepage 103 //?$><z><$xv<x><??t^^ Zi /i t V "~ V,, I, *, Z2 *2 V2, 1, V3, Z 4 k T v 4, i, Figure 4.9 Flow through stratified layers of soil Flow in the Vertical Direction When flow is in the vertical direction, the hydraulic gradients for each of the layers are different. Let these be denoted by i r z' 2,..., i n. Let h be the total loss of head as the water flows from the top layer to the bottom through a distance ofz. The average hydraulic gradient is h/z. The principle of continuity of flow requires that the downward velocity be the same in each layer. Therefore, h v = k v - = kj l =k 2 i 2 =--- = k n i n If /Zj, hj,..., h n, are the head losses in each of the layers, we have or = z ll +z z nn Solving the above equations we have k = Z (4.28) It should be noted that in all stratified layers of soils the horizontal permeability is generally greater than the vertical permeability. Varved clay soils exhibit the characteristics of a layered system. However, loess deposits possess vertical permeability greater than the horizontal permeability EMPIRICAL CORRELATIONS FOR HYDRAULIC CONDUCTIVITY Granular Soils Some of the factors that affect the permeability are interrelated such as grain size, void ratio, etc. The smaller the grain size, the smaller the voids which leads to the reduced size of flow channels and lower permeability. The average velocity of flow in a pore channel from Eq. (4.2b) is 8// 32// where d is the average diameter of a pore channel equal to 2R. (4.29)

18 104 Chapter 4 Eq. (4.29) expresses for a given hydraulic gradient /, the velocity of water in a circular pore channel is proportional to the square of the diameter of the pore channel. The average diameter of the voids in a soil at a given porosity increases practically in proportion to the grain size, D Extensive investigations of filter sands by Hazen (1892) led to the equation k(m/s) = CD 2 (4.30) where D e is a characteristic effective grain size which was determined to be equal to D 10 (10% size). Fig gives a relationship between k and effective grain size D 10 of granular soil which validates Eq. (4.30). The permeability data approximates a straight line with a slope equal to 2 consistent with Eq. (4.30). These data indicate an average value of C - 10~ 2 where k is expressed in m/s and D 10 in mm. According to the data in Fig. 4.10, Eq. (4.30) may underestimate or overestimate the permeability of granular soils by a factor of about 2. Further investigations on filter sands were carried out by Kenney et al., (1984). They found the effective grain size D 5 would be a better choice compared to D }Q. Fig gives relationships between D 5 and k. The sand they used in the investigation had a uniformity coefficient ranging from 1.04 to 12. Hydraulic Conductivity as a Function of Void Ratio for Granular Soils Further analysis of hydraulic conductivity in granular soils based on Hagen-Poiseuille's Eq. (4.2b) leads to interesting relationships between k and void ratio e. Three types of relationships may be expressed as follows. It can be shown that the hydraulic conductivity k can be expressed as k = kf(e) (4.31) 10- Silt Silty Sand Sand Fine Medium Coarse Gravel 10 ~ o 10" -Q o CJ o 10" 10" Hazen equation Jt= 1/100 )? 0 m/sec 10" D 10 (mm) 10 Figure 4.10 Hazen equation and data relating hydraulic conductivity and D 10 of granular soils (after Louden, 1952)

19 Soil Permeability and Seepage " Fine Sand Medium] Coarse Sand C,,= 1-3 T3 8 io o X 10-10' IO 1 D 5 (mm) Figure 4.11 Influence of gradation on permeability on granular soils (after Kenney et al., 1984) where k = a soil constant depending on temperature and void ratio e. F(e) may be expressed as F(e) = o 2e l + e (4.32) When e = 1, F(e) ~ 1. Therefore k represents the hydraulic conductivity corresponding to void ratio e - 1. Since k is assumed as a constant, k is a function of e only. By substituting in F(e), the limiting values, ;c = 0, x = 0.25, and x = 0.5, we get For Jc = 0, (4.33) x = 0.25, (4.34) x = 0.50 (4.35) F,(e) represents the geometric mean of F.(e) and F.( The arithmetic mean of the functions F^e) and F 3 (e) is = e 2 (4.36)

20 106 Chapter u o o o Void ratio function Figure 4.12 Relationship between void ratio and permeability for coarse grained soils Best Value for x for Coarse Grained Soils From laboratory tests determine k for various void ratios e of the sample. Then plot curves k versus 2e 2(1+x) /(l + e) for values of x = 0, 0.25, 0.5 and k versus e 2. The plot that fits well gives the best value of x. It has been found from experimental results that the function 2e 3 l + e (4.37) gives better agreement than the other functions. However, the function F 4 (e) = e 2 is sometimes preferred because of its simplicity and its fair degree of agreement with the experimental data. Fig present experimental data in the form of k versus functions of e. Figure 4.13 In situ permeability of soft clays in relation to initial void ratio, e o ; clay fraction; CF; and activity A (After Mesri et al., 1994)

21 Soil Permeability and Seepage Clay O Batiscon A Berthierville D St. Hilaire V Vosby Boston blue " 10,-8 Figure 4.14 Results of falling-head and constant-head permeability tests on undisturbed samples of soft clays (Terzaghi, Peck and Mesri, 1996) Fine Grained Soils Laboratory experiments have shown that hydraulic conductivity of very fine grained soils are not strictly a function of void ratio since there is a rapid decrease in the value of k for clays below the plastic limit. This is mostly due to the much higher viscosity of water in the normal channels which results from the fact that a considerable portion of water is exposed to large molecular attractions by the closely adjacent solid matter. It also depends upon the fabric of clays especially those of marine origin which are often flocculated. Fig shows that the hydraulic conductivity in the vertical direction, at in situ void ratio e Q, is correlated with clay fraction (CF) finer than mm and with the activity A (= I p /CF). Consolidation of soft clays may involve a significant decrease in void ratio and therefore of permeability. The relationships between e and k (log-scale) for a number of soft clays are shown in Fig (Terzaghi, Peck, and Mesri 1996). Example 4.5 A pumping test was carried out for determining the hydraulic conductivity of soil in place. A well of diameter 40 cm was drilled down to an impermeable stratum. The depth of water above the bearing stratum was 8 m. The yield from the well was 4 mvmin at a steady drawdown of 4.5 m. Determine the hydraulic conductivity of the soil in m/day if the observed radius of influence was 150m. Solution The formula for determining k is [Eq. (4.18)] 2.3 q k = xd 0 (2H-D 0 ) r 0 q = 4 m 3 /min = 4 x 60 x 24 m 3 /day D 0 = 4.5 m, H = 8 m, R. = 150 m, r Q = 0.2 m

22 108 Chapter 4 2.3x4x60x24 k = log 3.14x4.5(2x8-4.5) 0.2 = m/day Example 4.6 A pumping test was made in pervious gravels and sands extending to a depth of 50 ft, where a bed of clay was encountered. The normal ground water level was at the ground surface. Observation wells were located at distances of 10 and 25 ft from the pumping well. At a discharge of 761 ft 3 per minute from the pumping well, a steady state was attained in about 24 hr. The draw-down at a distance of 10 ft was 5.5 ft and at 25 ft was 1.21 ft. Compute the hydraulic conductivity in ft/sec. Solution Use Eq. (4.16) where where = 60 = ft 3 /sec k = = 10 ft, r = 25 ft, h = = ft, h = = 44.5 ft 2.3x12.683, 25 no 1A_ r/ log = 9.2 x 10 J ft/sec. 3.14( ) 10 Example 4.7 A field pumping test was conducted from an aquifer of sandy soil of 4 m thickness confined between two impervious strata. When equilibrium was established, 90 liters of water was pumped Test well Observation wells 1 2 ^ Impermeable stratum } iy.t -^ r, = 3 m ^^ ^-,..1 ". T /i, =2.1 m 1 /// T/i? = 2.7 m Confined aquifer \ 1 Impermeable Figure Ex. 4.7

23 Soil Permeability and Seepage 109 out per hour. The water elevation in an observation well 3.0 m away from the test well was 2.1 m and another 6.0 m away was 2.7 m from the roof level of the impervious stratum of the aquifer. Find the value of k of the soil in m/sec. (Fig. Ex. 4.7) Solution Use Eq. (4.24a) k, = <7, r 2 log q = 90 x 10 3 cm 3 /hr = 25 x KT 6 m 3 /sec 2.3x25xlO~ 6, 6 11/10 in_ 6. k = - log = x 10 6 m/sec 2x3.14x4( ) 3 Example 4.8 Calculate the yield per hour from a well driven into a confined aquifer. The following data are available: height of original piezometric level from the bed of the aquifer, H = ft, thickness of aquifer, H a ft, the depth of water in the well at steady state, h Q = ft, hydraulic conductivity of soil = ft/min, radius of well, r Q = 3.94 in. ( ft), radius of influence, R. = ft. Solution Since h Q is greater than H Q the equation for q (refer to Fig 4.7) is Eq. (4.24b) where k = ft/min = 4.74 ft/hr 2x3.14x4.74x16.41( ) ^1Mr, n ^^n Now q = ---- = ft 3 /hour «752 ft 3 /hour 2.3 log( ) Example 4.9 A sand deposit contains three distinct horizontal layers of equal thickness (Fig. 4.9). The hydraulic conductivity of the upper and lower layers is 10~ 3 cm/sec and that of the middle is 10~ 2 cm/sec. What are the equivalent values of the horizontal and vertical hydraulic conductivities of the three layers, and what is their ratio? Solution Horizontal flow ~(ki+k 2 +k 3 ) since z\ = Z 2 = 3

24 110 Chapter 4 k h = -( ) = -(2x 10~ ) = 4x 10-3 cm/sec Vertical flow Z 3 3 ililil _L _L _L.1 _L 3kik, 3xlO~ 3 xlo~ 2 1/l<a in_ 3. = 1.43 x 10 cm/sec 2k 2 + ki k h _ 4xlO~ 3 k v 1.43x10" = 2.8 Example 4.10 The following details refer to a test to determine the value of A; of a soil sample: sample thickness = 2.5 cm, diameter of soil sample = 7. 5 cm, diameter of stand pipe = 10mm, initial head of water in the stand pipe =100 cm, water level in the stand pipe after 3 h 20 min = 80 cm. Determine the value of k if e = What is the value of k of the same soil at a void ratio e = 0.90? Solution Use Eq. (4. 1 3) where, k = 2 ' ' 3aL log 4 (I) cm A = (7.5) 2 = cm 2 t= sec By substituting the value of k for e { = 0.75,, 2.3x0.785x2.5, 100 rtcv^ in,, k = k,= - x log - = x 10~ 6 cm/sec x For determining k at any other void ratio, use Eq. (4.35) i e l Now, k 2 = - - x x k { e For e 2 = (0.9V = 190 X " X 826 X 10 ' = l3146 X

25 Soil Permeability and Seepage 111 Example 4.11 In a falling head permeameter, the sample used is 20 cm long having a cross-sectional area of 24 cm 2. Calculate the time required for a drop of head from 25 to 12 cm if the crosssectional area of the stand pipe is 2 cm 2. The sample of soil is made of three layers. The thickness of the first layer from the top is 8 cm and has a value of k { = 2 x 10" 4 cm/sec, the second layer of thickness 8 cm has k 2 = 5 x 10~ 4 cm/sec and the bottom layer of thickness 4 cm has & 3 = 7 x 10~ 4 cm/sec. Assume that the flow is taking place perpendicular to the layers (Fig. Ex. 4.11). Solution Use Eq. (4.28) k = = 3.24xlO~ 20 4 cm/sec 2xlO~ 4 5xlO- 4 7xlO- 4 O O <4 ^^^^^ _l_ I Now from Eq. (4.13), or 2.3aL, h log n 2.3aL, h Q 2.3x2x20, 25 log = -log Ak /i, 24x3.24xlO~ 4 12 = 3771 sec = 62.9 minutes 8cm Layer 1 I ^ = 2 x 10" 4 cm/sec 8cm Layer 2 1 ^2 = 5 x 10" 1 cm/sec 3 = 7xl0^cm/sec Figure Ex Example 4.12 The data given below relate to two falling head permeameter tests performed on two different soil samples: (a) stand pipe area = 4 cm 2, (b) sample area = 28 cm 2, (c) sample height = 5 cm, (d) initial head in the stand pipe =100 cm, (e) final head = 20 cm, (f) time required for the fall of water level in test 1, t = 500 sec, (g) for test 2, t = 15 sec. Determine the values of k for each of the samples. If these two types of soils form adjacent layers in a natural state with flow (a) in the horizontal direction, and (b) flow in the vertical

26 112 Chapter 4 direction, determine the equivalent permeability for both the cases by assuming that the thickness of each layer is equal to 150 cm. Solution Use Eq. (4.13). 23aL h For test 2 k, = -log = 2.3x10 3 cm/sec 1 28x x4x5, x15 20 F/ovv in the horizontal direction Use Eq. (4.27) 3 = 76.7xlO~ 3 cm/sec F/ow in the vertical direction Use Eq. (4.28) = -(2.3 x x 150) x ID" 3 =39.5xlQ- 3 cm/sec,jl/w 300 Z fct 2.3x x10-3 = 4.46 x 10" 3 cm/sec 4.12 HYDRAULIC CONDUCTIVITY OF ROCKS BY PACKER METHOD Packers are primarily used in bore holes for testing the permeability of rocks under applied pressures. The apparatus used for the pressure test is comprised of a water pump, a manually adjusted automatic pressure relief valve, pressure gage, a water meter and a packer assembly. The packer assembly consists of a system of piping to which two expandable cylindrical rubber sleeves, called packers, are attached. The packers which provide a means of sealing a limited section of bore hole for testing, should have a length five times the diameter of the hole. They may be of the pneumatically or mechanically expandable type. The former are preferred since they adapt to an oversized hole whereas the latter may not. However, when pneumatic packers are used, the test apparatus must also include an air or water supply connected, through a pressure gage, to the packers by means of a higher pressure hose. The piping of a packer assembly is designed to permit testing of either the portion of the hole between the packers or the portion below the lower packer. The packers are usually set 50, 150 or 300 cm apart. The wider spacings are used for rock which is more uniform. The short spacing is used to test individual joints which may be the cause of high water loss in otherwise tight strata.

27 Soil Permeability and Seepage 113 Two types of packer methods are used for testing of permeability. They are: 1. Single packer method. 2. Double packer method. The single packer method is useful where the full length of the bore hole cannot stand uncased/ungrouted in soft rocks, such as soft sand stone, clay shale or due to the highly fractured and sheared nature of the rocks, or where it is considered necessary to have permeability values side by side with drilling. Where the rocks are sound and the full length of the hole can stand without casing/grouting, the double packer method may be adopted. The disadvantage of the double packer method is that leakage through the lower packer can go unnoticed and lead to overestimation of water loss. Single Packer Method The method used for performing water percolation tests in a section of a drilled hole using a single packer is shown in Fig. 4.15a. In this method the hole should be drilled to a particular depth desirable for the test. The core barrel should then be removed and the hole cleaned with water. The packer should be fixed at the desired level above the bottom of the hole and the test performed. Water should be pumped into the section under pressure. Each pressure should be maintained until the readings of water intake at intervals of 5 min show a nearly constant reading of water intake for one particular pressure. The constant rate of water intake should be noted. After performing the test the entire assembly should be removed. The drilling should then proceed for the next test section. Double Packer Method In this method the hole is first drilled to the final depth and cleaned. The packer assembly may be fixed at any desired test section as shown in Fig. 4.15b. Both packers are then expanded and water under pressure is introduced into the hole between the packers. The tests are conducted as before. Regardless of which procedure is used, a minimum of three pressures should be used for each section tested. The magnitude of these pressures are commonly 100, 200 and 300 kpa. (1,2 and 3 kg/cm 2 ) above the natural piezometric level. However in no case should the excess pressure be greater than about 20 kpa per meter of soil and rock overburden above the upper packer. The limitation is imposed to insure against possible heavy damage to the foundation. =" / Ground surface \ c &$^&\ s///$\ c ^ ^ /S7XXN E, Casing / ~T~ /- Packer n * Test section (a) Test section.j -Bottom of the hole ' 1 V N V ^ (b) 1 ^ Packer Perforated pipe Packer Figure 4.15 Test sections for single and double packer methods

28 114 Chapter 4 The formulae used to compute the permeability from pressure test data are (from US Bureau of Reclamation, 1968) g ' r - r» <438a) k- - sinrr 1 - for 10r n >L>r n fa TCM 27JLH 2r Q (4.3Sb) where k = hydraulic conductivity q = constant rate of flow into the hole L = length of the test section H = differential head on the test section r Q = radius of the bore hole 4.13 SEEPAGE The interaction between soils and percolating water has an important influence on: 1. The design of foundations and earth slopes, 2. The quantity of water that will be lost by percolation through a dam or its subsoil. Foundation failures due to 'piping' are quite common. Piping is a phenomenon by which the soil on the downstream sides of some hydraulic structures get lifted up due to excess pressure of water. The pressure that is exerted on the soil due to the seepage of water is called the seepage force or pressure. In the stability of slopes, the seepage force is a very important factor. Shear strengths of soils are reduced due to the development of neutral stress or pore pressures. A detailed understanding of the hydraulic conditions is therefore essential for a satisfactory design of structures. The computation of seepage loss under or through a dam, the uplift pressures caused by the water on the base of a concrete dam and the effect of seepage on the stability of earth slopes can be studied by constructing flow nets. Flow Net A flow net for an isometric medium is a network of flow lines and equipotential lines intersecting at right angles to each other. The path which a particle of water follows in its course of seepage through a saturated soil mass is called a flow line. Equipotential lines are lines that intersect the flow lines at right angles. At all points along an equipotential line, the water would rise in piezometric tubes to the same elevation known as the piezometric head. Fig gives a typical example of a flow net for the flow below a sheet pile wall. The head of water on the upstream side of the sheet pile is h t and on the downstream side h d. The head lost as the water flows from the upstream to the downstream side is h LAPLACE EQUATION Figure 4. 16(a) illustrates the flow of water along curved lines which are parallel to the section shown. The figure represents a section through an impermeable diaphragm extending to a depth D below the horizontal surface of a homogeneous stratum of soil of depth H.

29 Soil Permeability and Seepage 11 5 It is assumed that the difference h between the water levels on the two sides of the diaphragm is constant. The water enters the soil on the upstream side of the diaphragm, flows in a downward direction and rises on the downstream side towards the surface. Consider a prismatic element P shown shaded in Fig. 4.16(a) which is shown on a larger scale in (b). The element is a parallelepiped with sides dx, dy and dz. The x and z directions are as shown in the figure and the y direction is normal to the section. The velocity v of water which is tangential to the stream line can be resolved into components v x and v z in the x and z directions respectively. Let, dh i x =, the hydraulic gradient in the horizontal direction. dh i z =, the hydraulic gradient in the vertical direction. oz k x = hydraulic conductivity in the horizontal direction. k z = hydraulic conductivity in the vertical direction. If we assume that the water and soil are perfectly incompressible, and the flow is steady, then the quantity of water that enters the element must be equal to the quantity that leaves it. The quantity of water that enters the side ab = v x dzdy dv The quantity of water that leaves the side cd = v + dx dzdy x dx. The quantity of water that enters the side be = v z dxdy The quantity of water that leaves the side ad = v z + - dz dxdy dz Therefore, we have the equation, v x dzdy + v dxdy = v x + dx dzdy + v + - dz dxdy dx dz After simplifying, we obtain, dv r dv -^ + -^ = 0 (4.39) ox oz Equation (4.39) expresses the necessary condition for continuity of flow. According to Darcy's Law we may write, dh dh vx v = -k x r ^, v z = -k, T- ox l oz Substituting for v x and v z we obtain, d dh d dh dx * dx dz z dz d 2 h d 2 h ork x r -^- + k 7 -^- = Q (4.40) ox 2 z ' oz

30 116 Chapter 4 When k = k x, i.e., when the permeability is the same in all directions, Eq. (4.40) reduces to d 2 h ax. 2 - d 2 h = 0 (4.41) oz Eq. (4.41) is the Laplace Equation for homogeneous soil. It says that the change of gradient in the jc-direction plus the change of gradient in the z-direction is zero. The solution of this equation gives a family of curves meeting at right angles to each other. One family of these curves represents flow lines and the other equipotential lines. For the simple case shown in Fig. 4.16, the flow lines represent a family of semi-ellipses and the equipotential lines semi-hyperbolas. Anisotropic Soil Soils in nature do possess permeabilities which are different in the horizontal and vertical directions. The permeability in the horizontal direction is greater than in the vertical direction in sedimentary deposits and in most earth embankments. In loess deposits the vertical permeability is greater than the horizontal permeability. The study of flow nets would be of little value if this variation in the permeability is not taken into account. Eq. (4.40) applies for a soil mass where anisotropy exists. This equation may be written in the form d 2 h d 2 h ^ & 2 ^ (4-42) If we consider a new coordinate variable x c measured in the same direction as x multiplied by a constant, expressed by Eq. (4.42) may be written as (4.43) d 2 h d 2 h ~d^ +~d^ = (4-44) c Now Eq. (4.44) is a Laplace equation in the coordinates x c and z. This equation indicates that a cross-section through an anisotropic soil can be transformed to an imaginary section which possesses the same permeability in all directions. The transformation of the section can be effected as per Eq. (4.43) by multiplying the ^-coordinates by Jk z /k^ and keeping the z-coordinates at the natural scale. The flow net can be sketched on this transformed section. The permeability to be used with the transformed section is (4.45) 4.15 FLOW NET CONSTRUCTION Properties of a Flow Net The properties of a flow net can be expressed as given below: 1. Flow and equipotential lines are smooth curves. 2. Flow lines and equipotential lines meet at right angles to each other.

31 Soil Permeability and Seepage No two flow lines cross each other. 4. No two flow or equipotential lines start from the same point. Boundary Conditions Flow of water through earth masses is in general three dimensional. Since the analysis of three-dimensional flow is too complicated, the flow problems are solved on the assumption that the flow is two-dimensional. All flow lines in such a case are parallel to the plane of the figure, and the condition is therefore known as two-dimensional flow. All flow studies dealt with herein are for the steady state case. The expression for boundary conditions consists of statements of head or flow conditions at all boundary points. The boundary conditions are generally four in number though there are only three in some cases. The boundary conditions for the case shown in Fig are as follows: 1. Line ab is a boundary equipotential line along which the head is h ( 2. The line along the sheet pile wall is a flow boundary 3. The line xy is a boundary equipotential line along which the head is equal to h d 4. The line m n is a flow boundary (at depth H below bed level). If we consider any flow line, say, p 1 p 2 p 3 in Fig. 4.16, the potential head at p { is h ( and at p 3 is h d. The total head lost as the water flows along the line is h which is the difference between the upstream and downstream heads of water. The head lost as the water flows from p l to equipotential line k is Ah which is the difference between the heads shown by the piezometers. This loss of head Ah is a fraction of the total head lost. Flow Net Construction Flow nets are constructed in such a way as to keep the ratio of the sides of each block bounded by two flow lines and two equipotential lines a constant. If all the sides of one such block are equal, then the flow net must consist of squares. The square block referred to here does not constitute a square according to the strict meaning of the word, it only means that the average width of the square blocks are equal. For example, in Fig. 4.16, the width a l of block 1 is equal to its length b }. The area bounded by any two neighboring flow lines is called a/low channel. If the flow net is constructed in such a way that the ratio alb remains the same for all blocks, then it can be shown that there is the same quantity of seepage in each flow channel. In order to show this consider two blocks 1 and 2 in one flow channel and another block 3 in another flow channel as shown in Fig Block 3 is chosen in such a way that it lies within the same equipotential lines that bound the block 2. Darcy's law for the discharge through any block such as 1 per unit length of the section may be written as Ah a Aq = kia = a = kah b b where Ah represents the head loss in crossing the block. The expressions in this form for each of the three blocks under consideration are Aq { = kah, Aq 2 = kah 2 b \ b 2 In the above equation the value of hydraulic conductivity k remains the same for all the blocks. If the blocks are all squares then b 2