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# Craig's Soil Mechanics 7th Edition

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increasing the area through which seepage takes place. The extension may be uncased or supported by perforated casing depending on the type of soil. Another solution is to install within the casing a central tube perforated at its lower end and set within a pocket of coarser material. Expressions for the coefficient of permeability depend on whether the stratum is unconfined or confined, the position of the bottom of the casing within the stratum and details of the drainage face in the soil. If the soil is anisotropic with respect to permeability and if the borehole extends below the bottom of the casing (Figure 2.3(c)) then the horizontal permeability tends to be measured. If, on the other hand, the casing penetrates below soil level in the bottom of the borehole (Figure 2.3(d)) then vertical permeability tends to be measured. General formulae can be written, with the above details being represented by an ‘intake factor’ (F ). Values of intake factor F were published by Hvorslev [5] and are also given in BS 5930 [1]. For a constant-head test: k ¼ q Fhc For a variable-head test: k ¼ 2:3A Fðt2 � t1Þ log h1 h2 where k is the coefficient of permeability, q the rate of flow, hc the constant head, h1 the variable head at time t1, h2 the variable head at time t2 and A the cross-sectional area of casing or standpipe. The coefficient of permeability for a coarse soil can also be obtained from in-situ measurements of seepage velocity, using Equation 2.4. The method involves exca- vating uncased boreholes or trial pits at two points A and B (Figure 2.3(e)), seepage taking place from A towards B. The hydraulic gradient is given by the difference in the steady-state water levels in the boreholes divided by the distance AB. Dye or any other suitable tracer is inserted into borehole A and the time taken for the dye to appear in borehole B is measured. The seepage velocity is then the distance AB divided by this time. The porosity of the soil can be determined from density tests. Then k ¼ v 0n i 2.3 SEEPAGE THEORY The general case of seepage in two dimensions will now be considered. Initially it will be assumed that the soil is homogeneous and isotropic with respect to permeability, Seepage theory 37 the coefficient of permeability being k. In the x�z plane, Darcy’s law can be written in the generalized form: vx ¼ kix ¼ �k @h @x ð2:5aÞ vz ¼ kiz ¼ �k @h @z ð2:5bÞ with the total head h decreasing in the directions of vx and vz. An element of fully saturated soil having dimensions dx, dy and dz in the x, y and z directions, respectively, with flow taking place in the x�z plane only, is shown in Figure 2.4. The components of discharge velocity of water entering the element are vx and vz, and the rates of change of discharge velocity in the x and z directions are @vx/@x and @vz/@z, respectively. The volume of water entering the element per unit time is vx dy dzþ vz dx dy and the volume of water leaving per unit time is vx þ @vx @x dx � � dy dzþ vz þ @vz @z dz � � dx dy If the element is undergoing no volume change and if water is assumed to be incom- pressible, the difference between the volume of water entering the element per unit time and the volume leaving must be zero. Therefore @vx @x þ @vz @z ¼ 0 ð2:6Þ Figure 2.4 Seepage through a soil element. 38 Seepage Equation 2.6 is the equation of continuity in two dimensions. If, however, the volume of the element is undergoing change, the equation of continuity becomes @vx @x þ @vz @z � � dxdy dz ¼ dV dt ð2:7Þ where dV/dt is the volume change per unit time. Consider, now, the function �(x, z), called the potential function, such that @� @x ¼ vx ¼ �k @h @x ð2:8aÞ @� @z ¼ vz ¼ �k @h @z ð2:8bÞ From the Equations 2.6 and 2.8 it is apparent that @2� @x2 þ @ 2� @z2 ¼ 0 ð2:9Þ i.e. the function �(x, z) satisfies the Laplace equation. Integrating Equation 2.8: �ðx; zÞ ¼ �khðx; zÞ þ C where C is a constant. Thus, if the function �(x, z) is given a constant value, equal to �1 (say), it will represent a curve along which the value of total head (h1) is constant. If the function �(x, z) is given a series of constant values, �1, �2, �3, etc., a family of curves is specified along each of which the total head is a constant value (but a different value for each curve). Such curves are called equipotentials. A second function (x, z), called the flow function, is now introduced, such that � @ @x ¼ vz ¼ �k @h @z ð2:10aÞ @ @z ¼ vx ¼ �k @h @x ð2:10bÞ It can be shown that this function also satisfies the Laplace equation. The total differential of the function (x, z) is d ¼ @ @x dxþ @ @z dz ¼ �vz dxþ vx dz If the function (x, z) is given a constant value 1 then d ¼ 0 and dz dx ¼ vz vx ð2:11Þ Seepage theory 39 Thus, the tangent at any point on the curve represented by ðx; zÞ ¼ 1 specifies the direction of the resultant discharge velocity at that point: the curve therefore represents the flow path. If the function (x, z) is given a series of constant values, 1, 2, 3, etc., a second family of curves is specified, each representing a flow path. These curves are called flow lines. Referring to Figure 2.5, the flow per unit time between two flow lines for which the values of the flow function are 1 and 2 is given by �q ¼ Z 2 1 ð�vz dxþ vx dzÞ ¼ Z 2 1 @ @x dxþ @ @z dz � � ¼ 2 � 1 Thus, the flow through the ‘channel’ between the two flow lines is constant. The total differential of the function �(x, z) is d� ¼ @� @x dxþ @� @z dz ¼ vx dxþ vz dz If �(x, z) is constant then d� ¼ 0 and dz dx ¼ � vx vz ð2:12Þ Comparing Equations 2.11 and 2.12 it is apparent that the flow lines and the equi- potentials intersect each other at right angles. Consider, now, two flow lines 1 and ( 1 þ� ) separated by the distance �n. The flow lines are intersected orthogonally by two equipotentials �1 and (�1 þ��) Figure 2.5 Seepage between two flow lines. 40 Seepage separated by the distance �s, as shown in Figure 2.6. The directions s and n are inclined at angle to the x and z axes, respectively. At point A the discharge velocity (in direction s) is vs; the components of vs in the x and z directions, respectively, are vx ¼ vs cos vz ¼ vs sin Now @� @s ¼ @� @x @x @s þ @� @z @z @s ¼ vs cos2 þ vs sin2 ¼ vs and @ @n ¼ @ @x @x @n þ @ @z @z @n ¼ �vs sin ð�sin Þ þ vs cos2 ¼ vs Thus @ @n ¼ @� @s or approximately � �n ¼ �� �s ð2:13Þ Figure 2.6 Flow lines and equipotentials. Seepage theory 41 2.4 FLOW NETS In principle, for the solution of a practical seepage problem the functions �(x, z) and (x, z) must be found for the relevant boundary conditions. The solution is represented by a family of flow lines and a family of equipotentials, constituting what is referred to as a flow net. Possible methods of solution are complex variable techniques, the finite difference method, the finite element method, electrical ana- logy and the use of hydraulic models. Computer software based on either the finite difference or finite element methods is widely available for the solution of seepage problems. Williams et al. [10] described how solutions can be obtained from the finite difference form of the Laplace equation by means of a spreadsheet. Relatively simple problems can be solved by the trial and error sketching of the flow net, the general form of which can be deduced from consideration of the boundary condi- tions. Flow net sketching leads to a greater understanding of seepage principles. However, for problems in which the geometry becomes complex and there are zones of different permeabilities throughout