Soil Mechanics for Unsaturated Soils phần 4 potx

1M 7 STEADY-STATE FLOW produces a linear differential equation: d2hw dy2 - om (7.6) Equations similar to Eq. (7.6) can also be derived for one-dimensional flow in the x- and z-directions. Solution for One-Dimensional How The differential equation for one-dimensional steady-state flow through a homogeneous, saturated soil [Le., Eq. (7.6)] can be solved by integrating the equation twice. The result is a linear equation for the hydraulic head distribu- tion in the y-direction: (7.7) h, = c,y + c, where C,, C2 = constants of integration that can be deter- mined for specified boundary conditions y = distance in the y-direction. Figure 7.5 illustrates the case of one-dimensional steady- state flow through a homogeneous, saturated soil. A con- stant water pressure head is applied to the top of the soil column to establish a downward flow of water. The water coefficient of permeability is assumed to be constant throughout the column. The position of the water table at the base of the column is considered as the datum. The gravitational head distribution along the soil column is lin- ear, equal to hBI at the base of the column (e.g., h,] = 0), and h,, at the top of the column. The water pressure head distribution is also linear, equal to hpl at the base of the column (e.g., hpl = 0), and hpn at the top of the column. These distributions can be used to compute the hydraulic heads. - Water Saturatec soil table The hydraulic heads at the top and the base of the column constitute the boundary conditions for this problem: h,l = 0.0 h, = h,,, + hP, at y equal to 0.0 (base) at y equal to h,,, (top) (7.8) where hWl = hydraulic head at the base of the soil column h, = hydraulic head at the top of the soil column h,, = gravitational head at the top of the soil column h,,,, = pore-water pressure head at the top of the soil Substituting the boundary conditions specified in Eq. (7.8) into Eq. (7.7) gives the constants of integration, C, and C2: column. c, = 1 +- hPn h8, c, = 0.0. (7.9) The hydraulic head distribution can now be written as follows: h, = (1 + 2) y. (7.10) The hydraulic head can be seen to vary linearly with depth. It has a value of h, (i.e., h,, plus hp,) at the top of the soil column, and a value of zero at the bottom of the soil column (Fig. 7.5). If the column is divided into ten depth intervals, each interval represents a change in hy- draulic head of 0.1 h,. Therefore, points with equal hy- draulic heads can be plotted as a horizontal line at each depth. These lines are called equipotential lines. The pore- * hwn I 0 (+I Head, h 1 "W" Figure 7.5 One-dimensional steady-state water flow through a saturated homogeneous soil. 7.1 STEADY-STATE WATER FLOW 155 water pressure head, hp, distribution is linear under steady- state seepage conditions. The linearity in the hydraulic head and the pore-water pressure head distributions is the result of the conswt water coefficients of permeability. The equation for one-dimensional steady-state flow through an unsaturated soil [i.e., Eq. (7.4)] requires a more complex solution than that for a saturated soil. A numerical solution can be used as an alternative to a closed-form so- lution. The finite difference method will be used to illus- trate the solution to the flow equation for an unsaturated soil. Mite Dztfcrence Method The seepage differential equation can be written in a finite difference form. Consider the situation where a function, h( y), varies in space, as shown in Fig. 7.6. Values of the function at points along the curve can be computed using Taylor series: Ay2 d2h hi+l= hi + AY (s), + ($)i +$($)>,+. . ' = hi - Ay (t), +e (e) hi- 1 2! dy2 -&ye) + *. . 3! dy3 i (7.11) (7.12) where i - 1, i, i + 1 = three consecutive points spaced at in- Subtracting Eq. (7.12) from Eq. (7.11) and neglecting the higher order derivatives result in the first derivative of the function at point (i): crements, Ay. - hi+l - hi-1 ($)i - 2Ay ' (7.13) I Yi.1 Yi Yet1 Y Figure 7.6 Function h(y) shown in a finite difference form. .av'av. Summing Eqs. (7.11) and (7.12) and again neglecting the higher order derivatives gives the second derivative of the function at point (0: ($)i = hi+l + hi-1 - 2hi (7.14) AY Equations (7.13) and (7.14) are called the central differ- ence approximations for the first and second derivatives of the function, h( y), at point i. These approximations can be used to solve the differential equation. Similar approx- imations can be derived for a function, h(x), in the xdi- rection. The use of an iterative finite difference technique in solv- ing flow problems is illustrated in the following sections. One example involves the use of a head boundary condi- tion, while another illustrates the use of a flux boundary condition. Head Boundary Condition Steady-state evapomtion from a column of unsaturated soil is illustrated in Fig. 7.7. A tensiometer is installed near the ground surface to measure the negative pore-water pres- sure. One-dirnensional, steady-state flow is assumed when the tensiometer reading remains constant with respect to time. The pore-water pressure at the base of the column (i.e., the water table) is equal to zero. The hydraulic head distribution along the length of the column is given by Q. (7.4). This equation can be solved using the finite difference approximations in Eqs. (7.13) and (7.14). The column length is first discretized into (n) equally spaced nodal points at a distance Ay apart (Fig. 7.7). A central difference approximation is then applied to the hydraulic head and coefficient of permeability deriva- tives in Eq. (7.4). For example, Eq. (7.4) can be written in a finite difference form for point (i): (7.15) where kwy(r3, kV(i- kv(i+ = water coefficients of perme- ability in the ydirection at points (i), (i - l), and (i + l), respectively h,,,, h,(, - I), h,(i + 1) = hydraulic heads at points (i), (i - l), and (i + l), respec- tively. Equation (7.15) can be rearranged after assuming equal 156 7 STEADY-STATE FLOW Steadrstate Discretization Boundary conditions n o - hwn = hgn + hpn 0 0 @ (it11 1 (i-1) (0 m 0 i:: 0 0 0 i - hw’=O ’ t Datum Nodal points v P Water table hl Figure 7.7 One-dimensional, steady-state water flow through an unsaturated soil with a constant head boundary condition. Ay increments: -(8 kwy(i)} hw(b + (4 kwy(i) + kwy(i+~) - kwy(,-~)} * hw(i+ 1) + (4 kwy(i) + kwyci- I) - kwy(i+ I)) - hw(i- 1) = 0. (7.16) The hydraulic heads at the external points (he., points 1 and n) become the boundaq conditions. The hydraulic head at point 1 is zero. The elevation of point (n) relative to the datum, h,,, gives the gravitational head at point (n). The tensiometer reading near the ground surface indicates the negative pore-water pressure head at point (n) (Le., hp,). Therefore, the hydraulic head boundary condition at the top and the base of the soil column can be expressed math- ematically : hw(l) = 0.0 hw(,) = h,, + hpn at y equal to 0.0 (base) at y equal to h,, (top). (7.17) The finite difference seepage equation [i.e., Eq. (7.16)1 can be written for the (n - 2) internal points [i.e., points 2, 3, - - - , (n - l)]. As a result, there are (n - 2) equa- tions that must be solved simultaneously for (n - 2) hy- draulic heads at intermediate points. The finite difference scheme illustrated by Eq. (7.16) is called an implicit form. The equation is also nonlinear because the coefficients of permeability, kw, are a function of matric suction, which in turn is related to hydraulic head, hw. The nonlinear equa- tions require several iterations to produce convergence. During each iteration, each equation is assumed to be lin- ear by setting the water coefficients of permeability at each node to a constant value. For the first iteration, the kwy values at all points can be set equal to the saturated coefficient of permeability, ks. The (n - 2) linearized equations can then be solved simul- taneously using a procedure such as the Gaussian elimi- nation technique. The computed hydraulic heads are used to calculate new values for the water coefficient of perme- ability. The coefficient of permeability values at each point must be in agreement with the coefficient of permeability versus matric suction function. The revised coefficient of permeability values, kwy, are then used for the second it- eration. New hydraulic heads are computed for all depths. The iterative procedure is repeated until there is no longer a significant change in the computed hydraulic heads and the computed coefficients of permeability. Figure 7.8 illustrates typical distributions for the pore- water pressure and the hydraulic head along the unsatu- rated soil column. Flow is occumng under steady-state evaporation conditions. The nonlinearity of the flow equa- tion [Le., Eq. (7.16)] results in a nonlinear distribution of the hydraulic head and the pore-water pressure head. The equipotential lines are not equally spaced along the col- umn. This is different from the uniformly spaced equipo- tential lines for the homogeneous, saturated soil column. The difference is the result of a varying coefficient of permeability throughout the unsaturated soil column. The above analysis can similarly be applied to the steady-state downward flow of water through an unsaturated soil. Once again, the hydraulic head boundary conditions at two points along the soil column must be known. Eluw Boundary CondWn Infiltration into an unsaturated soil column is another ex- ample which can be used to illustrate the solution of the nonlinear differential flow equation (Fig. 7.9). Steady-state 7.1 STEADY-STATE WATER FLOW 157 L vw mure 7.8 Steady-state evaporation through an unsaturated soil column. infiltration may be established as a result of sprinkling ir- rigation. Let us assume a constant downward water flux of qW. Steady-state flow can be described using EQ. (7.4). The hydraulic head distribution can be determined by solv- ing the finite difference form of the steady-state flow equa- tion [Le., Eq. (7.16)]. The hydraulic head boundary con- dition at the ground surface is assumed to be unknown. However, the water flux, qW, is known, and is constant throughout the soil column for steady-state conditions. The soil column is first discretized into (n) nodal points with an equal spacing, Ay (Fig. 7.9). The water flux at Steady-state infiltration L I I i ; Water flux IO, i i i i i (4w) n Unsaturated soil point (i) can be expressed in terms of the hydraulic heads at points (i + 1) and (i - 1) using Darcy's law: where qW = water flux through the soil column during the steady-state flow; the flux is assumed to be posi- tive in an upward direction and negative in a downward direction A = cmss-sectional area of the soil column. Boundary Discretization . conditions "(Q (n - 1)~ @ 1 qw Iqw 1 @ -hwl=O Water table 't Datum I 1 @ -hwl=O Water table 't Datum I T t Nodal points Figure 7.9 One-dimensional steady-state water flow through an unsaturated soil with a flux boundary condition. 158 7 STEADY-STATE FLOW Equation (7.18) can be rearranged as follows: Substituting Eq. (7.19) into the flow equation for point (i) [Le., Eq. (7.16)] yields the following form: -{8kwy(t>l hw(0 + (4 kwy(i) + ky(i+l) - kwy(i-I)l + {4kwy(i) + kwy(i- I) - kwy(i+ 1)) hw(i - 1) = 0. (7.20) Equation (7.20) can now be' solved for the hydraulic head at point (i): hw(i) = hw(i- 1) (7.21) The finite difference Eq. (7.21) is in an explicit form. Therefore, the hydraulic heads can be solved directly start- ing from a known boundary condition. Point 1 (Fig. 7.9) has a zero hydraulic head. Therefore, the base of the soil column is a suitable point to commence solving for the heads. Hydraulic heads can subsequently be solved point by point, up to the ground surface. Equation (7.21) is non- linear since the coefficient of permeability, k,, is a func- tion of the hydraulic head, hw. The equation must be solved iteratively by setting the coefficients of permeability as constants for each iteration. '- Pore-water \ pressure head, h, \ \ '\ \ \ \ \ \\ \ \ \\ \ \ \: The coefficient of permeability at each node, kw, is as- sumed to be equal to the saturated coefficient of perme- ability, k,, for the first iteration. The computed hydraulic heads, and subsequently the negative pore-water pres- sures, are used to revise the coefficients of permeability for the second iteration. This iterative procedure is repeated until there is convergence with respect to the hydraulic heads and the coefficients of permeability. When comput- ing the hydraulic head at the ground surface, the kV(. + value can be assumed to be equal to the kwy(,,) value. Typical distributions of pore-water pressure and hy- draulic head during steady-state infiltration are illustrated in Fig. 7.10. The nonlinear distribution of the pore-water pressure and hydraulic head is produced by the nonlin- earity of Eq. (7.21). As a result, the equipotential lines are not uniformly distributed along the soil column. The above analysis is also applicable to steady-state, upward flow (e.g., evaporation from ground surface) where the flux, qwy, is known. In the case of a heterogeneous, saturated soil, the coef- ficients of permeability can be replaced by k, in Eq. (7.21): hw(i) hw(i- 1) Equation (7.22) becomes linear when the soil is homo- geneous : (7.23) Equation (7.23) defines a linear distribution of the hy- draulic head for a homogeneous, saturated soil column subjected to one-dimensional steady-state flow. I - hpn 0 -hwn 4 Water /. v table 'Datum - (+I qw I Head, h Figure 7.10 Steady-state infiltration thmugh an unsaturated soil. 7.1 STEADY-STATE WATER FLOW 159 7.1.3 Two-Dimensional Flow Seepage through an earth dam is a classical example of two-dimensional flow. Water flow is in the cross-sectional plane of the dam, while flow perpendicular to the plane is assumed to be negligible. Until recently, it has been con- ventional practice to neglect the flow of water in the un- saturated zone of the dam. The analysis presented herein assumes that water flows in both the saturated and unsat- urated zones in response to a hydraulic head driving poten- tial. The following two-dimensional formulation is an ex- panded form of the previous onedimensional flow equa- tion. The formulation is called an uncoupled solution since it only satisfies continuity. For a rigorous formulation of two-dimensional flow, continuity should be coupled with the force equilibrium equations. Formulation for Two-Dimensional Flow The following derivation is for the general case of a her- erogeneous, anisotropic, unsaturated soil [Fig. 7.2(b)J. The coefficients of permeability in the x-direction, k,, and the y-direction, k,, are assumed to be related to the matric suction by the same permeability function, k,(u, - u,), The ratio of the coefficients of permeability in the x- and y-directions, (kwJkv), is assumed to be constant at any point within the soil mass. A soil element with infinitesimal dimensions of dr, dy, and dz is considered, but flow is assumed to be two-di- mensional (Fig. 7.11). The flow rate, vWx, is positive when water flows in the positive x-direction. The flow rate, v,, is positive for flow in the positive y-direction. Continuity for two-dimensional, steady-state flow can be expressed as follows: v,) dr dz = 0 (7.24) vw+7dy dz Figure 7.11 Two-dimensional water flow through an unsatu- rated soil element. where v, = water flow rate across a unit area of the soil in Therefore, the net flux in the x- and ydirections is, the x-direction. (7.25) Substituting Darcy’s laws into Eq. (7.25) results in a nonlinear partial differential equation: = 0 (7.26) where k,(u, - u,) = water coefficients of permeability as a function of matric suction; the perme- ability can vary with location in the xdirection ah,/ax = hydraulic head gradient in the x-direc- tion. For the remainder of the formulations, kwx(u, - u,) and k,(u, - u,) are written as k, and k,,,., respectively, for simplicity. Equation (7.26) describes the hydraulic head distribution in the x-y plane for steady-state water flow. The nonlinearity of E& (7.26) becomes more obvious after an expansion of the equation: (7.27) where ak,/ax = change in water coefficient of permeability The spatial variation of the coefficient of permeability given in the thirrl and fourth terms in Eq. (7.27) produces nonlinearity in the governing flow equation. For the heterogeneous, isotropic case, the coefficients of permeability in the x- and y-directions ace equal (i.e., k,, = k, = k,). Therefore, Eq. (7.27) can be written as fol- lows: in the x-direction. a2h, ak, ah, ak, ah, ax ax ay ay k, (3 + T) + - - + - - = 0 (7.28) where k, = water coefficient of permeability in the x- and Table 7.1 summarizes the relevant equations for two-di- y-directions. mensional steady-state flow through unsaturated soils. 160 7 STEADY-STATE FLOW Table 7.1 Two-Dimensional Steady-State Equations for Unsaturated Soils Heterogeneous, Anisotropic Heterogeneous, Isotropic Eq. (7.29) Eq. (7.30) Seepage through a dam involves flow through the unsat- urated and saturated zones. Flow through the saturated soil can be considered as a special case of flow through an un- saturated soil. For the saturated portion, the water coeffi- cient of permeability becomes equal to the saturated coef- ficient of permeability, k,. The saturated coefficients of permeability in the x- and y-directions, k,, and ksy, respec- tively, may not be equal due to anisotropy. The saturated coefficients of permeability may vary with respect to lo- cation due to heterogeneity. A summary of steady-state equations for saturated soils under different conditions is presented in Table 7.2. Equations (7.31)-(7.34) are spe- cialized forms that can be derived from the steady-state flow equation for unsaturated soils [i.e., Eq. (7.27)]. There- Table 7.2 Two-Dimensional Steady-State Equations for Saturated Soils Anisotropic Isotropic Heterogeneous a’h, a’h, ksx + ksy - aY2 ak, ah, + ax ax ak, ah, + ax ax Eq. (7.31) Eq. (7.32) Homogeneous Eq. (7.33) Eq. (7.34) fore, steady-state seepage through saturated-unsaturated soils can be analyzed simultaneously using the same gov- erning equation. Solutions for Two-Dimensional Flow The differential equation describing two-dimensional steady-state flow through a homogeneous, isotropic satu- rated soil [Le., Eq. (7.34)] is called the Laplacian equa- tion. It is a linear, partial differential equation. The solu- tion of this equation describes the head at all points in a soil mass. The solution can be obtained using closed-form analytical methods, analog methods, or numerical meth- ods. Often, a graphical method referred to as drawing a “flownet” has been used to solve the Laplacian equation (Casagrande, 1937). The flownet solution results in two families of curves, referred to as flow lines and equipotential lines. The flow- net solution has been used extensively to analyze problems involving seepage through saturated soils, and is explained in most soil mechanics textbooks. Boundary conditions for the soil domain must be known prior to the construction of the flownet. Either the head or the flux is prescribed along the boundary. A boundary condition exception is the case of a free surface. A network of flow lines and equipotential lines is sketched by trial and emr in order to satisfy the boundary conditions and the requirement of right-angled, equidimensional elements. A head boundary condition or an impermeable boundary condition can readily be imposed for most saturated soils problems. For example, steady-state seepage beneath a sheet pile wall has the boundary conditions shown in Fig. 7.12(a). However, the conditions are more difficult to as- sign when dealing with unsaturated soils. Let us consider steady-state seepage through an earth dam [Fig. 7.12(b)]. In the past, the assumption has generally been made that the flow of water through the unsaturated zone is negligible due to its low permeability. In other words, the phreatic line is assumed to behave as an imper- vious, uppermost boundary when constructing the flownet. This uppermost boundary [i.e., line BC in Fig. 7.12(b)] is not only considered to be a phreatic line, but also an up- permost flow line. The uppennost boundary is referred to as a free surface under these special conditions (Freeze and Cherry, 1979). However, the position of the free surface is unknown, and it must be approximated prior to con- structing the flownet. The position of the free surface is usually determined using an empirical procedure (Casagrande, 1937). The as- sumption that the free surface is a phreatic line requires that the pore-water pressures be zero along this line. Equipo- tential lines must intersect the free surface at right angles since it is also an uppermost flow line. In other words, it is assumed that there is no flow across the free surface. The flownet can then be constructed. The flownet technique has been developed primarily to analyze steady-state seepage through isotropic, homoge- 7.1 STEADY-STATE WATER FLOW 161 Boundary conditions: BC and DE: qw = 0 AH and FG: qw = 0 AB: hw=H1 EF: hw = HZ HG: qw=O Assumed impervious -I impervious Y Equjpotential'lines Horizontal drain Boundary conditions: BC: free surface, it's location is unkown AB: hw=H1 CD: hw=O DA qw=O (b) Figure 7.12 Flownet constmctions to solve the Laplacian equation. (a) Steady-state seepage throughout a homogeneous, isotropic saturated soil; (b) steady-state seepage throughout a ho- mogeneous, isotropic earth dam. neous, saturated soils. The flownet technique becomes complex and difficult to use when analyzing anisotropic, heterogeneous soil systems. There is an inherent problem associated with applying the flownet technique to satu- rated-unsaturated flow. Freeze (1971) stated that, ", . .the boundary conditions that are satisfied on the free surface specify that the pressure head must be atmospheric and the surface must be a streamline. Whereas the first of these conditions is true, the second is not." Figure 7.13 compares two solutions of a saturated-un- saturated soil system. The flownet in Fig. 7.13(a) was drawn based on a numerical method solution for a satu- rated-unsaturated flow system. The flownet shown in Fig. 7.13(b) was constructed using an empirically defined free surface, thereby neglecting flow in the unsaturated zone. The free surface is a close approximation of the phreatic line from the saturated-unsaturated flow modeling. The in- correct assumption regarding the uppermost boundary con- dition can be avoided by realizing that there is flow be- tween the saturated and unsaturated zones (Freeze, 1971, Papagiannakis and Fnxllund, 1984). Steady-state flow in the saturated and unsaturated zones can be analyzed simultaneously using the same governing equation [Le., Eq. (7.26)]. Both zones are treated as a sin- gle domain. The water coefficient of permeability in the saturated zone is equal to k,. The water coefficient of permeability, k,,,, varies with respect to the matric suction in the unsaturated zone. The flownet technique is no longer applicable to saturated-unsaturated flow modeling when the governing flow equation is not of the Laplacian form. The general flow equation can be solved using a numerical tech- nique such as the finite difference or the finite element method. Figure 7.14 shows several typical solutions by Freeze (1971) involving saturated-unsaturated flow mod- eling. The following section briefly describes the fonnu- lation of the finite element method in analyzing steady-state seepage through saturated-unsaturated soils. Seepage Analysis Using the lcpnite Element Method The application of the finite element method requires the discretization of the soil mass into elements. Triangular and quadrilateral shapes of elements are commonly used for 162 I STEADY-STATE FLOW a) Homogeneous - Flow line __ = Equipotential line rn =Impervious boundary E E Freesurface I,\\'' D I::! I = Flow line - _ = Equipotential line (b) Figure 7.13 Steady-state seepage in a saturated-unsaturated soil system. (a) Flownet constructed from $arurare~-u~$a~ura~e~ flow modeling; (b) flownet constxuction by considering flow in the sat- urated zone (after Freeze and Cherry, 1979). two-dimensional problems. Figure 7.15 shows the cross section of a dam that has been discretized using triangular elements. The lines separating the elements intersect at no- dal points. The hydraulic head at each nodal point is ob- tained by solving the governing flow equation and applying the boundary conditions. The finite element formulation for steady-state seepage in two dimensions has been derived using the Galerkin principle of weighted residuals (Papagiannakis and Fred- lund, 1984): ]s{L}%wds = 0 (7.35) where {L} = matrix of the element area coordi- nates (i.e., {L, & &}) L,, &, 5 = area coordinates of points in the element that are related to the bl Cutoff c) Internal, basal e) Sloping core and Figure 7.14 Typical solutions for saturated-unsaturated flow modeling of various dam sections (from Fneeze, 1971). Cartesian coordinates of nodal points as follows (Fig. 7.16): "1 = 1/u((x2Y3 - x3Y2) + (Y2 k? = 1/u{(X3Yl - XlY3) + (Y3 - Y3)X + (x3 - X2)Y) - Y,)X + (x1 - X3)Y) - Y2)X + (x2 - Xl)Y) r, = 1/244{(XlY2 - X2Y1) + (Y1 xi, yi(i = 1, 2, 3) = Cartesian coordinates of the three nodal points of an element x, y = Cartesian coordinates of a point within the element A = area of the element [* :*-J = matrix of the water coefficients of permeability (Le., [k,]) {h,} = matrix of hydraulic heads at the nodal points, that is, 7.1 STRADY-STATE WATER Node number 1 Element number 1 8 195 Figure 7.15 Discretized cross section of a dam for finite element analysis. FLOW 163 - ow = external water flow rate in a direc- tion perpendicular to the boundary of the element Rearranging Eq. (7.35) yields a simplified form for the S = perimeter of the element. governing flow equation: 1“ [BIT[k,l[Bl dA {h,) - 1 tLITZ, = 0 (7.36) S where [B] = matrix of the derivatives of the area coordinates, which can be written as 3- 24 -[ (x3 - x2) (XI - x3) (x2 - XI) 1 (Yz’- Y3) (Y3 - Yl) (Y1 - Y2) Either the hydraulic head or the flow rate must be spec- ified at boundary nodal points. Specified hydraulic heads at the boundary nodes are called Dirichlet boundary con- ditions. A specified flow rate across the boundary is re- fed to as a Neuman boundary condition. The second term in Eq. (7.36) accounts for the specified flow rate measured in a dimtion normal to the boundary. For example, a spec- * = Nodal point yl i= 1 (XI, yl) -W Cartesian coordinates (1.0.0)- Area coordinates Figure 7.16 Area coordinates in relation to the Caltesian coor- dinates for a triangular element. ified flow rate, II,, in the vertical direction must be con- verted to a normal flow rate, Z,, as illustrated in Fig. 7.17. The normal flow rate is in turn converted to a nodal flow, Q, (Segerlind 1984). Figure 7.17 shows the computation of the nodal flows, QWi and QWj, at the boundary nodes (i) and (j), respectively. A positive nodal flow signifies that there is infiltration at the node or that the node acts as a “source.” A negative nodal flow indicates evaporation or evapotranspiration at the node and that the node acts as a “sink.” When the flow rate acmss a boundary is zero (e.g., impervious boundary), the second term in Eq. (7.36) dis- appears. The finite element equation [Eq. (7.36)] can be written for each element and assembled to form a set of global flow equations. This is performed while satisfying nodal com- patibility (Desai and Abel, 1972). Nodal compatibility re- quires that a particular node sharect by the sumunding ele- ments must have the same hydraulic head in all of the elements (Zienkiewicz, Desai 1975a). Equation (7.36) is nonlinear because the coefficients of permeability are a function of matric suction, which is re- lated to the hydraulic head at the nodal points, {h,) . The hydraulic heads are the unknown variables in Eq. (7.36). Equation (7.36) is solved by using an iterative method. For each iteration, the coefficient of permeability within an ele- ment is set to a value depending upon the average matric suction at the three nodal points. In this way, the global flow equations are linearized and can be solved simulta- neously using a Gaussian elimination technique. The com- puted hydraulic head at each nodal point is again averaged to determine a new coefficient of permeability from the permeability function, k,(u, - u,). The above steps are repeated until the hydraulic heads and the coefficients of permeability no longer change by a significant amount. The hydraulic head gradients in the x- and y-dimtions can be computed for an element by taking the derivative of the element hydraulic heads with respect to n and y, re- spectively: (7.37) [...]... equations for unsaturated soils For a saturated soil, the coefficient of permeability becomes equal to the saturated coefficient of permeability, k, A summary of three-dimensional, steady-state equations for suturuted soils cornsponding to various conditions is presented in Table 7 .4 The three dimensional steady-state flow equations can be Table 7 .4 Three-Dimensional Steady-StateEquations for Satumted Soils. .. Eq (7 .44 ) takes the following form: k , ak, + ay ah, ay ak ah, +a=o (7 .44 ) az az k, = change in water coefficient of permeability in the z-direction The fourth, fifth, and sixth terms in Eq (7 .44 ) account for the spatial variation in the coefficient of permeability In the case of two-dimensional flow, the hydraulic head gradient in the third direction is negligible (e.g., ah,/az = 0), and Eq (7 .44 ) reverts... through an unsaturated soil element I STEADY-STATE FLOW 1 74 Table 7.3 Three-Dimensional Steady-State Equations for Unsaturated Soils Heterogeneous, Anisotropic azh, kwxax2 Heterogeneous, Isotropic azh, azh, + k -+ k, aJ az2 kw Wy ak,, + ax ah, ax + -ak - + - - = o ak,, Z ah, ay ay a2h, a2h, a2h, ( + ay2 + 3) 3 ak, ak, + ah, + ah, ah, ax ax az az ay ay + -ak, o - = ah, az a t Eq (7 .46 ) Eq (7 .47 ) respectively,... stress to the soil The compressibilities of air, water, and air-water mixtures are first presented The compressibility of the soil structure is summarized in the form of a constitutive relationship for an unsaturated soil Equations which present the pore pressure as a function of the applied total stress require the use of the compressibility of the pore fluids Isotropic and anisotropic soils under various... pressure parameter for isotropic loading (i.e., du,/du3) a = isotropic (confining) total stress 3 Bo = pore-air pressure parameter for isotropic loading (i.e., du,/du3) Compressibility of water, Cw = 4. 68 x The compressibility of the pore fluid in an unsaturated soil [Le., Eq (8.13)] takes into account the matric suction of the soil through use of the B, and Bo parameters In the absence of soil solids, the... mixture is illustrated in Fig 8 .4 for various degrees of saturation The case considered has an initial absolute air pressure, Zm, of 202.6 kPa (Le., 2 atm) Values of B, and B, are assumed to be equal to 1.0 for all degrees of saturation This assumption may be unrealistic for low degrees (l/kPa) Compressibility, C (1/kPa) 181 Compressibility of air, (1/kPe) Ca = 4. 94 x Figure 8 .4 Components of compressibility... anomaly 1 84 8 PORE PRESSURE PARAMETERS Kelvin‘s equation Occ Iuded Water air bubbles ~ - - (microscopic) (a) (b) Figure 8.6 Microscopic and macroscopic models for a soil (a) Compressibility formulation for an air-water mixture; (b) effec- tive stress and intergranular stresses in a saturated soil can be best understood by first considering an increase in total stress applied to an unsaturitted soil specimen... deriving the pore pressure parameter equations for undruined loading is explained in Section 8.2.3 Consider an unsaturated soil specimen that undergoes one-dimensional, drained compression The stress state variables, (a - u,) and (u, u,), change as the soil is compressed As a result, the volume of the soil changes in accordance with the constitutive relation for the soil structure Volume change is primarily... compressibility of the soil structure (Skempton, 19 54) In other words, the tangent B parameter becomes equal to 1.0 (Le., B, = B, = 1.0) A convention in laboratory testing has been to assume complete saturation of a soil when the B pore pressure parameter reaches a value of 1.0 (Skempton, 19 54) It is possible for the secant B ' pore pressure parameter to be less than 1.0 even though the soil has reached... equation for the diffusing pore-air pressure distribution in the ydirection (i.e., ua = (y/L)uat) CHAPTER 8 Pore Pressure Parameters The mechanical behavior of unsaturated soils is directly affected by changes in the pore-air and pore-water pressures Two classes of pore pressure conditions may develop in the field First, there are the pore pressures associated with the flow or seepage through soils This . equations for two-di- y-directions. mensional steady-state flow through unsaturated soils. 160 7 STEADY-STATE FLOW Table 7.1 Two-Dimensional Steady-State Equations for Unsaturated Soils. with the force equilibrium equations. Formulation for Two-Dimensional Flow The following derivation is for the general case of a her- erogeneous, anisotropic, unsaturated soil [Fig permeability. The equation for one-dimensional steady-state flow through an unsaturated soil [i.e., Eq. (7 .4) ] requires a more complex solution than that for a saturated soil. A numerical solution