feaiget%2E27534%2E0010

10. Seepage and consolidation

10.1 SynopsisIn the previous chapters, analysis has been restricted to either drained or undrainedsoil conditions. While many problems can be solved making one, or a combination,of these two extreme conditions, real soil behaviour is often time related, with thepore water pressure response dependent on soil permeability, the rate of loadingand the hydraulic boundary conditions. To account for this behaviour, the seepageequations must be combined with the equilibrium and constitutive equations. Thischapter briefly describes the basis behind such a coupled approach and presents thefinite element equations. It is then shown how the steady state seepage equationscan be obtained from these general consolidation equations. The hydraulicboundary conditions relevant to geotechnical engineering are discussed afterwards.Some nonlinear permeability models are presented, followed by a short discussionon the numerical problems associated with unconfined seepage. The chapterfinishes by presenting an example of coupled finite element analysis.

10.2 IntroductionThe theory presented so far in this book has been restricted to dealing with eitherfully drained or undrained soil behaviour. While many geotechnical problems canbe solved by adopting such extreme soil conditions, real soil behaviour is usuallytime related, with the pore water pressure response dependent on soil permeability,the rate of loading and the hydraulic boundary conditions. To account for suchbehaviour it is necessary to combine the equations governing the flow of pore fluidthrough the soil skeleton, with the equations governing the deformation of the soildue to loading. Such theory is called coupled, as it essentially couples pore fluidflow and stress strain behaviour together.

The chapter begins by presenting the theory behind the coupled finite elementapproach. This results in both displacement and pore fluid pressure degrees offreedom at element nodes. If the soil skeleton is rigid, the soil cannot deform andthe coupled equations reduce to the steady state seepage equations. It is thereforea simple matter to establish the governing finite element equations for this situationfrom the more general coupled equations. Only pore fluid degrees of freedom ateach node are relevant for seepage analyses.

As the flow of water within the soil skeleton is now being considered, the

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306 / Finite element analysis in geotechnical engineering: Theory

hydraulic boundary conditions which control it must be accounted for. Theseboundary conditions consist of either prescribed flows or changes in pore fluidpressure. Some of the boundary conditions relevant to geotechnical engineering aredescribed in this chapter. In particular sources, sinks, infiltration and precipitationboundary conditions are covered. The latter option accounts for the finite capacityof soil to accommodate the entry of pore fluids from a boundary.

Although it is often assumed that the permeability of soil is constant, laboratoryand field tests show that this is not so. Fundamentally, one would expect thepermeability to depend on the size of the void space between the solid soil grainsand therefore depend on void ratio (or specific volume). Three nonlinearpermeability models are presented in this chapter. In one of these the permeabilityvaries with void ratio, whereas in the other two it varies with the mean effectivestress.

Two different types of pore fluidflow can be identified: those which donot involve a phreatic surface(confined flow) and those which do(unconfined flow), as shown in Figure10.1. Problems which involveunconfined flow require specialattention in numerical analysis, as it isnecessary to determine the position ofthe phreatic surface. This is notstraight forward and a brief discussionof how this may be achieved is given.

The chapter ends by presenting anexample of a coupled analysis.

;ic surface

Unconfined flow

V

Sand ^ ^ r

FlowClay

Confined flow

Figure 10.1: Examples of confinedand unconfined flow

10.3 Finite element formulation for coupled problemsWhen deriving the finite element equations in Chapter 2, it was assumed, whenevaluating the incremental strain energy in Equation (2.19), that the constitutivebehaviour could be written in terms of a relationship between increments of totalstress and strain:

{Acr} = [D]{A£} (10.1)

If the material behaviour is defined in terms of total stress, as for example in theTresca model, obtaining the constitutive matrix [D] is relatively straight forward.However, if the material behaviour is defined in terms of effective stress, which isthe preferred method in soil mechanics and follows from the principle of effectivestress, additional complications can arise. It has been shown in Chapter 3 how the[D] matrix can be obtained from the effective matrix [D'] in the special cases offully drained and undrained soil behaviour. If soil behaviour is somewhere betweenthese two extreme conditions, account must be taken of the time dependency of the


Seepage and consolidation / 307

changes in pore fluid pressure and effective stress. A procedure for achieving thisis described below.

Using the principle of effective stress Equation (10.1) becomes:

} + {Aay} (10.2)

where {Aaf}T={Apf, Apf, Apf9 0,0,0} and Apf is the change in pore fluid pressure.

In the finite element approach it is assumed that the nodal displacements andthe nodal pore fluid pressures are the primary unknowns. As before the incrementaldisplacements can be expressed in terms of nodal values using Equation (2.9). Inaddition, it is assumed that the incremental pore fluid pressure, Apf, can beexpressed in terms of nodal values using an equation similar to Equation (2.9):

{Apf}=[Np]{APf}n (10.3)

where [Ay is the matrix of pore fluid pressure interpolation functions, similar to[N]. The choice of [Np] will be discussed subsequently. However, [Np] is oftenassumed to be equivalent to [TV].

The analysis of time dependent consolidation requires the solution of Biot's(Biot (1941)) consolidation equations, coupled with the material constitutive modeland the equilibrium equations. The following basic equations have to be satisfiedfor a soil saturated with an incompressible pore fluid:

- The equations of equilibrium:

dx dx dy dzd<j'v dpf drYV drV7

dy dy ox dzdel dpf dr dr— - + -ZL + — s . + —y—dz dz dx dy

yz = 0

where yx, yy and yz are the components of the bulk unit weight of the soil actingin the x, y and z directions respectively.The constitutive behaviour, expressed in terms of effective stresses:

} (10.5)

The equation of continuity, see Figure 10.2:

i^+^+^_2=^ (10.6)dx dy dz ^ dt v }

where vx, vy and v, are the components of the superficial velocity of the porefluid in the coordinate directions, and Q represents any sources and/or sinks.



Generalised Darcy's law:

Kk

xy

kxz

Ky K

k kyy yz

Kyz Kzz

dhdxdhdydhdz

(10.7)

or:{v} = -[k]{Vh}

where h is the hydraulic head defined as:

(10.8)

Vector {iG}={iGx > k]y > hzV is the unit vector parallel, but in the oppositedirection, to gravity; ki} are the coefficients of the permeability matrix, [k], ofthe soil. If the soil is isotropic with a permeability k, then kxx= kyy= kz = k andk = k = k = 0n-xy ^xz ^yz yj-

Figure 10.2: Continuity conditions

As noted in Chapter 2 a more convenient form of the equations of equilibriumexpressed by Equation (10.4) can be found by considering the principle ofminimum potential energy which states that (see Equation (2.18)):

0 (10.9)

where AE is the incremental total potential energy, AW is the incremental strainenergy and AL is the incremental work done by applied loads. The incrementalstrain energy term, AW, is defined as:

= j - \ {A£}T{Acr} dVolVol.

(10.10)



Using Equation (10.2) this can be written in the following form:

= ± J [{As}T[D']{As} + {Aaf}{As}]dVol n0U)Vol

Noting that the second term in this equation is equivalent to Apf-Aev, gives:

( )Vol

The work done by the incremental applied loads AL can be divided intocontributions from body forces and surface tractions, and can therefore beexpressed as (see Equation (2.20)):

AL= j {Ad}T{AF} dVol + \ {Ad}T{AT} dSrf (l0 l3)Vol Srf '

Substituting Equations (10.12) and (10.13) into Equation (10.9) and followinga similar procedure to that outlined in Chapter 2 (i.e. Equations (2.21) to (2.25)),gives the following finite element equations associated with equilibrium:

[KG]{Ad}nG + [LG]{Apf}nG = {ARG} (10.14)

where:

[KG] = l [KE]t =t[\ [B]T[D'][B] dVol) (10.15)/=1 /=1 \Vol S j

= Z [LE], =±{\ {m}[B]T[Np] dVol] (10.16)

G} = i {ARE},=f, [N]T{AF}dVol\ + j [Nf{AT}dSrfVol J j \Srf

T

(10.17)

[m] = { 1 1 1 0 0 0 } (10.18)

Using the principle of virtual work, the continuity Equation (10.6) can bewritten as:

J [{v}1{V(APf)}+^- Apf]dVol-QApf =0 (10.19)Vol Ut

Substituting for {v} using Darcy's law given by Equation (10.7) gives:

voi f dt f f

Noting that {Vh} = (1/yj) Vpf+ {iG}, and approximating dejdt as AeJAt, Equation(10.20) can be written in finite element form as:

/}«G=[«c] + fi (10.21)



where:

Vol

dy

(10.22)

(10.23)

(10.24)

To solve Equations (10.14) and (10.21) a time marching process is adopted. Ifthe solution ({Ad}nG , {pf}tiG)i *s known at time tl9 then the solution ({Ad}nG ,{pf}nG)2 at time t2=tx+At is sought. To proceed it is necessary to assume:

J {\-/3){{pf}nG\\At (10.25)

This approximation is showngraphically in Figure 10.3. As {pf}nG

varies over the time step At, the ^integral on the left hand side ofEquation (10.25) represents the areaunder the curve in Figure 10.3 J

between /, and t2. However, themanner in which {pf}fiG varies (i.e theshape of the curve) is unknown, butthe value of ({pf}nG)\ is known while h hthe value of ({pf} tlG)2 is being sought. < •Equation (10.25) is therefore anapproximation of the area under the Figure 10.3: Approximation of porecurve. For example, if fi=\ the area is fluid integralessentially assumed to be {{pf}tlG)2At.Alternatively, if ft = 0.5 the area is approximated by 0.5 At [({pf}tiG)\+ ({P/},*;^]-In

order to ensure stability of the marching process, it is necessary to choose /?>0.5(Booker and Small (1975)). Substituting Equation (10.25) into (10.21) gives:

[LG]T{Ad}nG -/3At[0G]{Apf}nG = At (10.26)

Equations (10.14) and (10.26) may now be written in the following incrementalmatrix form:



10.4 Finite element implementationEquation (10.27) provides a set of simultaneous equations in terms of theincremental nodal displacements {Ad}nG and incremental nodal pore fluid pressures{Apf}nG. Once the stiffness matrix and right hand side vector have been assembled,the equations can be solved using the procedures described in Section 2.9.

As a marching procedure is necessary to solve for the time dependentbehaviour, the analysis must be performed incrementally. This is necessary evenif the constitutive behaviour is linear elastic and the permeabilities are constant. Ifthe constitutive behaviour is nonlinear, the time steps can be combined withchanges in the loading conditions so that the complete time history of constructioncan be simulated. The solution algorithms described in Chapter 9 can therefore beused.

In the above formulation the permeabilities have been expressed by the matrix[k]. If these permeabilities are not constant, but vary with stress or strain, thematrix [A:] (and therefore [0G] and [nG]) are not constant over an increment of ananalysis (and/or a time step). Care must therefore be taken when solving Equation(10.27). This problem is similar to that associated with nonlinear stress-strainbehaviour where [KG] is not constant over an increment. As noted in Chapter 9,there are several numerical procedures available for dealing with a nonlinear [KG],and, as demonstrated, some of these are more efficient than others. All theprocedures described in Chapter 9 (e.g. tangent stiffness, visco-plastic andNewton-Raphson) can be modified to accommodate nonlinear permeability. However, theAuthors' experience is that the modified Newton-Raphson scheme, with asubstepping stress point algorithm, is the most accurate.

In Equation (10.3) the incremental pore fluid pressure within an element hasbeen related to the values at the nodes using the matrix of pore fluid shapefunctions [Np]. If an incremental pore fluid pressure degree of freedom is assumedat each node of every consolidating element, [Np] is the same as the matrix ofdisplacement shape functions [N], Consequently, pore fluid pressures vary acrossthe element in the same fashion as the displacement components. For example, foran eight noded quadrilateral element, both the displacements and pore fluidpressures vary quadratically acrossthe element. However, if thedisplacements vary quadratically, thestrains, and therefore the effectivestresses (at least for a linear material),vary linearly. There is therefore aninconsistency between the variation ofeffective stresses and pore water • Displacement DOFpressures across the element. While o Displacement + pore fluid pressure DOFthis is theoretically acceptable, someusers prefer to have the same order ofvariation of both effective stresses Figure 10.4: Degrees of freedom forand pore water pressure. For an eight sn eight noded element


•a • o-Sand: nor-consc lidating

elei tients

Cla|y: con iolidatfngelei nents


noded element this can be achieved by only having pore fluid pressure degrees offreedom at the four corner nodes, see Figure 10.4. This will result in the [Np]matrix only having contributions from the corner nodes and therefore differingfrom [N\. Similar behaviour can be achieved by only having pore fluid pressuredegrees of freedom at the three apex nodes of a six noded triangle, or at the eightcorner nodes of a twenty noded hexahedron. Some software programs allow theuser to decide which of these two approaches to use.

It is possible to have someelements within a finite element meshwhich are consolidating and somewhich are not. For example, if asituation where sand overlies clay isbeing modelled, consolidatingelements (i.e. elements with porepressure degrees of freedom at their —nodes) might be used for the clay,whereas ordinary elements (i.e. nopore fluid pressure degrees offreedom at the nodes) might be usedfor the sand, see Figure 10.5. The p/gure IQ.5: Choice of elements forsand is then assumed to behave in a consolidating and non-consolidatingdrained manner by specifying a zero layersvalue for the bulk compressibility ofthe pore fluid, see Section 3.4. Clearly, care has to be taken to ensure the correcthydraulic boundary condition is applied to the nodes at the interface between clayand sand. Some software programs insist that the user decides which elements areto consolidate and which are not at the mesh generation stage. Others are moreflexible and allow the decision to be made during the analysis stage.

In the theory developed above, the finite element equations have beenformulated in terms of pore fluid pressure. It is also possible to formulate theequations in terms of hydraulic head, or in terms of excess pore fluid pressure. Insuch cases the hydraulic head or excess pore fluid pressure at the nodes willbecome degrees of freedom. It is important that the user is familiar with theapproach adopted by the software being used, as this will affect the manner inwhich the hydraulic boundary conditions are specified.

10.5 Steady state seepageIf the soil skeleton is assumed to be rigid, there can be no soil deformation andonly flow of pore fluid through the soil exists. Equation (10.14) is therefore notapplicable and Equation (10.21) reduces to:

-[<?hl{Pf}nG = [ % ] + 2 (10.28)

This is the finite element equation for steady state seepage. The only degrees of



freedom are the nodal pore fluid pressures. If the permeabilties are constant and theflow is confined, Equation (10.28) can be solved by a single inversion of the matrix[<PG]. As only pore fluid pressures are calculated, it is only possible to havepermeabilities varying with pore fluid pressure. If this is the case and/or if the flowis unconfmed, an iterative approach must be used to solve Equation (10.28).

If a particular piece of finite element software can deal with the coupledformulation given by Equation (10.27), but not the steady state formulation givenby Equation (10.28), it is still possible to use it to obtain a steady state seepagesolution. This is achieved by giving the soil fictitious linear elastic properties andapplying sufficient displacement constraints to prevent rigid body motion. Ananalysis is then performed applying the correct hydraulic boundary conditions andsufficient time steps for steady state conditions to be achieved. Once steady stateconditions have been reached, soil deformations are zero and the solution istherefore equivalent to that given by Equation (10.28).

10.6 Hydraulic boundary conditions10.6.1 IntroductionWith either coupled or steady state seepage analysis there are pore fluid pressuredegrees of freedom at the nodes, and for each node on the boundary of the mesh(or of that part of the mesh consisting of consolidating elements) it is necessary tospecify either a prescribed pore fluid pressure or a prescribed nodal flow. If acondition is not specified by the user, for one or several of the boundary nodes,most software packages will assume a default condition. This usually takes theform of zero nodal flow. Clearly, the user must be fully aware of the defaultcondition that the software assumes, and account for this when specifying theboundary conditions for an analysis. Boundary conditions can also be prescribedat internal nodes of the finite element mesh.

Prescribed values of incremental nodal pore fluid pressure affect only the left-hand side (i.e. {Apf}f]G) of the system equations. They are dealt with in a similarfashion to prescribed displacements. Prescribed nodal flow values affect the righthand side vector (i.e. Q) of the system equations. They are treated in a similar wayto prescribed nodal forces. They can be specified in the form of sources, sinks,infiltration and precipitation boundary conditions. It is also possible to tie nodalpore fluid pressures in a similar manner to that described for displacements inChapter 3. Such boundary conditions will affect the whole structure of the systemequations. The various hydraulic boundary condition options that are useful forg^Qtechnical engineering are discussed below.

10.6.2 Prescribed pore fluid pressuresThis option allows the user to specify a prescribed incremental change in nodalpore fluid pressure, {Apf}nG. As pore fluid pressure is a scalar quantity, local axesare irrelevant. Prescribed changes in pore fluid pressures are dealt with in a similarway to prescribed displacements, as described in Section 3.7.3.



Although it is the incremental change in pore fluid pressure that is the requiredquantity when solving Equation (10.27), it is often more convenient for the userto specify the accumulated value at the end of a particular increment. It is then leftto the software to work out the incremental change from the prescribed value,given by the user for the end of the increment, and the value stored internally in thecomputer, for the beginning of the increment. It is noted that not all softwarepackages have this facility. It should also be noted that some software packagesmay use change in head or excess pore fluid pressure instead of pore fluid pressure,as the nodal degree of freedom. Consequently, the boundary conditions will haveto be consistent.

As an example of the use ofprescribed pore fluid pressures,consider the excavation problemshown in Figure 10.6. Throughout theanalysis it is assumed that on the righthand side of the mesh the pore fluidpressures remain unchanged fromtheir initial values. Consequently, forall the nodes along the boundary AB,a zero incremental pore fluid pressure(i.e. Apf = 0) is specified for everyincrement of the analysis. The firstincrements of the analysis simulateexcavation in front of the wall and it is assumed that the excavated surface isimpermeable. Consequently, no pore fluid pressure boundary condition isprescribed along this surface and a default condition of zero nodal flow is imposed.However, once excavation is completed, as shown in Figure 10.6, the excavatedsoil surface is assumed to be permeable, with a zero pore fluid pressure. Therefore,for the increment after excavation has been completed, the final accumulated value(i.e. pf= 0) is specified along CD. As the program knows the accumulated porefluid pressure at the nodes along this boundary at the end of excavation, it canevaluate Apf. For subsequent increments the pore fluid pressure remains at zeroalong CD and consequently Apf = 0 is applied.

Figure 10.6: Prescribed pore fluidboundary conditions

10.6.3 Tied degrees of freedomThis boundary condition option allows a condition of equal incremental nodal porefluid pressure to be imposed at two or more nodes, whilst the magnitude of theincremental nodal pore fluid pressure remains unknown. This concept is explainedin detail for displacements in Section 3.7.4 and therefore is not repeated here.Because pore fluid pressure is a scalar quantity, there is only one tying option,compared to the several that are available for displacement which is a vector, seeFigure 3.20.

As an example of the use of tied pore fluid pressure, consider the example oftwo consolidating layers of soil, separated by interface elements shown in Figure


Seepage and consolidation / 31 5

^ooo^Layer 1

Interfaceelements

Layer 2

trtrxrc

Pf=Pi Pf ~

10.7. Because interface elements havezero thickness, they are not usuallyformulated to account forconsolidation. In the situation shownin Figure 10.7, there is a set of nodesalong the underside of soil layer 1 andanother set on the upper surface ofsoil layer 2, corresponding to theupper and lower side of the row ofinterface elements respectively.Because the interface elements do notaccount for consolidation, there is noseepage link between these two rowsof nodes and, unless a boundarycondition is specified for these nodes,most software programs will treat each row as an impermeable boundary (i.e. zeroincremental nodal flow). If the interface is to be treated as a permeable boundary,the solution is to tie the incremental pore fluid pressures of adjacent nodes acrossthe interface elements. For example, tie the incremental pore fluid pressures fornodes AB, CD, ..., etc.

Figure 10.7: Tied pore fiuidpressures

Rainfall intensity = qn

I I I U I I I I

10.6.4 InfiltrationWhen it is necessary to prescribe pore fluid flows across a boundary of the finiteelement mesh for a particular increment of the analysis, infiltration boundaryconditions are used. These flows are treated in a similar fashion to boundarystresses as described in Section 3.7.6.

An example of an infiltrationboundary condition is shown inFigure 10.8, where it is assumed thatrainfall provides a flow rate qn on thesoil surface adjacent to theexcavation. In general, the flow ratemay vary along the boundary overwhich it is active. To apply such aboundary condition in finite elementanalysis, the flow over the boundarymust be converted into equivalent

nodal flows. Many finite element Fjgure W.8: Example of infiltrationprograms will do this automatically boundary conditionsfor generally distributed boundaryflows and for arbitrary shaped boundaries.

The nodal flows equivalent to the infiltration boundary condition aredetermined from the following equation:



Qinfi,} =Srf

gn dSrf (10.29)

where Srfis the element side over which the infiltration flow is prescribed. As withboundary stresses, this integral can be evaluated numerically for each element sideon the specified boundary range, see Section 3.7.6.

Extraction well / /

ft/I

// Z

Injection well

10.6.5 Sources and sinksA further option for applying flow boundary conditions is to apply sources (inflow)or sinks (outflow) at discrete nodes, in the form of prescribed nodal flows. Forplane strain and axi-symmetric analyses these are essentially line flows actingperpendicularly to the plane of the finite element mesh.

An example of a source and sinkboundary condition is shown inFigure 10.9 in the form of a simpledewatering scheme, involving a rowof extraction wells (sinks) within anexcavation and, to limit excessivesettlements behind the retaining wall,a row of injection wells (sources).The effect of the extraction wellscould be modelled by applying a flowrate equivalent to the pumping rate atnode A, and the effect of the injectionwells could be modelled by applyinga flow rate equivalent to the injectionrate at node B.

Figure 10.9: Example of sourcesand sinks boundary conditions

10.6.6 PrecipitationThis boundary condition option allows the user to essentially prescribe a dualboundary condition to part of the mesh boundary. Both an infiltration flow rate, qm

and a pore fluid pressure, pfh, are specified. At the start of an increment, each nodeon the boundary is checked to see if the pore fluid pressure is more compressivethan pfh. If it is, the boundary condition for that node is taken as a prescribedincremental pore fluid pressure Apf, the magnitude of which gives an accumulatedpore fluid pressure equal to pfh at the end of the increment. Alternatively, if thepore fluid pressure is more tensile than/?^, or if the current flow rate at the nodeexceeds the value equivalent to qn, the boundary condition is taken as a prescribedinfiltration with the nodal flow rate determined from qn. The following twoexamples show how this boundary condition may be used.



Tunnel problem

flow from tunnel

a) Prescribed pore fluid pressureas boundary condition in short term

Because pf more tensilethan/7^, zero flowboundary conditions adopted

b) Precipitation boundaryconditions in short term

/^more compressive than/?^:pore fluid boundaryconditions adopted

PjmoxQ tensile than/?^:flow boundaryconditions adopted

c) Precipitation boundary conditionsat intermediate stage

pf at all nodes morecompressive than j ^ :pore fluid boundaryconditions adoptedat all nodes

d) Precipitation boundaryconditions in long term

Figure 10.10: Precipitation boundary conditions in tunnel problem

After excavation for a tunnel, assuming the tunnel boundary to be impermeable,the pore fluid pressure in the soil adjacent to the tunnel could be tensile, seeVolume 2 of this book. If for subsequent increments of the analysis (tunnelboundary now permeable) a prescribed zero accumulated pore fluid pressureboundary condition is applied to the nodes on the tunnel boundary, flow of waterfrom the tunnel into the soil would result, see Figure 10.10a. This is unrealistic,because there is unlikely to be a sufficient supply of water in the tunnel. Thisproblem can be dealt with using the precipitation boundary option with qn = 0 andp/h = 0. Initially (after excavation), the pore fluid pressures at the nodes on thetunnel boundary are more tensile thanpjh, consequently a flow boundary conditionwith qf = 0 (i.e. no flow) is adopted, see Figure 10.10b. With time the tensile porefluid pressures reduce due to swelling and eventually become more compressivethanpjh. When this occurs, the pore fluid stress boundary condition is applied, witha magnitude set to give an accumulated pore fluid pressure at the end of theincrement equal topfh, see Figure lO.lOd. The pore fluid pressure checks are madeon a nodal basis for all nodes on the tunnel boundary for each increment. Thisimplies that the boundary condition can change at individual nodes at differentincrements of the analysis. At any one increment some nodes can have a prescribedpore fluid stress boundary condition, while others can have a flow condition, seeFigure 10.10c.



Rainfall infiltrationIn this case the problem relates to aboundary which is subject to rainfallof a set intensity. If the soil is ofsufficient permeability and/or therainfall intensity is small, the soil canabsorb the water and a flow boundarycondition is appropriate, see Figure10.11a. However, if the soil is lesspermeable and/or the rainfall intensityis high, the soil will not be able toabsorb the water, which will pond onthe surface, see Figure 10.1 lb. Thereis a finite depth to such ponding, •which is problem specific, andconsequently a pore fluid pressureboundary condition would beapplicable. However, it is not alwayspossible to decide which boundarycondition is relevant before an

uuuuj \ j \ / \ /

Low rain fall intensitycompared with permeability of soil;no surface ponding

11111111.

b)

High rain fall intensitycompared with permeability of soil;surface ponding

Figure 10.11: Rainfall infiltrationboundary conditions

analysis is undertaken, because the behaviour will depend on soil stratification,permeability and geometry. The dilemma can be overcome by using theprecipitation boundary condition, with qn set equal to the rainfall intensityset to have a value more compressive (i.e. equivalent to the ponding levelthe initial value of the pore fluid stress at the soil boundary. Because pf] is moretensile than/?^, a flow boundary condition will be assumed initially. If during theanalysis the pore fluid pressure becomes more compressive than/?/A, the boundarycondition will switch to that of a prescribed pore fluid pressure.

10.7 Permeability models10.7.1 IntroductionWhen performing coupled (or steady seepage) analysis it is necessary to inputpermeability values for the soils undergoing seepage. For coupled analysis it is alsonecessary to input the constitutive behaviour. Several options exist for specifyingpermeabilities. For example, the soil can be assumed to be isotropic or anisotropic,the permeabilities can vary spatially, or they can vary nonlinearly as a function ofvoid ratio or mean effective stress. Some of the models that the Authors find usefulare briefly described below.

10.7.2 Linear isotropic permeabilityThis model assumes permeability to be isotropic and, at any point, defined by asingle value of k. However, in most soils the permeability varies with void ratio



and therefore mean effective stress, or 30depth. It is therefore convenient tohave the option to vary k spatially.For example, options are often 20 -available for varying A: in a piece wise ^linear fashion across a finite element w

mesh. This option can be used to .0simulate a permeability varying with j>depth, see Figure 10.12 (note: 3coefficient of permeability is plottedon a log scale). With this model, thepermeability at an integration pointremains constant throughout ananalysis. -10

Thames gravel

Weathered London clay i

Unweathered London clay /

L

Woolwicji & Reading bed cla[y

Woolwich & Reading bed sand

101 101 io-8

Coefficient of permeability, k (m/s)

Figure 10.12: Permeability profilefor London clay

10.7.3 Linear anisotropicpermeability

In this model the permeability isassumed to be direction dependent. Aset of permeability axes are defined(xm , ym , zm) and values for the coefficient of permeability in each directionspecified (kxm, kym, kzm). This enables the permeability matrix in Equation (10.7) tobe obtained. It should be noted that this matrix is associated with the globalcoordinate axes and therefore, if the material and global axes differ, thepermeability coefficients must be transformed. This is usually performedautomatically by the software. Again, it is useful to have the option of varying thepermeability values spatially.

10.7.4 Nonlinear permeability related to void ratioIn this model the permeability is assumed to be isotropic, but to vary as a functionof the void ratio. The following relationship between the coefficient ofpermeability, k, and void ratio, e, is assumed:

+be)k = (10.30)

where a and b are material parameters. With this model it is necessary to know thevoid ratio at any stage of the analysis. This implies that the initial value, at thebeginning of the analysis, must be input into the finite element program.

From a fundamental point of view, allowing the permeability to vary with voidratio makes sense. However, there is often little laboratory or field data availableto determine the parameters a and b. Consequently, it is often convenient to adoptan expression in which k varies with mean effective stress. Two such models arepresented below.



10.7.5 Nonlinear permeability related to mean effective stressusing a logarithmic relationship

Again the permeability is assumed to be isotropic, but to vary as a function ofmean effective stress,/?', according to the following relationship:

(10.31)

where ko is the coefficient of permeability at zero mean effective stress and a is aconstant, incorporating the initial void ratio at zero mean effective stress and thecoefficient of volume compressibility, mv. The derivation of the logarithmic lawtherefore incorporates the assumption that mv is constant (Vaughan (1989)).

10.7.6 Nonlinear permeability related to mean effective stressusing a power law relationship

This model has been derived assuming that the compression index, Cc, remainsconstant (Vaughan (1989)). The relationship between permeability and meaneffective stress takes the form:

(10.32)

where again ko and a are material parameters.

10.8 Unconfined seepage flowSituations which involve unconfined flow, where it is necessary for the softwareto determine a phreatic surface, can be problematic, because there does not appearto be a robust algorithm for finding and accommodating the phreatic surface. Thereare several different algorithms available. Some of these involve adjustments to thefinite element mesh so that the phreatic surface follows a mesh boundary. Suchmethods are not applicable to coupled consolidation analysis.

A more general approach is to reduce the permeability when the soil sustainsa tensile pore fluid pressure associated with a position above the phreatic surface.A typical variation of permeabilitywith pore fluid pressure is shown inFigure 10.13. If the accumulated porefluid pressure is more compressivethan pfu the soil's normalpermeability (e.g. given by one of themodels described above) is adopted.However, if the accumulated porefluid pressure becomes more tensilethan pj2 , the soil is assumed to be Figure 10.13: Variation ofabove the phreatic surface and the permeability with pore fluid pressure

Log k = log kn* 5 f clogk

\ogR/—

Pn

KJR

(+ve)Pore pressure



normal permeability is reduced by a large factor, R. Usually R takes a valuesomewhere between 100 and 1000. If the accumulated pore fluid pressure isbetween/fy and/?/2, the permeability is found using a linear interpolation betweenthe two extreme values. Clearly, this approach requires an iterative algorithm. Itis therefore well suited to a nonlinear solution strategy of the modified Newton-Raphson type.

Analysis involving a phreatic surface also require the use of the precipitationboundary option, so that boundary nodes can automatically switch from aprescribed pore fluid pressure to a prescribed flow.

The Authors' experience is that numerical instability can sometimes arise withthis approach and that more research is required to obtain a robust algorithm.

10.9 Validation exampleClosed form solutions forconsolidation problems are not easyto obtain. This is especially true whendealing with elasto-plastic materials.Exact solutions have been found onlyto problems involving linear elasticmaterials, with simple geometry andsubjected to simple boundaryconditions.

One such problem is that of aporous elastic half-space, subjected toa load of intensity q over a width 2a,under conditions of plane strain. Thisproblem was used by the Authors asone of a series of validation exerciseswhen coupled analysis was first codedinto ICFEP. Some of the results fromthese analyses are now presented. " "

Originally, the finite element meshshown in Figure 10.14 was used. The Fi9ure 10-14: FE mesh for

boundary conditions employed are consolidation problem -1

noted on this figure. The water table was assumed to be at the ground surface.results are expressed in terms of the adjusted time factor, T:

II

Free,

X

Smoo

th, p

erm

eabl

e bo

unda

ryimpe

Fixed, i

rmeablebound

a

ary

mpermeable boundary

b = 6a

fB

The

rp _ Ct

and the adjusted coefficient of consolidation, c:

-_ 2Gk

(10.33)

(10.34)



Schiffinanefa/., 1969FE prediction, b=X8aFE prediction, b-6a

where G is the elastic shear modulus, k is the coefficient of permeability, yf\s thebulk unit weight of the pore fluid, and t denotes time.

The load intensity q was applied in a very small time step (expressed in termsof the adjusted time factor, AT", it was 1.15xlO"5), resulting in an undrainedresponse. Logarithmic time increments, typically five per 'log' cycle, were usedthereafter. The response during the first iog' cycle was essentially undrained, andthe results for this iog' cycle are not presented.

The variation of normalisedexcess pore fluid pressure, pexcess /q, °beneath the centre of the loaded areais given in Figure 10.15 for thespecific case of T= 0.1 andPoisson'sratio ju=0. It can be seen that thefinite element prediction (opensquares) over predicts the excesspore fluid pressure, particularly atgreater depths beneath the loadedarea. Because the closed formsolution, presented by the full line inFigure 10.15 (Schiffman et al(1969)), was derived for an infinitehalf-space, it was suspected that thisdiscrepancy was due to the closeproximity of the bottom and/or lateral boundaries in the finite element mesh. Theeffect of the position of both boundaries on the predictions was investigated and,surprisingly, it was found that the position of the lateral boundary has the greatestinfluence. Results from the mesh shown in Figure 10.16, which has a similar depthto that given in Figure 10.14, but three times the width, are shown in Figure 10.15as solid squares. These results are in excellent agreement with the closed formsolution, except for a slight discrepancy at the base of the mesh.

In Figure 10.17 the variation of normalised excess pore fluid pressure, pexcess/q,with adjusted time factor, T, is given for two specific points in the half-space, seeFigure 10.16. Two predictions were made, both using the mesh shown in Figure10.16, with different forms of the element pore fluid pressure interpolation matrix,[7V/;]. Eight noded isoparametric elements were employed and in one analysis,labelled as code 8, all eight nodes had a pore fluid pressure degree of freedom. Inthe other analysis, labelled code 4, only the four corner nodes had pore fluidpressure degrees of freedom. Consequently, in the code 8 analysis there was aquadratic variation of pore fluid pressure across each finite element, whereas in thecode 4 analysis the distribution was linear. Also shown in Figure 10.17 is theclosed form solution. It can be seen that the predictions from both finite elementanalyses are in excellent agreement with the closed form solution. In addition,there is very little difference between the predictions from the two finite elementanalysis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Normalized excess pore pressure, pexcess Iq

Figure 10.15; Variation ofnormalised excess pore pressure



B:y/a =x/a = 0.0 >

A: y/a = 0.5x/a =1.0 '

Free, permeable boundary

Fixed, impermeable boundary

Z>=18a

Figure 10.16: F.E. mesh for consolidationproblem - II

yla = 0.5B, Schiffinanefa/., 1969A,Schiffinan£tfa/., 1969B, FE prediction, code 4A, FE prediction, code 4B, FE prediction, code 8A, FE prediction, code 8

0.0001 0.001 0.01 0.1Adjusted time factor, T

Figure 10.17: Variation of normalised excesspore pressure

10.10 Summary1. This chapter has considered the modifications to the finite element theory that

are necessary to enable time dependent soil behaviour to be simulated. Thisinvolves combining the equations governing the flow of pore fluid through thesoil skeleton, with the equations governing the deformation of the soil due toloading.

2. The resulting finite element equations involve both displacement and porefluid pressure degrees of freedom at element nodes. In addition, to enable asolution to be obtained, a time marching algorithm is necessary. This involves



an assumption of the magnitude of the average pore fluid pressures over eachtime step. It is assumed that, at each node, the average pore fluid pressureduring the time step is linearly related to the values at the beginning and endof the time step. This involves the parameter/?. For numerical stability /?> 0.5.

3. Even for linear material behaviour analysis must be performed incrementally,to accommodate the time marching process.

4. It is possible to have some elements within a finite element mesh which areconsolidating and some which are not.

5. If the soil skeleton is rigid, there can be no deformation and only flow of porefluid through the soil occurs. This results in a considerable reduction in thecomplexity of the governing finite element equations. Only pore fluid pressuredegrees of freedom need to be considered.

6. In coupled analysis hydraulic boundary conditions must be considered. Theseconsist of prescribed nodal pore fluid pressures, tied pore fluid freedoms,infiltration, sinks and sources and precipitation.

7. For all nodes on the boundary of the finite element mesh either a prescribedpore fluid pressure or a flow boundary condition must be specified. If aboundary condition is not set for a boundary node, most software packageswill implicitly assume that the node represents an impermeable boundary (i.ea zero nodal flow condition).

8. It is also necessary to input the coefficients of soil permeability. Thepermeability can take several different forms: it can be isotropic or anisotropic,it can vary spatially, or it can vary as a function of void ratio or mean effectivestress. In the latter case the permeability at each integration point will varyduring the analysis and the software must have the appropriate algorithms tocope with this.

9. Problems involving unconfined seepage, in which the analysis has todetermine the position of a phreatic surface, can be accommodated. However,at present the algorithms that are available are not robust, which can lead tonumerical instability. Further research is required.

10. The coupled theory presented in this chapter assumes the soil to be fullysaturated. Further complications arise if the soil is partly saturated.


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