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A meshless method for axisymmetric problems in continuously nonhomogeneous saturated porous media

A meshless method for axisymmetric problems in continuously nonhomogeneous saturated porous media
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  A meshless method for axisymmetric problems in continuouslynonhomogeneous saturated porous media  J. Sladek a, ⇑ , V. Sladek a , M. Schanz b a Institution of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia b Institute of Applied Mechanics, Graz University of Technology, Graz, Austria a r t i c l e i n f o  Article history: Received 15 October 2013Received in revised form 6 June 2014Accepted 8 July 2014Available online 2 August 2014 Keywords: Solid and fluid phasesCoupled problemMeshless local Petrov–Galerkin method(MLPG)Moving least-squares approximation3D axisymmetric problemBorehole a b s t r a c t A meshless method based on the local Petrov–Galerkin approach is proposed to analyze 3-d axisymmet-ricproblemsinporousfunctionallygradedmaterials. Constitutiveequationsfor porousmaterialspossessa coupling between mechanical displacements for solid and fluid phases. The work is based on the u–uformulation and the incognita fields of the coupled analysis in focus are the solid skeleton displacementsandthefluiddisplacements. Independentspatial discretizationisconsideredforeachphaseofthemodel,rendering a more flexible and efficient methodology. Both displacements are approximated by themoving least-squares (MLS) scheme. The paper presents in the first time a general meshless methodfor the numerical analysis of axisymmetric problems incontinuously nonhomogeneous saturated porousmedia. Numerical results are given for boreholes in continuously nonhomogeneous porous mediumwithprescribed misfit and exponential variation of material parameters in the excavation zone.   2014 Elsevier Ltd. All rights reserved. 1. Introduction Poroelasticity belongs to the continuummechanics with two ormore phases. The one-dimensional theory of the consolidation of awater saturated elastic porous geomaterial was first developed byTerzaghi [1]. Later Biot [2] formulated a theory for multidimen- sional problems of porous materials saturated by a viscous fluid.The generalized three-dimensional theory of poroelasticity inanisotropic porous materials has been developed by Biot [3] too.Theory of poroelasticity has been successfully applied in the studyof variety of problems in geomechanics, biomechanics, materialsengineering, environmental geomechanics and energy resourcerecovery from geological formations [4,5]. The extension to a nearly saturated poroelastic material has been presented by Aifan-tis [6] and Wilson and Aifantis [7] for the quasi-static case. The dynamic extension of Biot’s theory to three phases has been pub-lished by Vardoulakis and Beskos [8]. A state of the art overview on the theory of dynamic poroelasticity, its numerical approxima-tion, and applications may be found in Schanz’ review paper [5].Since the coupled differential equations are generally difficultto solve exactly, it appears that numerical approaches have to beadopted to attain solutions. Analytical methods are restricted tosimple boundaryvalue problems. Anice reviewis givenby Selvad-urai [9]. Despitethe universalityandgreat successof the finiteand boundary element methods in their applicability even to multi-field problems, there are some restrictions leading to exclusion of the finite elements with equal order interpolation for pressureand displacements in poroelastic problems [10–13]. The dynamic Green’s function of homogeneous poroelastic half-plane has beenderived by Senjuntichai and Rajapakse [14] and later applied to avertical vibration of an embedded rigid foundation in a poroelasticsoil[15].Theboundaryelementmethod(BEM)hasbeendeveloped for transient and time harmonic analysis of dynamic poroelasticityproblems [16]. Later asimple BEM formulation for poroelasticity via particular integrals has been developed by Banerjee [17]. Time domain BEM has been applied for axisymmetric quasi-staticporoelasticity [18]. Dynamic Green’s functions for poroelastic and layered poroelastic half-spaces have been derived in [19] and[20]. In general, material coefficients in poroelasticity are aniso-tropic [21] and spatially variable.Axisymmetricproblemshavereceivedconsiderableattentioninthe past due to their close relevance to geotechnical and rocktesting methods such as uni-axial and tri-axial compression tests,double-punchtestsandpointloadstrengthtests.Inaddition,stressanalysis of cylinders is also relevant to applications involving bio-medical andmechanical engineering. Acylindrical boreholedrilledin a soil/rock medium is commonly found in the petroleum indus-try. Stabilityofboreholeis importantbecauseitis theoneofmajor   2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: (J. Sladek).Computers and Geotechnics 62 (2014) 100–109 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage:  problems in oil and gas industries. In the past, the classical theoryof elasticity has been used extensively to analyze various elasto-static and elastodynamic problems involving cylinders and bore-holes [22]. However, geological materials are normally two-phase materials consisting of a solid skeleton with voids filled withwater. Such materials are commonly known as poroelastic materi-als and widely considered as much more realistic representationfor natural soils and rocks than ideal elastic materials [23]. Thegoverningequationsfor a poroelastic material undergoing axisym-metric deformations are given by Rice and Cleary [24]. Amisfit of  the radial displacement with respect to the borehole diameter isconsidered here. A cylindrical borehole in a poroelastic mediumwith consideration of excavation disturbed zone is considered.Shearmodulusandpermeabilitycoefficientareassumedtobecon-tinuously non-homogenous in radial direction for the disturbedzone. Vrettos [25] derived Green’s functions for vertical point loadon a non-homogeneous medium with anexponential variation of the shear modulus decreasing with depth.Inspiteof thegreatsuccessof domainandboundarydiscretiza-tion methods for the solution of general boundary value problems,there is still a growing interest in the development of newadvanced computational methods. The finite element method(FEM) can be successfully applied to problems with an arbitraryvariation of material properties by using special graded elements.In commercial computer codes, however, material properties areconsidered to be uniform within each element. In recent years,meshless formulations are becoming popular due to their highadaptability and easier preparation of input and output data innumerical analyses. The moving least squares (MLS) approxima-tion is generally considered as one of many schemes to interpolatediscretedatawithareasonableaccuracy.Theorderofcontinuityof the MLS approximation is given by the minimum between theorders of continuity of the basis functions and that of the weightfunction. Thus, continuity can be tuned to a desired value. Inconventional discretization methods, however, the interpolationfunctions usually result in a discontinuity in the secondary fields(gradients of primary fields) on the interfaces of elements. Formodeling coupledfields the approach based onpiecewise continu-ous elements can bring some inaccuracies. Therefore, a modelwhich is based on C 1 -continuity, such as the meshless method, isexpected to be more accurate than conventional discretizationtechniques. A drawback of meshless methods is higher CPU timecompared to regular FEM. However, this drawback can be over-come. Recently, the authors [48–51] have developed a modifiedMLPG formulation, where Taylor series expansions and analyticalintegrations over the local sub-domains in two-dimensional elas-todynamics are applied.Avarietyofmeshlessmethodscanbederivedfromaweak-formformulationeither on the global domainor on a set of local subdo-mains. In the global formulation, background cells are required forthe integrationof theweak-form. Inmethodsbasedonlocal weak-form formulation, on the other hand, no background cells arerequired. The meshless local Petrov–Galerkin (MLPG) method is afundamental base for the derivation of many meshless formula-tions,sincethetrialandtestfunctionscanbechosenfromdifferentfunctionalspaces[26–29].TheMLPGmethodwithaHeavisidestep function as the test function [29] has been successfully applied tosolve various 3-d axisymmetric problems [30–32]. The MLPG has been successfully applied to porous problems [33,52,53]. Many meshless formulations in poroelastic media have been applied toanalyze consolidation problems [54–57]. The MLPG has beenapplied also to dynamic poroelastic problems, however up to dayonly as two-dimensional analyses [33,52]. In all early published papers basedonthe meshlessformulations homogeneousmaterialproperties are considered. A meshfree algorithm based on theGalerkinapproachisproposedforthefullycoupledanalysisofflowand deformation in unsaturated poroelastic media by Khoshghalband Khalili [34]. Temporal discretization is achieved there using a three-point approximation technique with second order accuracy.Sheu[35] analyzedthepredictionof probabilisticsettlementswiththe uncertainty in the spatial variability of Young’s modulus toillustrate the preliminary development of a spectral stochasticmeshless local Petrov–Galerkin (SSMLPG) method. Generalizedpolynomial chaos expansions of Young’s moduli and a two-dimensional meshfree weak–strong formulation in elasticity arecombined to derive the SSMLPG formulation.In the present paper, the MLPG is developed for an axisymmet-ric 3D boundary value problem in a porous material with continu-ously varying material properties. It is the first meshlessapplication to such a problem. Because of the axial symmetry,the analyzed domain is the cross-section of the considered bodywith the plane involving the axis of symmetry. Both governingequations for the balance of momentum in solid and fluid phasesare satisfied in a weak form on small fictitious subdomains in thepresent paper. Nodal points are introduced and spread on the ana-lyzed domain and each node is surrounded by a small circle forsimplicity; but in general, it can be of an arbitrary shape. The spa-tial variations of the displacements in solid and fluid phases areapproximated by the moving least-squares scheme [36,37]. After performing the spatial integrations, one obtains a system of ordin-ary differential equations (ODE) for temporal variations of certainnodal unknowns. The backward difference method is applied forthe approximation of ‘‘velocities’’ and the Houbolt method [38] isappliedfortheaccelerationsintheODE. Theinfluenceofthemate-rial gradation on displacements and stresses in porous mediumaround the borehole is investigated. 2. Local boundary integral equations In Biot’s theory a fully saturated material is considered, i.e., anelastic skeleton with astatistical distribution of interconnectedpores is modeled. The porosity is denoted by /  ¼  V   f  V   ð 1 Þ where  V   f  is the volume of the interconnected pores contained in asample of bulk volume  V  . The sealed pores are considered as partof the solid. Full saturation is assumed leading to  V   = V   f  + V  s with V  s the volume of the solid. There are several possibilities to writegoverning equations for porous materials:(i) To use the solid displacement  u si  and the fluid displacement u  f i  with six (four) unknowns in 3-d (2-d) problems [39].(ii) Alternatively the solid displacement  u si  and the seepage dis-placement w i  ( w i  ¼  / ð u  f i   u si Þ )withalsosix(four)unknownsin 3-d (2-d) can be used [40]. Beside the seepage displace- ment also sometimes the seepage velocity, i.e., the timederivative of   w i  is applied.(iii) A combination of the pore pressure  p  and the soliddisplacement  u si  with four (three) unknowns in 3-d (2-d)can be established. As shown by Bonnet [41], this choice is sufficient.In the present paper we use the first approach based on solidand fluid displacements. If the constitutive equations are formu-lated for the elastic solid and the interstitial fluid, apartial stressformulation is obtained [2,3] r sij  ¼  2 G e sij  þ  K    23 G þ  Q  2 R  ! e skk d ij  þ  Q  e  f kk d ij ;  ð 2 Þ r  f  ¼  /  p  ¼  Q  e skk  þ  R e  f kk ;  ð 3 Þ  J. Sladek et al./Computers and Geotechnics 62 (2014) 100–109  101  where strain tensors are given by e sij  ¼  12  u si ;  j  þ  u s j ; i    and  e  f kk  ¼  u  f k ; k : The respective stress tensors are denoted by  r sij  and  r  f  d ij . Theelastic skeleton is assumed to be isotropic and homogeneous withthe two elastic material constants bulk modulus  K   and shearmodulus  G . The couplingbetween the solid and the fluid is charac-terized by the two parameters  Q   and  R . Considerations of constitu-tive relations at micro mechanical level as given by Detournay andCheng[42]showsthattheseparameterscanbecalculatedfromthebulk moduli of the constituents by R  ¼  / 2 K   f  K  s K  s K   f  ð K  s   K  Þ þ  / K  s ð K  s   K   f  Þ ;  ð 4 Þ Q   ¼  / ½ð 1   / Þ K  s   K   K   f  K  s K   f  ð K  s   K  Þ þ  / K  s ð K  s   K   f  Þ ;  ð 5 Þ where  K  s denotes the bulk modulus of the solid grains and  K   f  thebulk modulus of the fluid.AnalternativerepresentationoftheconstitutiveEqs. (2)and(3)is used in Biot’s earlier work [2]. There, the total stress r ij  ¼  r sij  þ r  f  d ij  is introduced and with Biot’s effective stresscoefficient a  ¼  1   K K  s  ¼  /  1 þ  Q R   ;  ð 6 Þ the constitutive equation can be written as r ij  ¼  2 G e sij  þ  K    23 G   e skk d ij   a d ij  p  ¼  r effecti v  eij   a d ij  p :  ð 7 Þ The governing equations for porous materials can be written asthe balance of momentum for total stresses and for fluid [39]: r ij ;  j ð  x  ; s Þ þ ð 1   / Þ  f  si  þ  /  f   f i  ¼ ð 1   / Þ q s € u si ð  x  ; s Þ þ  / q  f  € u  f i ð  x  ; s Þ ;  ð 8 Þ r  f  ; i ð  x  ; s Þ þ  /  f   f i  ¼  / q  f  € u  f i ð  x  ; s Þ þ q  A  € u  f i ð  x  ; s Þ   € u si ð  x  ; s Þ h i þ  / 2 j  _ u  f i ð  x  ; s Þ   _ u si ð  x  ; s Þ h i ;  ð 9 Þ where  f  si  and  f   f i  are denote the volume densities of body forces insolid and fluid medium, respectively. Further, the respective massdensities are denoted by  q s  and  q  f  . The apparent mass density  q  A is introduced [39] to describe the dynamic interaction betweensolid and fluid phases. Usually it has the following form,  q  A  = C  / q  f  , where  C   is a factor depending on the geometry of the poresandthefrequencyofexcitation.Theterm / 2 / j isthedampingfactorvalid for circular pores, when  j  denotes the permeability.Foranaxisymmetricprobleminaporouspiezoelectricmaterial,it is convenient to use cylindrical coordinates  x    ð r  ; u ;  z  Þ . The axisofsymmetryisidentical withthe  z  -axis. Theangularcomponentof the displacements vanishes and all physical fields as well asmaterial coefficients are independent of the angular coordinate u . A cylinder of height  h  and radius  a  is generated by the rotationof the planar domain  X  bounded by the boundary  C  around theaxis of symmetry as shown in Fig. 1.Thus, in the cylindrical coordinates the nonzero strains aregiven as e srr   ¼  u sr  ; r  ; e s uu  ¼  u sr  = r  ;  e srz   ¼ ð u sr  ;  z   þ  u s z  ; r  Þ = 2 ;  e s zz   ¼  u s z  ;  z  ; e  f kk  ¼  u  f r  ; r   þ  u  f  z  ;  z   þ  u  f r  = r  : Generally, material properties are varying with Cartesian coor-dinates. In such a case matrices  C ,  Q   and  R   in the following consti-tutive equations are functions of ( r  ,  z  ) r srr  r s uu r s zz  r srz  0BBB@1CCCA ¼ c  11  c  12  c  13  0 c  12  c  11  c  13  0 c  13  c  13  c  11  00 0 0  c  44 2666437775 u sr  ; r  u sr  = r u s z  ;  z  u sr  ;  z   þ  u s z  ; r  0BBB@1CCCA þ Q Q Q  0 2666437775 e  f kk ;  ð 10 Þ r  f  ¼  Q Q Q  ½  u sr  ; r  u sr  = r u s z  ;  z  0B@1CA þ  R R R ½  u  f r  ; r  u  f r  = r u  f  z  ;  z  0BB@1CCA ;  ð 11 Þ r ab  ¼  r sab  þ  d ab r  f  ;  ð 12 Þ where c  11  ¼  43 G þ  K   þ  Q  2 R  ;  c  12  ¼  c  13  ¼  K    23 G  þ  Q  2 R  ;  c  44  ¼  G : The following essential and natural boundary conditions areassumed for the mechanical field in the solid phase u si ð  x  ; s Þ ¼  ~ u si ð  x  ; s Þ ;  on  C u ; t  i ð  x  ; s Þ   r ij n  j  ¼  ~ t  i ð  x  ; s Þ ;  on  C t  ;  C  ¼  C u  [ C t  ; whilst for the fluid phase u  f i ð  x  ; s Þ ¼  ~ u  f i ð  x  ; s Þ ;  on  C  f u ; r  f  ð  x  ; s Þ ¼  ~ r  f  ð  x  ; s Þ ;  on  C  f t  ;  C  ¼  C  f u  [ C  f t  : where C u  is the part of the global boundary C with prescribed dis-placements, while on C t   the traction vector is given.The initial conditions for mechanical displacements areassumed as u si ð  x  ; s Þ  s ¼ 0  ¼  u si ð  x  ; 0 Þ  and  _ u si ð  x  ; s Þ  s ¼ 0  ¼  _ u si ð  x  ; 0 Þ u  f i ð  x  ; s Þ  s ¼ 0 ¼  u  f i ð  x  ; 0 Þ  and  _ u  f i ð  x  ; s Þ  s ¼ 0 ¼  _ u  f i ð  x  ; 0 Þ  in  X Assuming for vanishing body forces, the governing Eqs. (6) and(7) in the cylindrical coordinate system take the form r rb ; b ð r  ;  z  ; s Þ þ 1 r   ½ r rr  ð r  ;  z  ; s Þ  r uu ð r  ;  z  ; s Þ  ð 1   / Þ q s ð  x  Þ € u sr  ð r  ;  z  ; s Þþ q  f  ð  x  Þ / € u  f r  ð r  ;  z  ; s Þ ¼  0 ;  ð 13 Þ 1=r2ha3=z Γ Ω Fig. 1.  Geometryandgeneratingsectionofacircularcylinderofradius a andheight h .102  J. Sladek et al./Computers and Geotechnics 62 (2014) 100–109  r  zb ; b ð r  ;  z  ; s Þ þ 1 r   r rz  ð r  ;  z  ; s Þ  ð 1   / Þ q s ð  x  Þ € u s z  ð r  ;  z  ; s Þþ q  f  ð  x  Þ / € u  f  z  ð r  ;  z  ; s Þ ¼  0 ;  ð 14 Þ r  f  ; b ð r  ;  z  ; s Þ   / q  f  ð  x  Þ € u  f b ð r  ;  z  ; s Þ  q  A ð  x  Þ½ € u  f b ð r  ;  z  ; s Þ   € u sb ð r  ;  z  ; s Þ  / 2 j  ½ _ u  f b ð r  ;  z  ; s Þ   _ u sb ð r  ;  z  ; s Þ ¼  0 ;  ð 15 Þ with the summation convention being assumed for repeatedsubscript  b  e  { r  ,  z  }.The local weak form of the governing Eqs. (13)–(15) can bewritten as Z  X s r rb ; b U  1 d X þ Z  X s 1 r   ð r rr    r uu Þ U  1 d X  Z  X s ð 1  / Þ q s € u sr  ð r  ;  z  ; s Þ U  1 d X þ Z  X s q  f  / € u  f r  ð r  ;  z  ; s Þ U  1 d X ¼ 0 ;  ð 16 Þ Z  X s r  zb ; b U  2 d X þ Z  X s 1 r   r rz  ð r  ;  z  ; s Þ U  2 d X  Z  X s ð 1   / Þ q s € u s z  ð r  ;  z  ; s Þ U  2 d X þ Z  X s q  f  / € u  f  z  ð r  ;  z  ; s Þ U  2 d X  ¼  0  ð 17 Þ Z  X s r  f  ; b U  3 d X  Z  X s / q  f  € u  f b ð r  ;  z  ; s Þ U  3 d X  Z  X s q  A  € u  f b ð r  ;  z  ; s Þ   € u sb ð r  ;  z  ; s Þ h i U  3 d X  Z  X s / 2 j  _ u  f b ð r  ;  z  ; s Þ   _ u sb ð r  ;  z  ; s Þ h i U  3 d X  ¼  0 ;  ð 18 Þ where  U  k (  x  ) are test functions and  o X s  is the boundary of the localsubdomain which consists of three parts  o X s  = L s [ C st  [ C su  [37].Here,  L s  is the local boundary which is totally inside the globaldomain, C st   is the part of the local boundary which coincides withthe global traction boundary, i.e.,  C st   = o X s \ C t  . Similarly  C su  isthe part of the local boundary that coincides with the global dis-placement boundary, i.e., C su  = o X s \ C u .Applying the Gauss divergence theorem to the first domainintegrals in Eqs. (16)–(18) and assuming the test functions to begiven by the Heaviside unit step functions within the subdomain,onecanrecasttheaboveequationsintothefollowinglocalintegralequations Z  @  X s r rb ð r  ;  z  ; s Þ n b d C þ Z  X s 1 r   ð r rr    r uu Þ d X  Z  X s ð 1  / Þ q s € u sr  ð r  ;  z  ; s Þ d X þ Z  X s q  f  / € u  f r  ð r  ;  z  ; s Þ d X ¼ 0 ;  ð 19 Þ Z  @  X s r  zb ð r  ;  z  ; s Þ n b d C þ Z  X s 1 r  r rz  ð r  ;  z  ; s Þ d X  Z  X s ð 1  / Þ q s € u s z  ð r  ;  z  ; s Þ d X þ Z  X s q  f  / € u  f  z  ð r  ;  z  ; s Þ d X  ¼  0 ;  ð 20 Þ Z  @  X s r  f  n b d C  Z  X s / q  f  € u  f b ð r  ;  z  ; s Þ d X  Z  X s q  A  € u  f b ð r  ;  z  ; s Þ  € u sb ð r  ;  z  ; s Þ h i d X  Z  X s / 2 j  _ u  f b ð r  ;  z  ; s Þ  _ u sb ð r  ;  z  ; s Þ h i d X ¼ 0 :  ð 21 Þ Inthepresentpaperthetrial functionsareapproximatedbytheMLS method on a number of nodes spread over the influencedomain. According to the MLS [36,37] method, the approximationof physical fields  u (  x  , s ) (displacements for both phases) over anumber of randomly located nodes {  x  a },  a  =1,2, . . . ,  n , is given by u ð  x  ; s Þ ¼  p T  ð  x  Þ a ð  x  ; s Þ ;  ð 22 Þ where p T  (  x  )=[ p 1 (  x  ), p 2 (  x  ), . . . , p m (  x  )] is a completemonomial basisof order  m ; and a (  x  , s ) is a vectorcontainingthecoefficients  a  j (  x  , s ),  j  =1,2, . . . , m  and  x   ( r  ,  z  ). For the considered axis-symmetric case,the monomial basis on the cross section is expressed by the radialand axial coordinates ( r  ,  z  ) p T  ð  x  Þ ¼ ½ 1 ; r  ;  z   ;  for linear basis  m  ¼  3 ;  ð 23 Þ p T  ð  x  Þ ¼ ½ 1 ; r  ;  z  ; r  2 ; rz  ;  z  2  ;  for quadratic basis  m  ¼  6 :  ð 24 Þ The coefficient vector  a (  x  , s ) is determined by minimizing aweighted discrete  L 2 -norm defined as  J  ð  x  Þ ¼ X na ¼ 1 w a ð  x  Þ  p T  ð  x  a Þ a ð  x  ; s Þ   ^ u a ð s Þ   2 ;  ð 25 Þ where  n  is the number of nodes used for the approximation. It isdetermined by the weight function  w a (  x  ) associated with the node a . A 4th order spline-type weight function is applied in the presentwork. The symbol  ^ u a ð s Þ  stands for the fictitious nodal values, butnot the nodal values of the unknown trial functions in general. Thestationarityof   J  inEq.(25)withrespectto a (  x  , s )leadstothefollow-ing linear relationbetween  a (  x  , s ) and  ^ u ð s Þ ¼ ½ ^ u 1 ð s Þ ;  . . .  ;  ^ u n ð s Þ T   A ð  x  Þ a ð  x  ; s Þ  B ð  x  Þ ^ u ð s Þ ¼  0 ;  ð 26 Þ where  A ð  x  Þ ¼ X na ¼ 1 w a ð  x  Þ p ð  x  a Þ p T  ð  x  a Þ ; B ð  x  Þ ¼  w 1 ð  x  Þ p ð  x  1 Þ ; w 2 ð  x  Þ p ð  x  2 Þ ; ... ; w n ð  x  Þ p ð  x  n Þ   :  ð 27 Þ ThesolutionofEq.(26)for a (  x  , s )andasubsequentsubstitutioninto Eq. (22) gives the approximation formulas for the displace-ments in solid and fluid phases [37] u s ð  x  ; s Þ ¼ X na ¼ 1 N  a ð  x  Þ ^ u sa ð s Þ ; u  f  ð  x  ; s Þ ¼ X na ¼ 1 N  a ð  x  Þ ^ u  fa ð s Þ ;  ð 28 Þ where the nodal values  ^ u sa ð s Þ ¼ ð ^ u sar   ð s Þ ;  ^ u sa z   ð s ÞÞ T  and  ^ u  fa ð s Þ ¼ ^ u  far   ð s Þ ;  ^ u  fa z   ð s Þ   T  are fictitious parameters for the displacements insolid and fluid phases, respectively, and  N  a (  x  ) is the shape functionassociatedwithnode a . Thenumberofnodes n  usedfortheapprox-imation is determined by the weight function  w a (  x  ).A 4th order spline-type weight function is applied in the pres-ent work as below w a ð  x  Þ ¼  1  6  d a r  a   2 þ 8  d a r  a   3  3  d a r  a   4 ;  0 6 d a 6 r  a 0 ;  d a P r  a ; 8<: ð 29 Þ where  d a = k  x    x  a k  and  r  a is the size of the support domain. It isseen that the  C  1 – continuity is ensured over the entire domain,and therefore the continuity of gradients of approximated fields of displacementsandporepressureissatisfied.IntheMLSapproxima-tion the rate of the convergence of the solution may depend uponthe nodal distance as well as the size of the support domain [43].Itshouldbenotedthatasmallersizeofthesubdomainsmayinducelarger oscillations in the nodal shape functions [37]. A necessary condition for a regular MLS approximation is that at least  m  weightfunctionsarenon-zero(i.e. n P m )foreachsamplepoint  x   e X .Thiscondition determines the size of the support domain.Substituting the approximations given by Eq. (28) into the localintegral Eqs. (19)–(21), we obtain a system of ordinary differential  J. Sladek et al./Computers and Geotechnics 62 (2014) 100–109  103  equations (ODE) for the fictitious unknown parameters ^ u sar   ;  ^ u sa z   ;  ^ u  far   ;  ^ u  fa z  n o : X na ¼ 1 ^ u sar   ð s Þ Z  @  X s ð c  11 ð  x  Þþ Q  ð  x  ÞÞ n r  ð  x  Þ N  a ; r  ð  x  Þþ c  12 ð  x  Þþ Q  ð  x  Þ r  n r  ð  x  Þ N  a ð  x  Þ  þ c  44 ð  x  Þ n  z  ð  x  Þ N  a ;  z  ð  x  Þ io d C þ Z  X s 1 r  ð c  11 ð  x  Þ c  12 ð  x  ÞÞð N  a ; r  ð  x  Þ 1 r N  a ð  x  ÞÞ   d X    X na ¼ 1 €^ u sar   ð s Þ Z  X s ð 1  / Þ q s ð  x  Þ N  a ð  x  Þ d X þ X na ¼ 1 ^ u sa z   ð s Þ Z  @  X s ð c  13 ð  x  Þþ Q  ð  x  ÞÞ n r  ð  x  Þ N  a ;  z  ð  x  Þþ c  44 ð  x  Þ n  z  ð  x  Þ N  a ; r  ð  x  Þ h i d C þ X na ¼ 1 ^ u  far   ð s Þ Z  @  X s ð Q  ð  x  Þþ R ð  x  ÞÞ n r  ð  x  Þð N  a ; r  ð  x  Þþ 1 r N  a ð  x  ÞÞ d C þ X na ¼ 1 ^ u  fa z   ð s Þ Z  @  X s ð Q  ð  x  Þþ R ð  x  ÞÞ n r  ð  x  Þ N  a ;  z  ð  x  Þ d C þ X na ¼ 1 €^ u  far   ð s Þ Z  X s q  f  ð  x  Þ / ð  x  Þ N  a ð  x  Þ d X ¼ 0 ;  ð 30 Þ X na ¼ 1 ^ u sa z   ð s Þ Z  @  X s ð c  11 ð  x  Þþ Q  ð  x  ÞÞ n  z  ð  x  Þ N  a ;  z  ð  x  Þþ c  44 ð  x  Þ n r  ð  x  Þ N  a ; r  ð  x  Þ h i  d C þ Z  X s c  44 ð  x  Þ r  N  a ; r  ð  x  Þ d X   X na ¼ 1 €^ u sa z   ð s Þ Z  X s ð 1  / Þ q s ð  x  Þ N  a ð  x  Þ d X þ X na ¼ 1 ^ u sar   ð s Þ Z  @  X s ð c  44 ð  x  Þ n r  ð  x  Þ N  a ;  z  ð  x  Þþð c  13 ð  x  Þþ Q  ð  x  ÞÞ n  z  ð  x  Þð N  a ; r  ð  x  Þ  þ 1 r  N  a ð  x  ÞÞÞ d C þ Z  X s c  44 ð  x  Þ r  N  a ;  z  ð  x  Þ d X  þ X na ¼ 1 ^ u  far   ð s Þ Z  @  X s ð Q  ð  x  Þþ R ð  x  ÞÞ n  z  ð  x  Þð N  a ; r  ð  x  Þþ 1 r  N  a ð  x  ÞÞ d C þ X na ¼ 1 ^ u  fa z   ð s Þ Z  @  X s ð Q  ð  x  Þþ R ð  x  ÞÞ n  z  ð  x  Þ N  a ;  z  ð  x  Þ d C þ X na ¼ 1 €^ u  fa z   ð s Þ Z  X s q  f  ð  x  Þ / ð  x  Þ N  a ð  x  Þ d X ¼ 0 ;  ð 31 Þ X na ¼ 1 ^ u sar   ð s Þ Z  @  X s n b ð  x  Þ Q  ð  x  Þ  N  a ; r  ð  x  Þ þ 1 r  N  a ð  x  Þ   d C þ X na ¼ 1 ^ u a z  ð s Þ Z  @  X s n b ð  x  Þ Q  ð  x  Þ N  a ;  z  ð  x  Þ d C þþ X na ¼ 1 ^ u  far   ð s Þ Z  @  X s n b ð  x  Þ R ð  x  Þ  N  a ; r  ð  x  Þ þ 1 r  N  a ð  x  Þ   d C þ X na ¼ 1 ^ u  fa z   ð s Þ Z  @  X s n b ð  x  Þ R ð  x  Þ N  a ;  z  ð  x  Þ d C  X na ¼ 1 €^ u  fab  ð s Þ   €^ u sab  ð s Þ h iZ  X s q  A ð  x  Þ N  a ð  x  Þ d X  X na ¼ 1 _^ u  fab  ð s Þ   _^ u sab  ð s Þ h iZ  X s / 2 j  ð  x  Þ N  a ð  x  Þ d X  X na ¼ 1 €^ u  fab  ð s Þ Z  X s / q  f  ð  x  Þ N  a ð  x  Þ d X  ¼  0 ;  b  2 f r  ;  z  g :  ð 32 Þ Eqs. (30)–(32) are considered in the subdomains  X s  aroundeach interior node  x  s and at the boundary nodes with prescribednatural boundaryconditions ( C t   and C  f t  ). On the parts of the globalboundary C u  and C  f u  withprescribedmechanicaldisplacementsthecollocation Eq. (28) are applied. X na ¼ 1 N  a ð f Þ ^ u sa ð s Þ ¼  ~ u s ð f ; s Þ  for  f  2  C u ;  ð 33 Þ X na ¼ 1 N  a ð f Þ ^ u  fa ð s Þ ¼  ~ u  f  ð f ; s Þ  for  f  2  C  f u ;  ð 34 Þ By collecting the discretized local integral equations togetherwith the discretized boundary conditions for the displacementsin solid and fluid phases, we arrive at a complete system of ordinary differential equations. The backward difference methodis applied for the approximation of ‘‘velocities’’ in the ODE (30)–(32). The Houbolt method [38] is applied for the accelerations in the same ODE. 3. Numerical examples In the first example, a finite cylinder (Fig. 1) with porous prop-ertiesisanalyzed. Thecylinderisunderauniformpressureloadonthe top surface,  r 33  =  10 4 Pa with Heaviside time variation. Thebottom surface of the cylinder is fixed in the axial direction, u s 3  ¼  u  f  3  ¼  0. On the lateral sides of the cylinder, i.e., at  r   = a , theaxial component of thetractionvector andthe radial displacementare vanishing. In such a case the axial symmetric problem corre-sponds to 1-d problem, where analytical solution is available[44]. Analytical solution can be derived in the Laplace transformdomain. A numerical inverse transformation (CQM) is applied tothe analytical solution. The radius of the cylinder  a  =1m and itsthickness  h  =1m are considered. The mechanical displacementsfor both solid and fluid phases on the cross section X of the cylin-der areapproximatedbyusing121(11  11) equidistantlydistrib-utednodes. Thelocal subdomainsareselectedtobecircularwitharadius  r  loc   =0.08m. The material coefficients correspond to Bereasandstone [5]: K   ¼  8 : 0  10 9 N m  2 ;  G  ¼  6 : 0  10 9 N m  2 ;  R  ¼  4 : 7  10 8 N m  2 ; q s  ¼  2800 kg m  3 ;  q  f   ¼  1000 kg m  3 ; a  ¼  0 : 8 ;  /  ¼  0 : 19 ;  j  ¼  1 : 9  10  10 m 4 = Ns : Coefficient  Q   is computed from Eq. (6). Firstly, homogeneousmaterial properties are considered. We have tested computer codefor various values of the coupling parameter  Q   in Berea sandstonematerial. Other material parameters correspond to the normalBerea sandstone given above. Fig. 2 presents a temporal variationof the axial displacement on the top of cylinder for  Q   =0. Onecan observe a good agreement of the present MLPG and analyticalresults. Time step D t   =0,1  10  5 s has been selected in numericalanalyses. It satisfies the Courant criterion for the time step D t  6 D  x = c  1 , where  c  1  is the velocity of propagation of dilatationalwaves in the solid  ð c  1  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð 3 K   þ 4 G Þ = 3 q s p   ¼  2 : 39  10 3 ms  1 Þ  and D  x  ¼  a = 11   0 : 1m is the distance between two nodes in the spa-tial discretization.In the next example we have considered the Berea sandstonewith  Q   =1.675  10 9 Nm  2 , which corresponds  a =0.867. Numeri-cal results are presented in Fig. 3.Finally,wehaveanalyzedalsonormalBereasandstonecylinder.MLPG and analytical results are presented in Fig. 4. In all materialmodel cases one can observe a good agreement of results. Fig. 5presents a comparison of axial displacements for various bulkmodulus  Q  . The largest axial displacement is observed for incom-pressible material model ( Q   =0). With increasing value of the bulkmodulus  Q   the axial displacement decreases and frequency of oscillations increases due to higher elastic wave velocity.In the next numerical example we analyze the water saturatedsoil cylinder, where material properties are following [45]: K  0  ¼  2 : 1  10 8 N m  2 ;  G 0  ¼  9 : 8  10 7 N m  2 ; R 0  ¼  1 : 206  10 9 N m  2 ;  Q  0  ¼  1 : 259  10 9 N m  2 ; q s  ¼  2700 kg m  3 ;  q  f   ¼  1000 kg m  3 ; 104  J. Sladek et al./Computers and Geotechnics 62 (2014) 100–109
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