POROMECHANICS of POROUS and FRACTURED RESERVOIRS Jishan Liu, Derek Elsworth KIGAM, Daejeon, Korea May 15-18, 2017

POROMECHANICS OF POROUS AND FRACTURED RESERVOIRS Jishan Liu, Derek Elsworth KIGAM, Daejeon, Korea May 15-18, 2017

1. Poromechanics – Flow Properties (Jishan Liu)

1:1 Reservoir Pressure System – How to calculate overburden stress and reservoir pressure Day 11 1:2 Darcy’s Law – Permeability and its changes, reservoir classification 1:3 Mass Conservation Law – flow equations 1:4 Steady-State Behaviors – Solutions of simple flow problems 1:5 Hydraulic Diffusivity – Definition, physical meaning, and its application in reservoirs 1:6 Rock Properties – Their Dependence on Stress Conditions Day 1

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2. Poromechanics – Fluid Storage Properties (Jishan Liu) 2:1 Fluid Properties – How they change and affect flow Day 2 2:2 Mechanisms of Liquid Production or Injection 2:3 Estimation of Original Hydrocarbons in Place 2:4 Estimation of Ultimate Recovery or Injection of Hydrocarbons 2:5 Flow – Deformation Coupling in Coal 2:6 Flow – Deformation Coupling in Shale Day 2

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3. Poromechanics –Modeling Porous Medium Flows (Derek Elsworth) 3:1 Single porosity flows - Finite Element Methods [2:1] Lecture Day 3 3:2 2D Triangular Constant Gradient Elements [2:3] Lecture 3:3 Transient Behavior - Mass Matrices [2:6] Lecture 3:4 Transient Behavior - Integration in Time [2:7] Lecture 3:5 Dual-Porosity-Dual-Permeability Models [6:1] Lecture Day 3

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4. Poromechanics –Modeling Coupled Porous Medium Flow and Deformation (Derek Elsworth) 4:1 Mechanical properties – http://www.ems.psu.edu/~elsworth/courses/geoee500/GeoEE500_1.PDF Day 4 4:2 Biot consolidation – http://www.ems.psu.edu/~elsworth/courses/geoee500/GeoEE500_1.PDF 4:3 Dual-porosity poroelasticity 4:4 Mechanical deformation - 1D and 2D Elements [5:1][5:2] Lecture 4:5 Coupled Hydro-Mechanical Models [6:2] Lecture Day 4

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1 All sessions nominally1h:15m

POROMECHANICS OF POROUS AND FRACTURED RESERVOIRS Jishan Liu, Derek Elsworth KIGAM, Daejeon, Korea May 15-18, 2017

1. Poromechanics – Flow Properties (Jishan Liu)

1:1 Reservoir Pressure System – How to calculate overburden stress and reservoir pressure Day 12 1:2 Darcy’s Law – Permeability and its changes, reservoir classification 1:3 Mass Conservation Law – flow equations 1:4 Steady-State Behaviors – Solutions of simple flow problems 1:5 Hydraulic Diffusivity – Definition, physical meaning, and its application in reservoirs 1:6 Rock Properties – Their Dependence on Stress Conditions Day 1

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2 All sessions nominally1h:15m

Summary of Notation – Diffusion Equation

! $ !!! Tensor: %&$'"# $%&'" ()* !! ! + !"# # ! , , ! Matrix: %$ & $ ' $%&' " ()* #

Indicial:

$$$$!!! $$! $ %&"#!!!'./% & "! # './%$&$'./% ---( &*-01/(*$'" # +!!!)!* +!- +

$$$$!!! Epanded: %&"#' +!!"!!!

Advective-Diffusive Flows

$ %&$'"# ! $ +

Momentum Transfer - Fluid Mechanics – Navier-Stokes Equations (Incompressible)

Tensor:

! !!!!! "#!!"2 !$%&' !! ! + !"# ! +

Expanded:

003! , ! 0 !! !!!! 00"!-., 0 " 003 #"" "2- !!!.0" +!"# !"# 00## 3 , # 0# # 000 -.!#" + !"# 3:1 Single porosity flows - Finite Element Methods [2:1] https://youtu.be/46QiOhaC47c

[2:1]!Fluid!Flow!and!Pressure!Diffusion! Recap!of!FEM! Comsol!Applied!to!Flow!! 1D!Element! ! !

3:2 2D Triangular Constant Gradient Elements [2:3] http://youtu.be/a_8VZejdgTA [2:3]!Fluid!Flow!and!Pressure!Diffusion! Recap! 2D!Triangular!(Constant!Gradient)!Elements! ! Derivation! ! Example! ! EGEEfem! ! ! ! !

3:3 Transient Behavior - Mass Matrices [2:6] https://youtu.be/fl5AVItg51E [2:6]!Fluid!Flow!and!Pressure!Diffusion! Recap!

Kh + Sh! = q Transient!Behavior!!!!!! !! ! “Mass”!matrices! ! ! ! ! ! ! ! ! ! ! ! !

3:4 Transient Behavior - Integration in Time [2:7] https://youtu.be/no5Y6dCXcyE [2:7]!Fluid!Flow!and!Pressure!Diffusion! Recap!

Kh + Sh! = q Transient!Behavior!!!!!! !! ! Time!integration! ! EGEEfem! ! ! ! ! ! ! ! !

3:5 Dual-Porosity-Dual-Permeability Models [6:1] Lecture https://youtu.be/key0TuOSDBU [6:1]!Linked!Mechanisms! Dual!Porosity/Dual!Permeability! ! Concept! ! Dual!permeability!! ! ! Heuristic!derivation! ! ! Comsol!implementation! ! ! !

POROMECHANICS OF POROUS AND FRACTURED RESERVOIRS Jishan Liu, Derek Elsworth KIGAM, Daejeon, Korea May 15-18, 2017

1. Poromechanics – Flow Properties (Jishan Liu)

1:1 Reservoir Pressure System – How to calculate overburden stress and reservoir pressure Day 13 1:2 Darcy’s Law – Permeability and its changes, reservoir classification 1:3 Mass Conservation Law – flow equations 1:4 Steady-State Behaviors – Solutions of simple flow problems 1:5 Hydraulic Diffusivity – Definition, physical meaning, and its application in reservoirs 1:6 Rock Properties – Their Dependence on Stress Conditions Day 1

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3 All sessions nominally1h:15m

4:1 Mechanical properties http://www.ems.psu.edu/~elsworth/courses/geoee500/GeoEE500_1.PDF

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b. t. W Snrtn vol. 15, Na }l5, tD. ailL5l?. 196 C ItS hdftu bt Ebrir Scic Lrd. Allrifbrmcd hraud u Gat Bnuu @*'^" fIlF 6t3/93it+fmtErrct

A TRIBI.TTETO MAURICE A. BIOT

MauriccA. Biot (190!19t5)

In presentingMauricc Anthony Biot with the TimoshenkoMedal in 1962,R. D. Mindlin, "Fundamentally, the eminentProfessor at ColumbiaUniversity, wrote: Tony Biot has a strongconsciousness of the physical world aroundhim. He hasa kecninsight which enables him to recognizethe cssentialfeatures of a physical phenomenonand build them into a mathematicalmodel without blindly including non-essentials. Then he has, at hisfingertips, a vastarray of the toolsof mathematicalanalysis and analyticalmethods of approximation whichhe uses skillfully to extract,from themodel, predictions of thehitherto unpredictable. They are all too few suchmen thesedays." These words by Mindlin accuratelyportrayed M. A. Biot as an intuitive engineer,who could masterthe advancedtools of a physical scientist,and asa scientistwho did not losesight of the physicalworld. MauriccBiot wasborn in Antwerp,Belgium on May 25th 1905.The war in l9ltl-I918 and the siegeof Antwerp causedthe Biot family to travel first to london, then Paris,and finally settlingin Chamb€ry,France. Thesc moves matured the young Biot and exposed him to scverallanguages. I:ter returningto Antwerp,M. Biot concludedhis sccondary school. In 1923he enrolled at a schoolin Brusselsfor preparatoryoourses in mathematics,and in 1924was admitted at the Universit6catholique de Louvain.It wasat this time that Biot showedhis insatiable appetitiefor knowledge.While pursuinghis studiesin Engineering,Biot also attended ooursesin Philosophy(he was awardeda Bachelordegree in Philosophyin 1927)and Economics.He obtaincda Mining Engineeringdegrce in 1929,and an ElectricalEngineering degreein 1930. "Theoretical After defendinghis thesiscntitled studieson induced electricalcurrents", Biot was awardeda Doctor of Scienccdegree in 1931.The sponsorshipof the Belgian AmericanEducational Foundation allowed Biot to spcndthe next two yearsin the U.S. (1931-1933)at the California Institute of Technologyin Pasadena.It was at Cal Tech wherehe first met and workedwith Theodorevon Krirm6n,who had arrivedin the U.S.in 1929.Biot acquireda Ph.D. in AeronauticalSciences in 1932by defendinghis work "Conceptof responscspectrum based on the earthquakeacceleration." The methodology broughtgreat simplifications to the analysisof structuresunder transient loading and has sincebeen used as a tool in earthquake-proofdesign. It wasduring the sameperiod that he publishedhis first paperson a newapproach to thenonlinear theory ofelasticity accounting for the effectof initial stress.By that time he had publishedabout a dozenrientific papers and patentedhis first threeinventions.

.l5l5 45t6 A tributc to Mauricc A. Biot After a few monthsat the Universityof Michigan,Biot returnedto Europe.In 1933and 1934,the BelgianNational ScientificResearch Foundation grantedhim the opportunity to travelto Delft, Gdttingen,and Zurich. With suchsharp intelligence he wassoon recognized by the university community. [n 1934,Biot started his academiccareer as a teacherof appliedmathematics at Harvard University.In June 1935,he rcturnedto Pasadenaas an AdvancedFellow of the BclgianAmerican Educational Foundation. By 1936,Biot was electedto the faculty at his Alma Mater, the Universit€catholique de Louvain, where he taught Elasticityand AnalyticalMechanics. From 1937to 1946,Biot wasa Professorof TheoreticalMechanics and PhysicalMathematics at Columbia Univenity. In 1946Brown University offeredhim the position of Professorin Applied Physicsand Scienoes,which he helduntil 1952. It was in l9zt0that the monograph MathematicalMethods in Engineeringwas written with Th. von Kdrmdn. Its translation into nine languagesis evidenceof its influenceon scveralgenerations of engineers.Later in his careerhe wrote two more books: Mechanics of IncrementalDeformations (1955) and VariationalPrinciples in Heat Transfer(1970). The U.S. fascinatedBiot, who found in it an environmentconducive for research.Biot becamean Americancitizen in 1941.The war in Europecame to Biot as a major distress and he took an activerole in it. On lcavefrom ColumbiaUniversity, he workedat the Cal Tech AeronauticalLaboratory on problemsof vibration and flutter, on the dynamic stability of projectilesand also on anti-submarineshell impact. During the war, as a LieutenantCommander in the U.S.Navy, Biot headedthe StructuralDynamics Scction of the Bureauof Aeronauticsin Washington,D.C. (1943-1945),and laterconducted technical missionsin Europewith combattroops. By 1951,Biot had produceda largenumber of scientificworks for ShellDevelopment Co.,Cornell Aeronautical Laboratory, and for the U.S.fur Force.After 1952Biot worked largelyalone as a consultantfor variousgovernmental agencies and industriallaboratories. From 1969to 1982,Biot was a consultantfor Mobil Researchand DevelopmentCor- porationin Dallas,in the areaof Rock Mechanics. Relocatedin Brusselssince 1970, Maurice Biot continuedhis researchuntil his lastday. It wason one of his lasttrips to the U.S.that Biot felt the earlysigns of his illnessthat wouldsuddenly depnve him of hislife on the l2th of Septembcr1985, at theage of eighty. The work and originalcontribution which distinguished Biot's career cover an unusually broad rangeof scienceand technologyincluding applied mechanics, sound, heat, thermodynamics,aeronautics, geophysics, and electromagnetism.The level of his work has rangedfrom the highlytheoretical and mathematical to practicalapplications and patented inventions. Aeronauticalproblems and fluid mechanicswere the objectsof mostof hisefforts during the 1940s.He developedthe thrcedimensionalaerodynamics theory of oscillatingairfoils along with new methodsof vibration analysisbased on matrix theory and generalized coordinates.This led to widelyapplied design proccdures of complexaircraft structures in orderto preventcatastrophic flutter. He alsopatented an electrical analogue flutter predictor basedon a simplecircuit designwhich simulatesaerodynamic forces. After the war he continuedwork on non-stationaryaerodynamic instability of thin supersonicwings, and on the first evaluationofthe transonicdrag ofan acceleratedbody. In the 1950s,Biot's work was concernedprimarily with problemsin solid mechanics, porousmedia, thermodynamics, and heat transfer.He developeda new approachto the thermodynamicsof irreversibleproccsses by introducing a generalizedform of the free energyas a key potential.The formulationwas associated with new variationalprinciples and Lagrangian-typeequations. The resultswith the introductionof internalcoordinates providedthe thermodynamicsfoundation of a generaltheory of anisotropicviscoelasticity and thermoelasticity.He latergave a systematicpresentation of this work in a monograph VariationalPrinciples in Heat Transferin 1970and indicatedits applicabilityto many other problemssuch as those of elasticaquifers or neutrondiffusion in nuclearreactor design. Biot'sinterest in the mechanicsof porousmedia dated back to 1940with a fundamental paperin soil mechanicsand consolidation.He returnedto the subjectin the 1950sin the more generalcontext of rock mechanicsin connectionwith problemsin the oil industry. A uibua rc Mruricc A. Biot 45t7 OD thc basisof his earlier work ia thcrmodynamics,he extcndedhis thcory to the acoustics in porousmedia and showcdthat thcre cxistcdthrcc typcsof acousticwavcs in suchmedia. In anothcr contribution hc was the first personto corrcctly provide the solution of the so- callcd Stoncleywave, i.c. an interfacewavc betweena fluid ovcrlayingan elasticsolid half- rpacewhich, as somehave argued, should more appropriatclybc namedafter him. For a short period in thc middle 1950sBiot bccameinvolved with rocket radio-guidancc problems and the question of disturbancefrom ground rtflections. He showedthat the rcflectionof elcctromagreticand acousticwaves from a rough surfacemay bc rcplacedby a smooth boundary condition. He also introduced a new approach to pulsc generated transientwaves based on a continuousspoctrum of normal coordinates.The combination of thc two methodsprovidcd the only practical solution at that time of rcme important problems. In a scriesof papersstarting in 1957,Biot extendedhis carlier work in the mechanicsof initially stresscdsolids, developing a mathematicaltheory of folding instability of stratified viscousand viscoclasticsolids. He verified the rcsultsin the laboratory and applied them to cxplain the dominant fcaturesof geologicalstructures. In particular, he brought to light the phenomenonof internal buckling of a confinedanisotropic or stratifiedmedium under compressivestress and provided a quantitative analysis.He applied the theory with the samesuooess to problemsof gravity instability and salt dome formation. ln a later period he presenteda systematictreatment of the mechanicsof initially stressedcontinua in the monogtaphMechanics of IncrementalDeformations, published in 1965.In the 1970sBiot's formulationof the variationalprinciple of virtual dissipationin the thermodynamicsof irreversibleprocesses along with a new approach to open systemsled to a synthesisof classicalmechanics and irreversiblethermodynamics. He applied thesenew theoriesto directlyobtain the field equationsin systemswhere deformations are coupledto thermo- moleculardiffusion and chemicalreactions. On this basishe also further developedthe theoryof porousmedia including heat and mass transport with phasechanges and adsorp- tion effects. The honorsthat Biot reccivedduring his lifetimcincluded the TimoshcnkoMedal of the American Societyof MechanicalEngineers (1962), the Th. von KArmdn Medal of the AmericanSociety of Civil Engineers(1967), and an Honorary Fellow of the Acoustical Societyof America (1983).He was also a memberof the U.S. National Academyof Engineering. (Compilationbased on the "Note on MauriceAnthony Biot" by A. Delmerand A. Jaumottepublished in 1990by the Acad€mieRoyale de Belgique,and on materialsupplied by MadameM. A. Biot.) A. H.-D.Crcnc E. Derotnxrv Y. AsouslerurN General Theory of Three-Dimeusional consolidation*

Mevnrcr A. Bror Columlio (Jnhtcrsitl, Ncu York, Nctts Yorh (Received October 25, 1940)

whose The settlement of soils under load is caused by a phenomenon called consolidation, water out of mechanism is known to be in nany casesidentical with the processof squeezing viewpoint are an etastic porous medium. The mathematicat physical consequencesof this determine the established in the present paper. The number of physical constants necessaq'to prediction of settle- properties of the soil is derived along with the general equations for the as examples' mentr and stressesin three-dimension-alproblems. Simple apptications are treated The operational calculus is shown to be a powerful method of solution of consolidation rpor temperature problems,

ted at a small INrnooucrtoN possible: the extension to the three-dimensional case.and the establishment of equations valid for mechanism of T is well known to engineering practice that a The f . any arbitrary load variable rvith time' I an instan- ented must be soil under load does not assume theory was first presented by the author in rather load, but settles sparking time taneous deflection under that abstract form in a previous publication'2 The settlement is so, the form of gradually at a variable rate. Such present paper gives a more rigorous and complete sands saturated with :sts that for a very apparent in clal's and treatment of the theory which leads to results by a gradual :athode sparks water. The settlement is caused more generalthan those obtained in the previous variation. This unt of energy adaptation of the soil to the load paper. A simple .hodesparking process is known as soil consolidntion The following basic properties of the soil are phenomenon was first due either to mechanism to explain this assumed: (1) isotropy of the material, (2) re- assumes that the lating because proposed by K. Terzaghi.t He versibility of stress-strain relations under final the soil are more ting produced grains or particles constituting equilibrium conditions, (3) linearity of stress- by certain molecular nth. The fact or less bound together strain relations, (4) small strains, (5) the water porous material with rtion of every forces and constitute a contained in the pores is incompressible,(6) the voids of the elastic skel- ually towards elastic properties. The water may contain air bubbles, (7) the water water. A good example of rut the possi- eton are filled with flows through the porous skeleton according to saturated with of the total such a model is a rubber sponge Darcy's lau'. this system will produce o an unstable water. A load applied to Of these basic assumptions (2) and (3) are on the rate at he transition a gradual settlement, depending rnost subject to criticism' Hos'ever, we should squeezedout of the which the water is being keep in mind that they also constitute the basisof these concepts to the ely with tJre voids. Terzaghi applied Terzaghi's theory, which has been found quite of a column of soil s in mercury analy'sisof the settlement satisfactory for the practical requirements of and prevented from lateral same experi- under a constant load engineering. In fact it can be imagined that the successof this theory ng in various expansion. The remarkable grains composing the soil are held together in a gettlement for many types of omena can be in predicting the certain pattern by surface tension forces and tend strongest incentives in :trical charac- soils has been one of the to assume a configuration of minimum potential of soil mechanics. phase of the the creation of a science energy. This would especially be true for the however, is restricted to r publication. Terzaghi's treatment, colloidal particles constituting clay. It seems problem of a column under a the one-dimensional reasonable to assume that for small strains, when viewpoint of mathe- constant load. From the the grain pattern is not too much disturbed, the generalizations of this are matical physics two assumption of .reversibility will be applicable' is not essentialand I Publication assisted bv the Ernest Kempton Adams The assumption of isotropy University' Fund- fc Phvsical Researchof Columbia t "Le des I Erilboumcclanih ouf Bofunphysikalischcr M. A. Biot, problime de la .Consolidltion f . Terz.Lhi, "Principle Ann' Soc' Sci' Gruidlopc tt-Aoiie F. Deuticke, 1925); of soil Maiieres:.isil.l"et "dut une charge," mechani-cs,'''Etig.-New. Record (1925),a seriesof articles' Bruxelles855, 110-113(1935). 1(< ED KYSICS VoLUME 12, FEBRUARY, 1941 anisotropy can easily be introduced as a re6ne- sametime smallenough, compared to the scaleof tional varia ment. Another refinement which might be of the macroscopic phenomena in which we are pores.We t practical importanceis the inffuence,upon the interested, so that it may be considered as water volun stress distribution and the settlement, of the infinitesimal in the mathematical treatment. quantity ti state of initial stressin the soil beforeapplication The averagestress condition in tJle soil is tien incrcmcnl o; of the load. It was shown by the presentauthorr represented by forces distributed uniformly on I*t us cr that this influenceis greater for materialsof low the facesof this cubic element.The corresponding water press modulus.Both refinements elastic will be left out stresscomponents are denoted by I .1. uniform thr of the present theory in order to avoid undue rf: the element o2 T' T' heavinessof presentation. 6. case, provi< r, crr 12 (1.1) E The first and secondsections deal mainly witJr slow rate 7J 7' the mathematical formulation of the physical C7 negligible. propertiesof the soil and the numberof constants They must satisfy the well-known equilibrium 't.. It is clear necessary to describe these properties. The conditionsof a stressfield. soil to occu number of these constants including Darcy's scopic conc 0o, 0r, 0r" permeabilitycoefficient is found equal to five in -+-+-:0, function of the most general case. Section 3 gives a dis- 6x 0y 0z i.e.,the sev cussionof the physical interpretation of these 6r, 0o" 0r, e: various constants. In Sections 4 and 5 are _+_+_:0, (r.2) establishedthe fundamental equations for the Ex 0y 0z must be de: consolidationand an application is made to the 0ru 0r, 6o, 0: one-dimensionalproblem corresponding to a _+_+_:0. standardsoil test. Section 6 gives the simplified 0x 6y 0z Furthermor theory for the casemost important in practiceof variationsi: Physically rve may think a soil completely saturated with water. The of these stressesas the relatior composedof two parts; one which is equationsfor this casecoincide with thoseof the causedby may be ta) the hydrostatic pressureof the water previous publication.2 In the last section is filling the We first cc pores,the other causedby the average shownhow the mathematicaltool known as the stressin the particu the skeleton.In this sensethe stressesin operalionalcalculus can be applied most con- the soil ponentsof s are said to be carried partly by venientlyfor the calculationof the settlement the water and stresscomp partly by the solidconstituent. without having to calculateany stressor water soil to hav pressuredistribution insidethe soil.This metlod must redu< '! 2. SrnerN Rer,erep ro Srnrss awo Hooke'sla, I of attack constitutesa major simplificationand it 'I \\IerBR Pnrssune theory of e' :r provesto be of high value in the solution of the .1 more complextwo- and three-dimensionalprob- We now call our attention to the strain in the lems. In the present paper applications are soil. Denoting by a, v, w the componentsof tle restrictedto one-dimensionalexamples. A series displacementof the soil and assumingthe strain of applicationsto practical casesof two-dimen- to be small, the valuesof the strain components sional consolidationwill be the object of subse- are quentpapers. 6u 0w 0v Cr:-t 'tr--+- 6x 0y 0z 1. Soll Srnessrs 0a 0u 0w Considera small cubic element of the con- €v:--, ?r:1-* ,-, (2,1) solidating soil, its sidesbeing parallel with the 0y 0z 0x coordinateaxes. This elementis taken to be large 0w 0v Eu enoughcompared to the sizeof the poresso that Az:-, ?r:--l--, 0z it may be treated as homogeneous,and at the - 0x _0y In order In thesere I M. A. Biot, "Nonlinear theory of elasticitl' and the to describecompletely the macroscopic linearizedcase for a body under initial stress." condition of the soil we must consideran addi- interpreted

156 JouRNAt oF APPLIEDpsxsrcs VOLUrrrnf: shear modulus and Poisson's ratio for the I to the scaleof tional variable giving tie amount of water in the the There are only two distinct which we are oores.We thereforedenote by d the increment of solid sketeton. becauseof the relation considered as iater volume per urtit volume of soil and call this constants treatment. quantity the sorietion itt water contenl. The E the soil is then inrremenl oJ wotcr pressurewill be denoted by a' G--. (2.3) 2(r*v) I uniformly on I-et us consider a cubic element of soil. The water pressurein the poresmay be consideredas r corresponding Supposenow that the effect of the water pressure uniform throughout, provided either the size of c is introduced. First it cannot produce any the elementis small enoughor, if this is not the shearingstrain by reasonof the assumedisotropy case, provided the changesoccur at sufficiently (1.1) of the soil; secondfor the same reasonits effect slow rate to render the pressure differences must be the same on all three componentsof negligible. strain e, c, c,. Hence taking into account the It is clear that if we assumethe changesin tJre rn equilibrium influenceof o relations (2.2) become soil to occur by reversible processesthe macroscopic condition of the soil must be a definite 63yo function of the stressesand the water pressure e,:---(o"*- o,)*=-, E E.. 3H i.e.. the sevenvariables

'Yr 'f, 6J ', d e, Clt e, 7x 0 | \r ey:=-=\otto,)-t7-, (2.4) (1.2) must be definitefunctions of the variables: EE 3H

67 6t 6t 7t 7! Tt O' o. t e,:---(o"+oy)+-_, Furthermore if we assumethe strains and the EE 3H variationsin water contentto be smallquantities, 1t: r''/G' tse stresses as the relation betweenthese two sets of variables h is causedby may be taken as linear in first approximation' 'Yy: ry/G, ater filling the We first considerthese functional relations for ^Y':T'/G' eragestress in the particular casewhere o:0. The'six com- ssesin the soil ponentsof strainare then functionsonly of the six where I/ is an additional physical constant. the water and stresscomponents 62 cy c2 r, r" r,' Assumingthe Theserelations express the six strain components soil to have isotropic propertiesthese relations of the soil as a function 6f the stressesin the soil must reduce to the well-known expressionsof and the pressureof the water in the pores.We ESS AND Hooke's law for an isotropic elastic body in the still have to consider the dependenceof the theoryof elasticity;we have increment of water content 0 on these same e strain in the variables.The most generalrelation is )onentsof the ,,:2-L6o*o,), EE 0: agr*azo"*o$rloar, ring the strain *o{"*osr"*ozo' (2.5) n components 6r'r,\ --\6 material a ee:--EE ttox) t Now because of the isotropy of the change in sign of r' r" t, cannot affect the water content, therefore or:oL:ol:0 and the effect ,,:2-L6,*o"), (2.2) of the shear stress components on d vanishes. EE Furthermore all three directions r, y, z must have (2.r) 'Yr:7./G, . equivalent properties o1:@2:sr. Therefore relation (2.5) may be written in the form 'ly: rt/G, 1o 'Y': f '/G' 0:-:(o,*a"{a,)*-;, (2.6) 3Hr J< e macroscoplc In theserelations the constantsE, G, I may be where Ilr and R are two physical constants. sider an addi- interpreted, respectively,as Young's modulus, 157 ,IED PSY$CS Vol,uuB lz, FEBRUARY,t9+r Relations(2.4) and (2.6) contain fivc distinct sideredas a function of the two variables c. 0. 3. PuYslc physical constants.We are now going to prove Now we must have that this number may be reduced to four; in aa au The constat fact that H:Ht if we introduce the assumption _:(rb _:6. as Yc of the existenceof a potential energy of the soil. 0e A0 meaning This assumptionmeans that if the changes.occur Hence and t}re Poissc ?ot time at an infinitely slow rate, the work done to bring _0o provided to squet the soil from the initial condition to its 6nal state A0 Ee water of strain and water content,is independentof the or consideredas t *'ay by which the 6nal state is reached and is a solid skeleton. sincr definitefunction of the six strain componentsand EA HTA constants the water content.This assumptionfollows quite Assume,for er naturally from that of reversibility introduced We have thus proved that If:I/r and we may ports an axia above, since the absenceof a potential energy write expand freely to applied long would then imply that an indefinite amount of 0--(o,*ou*o,)*-. (2.10) energ'ycould be drawn out of the soil by loading 3HR settlementis rt and unloadingalong a closedcycle. beensqueezed The potentialenergy of the soil per unit volume Relations12.4) and (2.10)are the fundamental is, accordingt' is relationsdescribing completely in 6rst approximation the propertiesof the soil, for strain and U:i(o"e,* ore"* o,a,* r,"f, u'ater content, under equilibrium conditions. *r"'y"*r,1,*o0). (2.7) They contain four distinct ph1'sicalconstants and the latera G, v, H and R. For further useit is convenient In order to prove that H:Hr let us consider a to expressthe stressesas functionsof the strain particular condition of stress Such that and the water pressureo. Solving Eq. (2.4) with

6z: 6y= 6 z: Otr respectto the stresseswe find T": Ty: T r:0, The coefficien . / ve \ Then the potential energy becomes o.=ZGle.*:---.-l-"o, bulging to th \ 1-2v/ librium condir g:|(oye*a0) with e:e.!erle, To interpre / vc \ ou:2Gl q*- l-oo, sampleof soil and Eqs. Q.$ and (2.6) \ | -2v/ t that the stres i t 3(l - 2v) o us drain the r i c:------or-F:;, 0:ot/Ht*o/R. (2.8) o,:zG(e,*;r)-",, (z.tr) tube passing Li EH negative pres i Tx:G^fz, certain amoul The quantity € represents the volume increase of . : amount is gir the soil per unit initial volume. Solving for or rr:G7r, and o r':G'Y' e0 u'ith Ort:---, 2(r*v) G RA HA The correspo: 3(L-2v) H given by (2.4 - € 3(l-2v)0 . (r-_*_, (2.g) In the same way we may expressthe variation in H,A EA water content as

3(t -2v) 1 0:ae*a/Q, (2.r2) The A:- where coefficier .ER HHT 1ls pressibility o pressure, wh The potential energy in this case may be con- O R H water conten

158 JorrRNAr,oF APPLIEDPEySIcs VoLU!trE 12, two elastic constantsand the constants ariables c, 0. 3. Psvslcel INrenPnBrlrIoN oF THE sure.The four distinct constantswhich Sott- CoNst.lurs I/ and R are the under our assumption define completely the The constantJ E, G and Y have the same physical proportions of an isotropic soil in the modulus meaning as Young's modulus the shear equilibrium conditions. and the Poisson ratio in the theory of elasticity Other constants have been derived from tlese provided time has been allowed for the excess four. For instance c is a coefficient defined as water to squeeze out' These quantities may be constants of the 2(r*v) G considered as the average elastic "--ift-2,l,E (3.s) solid skeleton' There are only two distinct such constants since tlrey must satisfy relation (2'3)' Assume, for example, that a column of soil sup- According to (2.12) it measures the ratio of the to and we may ports an axial load ps:-ot while allowed water volume squeezedout to the volume change been "*pand freely laterally. If the load has of the soil if the latter is compressed while applied long enough so that a final state of allowing the water to escape (o:0)' The coeffi- . (2.10) has ,"ttl"-"nt is reached,i.e., all the excesswater cient l/Q defined as been squeezedout and c:0 then the axial strain fundamental is, according to (2.4), (3.6) first approxi- R H 'or Po O strain and ,":-E (3.1) which can be r conditions. is a measureof the amount of water while the cal constants forced into the soil under pressure and the lateral strain It is quite convenient to volume of the soil is kept constant' wiil be of ,he strain and vPo obvious that the constants a and Q er:e!:-: -ye.. (3.2) soil not completely saturated 1. (2.4) with significance for a wlth water and containing air bubbles' In that a and can take values The coefficient/ measuresthe ratio of the lateral case the constants Q degree of saturation of the soil' bulging to the vertical strain under final equi- depending on the test suggests the derivation librium conditions. The standard soil A column of soil supports To interpret the constantsIf and R considera of additional constants. bag so a load po: - o zandis confined laterally in a rigid 6, sampleof soil enclosedin a thin rubber no lateral expansion can occur' thar the stressesapplied to the soil be zero' Let sheath so that to escape for instance by us drain the water from this soil through a thin The water is allorved the load through a porous slab' When o, (2.11) tube passingthrough the walls of the bag' If a applying has been squeezed out the negative pressure-o is applied to the tube a all the excess water given by relations (2'11) in which certainamount of u'aterwill be suckedout' This aiial strain is u'rite amountis given bv (2.10) we put o:0. We er: -prfr. (3.7) o f:-- (3.3) The coefficient R 'l-2v o:- (3.8) The correspondingvolume change of the soil is . 2G(r- v) given by (2.4) will be called thefirc| compressibility' e variation in o €: -- (3.4) If we measurethe axial strain jusf after the H load has been applied so that the water has not (2.12) had time to flow out, we must Put 0:0 in The coefficiefi l/H is a measure of the com- relation (2.I2). We deduie the value of the water pressibility of the soil for a change in water pr.*ur", while l/R measures the change in Pressure c: - aQe,. (3'9) water content for a given change in water pres- 159 TED PSYSICS Vor,ulrB 12, FEBRUARY, t9lt substituting this value in (2.11) we write There are three equations with four unknowns where o is th' ta,o, rn, 2r: _p&;. (3.10) a. In order to have a completesystem we constant. Fr< The coefficient need one more equation.This is done by intro-' o ducing Darcy's law governing the flow of water o,r:7;;5 (3.11) in a porous medium. We consider again an and from (2. elementary cube of soil and e.ll V, the volume of will be called the instanta.ncouscompressibitrity. water flowing per secondand unit area through The physical constants consideredabove refer the face of this cube perpendicular to the r axis. Note that E to the properties of the soil for the state of In the sameway we define V, and I/,. According equilibrium when the water pressureis uniform to Darcy's law these three components of the throughout. We shall seehereafter that in order rate of flow are related to the water pressureby relatior to study the transient state we must add to the the relations This four distinct constants above tie so-called cofficientof permeabililyof 6a 0o the soil. -k-:, 0a V.: Vo: -k-., V,: -p--. (4.2) with 4. GBwrul Eguerrows GovrnNrNc 0x 0y 0z ConsorroerroN The physical constant & is called the coefi.cientof We now proceedto establishthe differential permeability of the soil. On the other hand, if we The constar equationsfor the transient phenomenonof con- assumethe water to be incompressiblethe rate of water Pressl solidation,i.e., those equations governing the dis- watercontent of an elementof soilmust be equal boundary a: tribution of stress,water content,and settlement to the volume of water entering per second Taking tl as a functionof time in a soil undergiven loads. through the surfaceof the element,hence Substituting expression(2.11) for the stresses into the equilibriumconditions (1.2) u'e find dd 0v, 0v, av" _: (4.3) t_, Oe ct6 at 0r 0y 0z GV2u+_ __d_:0, 1-2v 0r 0x L Combining Eqs. (Zf) (4.2) and (4.3) we obtain The 6rst cc G0e0o ability of ti GVUa- --a-:0, 0e t6o soil. The se -. l-2v Ey 0y hY2o:a-*- (4.4) The initi, (4.1) dt Qat G0e0c water musl GVfu+- --d-:0, | -2v 0z 0z The four differential Eqs. (a.1) and (4.4) are the basic equations satisfied by the four unknowns pz gs2 : Qz/ | 62f 0y2{ 62/ l7z. u, v, w, o. Carrying t 5. Appr,rceuoN To e SreNpeno Sotl Tesr

Let us examinethe particular caseof a column of soil supporting a load ps:-otand confined Iaterally in a rigid sheathso that no lateral expansioncan occur. It is assumedalso that no water can escapelaterally or through the bottom while it is free to escapeat the upper surfaceby applying the where or al load through a very porous slab. The solt Take the z axis positive downward; the only component of displacement in this case will be zr. condition Both u and the water pressurea will dependonly on the coordinate z and the time l. The differential Eqs. (4.1) and (4.4) become .1- I Afu Ap/' --_4_:Q, (s.1) a 022 dz The set 02a A\n l 0o h-- d-+-.:-_, /( ?) 022 0z0t Q at

160 JoURTYALOF APPL,ED PEySrcS VoLure 1'

is a defin"rl by (3.8). The stressc, throughout the loaded column r four unknowns whereo is the final compressibility From-(2'l1) we have nplete systemwe: constant. t&o is done by intro- ps: -ct: --1-*aa (s.3) :he flow of water odz nsider again an and from (2'12) I/, A7o 6 the volume of 0:o--*-. nit area through ozO rlar to the * (5'1) and that axis. Note that Eq. (5.3) implies od V,. According t Ah) 0o mponents of the a 0z0t At .'ater pressureby This relation carried into (5'2) gives 02a | 0o -:- -, (s.4) 022 c 0t '': 0o -ft-. (4.2) with tal Ez --crl*-. (s.s) c kQk l the coeficient of (5.4)shows the important result that the rther hand, if we The constantc is called the consolid,ationconslanl. Equation heat conduction'This equation along with the :ssiblethe rate of water pressuresatisfies the well-known equation of solution of the problem of consolidation' cil must be equal boundaryand the initial conditionsleads to a complete the top we have the boundary conditions :'ing per second Taking the height of the soil column to be L and z:0 at :nt, hence o:0 for z:0,

Ea (s.6) )v, -:0 f,orz:h. ::-. (4'3) ctz 6z the load is zerobecause the perme- The first condition expressestlat the pressureof the water under (4.3) we obtain to be largewith respectto that of the ability of the slab through which the load is appliedis assumed that no water escapesthrough the bottom' T soil.ihe secondcondition expresses zerowhen the load is appliedbecause the -. (4.4) The initial conaitionirihat the changeof water content is (2'12) water must escapewith a finite velocity' Hence from 0wc and (4.4) are t}re o:,--*6:o forl:o' : four unknowns pressure Carrying this into (5.3) we derive the initial vdlue of the water rt | \ &-oi (s.7) o:po/ for l:0 or c--Po, -a' and confined \*O*") hat no water can definedbv (3'8) and (3'11)' wherear and a are the instantaneousand fnal compressibilitycoefficients by applying the conditions (5.6) and the initial The solution of the difierentiat equation (5.4) with the boundary be written in the form of a series s case will be u. condition (5.?) may . The differential " rz 1 r 13r\2 1 -*'3tz '' I (s.8) o:!o-o, ,"loo* | -( \' -'J,rt sin "lsin r o,-ol L \zrl i*1"*o L- \a) |' (s'1) The settlement may be found from relation (5'3)' We have (s.2) !Ur-ana-oPo. (s.e)

161 ,PLIED PETSICS Vorvur 12, tr'EBRUARY,l9+l 'I" 'lt\

The total settlementis -a'. 6. SIM?LIFIED ct|w 8 For a comP 7oo:- : 1 1 fen*t)r-1, I | -dr:- 1o-o)hp,t=+op.- .|_l: alaotpo. test showstha .to 6z 12' ;(Znll)z--1 l (s.10) L zn J.-|,,",.' be taken equ Immediatelyafterloading(l:0),tlredeflectionis comPressibilit the soil is equr 8o[ i Accordin ys - -(a out. i: - o;)hpo"."7(2n!l)z''E ohp. .n2' .-_ .=:* Taking into account that reducest o[T2 This soil to tl ui:oih\o' (s.11) the T 1r"*tyr:;' permeabilitY. deduce which checkswith the result (3.10)above. The final deflectionfor r: .o is w-:ahfo. (s.12) It is of interestto find a simplified expressionfor'the law of settlementin the periodof time immedi- from (5 ately and after loading.To do this we first eriminatethe initial deflectionzoi by considering .f: - constanttakt 'F-ia 8 * 1 701: J-6- .. 1 f /(Zn+t)r\, .T 7D6-tp;: a;)hp, t - *o (#) r,ll (s.1 3) ji + #{ L_ r. (2.1 t Relation rfis that part of the deflection which is causedby consolidation.we then considerthe rare oIlxlrelses setuement. The general dw,2c(a-a;) : 1 fentlzrl, I I exn (4.4) are sim d,:--T-?o l-L " ]"1 (s.14) For l:0 this seriesdoes not converge; which meansthat at the first instant of loadingthe rate of settlementis infinite. Hence-the curve representingthe settlementzr, as a function of time starts with a verticalslope and tendsasymptotically toward the value (a-a;)hptas shownin Fig. 1 (curve 1)' It is obviousthat during the initial periodof settlementthe heightD of the columncannot have any-influenceon the phenomenon becausethe waterpressure at the?epth z:hhasnot yet had time to change'Therefore in order to find the nature of the settlementcurve in the vicinity of l:0 it is enoughto considerthe caseu'here h: q .In this casewe put i n/h: E, l/h: dg and write (5.14) as dw, € - : - 2c(a a) p o D exp [ - nr({ + * A)rct)AE By addingt of Eqs. (6.5 for h: o. The rate of settlementbecomes the integral

dw, f- c(a-o;)ps -: Zc(a- a;)?o I exp (-zr2f2ct)d,g:-. (s.1s) wherea is t at J o (trct)l From (6. The value of the settlement is obtained by integration

dw, ft --dl:2(a-a)pol:l tct\l w,: | _\o . (s.16) Jo dt / Hence the It follows a paraboliccurye as a function of time (curve 2 in Fig. 1). equation o 162 JorrRNAr oF AppLED psysrcs Voruur r CLAy Equations (6.5) and (6.8) are the-fundamental 6. Srupnrrpo TgBonv FoR A Satunetso "qu"bont governing the consolidation of a com- For a completely saturated clay the standard pletely saturated clay. Becauseof (6'4) the initial may )s. (5.10) test showsthat the initial compressibility o; condition 0:0 becomesc:0, i.e., at the instant final Uu t"t utt equal to zero compared to the of loading no volume changeof the soil occurs' of compressibility a, and that the volume change This conditionintroduced in Eq' (6'7) showsthat the soil is equal to the amount of water squeezed at the instant of loading the water pressurein the out. According to (2.12) and (3'11) this implies poresalso satisfiesLaplace's equation' (6.1) Q: *, a:1. Vo2:0. (6'9) of This reducesthe number of physical constants The settlementfor the standardtest of a column the the soil to the two elastic constants and of clay of height lz under the load 2ois given b1' (s.11) (3'6) permeability.From relations (3'5) and we (5.13)bV putting&t:0. deduce 2Gg*v) (('2) 9.1 n-n:fia,1 w,---ohpoL;- - (s.12) T2 o (2nll)2 time immedi- and from (5'5) the value of the consolidation c constant takes the simPle form x{r-"*n?e#)",]} (610) c: k/a. (6.3) (s.13) From (5.16)the settlementfor an infinitelyhigh Relation (2.12) becomes columnis 0: e. (6.4) consider the ,,:zoor(L)t. (6.11) The general differential equations (4.1) and (4.4) are simPlified, (s.14) It is easyto imaginea mechanicalmodel having G0e6o Con- ---:0, the propertiesimplied in these equations' GV2u+- great number of small rg the rate of l-2v 61 0x sidei a .ytt"* made of a ,f time starts rigid paiticles held together by tinv helical G0e0o will be elasticallydeformable Fig. 1 (curve ---:0, (6.s) rp.i"g.. tttis system Gv?1+- If we cannot have | -2v 0Y 0Y "na *itt possessaverage elastic constants' yet had time fill completelywith water the voids betrveenthe ' of l:0 G0e0c it is Gvrl)+- ---:0, t-2v 0z 0z /' 0e 1t/, kVa2 -- (6.6) 2.t' | ..- at l./ t/ l-/ By adding the derivativeswith respectto r, y, z of Eqs. (6.5), resPectively,we find

Ye2:oYo2, (6.7)

(s.1s) whereo is the final compressibilitygiven by (3'8)' From (6.6) and (6.7) we derive - 1dc V €2:-- (6.8) c0t 1. Settlement causedby consolidation as a function (s.16) Frc.-tio".'C-u*e of of I representi tlie settlement of a column Hence the volume change of the soil satisfies the t'.ilf,i i itiii" 6iapi' cu*" 2 representsthe settlement equation of heat conduction. forln infinitely high column. 163 o Pgrsrcs VoLUME 12, FEBRUARY, l9+t particles, we shall have a model of a completely Eqs. (5.1) become saturated clay. 1 Obviously such a system is incompressible A\o 0c 02o 0u if no __:_, k_:O_. water ' (7.1) Dc1 is allowed to be squeezedout (this corre_ o 02, 0z 02, dz Rcswcl spondsto the condition 0= -) and the change A solution of these equations which vanishes of volume is equal to the volume of water at infinity is l squeezedout (this correspondsto tie condition a:1). If the systems 7!- CP-'btdl, col containedair bubblesthis @l would not be the case and (7.2) we would have to 1tbtl clc consider the general case where is 6nite o: th, e and Cr--l'- | Cre-,br"tr a*1. o\c / for sp' Whether this model representsschematically The boundary conditions are for s:0 the actual constitution of soilsis uncertain.It is quite possible,however, l0w that the soil particles are (f - :- -, held t: 1 o:0. togetherby capillary forceswhich behavein a oz fT has be, pretty Hence _*- much the sameway as the springsof the i) I thousand R.. model. h evacuatedtr ,r:"(;)' , Cz:1. gradients to 7. OrnaartoN.lr Cercur.us Appr,lro ro is concerned The settlement CoNsolroerroN u,, at the top (z:0) causedby the in the devel sudden applicationof a unit load is radiographs The calculationof settlementunder a suddenly appliedload leadsnaturally to the applicationof -.:'(i)"u, operationalmethods, developed by Heavisidefor Since the the anal),sisof transientsin electric circuits. As The meaning of this symbolic expression is purposes,a an illustration of the power and simplicity derived from the operationalequationr equipment introduced by the operational calculus in the toward sho. treatment of consolidation problem we shall (7.3) sulted somt derive by this procedure the settlement of a ,!,tur-'(:)' and equipm completelysaturated clay column alreadv catcu_ twentieth t, lated in the previous section. In subsequent The settlement as a function of time under the load po fast to sto articles the operational method will be used is therefore parts of tl extensivelyfor the solution of various consotida_ t'flash" tec tion problems.We consider the case of ar:r"Or(l). (7.4) a clay stimulated columninfinitely high and take as beforethe top exposures. to be the origin of the vertical coordinatez. This coincideswith For a the value (6.11) above. Oosterkam completelysaturated clay a: l, e: o and with . V. Arrh, O-pcrotional this goal. the operationalnotations, reptacing E/0t p, Circuit Anotyis (John Wile1,, by New York, tgZg),p. tCZ. discharge t rents, but position pl temperatul mercury. ( amp. emis temperatu melti g P, procedure. r MaxStet &lnSiauas- Chapter 4, I tI{. H. K 128 (1938). I W.J.Oo

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1., (z'r\ / )l l Y,.t = ( Uvu, -l tJ / Jy ,L

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I't [,,J67,a-r&..;,1..1. 1r also , r.(*/k1 ba +Ls- se'\/c.'\ s fivstet , ir, C+i,'--?f-'t- \J

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(bacv (ca-se! r,Lo r,.ei ,r(-.< c|r,-ngp .'. Ao u-i<--r M.e {ro,^ v

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a

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Ar r 1 -', tt - t / ,\P P1 +-z J';;"l'.i f1 F I

t-, (,-,,r1;,"l €r&*tr', ?f ir'-:t, & (h.J | ,

L(.= iLCr, + d: €: + {t€z + T'.5YU

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| ('.t

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= (r-zv)f + (.2 )) €v LJ Z tl

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tl A

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dr.= tv € s0 -zv)

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Lq tY + L5 Y7" ,-= f9 0- zu) t* lyr.rg-Sp S'(a/s" tq{of'^: I e"1,u^Q,7)

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3 ,-e- 14,,:^4- E a'-1r':!LA t1 ?

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t, -- .J ! )v., ) { -r- G o C.{ J7- .*'

e qu' r':(i ta : (+ t) 3 nf^^-t t-6, l: V ,,rv, A"o ' *J d 5. ON€ - br^terJS tot'JAt- cd.rscrL( DA-11/Y,) I

An loooL"l 3 fi,p wI Cr.rl"e SSrLQ &-aJ khJe' k .

O,^a eeyo:+t.'^ Q'f) n7,esa'7< q-il 1Lr'<'*"

atJ' CC..Jhl-.7- ;a LaJa-*l'a{h"f

oO n0 1 d- dP uL +C, o (/*'.4*tr) I G,D qv- -,-l -: .l'f (t -zv) dL

ft g'o-., rtLG UX 62-no/' Nra-t,-eti - ,-

\? a2 ,{ n) o|/L v 0ut e* (f'2) a ( r'2") a' 1r C / ^ -\ fr oc ll - Lv \ VO at I\,/ l- Lv ) \/

n. y'/, ., \ )^ /<.r\ (,t l-'t/ lg, r-tI '. l d-u+ d "r /^) \- -/ ,/ ) r 1-7- z)7 c I J- ).r l VUA} {\/ | L-v )

JT

/- -r? 1< ,/ \ d ct-z /+ (i''.*\-/ l aL' AL

ecVu^}- (+ r; Tp"*t eaM4*^,1 k'

,1 d ,t r'r, e r Lak = * dut J. aj \ DL' aTat q-/1 dt t+,

fA- 6"us1iuA"o- e7 ^12\ /4 $:r'"i Fno*,,e%r^An-^Q,L() 4 o7gr"fo ' - (> "l fi.-!-1ar^t' s&et of?fu4ol t4^* , t

- dr = -2q(';'H, *.i,,'#\ + q? (r+;

- 61 : '>q ( -_") )r, + cP ( set lCzu; at

/ -dz nl rJ&fr'h",&t

t''i Co-("'ttn^ ]o citv-a Dr'p p'*'!Scr'e 11" J;J

o(u,rl 'i'i>, 'Zt I i,,r.,..,/ .. {r,,. +-* v{,(u,,.- i1,^"n .- / 1x* ,2 1 it'\l "u# e = l,f, + P (s'to; u<_a)

/1...{ y'nr,{^1 Qcir,r.to;('t,'^(f'Zl \aZ, No1^ b: t,*t

Dq = -_l E'u, + d+ (:,'i; fZ a- az. aL

v'."'' cc. L ,"*f n,^j rn,A q; t::f? e::n<-c{t 2t' wr> € 2{r = c 4 ,i c!. l-L -'tz*: la.-u d.-e s^} t (*i i 7 r/d2

Ta t J "J < o (/*t t-t 6'< S-e' JZ ' .J't(

Alb k,q\ a^/ o7r4\ av\ *;fi^, c 5,,^'--(".t n,4. \-) 1(tet

i7, 1r avt I )tu. + J- AP (s,rzJ -: /1 \, ^^ ,at (/v dOd a >t

-t r? \ d'uv ddD (r r:; C,{ L \ I 6- ?:"-"* at

-rl, - .4, I tl I ( Q4 rt"A 1 Qr 1 ce e-t.d\ ffufa( b,^- (t'P)*a(' I tn r-n" s^A+-4tt U L. ' .\J a I t t / t- Ot t | . -\ /r--t7\tJ rn(C J 'lr', "e a.L- c'-=11^ \ ) J, !c,er S l-Lfvh.'\', ( i' ) ')I J-- {7 \ rf r\ t I' / ^ I .].-'t iu )< d? o( t\OL u/ + ' Q/ i ) l -1 a) r- ^1 | \r<- /^J f vv o|

I l ^ll I Lr' | ..J2-4^ .x .^ i E 0k/ o: '' I

l" (, ""g-(rrl ,rA-', g.p

Ncq.{- "-t ,i! urxJ tn -A,-( l, ,'t{

) 7 LV ( Y F,,r)

,- : t1 ?p v yl J =l) ?z )

|^ , k i-i (( ,-,,1, st,-,*, :f Ij y"t it," F. s l'JL

lr -tJ lor,tr " vrl<,-..-z cAa^&L lntk-.|- , y e

.\ - /n Lt).,\ -= *d_Jz + :Q (g ti) +-,4.- ll' e t-P; (L/1 AT ; f.=C 2ur (s,,e t6 _._9,i\ \ S.Lslal"L c,r(Z e17"

-ft = +J ?; + *P. ({ z, ea*

6z /( ,,)

?- rl \ +J- (rzzr /t= ./)

r- (f-.rC) 7.^, J* sc.,',zr{ (s,rs) S.L.,eJ --/ *t"/ f 2Z ! t,'., J -L-c-7 n rar),YssrOiLti '\ 11, ( lt i rt:-t),?J S :!u-4 c-, ,, 1.r. !! 'n,1 orT I /,.in , L',1' /"t,tla*-''i-'J rl r !'-+ s. Ls p / l -- 1c l-i' \a-'..\ ! ! I ,,-).'r',h,-r, ex,,-L, *1 rtt€ff LL .tU I ,'\. --cr ec ;{ rn --h n-l- n'{: r-{ (L r -', il lco/o-+,"-tv Dt$.oC . +

,n,/ / i,n,4,.-i, cf/sa+,

c-t nJn Ca"-7,c-sJt 4 Cr^-r/--d*ott laoJ. eo-ru'/ ** '|+'i ' sc'r4- fu 4f

4:3 Dual-porosity poroelasticity

Flow-Deformation Response of Dual-Porosity Media

By Derek Elsworth,1 Member, ASCE, and Mao Bai2

ABSTRACT: A constitutive model is presented to define the linear poroelastic response of fissured media to determine the influence of dual porosity effects. A stress-strain relationship and two equations representing conservation of mass in the porous and fractured material are required. The behavior is defined in terms of the hydraulic and mechanical parameters for the intact porous matrix and the surrounding fracture system, allowing generated fluid pressure magnitudes and equilibration rates to be determined. Under undrained hydrostatic loading, the pore pressure-generation coefficients B, may exceed unity in either of the porous media or the fracture, representing a form of piston effect. Pressures generated within the fracture system equilibrate with time by reverse flow into the porous blocks. The equilibration time appears negligible for permeable sandstones, but it is significant for low-permeability geologic media. The constitutive model is represented in finite element format to allow solution for general boundary conditions where the influence of dual-porosity behavior may be examined in a global context.

INTRODUCTION

The linear flow-deformation behavior of geologic media is governed by the theory of single-porosity poroelasticity, as expounded originally by Biot (1941). Where, as in the case of fissured rock and soils, the medium com- prises discrete fractions of differing solid compressibilities and permeabilities, a dual-porosity approach appears more appropriate. The dual-porosity approach has been extensively developed to represent single-phase and multiphase flow in petroleum reservoirs. The original char- acterization of naturally fractured reservoirs by Warren and Root (1963) has been developed for radial flow in both blocky (Odeh 1965) and tabular reservoirs (Kazemi 1969) using analytical approaches and extended to multiphase flow using numerical techniques (Yamamoto et al. 1971; Kazemi et al. 1976; Kazemi and Merrill 1979; Thomas et al. 1983). Interest in single- phase behavior within dual-porosity reservoirs has been concerned with accurate representation of the flux fields within the porous and fractured components (Huyakorn et al. 1983) for application to mass (Bibby 1981) and thermal transport (Pruess and Narasimhan 1985; Elsworth 1989). In all of these applications it is assumed that total stresses remain constant with time, and therefore poroelastic effects are not incorporated. The theory presented by Aifantis (1977, 1980) and Khaled et al. (1984) provides a suitable framework in which the flow-deformation behavior of dual-porosity media may be fully coupled to the deformation field as a multiphase continuum. Multiphase poroelasticity requires that the conditions of flow continuity within the different solid phases and the exchange between the phases are superimposed on the elastic displacement behavior of the fissured mass as body forces. As initially defined (Aifantis 1977), the constitutive coefficients describing the behavior of the aggregated medium

Associate Professor, Dept. of Mineral Engrg., Pennsylvania State Univ., Uni- versity Park, PA 16802. 2Research Associate, Dept. of Mineral Engrg., Pennsylvania State Univ., Uni- versity Park, PA 16802. Now at School of Petroleum and Geological Engineering, University of Oklahoma, Norman OK 73019-0628. Note. Discussion open until June 1, 1992. To extend the closing date one month, Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on May 11, 1989. This paper is part of the Journal of Geotechnical Engineering, Vol. 118, No. 1, January, 1992. ©ASCE, ISSN0733-9410/92/0001-0107/$1.00 + $.15 per page. Paper No. 26530. 107

J. Geotech. Engrg., 1992, 118(1): 107-124 defy direct physical interpretation. Although phenomenological coefficients describing both load-deformation and fluid-percolation response may be determined directly from laboratory and field testing of the fractured systems (Wilson and Aifantis 1982), these coefficients may be recovered more conveniently from basic knowledge of modulii and permeability of the components comprising matrix and fissure porosities. Defining the response of the system directly in terms of the elastic and permeability properties of the components, with due regard for fissure geometry, offers the further advantage of ensuring that material nonlinearities of unknown magnitude are not inadvertently included within data reduced from field testing. This is an important factor, given long-standing knowledge on the nonlinear load-deformation behavior of interfaces (Goodman 1974) and the strong aperture dependence of fluid transmission in fissures (Iwai 1976). Indeed, the conditions needed to satisfy requirements for a linear theory of dual poroelasticity for fissured media may be so restrictive that, in all practicality, the phenomenon must be viewed as intrinsically nonlinear. This argument aside, only linear phenomena are considered in the following. Where the structure of the fissured mass is well defined, as in the case of regularly jointed rocks and fissured soils, the contribution of fissure and matrix components to the overall flow and deformation response of the medium are readily apparent. Indeed, where elastic and flow properties of the fissures and matrix are known a priori, it is reasonable to develop the governing equations directly from this constituent basis. This exercise is completed in the following to develop the macroscopic continuum equations of linear poroelasticity on the basics of known fluid compressibility and defined fissure stiffness, porosities, and permeabilities.

CONSTITUTIVE EQUATIONS The behavior of the dual porosity aggregate may be defined in terms of component equations representing the solid deformation and coupled fluid- pressure response. The morphology of the continuum is represented in Fig. 1. Solid Deformation The relationship between changes in total stresses (da) and intergranular stresses (do-') are governed by the Terzaghi (1928, 1943) relationship of the form Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved.

FIG. 1. Morphology of Porous-Fractured Aggregate

108

J. Geotech. Engrg., 1992, 118(1): 107-124 dax = dcrj + mdp1 (la)

da2 = da2 + mdp2 (16) where dp = change in fluid pressure. The effect of grain compression on the intergranular stresses are neglected. Stresses and strains are positive in compression. Subscripts 1 and 2 refer to the porous and fractured phases, - respectively, and are represented as do! = 3[cr.vx, ayy, erzz, uxy, crvz,

ScTx = da2 = da (2) The linear constitutive relationships for the separate phases are defined as da[ = Djdei (3a)

da2 = D2dE2 (3b) where e = solid strain within each of the two materials, namely porous and fractured material, and the inverse relations recovered from (3a,b) are

de1 = C^CTI (4a)

dz2 = C2dcr2 (4b) where D„ = the elasticity matrix for phase n with all other phases assumed rigid, and C„ = the compliance matrix under a similar arrangement, such that D,7X = C„. Since D„ and C„ are written to represent the elastic behavior of phase n only, and are stated at a macroscopic level to contain a representative sample of matrix and fissure geometry, the total strain due to deformation in each of the phases is

de = del + de2 (5) or, substituting (1) into (4) and recording the result into (5) gives

de = (Ci + C2)3o- - C1mdp1 - C2mdp2 (6) or Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. da = D12(de + Cjmd/?! + C2mdp2) ' (7) x where D12 = (Cx + C2)^ and dividing through by time increment dt allows (7) to be recorded as 109

J. Geotech. Engrg., 1992, 118(1): 107-124 =

Fluid Pressure Response Conservation of fluid mass must be maintained for each of the two porosity types, with appropriate transfer being possible between the two. The basic statement of continuity of flow requires that the divergence of the flow- velocity vector be equal to the rate of fluid accumulation per unit volume of space, i.e., Vrv = rate of accumulation. Equating the continuity con- straint for the porous phase with changes in fluid mass due to all possible mechanisms gives the rate of fluid accumulation as the summation of: (1) Change in total body strain resulting in fluid expulsion; (2) change in fluid pressure precipitating changes in volumetric pore fluid content as a result of fluid and grain compressibility; and (3) volumetric transfer between the porous blocks and the fractures under differential pressures. These three source terms represent the right-hand side of the continuity relationship as r V vx = m^! - c^ + K(Pl - p2) (9) r where V = (d/dxl, . . . , d/dx,) in the case of ;' dimensional geometry; Eulerian flow velocity vf = [vvl, . . . , vvi] and ax = [nJKf + (1 - n^/K,], where nx = the porosity of phase 1; Kf = the fluid bulk modulus; and ks = the solid grain bulk modulus. Transfer between the porous block and the surrounding fractures may be quantified by the assumption of a quasi- steady response governed by the coefficient K and the instantaneous pressure differential (/?! — p2) (Warren and Root 1963). T In (9), the volume strain within the porous medium, m e1, refers to the drained condition where excess pore pressures are zero throughout the loading process. The proportion of the volume strain manifest within the porous T phase may be determined from the total volume strain, m e1, by substituting (7) into (1) and the result into (4). This gives following division by dt, which T T r m e1 = mCJ)12(e + C^mpt + C2mp2) - mC1m/J1 (10)

which may be reduced to the drained response by requiring that p1 = p2 = 0. [The requirement of using drained solid body strain may be confirmed by noting the form of (40) where pressure changes are set to px = p2 = 0 leaving the only mechanism for fluid expulsion into the separate phases through displacement rates, ii.] The truncated (drained) form of (10) may then be substituted into the continuity relationship of (9) to yield T V vt = m^CJ^e - axA - K{Pl - p2) (11) to give the final form of the continuity relationship for phase 1. Again, it is noted that the influence of grain compressibility as a result of changes in intergranular stresses has been neglected. This effect may, however, be incorporated by substituting the vector mt for m in (11), where m1 accom- modates the intergranular stress relationship of (1). This effect is small for the stress levels of interests within geotechnical engineering and may be neglected. Repeating the same process for the fractured material gives a second Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. continuity relationship r r V v2 = mC2D12e - ct2p2 + K(Pl - p2) (12)

where the fracture compressibility term is defined as a2 = n2/Kf, with n2 110

J. Geotech. Engrg., 1992, 118(1): 107-124 representing the fracture porosity. Again the influence of grain compressibility on the intergranular stress relationship is neglected, but it may be readily included by substituting m2 for m.

DUAL-POROSITY LOAD RESPONSE For the case of general loading, the dual-porosity response may be fully described by (8), (11), and (12) to determine the magnitude of instanta- neously generated displacements and fluid pressures and their modification with time. Although these relationships are entirely general, it is instructive to consider the behavior under purely hydrostatic loading and examine the response in normalized format. This will allow us to determine the important differences between true dual-porosity systems and their representation by an equivalent single-porosity system.

Hydrostatic Behavior Where a hydrostatic load is applied to the dual-porosity medium, the three-dimensional form of (7) that contains a total of six subordinate equations may be reduced to a single component. For a ubiquitously jointed medium containing orthogonal fractures of uniform spacing, s, the strain components may be reduced on noting that mTe = 3e (13) and the compliance matrices are represented as

C-<^ <14„) c> = h <146> where k„ = the joint normal stiffness and the stiffness matrix reduces to 1 D12 = (Q + cy- (15) Rearranging (8), (11), and (12) for an applied hydrostatic stress magnitude cr yields

6- = D12s + DuQp! + D12C2p2 (16)

K(Pi ~ p2) = lDl2Cxi - axpx (17)

-K(Pl - p2) = 3D12C2e - a2p2 (18) T T where a zero external flux condition (V \1 = V y2 = 0) has been applied as a further boundary condition. The behavior is most conveniently decom- posed into: the undrained loading stage when fluid pressures pl0 and p2a are generated; and the dissipation of pressures through fluid exchange in the subsequent drained behavior. The undrained pore pressures provide initial conditions to the initial value problem posed in (16)—(18). The magnitudes of undrained pore pressures may be evaluated, as discussed subsequently. Following undrained loading, dissipation begins under constant total stresses

Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. at the boundary where & = 0. Interest is restricted to the continuum level, where stress gradients are defined to be zero. Spatial changes in total stress that may occur within the volume, even under unchanged boundary stresses [see Mandel (1953)], are still admissible, but they require the additional

111 ,

J. Geotech. Engrg., 1992, 118(1): 107-124 step of incorporating the equilibrium relationship introduced in (34). These considerations are in addition to the constitutive equations and our interest, for the moment, is merely in the constitutive relations. With some rearrangement, the governing equations [(16)-(18)] may be represented as

s = - C1p1 - C2p2 (19)

P^= -^(Pi-^ (20)

p2 = — (Pi - P2) (21) 7l2 where ^=^(t+i) <22«> *""5^fe + 1) -<22t) 7i2 = (an a 2 2 - QC2) (22c) a"=l3lfe+Cl) ^ °= = Ufe + C') ^ Eqs. (20) and (21) are independent of the strain field and may be solved in time where boundary conditions are supplied directly. Solution to (20) and (21) may be determined as

Pi(t) =Po~ (Po ~ pl0)e-^ti+722)^i2 (23a) + Pi(t) = Po ~ (p0 - P2Je-<-™ y*>

D e 3D c 1 i " ° = (i 1 ^ " i ^ g i y _ (24) v \ oil a2 / px ' Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. and the normalized pressure response is returned as

^ = ^ = 5l (25)

112

J. Geotech. Engrg., 1992, 118(1): 107-124 ^ = ^ = B2 (26)

where these also are equivalent to Skempton's pore pressure coefficients, #! and B2, written separately for the fluid in the pores and fractures. With some rearrangement, the pore pressure parameters may be defined as

B1 = -^ (27) i _|—i_^ a2 C\ + 3C' 11 S + Q and ^ = -Si (28) 3C, -I The pore pressure parameters are no longer bounded between zero and unity, as is the magnitude of the instantaneous normalized strain, which represents the aggregated influence of effective stresses in the two phases. A large magnitude of $x suggests that the material will exhibit a small instantaneous strain, although the fluid pressures generated are further controlled by the magnitude of the porous or fracture compliances and the compressibility of the fluid as embodied in a± and a2. r The long-term response of the system where V ^ = Vv2 = 0 results in an equilibrium pore pressure distribution with px = p2 and a steady magnitude of normalized strain. These long-term equilibrium parameters [pi(°°), co p2( ) and e(°°)] are most easily recovered from solving the equivalent single- porosity problem, as follows.

EQUIVALENT SINGLE-POROSITY RESPONSE The equations developed to describe the dual-porosity response may be modified to represent the case where it is assumed that pressures within both the porous and fractured material remain in equilibrium (i.e., p± = p2)- This is the assumption made when real porous-fractured systems are represented by a simple equivalent phase system. Requiring px = p2 for all times, including initial pressure pl0 and p2o, then (16) reduces to

6- = D12e + p (29)

and adding (17) and (18) under a similar requirement that px = p2, gives

3E = («! + a2)p (30) Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. where these represent the basic constitutive equations under hydrostatic loading. Implicit within (17) and (18), and by inference therefore in (30), is the requirement that Vrv = 0. Since no drainage is allowed, e is zero 113

J. Geotech. Engrg., 1992, 118(1): 107-124 following loading with the result that fluid pressures are maintained at their initially induced magnitude. This differs from the true dual-porosity case where there is an interchange of fluid between the pores and fractures. In the usual manner, (29) and (30) may be premultiplied by dt and the limit taken as dt —> 0 to recover the instantaneous normalized strain and pore pressure magnitudes as

^ 2 = A + _ 3- V' = - '.(3D a V ^12(01 + a2 )/ fe and

Po _ 3 1 = 5 (32)

where again B is bounded by zero and unity and incorporates the compliance of the porous solid, the fracture, and the fluid together with the appropriate distribution of porosities within each of the phases.

PARAMETRIC RESPONSE

Under undrained loading, the instantaneous generation of pore pressures is controlled by Bx and B2 in the true dual-porosity system, and by 50 in the pseudo dual-porosity system. For dual porosity, the mismatch in medium stiffness and porosity between the porous body and the fracture result in differential pressure generation that will diffuse with time to an equilibrium configuration. Where external drainage is controlled, the equilibrium pressure is given by plu = B0 where this magnitude is intermediate to the instantaneous pore and fracture pressures. Of significance are the magnitudes of instantaneous pore pressures in the pore and fracture as controlled by B1 and B2 of (25) and (26). Unlike B [(32)] for the pseudo dual-porosity system, the magnitude of Bl and B2 are not confined between zero and unity. Rather, one may be greater than unity and the complementary parameter less than unity. This behavior is representative of a piston effect, whereby pressures are amplified in the low stiffness porosity. According to (25) and (26), 5, is largest for large a,/oty- and small CJCj for r = 1,2;/ = 1,2 and i + /, and also is controlled by the ratio a,/Q. Physically, pore pressure magnitudes are increased as porosity increases or as solid stiffness decreases. The influence is illustrated for a variety of sedimentary and crystalline rocks in Table 1. Porosities and stiffnesses have been added to augment available data for poroelastic behavior of single-porosity solids. Regardless of fracture spacing, all materials return B2 magnitudes close to or greater than unity for the fracture pressures and B1 magnitudes of negligible proportion. Pore pressure parameter magnitudes for the aggregated system given by B in the table remain close to unity. As the fracture spacing is increased (from 0.1 m to 0.5 m), the aggregated B decreases uniformly due to the net increase in stiffness of the system. However, no Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. similar generalization is possible for the coefficients Bt and B2. The instantaneous pore pressure regime in the closed system (VT • v = 0) is modified with time by diffusive exchange between the porous body and the fracture. For realistic parameter estimates, given in Table 1, dif- 114

TABLE 1. Material Properties

Ruhr Tennessee Charcoal Berea Westerly Weber Parameter sandstone marble granite sandstone granite sandstone (1) (2) (3) (4) (5) (6) (7) £(GPa)a 29.8 60.0 47.5 14.4 37.5 28.1 v" 0.12 0.25 0.27 0.20 0.25 0.15 k„(GPa • m- 1)b-d 12.1 83.0 83.0 12.1 83.0 12.1 Ks(GPa)* 36.0 50.0 45.4 36.0 45.4 36.0 J. Geotech. Engrg., 1992, 118(1): 107-124 ^(GPa)" 3.3 3.3 3.3 3.3 3.3 3.3 k^mf 2 x 10"16 1 x 10-19 1 x 10-19 1.9 x lO-1 3 4 x 10-19 1 x 10-15 (jt(GPa • s) 1 x 10-1 2 1 x 10"12 1 x lO'1 2 1 x 10~12 1 x 10~12 1 x 10-12 n\ 0.032 0.0052 0.014 0.064 0.00106 0.176 f c f c " 2 0.0032° 0.052 0.05 0.0064 0.0106 0.0176 s(m) 0.1 0.1 0.1 0.1 0.1 0.1 B 0.986 0.912 0.905 0.982 0.941 0.969 4 2 2 3 2 3 Bt 8.5 x 10" 5.2 x 10" 4.9 x 10- 2.3 x lO" 1.9 x 10" 2.2 x 10" B2 1.031 1.018 1.029 1.050 1.104 1.028 2 1 1 5 3 «50(S) 1.96 x lO' 2.08 x 10 2.44 x 10 2.40 x 10" 4.91 x 10 ° 8.06 x 10" 2 1 2 4 1 2 *9s(s) 8.47 x 10' 8.98 x 10 1.05 x 10 1.06 x 10- 2.13 x 10 3.48 x 10- s(m) ' 0.5 0.5 0.5 0.5 0.5 0.5 B 0.938 0.723 0.711 0.929 0.818 0.875 3 1 2 1 2 Bi 4.72 x lO' 2.47 x 10-' 2.38 x 10" 1.34 x 10" 1.19 x lO" 1.20 x 10"

B2 1.151 0.974 1.008 1.243 1.415 1.136 1 2 2 4 1 'so(s) 3.72 x 10' 3.42 x 10 3.77 x 10 4.11 x 10" 5.88 x 10 1.49 x 10-' 3 3 3 2 f95(s) 1.61 x 10° 1.43 x 10 1.63 x 10 1.78 x 10~ 2.54 x 10 6.36 x 10-' aFrom Rice and Cleary (1976). bFrom Witherspoon et al. (1980). Trom Touloukian et al. (1989). dFrom Ryan et al. (1977).

°n2 = O.ln, for sedimentary rock. f n2 = 10.0/ij for crystalline and metamorphic rock. fusion is most likely from the fractures into the circumscribed blocks. The rate of this process may be considered as an indication of the relative importance of the dual-porosity effect. The equilibration rate is controlled by a dimensionless time given in (23A) and (23b) as tD = (yu + -y22)f/7i2, whereby the time to 50% and 95% equilibration of pressures are given by t5S = 0.6931 and P£ = 2.9957. In terms of the physical parameters of the system

*(%+ % + ' ) '

3£>12(aua22 - QQ where, in addition to the material coefficients controlling deformation, the block permeability, kh and appropriate diffusion lengths are incorporated. The behavior of representative sandstones in Table 1 are characterized by very rapid response times, reaching t9S in less than a second. The low- permeability matrix of the crystalline rock impedes equilibration that, de- pendent on permeability and fracture spacing, may extend over thousands of seconds. Characteristic responses are illustrated in Fig. 2 to illustrate the form of the response. With drainage potential proportional to fracture spacing s2, flatter responses in time are elicited where the path length is increased. Although parametric analyses previously were limited to sandstones and crystalline rock, the major trends in behavior are apparent. Firstly, fluid pressures developed in compliant fractures may considerably exceed those developed in the porous body. These differential pressures will only be significant, however, if matrix permeabilities are sufficiently low that the pressure differential may be sustained in time. From the time scales apparent for permeable (sandstones) and impermeable (crystalline) materials, it appears that dual-porosity effects may be of little consequence in permeable media, but drainage time scales are sufficient in low-permeability rocks to cause noticeable effect. Where permeability controls behavior, clays and low-permeability silts may be subject to dual-porosity effects.

T 1 1 r

FIG. 2. Undrained Pore Pressure Response for Fractured Sandstone and Frac- tured Marble

116

J. Geotech. Engrg., 1992, 118(1): 107-124 GLOBAL BEHAVIOR It is possible to solve a broader range of problems only if the previously developed constitutive behaviors are globally framed to represent spatial interaction. This requires that the constitutive, load-deformation relationship of (7) is substituted into an equilibrium statement and that the flow continuity (11) and (12) have an appropriate constitutive equation applied. The constitutive relationship, in this instance, is supplied by Darcy's law (Bear 1972). For the general representation of heterogeneous, mixed-initial value problems, the finite element method is chosen. The equilibrium statement in its most general form is given as

BTda dV - df = 0 (34) ( where B = the strain-displacement matrix; df = an incremental vector of applied boundary tractions, and the integration is completed over the volume of the domain (dV). Defining all quantities in terms of nodal variables de = BSu (35a)

dp1 = N3pi (355)

dp2 = Ndp2 (35c) where du = a vector of incremental nodal displacements; dp = a vector of nodal pressures; and N = a vector of shape functions interpolating fluid pressures. Substituting (7) and (35) into (34), dividing by dt, and rearranging gives the incremental equation in finite element format as

r J BD12B dV u + J B^D^dmN dV p,

r + J BD12C2mN dV p2 = / (36)

where a superscript dot identifies time derivative. Darcy's law must be added to the continuity requirements already imposed in (11) and (12) and may be simply stated as

vi = — V(px + 7Z) (37)

where 7 = the unit weight of the fluid; a. = the dynamic viscosity; kt = the porous medium permeability with the fracture permeability set to zero; and Z = the elevation of the control volume. Substituting (37) into (11) and applying the Galerkin principle results in

r - - f A^A dVVl + I NWQD^B dVu - at \ NN dV p1 (X Jv Jv Jv

for the porous phase and for the fractured medium 117

J. Geotech. Engrg., 1992, 118(1): 107-124 r r r r - - J Ak2A dV p2 + J Nm C2D12B dV li - a2 J NN dV p2

r r + K J NN dV(fix - p2) = - J Ak2A dV Z (39)

where k2 = a matrix of fracture permeabilities. Eqs. (36), (38), and (39) may be written at any time level (say, t - Af) and are most conveniently represented in matrix form as 0 0 ~u~ r-t-Ar (-Ki - S3) s3 Pi ( -K2 - S3)J -P2J t+\t F Gx G2 r u i r f -1 K Z (40) + Ex -Sj 0 Pi = l7 L E 2 0 -S2 .i»2. LK27ZJ where all submatrices are defined in Appendix I. These coupled equations remain symmetric, but they must be rearranged prior to solution in time. All terms on the right-hand side are known. The matrix relation may be integrated in time by using any convenient representation of the time derivatives. Using a fully implicit scheme, such that

(u'+Ar - u') (41a) At'+M 1 Pi+ ' (Pi+A' - Pi) (416) Af

1 +A F»2+A' (P' ' - p) (41c) Af 2 and substituting (41) into (40) gives

F Gt G2 1 +Ar Ex (-K1Af' - St - AfS3) AfS3 Af +A E2 ArS3 -K2Ar' ' - S2 - A/S3)J "u" 1 / 1 F G G2 r 1 Ei -Si 0 Pi + K l7Z (42) Af L E 2 0 - S2 -P2. LK27Zj where the matrix comprising the left-hand side is time invariant if a constant time-step magnitude is chosen. Initial undrained conditions are recovered if Kj = K2 = 0 and K = 0 allowing (42) to be solved for u, px and p2. A two-dimensional formulation is chosen to illustrate the potential of the dual-porosity formulation. Suitable shape functions must be chosen to represent the quadratic displacement field and bilinear fluid pressure variation present within individual elements. A four-node quadrilateral isoparametric

Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. element with quadratic-incompatible modes is used. The incompatible modes are condensed out at the element level. The broad range of material parameters that influence the pressure generation and dissipation response makes meaningful representation of transient behavior difficult. The versatility of the proposed technique is illus-

118

J. Geotech. Engrg., 1992, 118(1): 107-124 trated for a one-dimensional column for the parameters reported in Table 2. The fluid pressure response with depth is illustrated in Fig. 3 for a fracture spacing, s, of 0.1 m. Fluid pressures generated within the fractures are considerably greater than the matrix pore pressures. With time, the fracture pressures dissipate into the porous blocks until an equilibrium is reached. In this example, following the establishment of an equilibrium pressure distribution, the consolidation process will continue by drainage from the top of the layer. The surface settlement associated with the expulsion of fluid from the dual-porosity system is illustrated in Fig. 4. Fracture spacings of 0.025, 0.05, and 0.1 m are used to demonstrate the differing responses. Where spacing is decreased, the dissipation process is accelerated. The dual- porosity response is contrasted with the behavior of a single-porosity system, where the different form of settlement-versus-time behavior is apparent. A two-dimensional geometry, represented in Fig. 5, also may be used to illustrate the essential differences between single-porosity and dual-porosity effects. The response to a load of finite extent is illustrated in Fig. 6. As fracture spacing is decreased, the magnitude of initial normalized displacement increases, reflecting the dominant influence of fracture stiffnesses on the deformation behavior. The small block dimensions present within the

TABLE 2. Example Coefficients Parameter Definition Magnitude Units (1) (2) (3) (4) E modulus of elasticity 1.0 MN/m2 V Poisson's ratio 0.15 fissure stiffness 0.1 MN/m2/m K 2 Kf fluid bulk modulus 0.1 MN/m « i matrix porosity 0.1 n2 fissure porosity 0.05 3 4 ktl\i. matrix permeability 0.01 x 10- m /(MN • s) 4 k2/\i. fissure permeability 0.1 m /(MN • s) s fissure spacing 0.025, 0.05, 0.1 m Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. FLUID PRESSURE (P/u) -

FIG. 3. Pore Pressure Equilibration Response for One-Dimensional Column Loaded Axially

J. Geotech. Engrg., 1992, 118(1): 107-124 •A—;va—r.-*H

o—- Blot(E=0.1)

— -a— Blot(E=1)

—••*•— Blot(E=10)

* — Dual (s=0.05)

» — Dual(s=0.1)

• — Dual (s=0.2)

10' 10° 10' 10n53 10in6D 10in7' 10° lO* 10'"1010,n 11,„12' 'lO'^Ii O

Time (sec) FIG. 4. Surface Displacement Response for One-Dimensional Column Loaded Axially

Htl $h 4

*,» ^ < *•*"

X 6n dn

FIG. 5. Two-Dimensional Geometry

closely fractured medium result in an acceleration of the consolidation process. As bulk permeability magnitude is correspondingly increased, the dual- Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. porosity behavior exhibits a characteristic acceleration in the early time response and a deceleration in the late time response over the equivalent single-porosity behavior. The former characteristic results from the rapid equilibrating of fluid pressures within the fractures where dissipation in-

120

J. Geotech. Engrg., 1992, 118(1): 107-124 creases the magnitude of effective stresses. The high compressibility of the fractures correspondingly yields a magnified response. At later times, the dual porosity exhibits a more sluggish response over the equivalent single- porosity behavior as a result of delayed changes in effective stresses within the porous blocks. This behavior is slightly magnified in time as block dimension is increased, resulting in a longer dissipation tail, as evident in Fig. 6.

CONCLUSIONS A constitutive model for the coupled flow-deformation response of dual- porosity media has been presented. The formulation is stated in component terms, where behavior is controlled by the mechanical and hydraulic response of the individual porous and fractured phases. Representation in this form allows the potential importance of dual-porosity effects to be evaluated in controlling the coupled flow-deformation behavior of fractured geologic media. Pore pressure coefficients B1 and B2 may be defined for the dual-porosity response, where the magnitudes are not necessarily confined to the range between zero and unity. For realistic material parameters, the magnitude of the pore pressure coefficient for the fractures is commonly greater than unity, and for the porous medium, less than unity. With the generation of this pressure differential, fluid transfer between the porous medium and the fracture will attempt to equilibrate the pressures. Equilibration times are controlled by the compliances of the porous medium and the fractures, together with porosities and the block-fracture fluid-transfer coefficient K. This latter component is a function of block size and block permeability. Where undrained loading is prescribed, the times to 50% and 95% equilibration may be determined explicitly. Where general boundary conditions are applied to the dual-porosity system, a numerical solution method must be employed, Representation as a finite element system is particularly appropriate, allowing spatial variability in parameters and general boundary conditions to be readily accommodated.

z 1-° .

< O.B DUAL POROSITY a. s= 0.1m—~~^/^ s: 0.05 mO^C

Io 0.6 / / J ^SINGLE POROSITY N O^s-.O.Im ^^6= 0.05 m 1 °-4 „^d£"Ss

0.2 < i """I 11 mini ni i • '"I'll "I ' » " " " 1 ' 1 1 " ' " I 1 2 108 Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. 10 10 103 10" 105 1Q6 1Q7 ELAPSED TIME (SECONDS)

FIG. 6. Surface Displacement Response at Centerline for Single-Porosity and Dual-Porosity Systems (parameters Given in Table 2)

121

J. Geotech. Engrg., 1992, 118(1): 107-124 The procedure may be applied to a variety of engineering situations where the accurate determination of parameters associated with dual-porosity systems is important. These include the performance of fractured clay tills used as barriers to contaminated ground water and waste leachate and the behavior of fractured rocks deforming above mined underground openings. In either instance, the behavior of the dual-porosity medium is distinctively different from that of a single-porosity continuum, and it is important to recognize this differentiation if serviceable designs are to be commissioned. For example, large initial deformations are apparent in the dual-porosity medium under undrained loading that result in considerably different transient response to anything that can be explained using single-porosity representation. Thus, use of an incorrect phenomenological model in describing field data would erroneously predict the performance of the prototype. For this reason, it is imperative that a correct and adequate model is initially selected.

ACKNOWLEDGMENTS Support provided by the Standard Oil Center for Scientific Excellence and the Waterloo Centre for Ground Research is most gratefully acknowl- edged.

APPENDIX I. SUBMATRICES

E, = J NWC^B dV (43)

E2 = J NWQAzB dV (44)

F = J BT)12B dV (45)

r Gt = J BD12CimN dV (46)

r G2 = J BD12C2mN dV (47)

H = J NrN dV (48)

Kj = - I ArkiA dV (49)

U. Jv

r

Si = «xH (51)

52 = a2H (52) 53 = KEL (53) 122

J. Geotech. Engrg., 1992, 118(1): 107-124 APPENDIX II. REFERENCES Aifantis, E. C. (1977). "Introducing a multi-porous medium." Developments in mechanics, 8, 209-211. Aifantis, E. C. (1980). "On the problem of diffusion in solids." Acta Mechanica, 37, 265-296. Bear, J. (1971). Dynamics of fluids in porous media. American Elsevier, New York, N.Y. Bibby, R. (1981). "Mass transport of solutes in dual porosity media." Water Resour. Res., 17, 1075-1081. Biot, M. A. (1941). "General theory of three-dimensional consolidation." /. Appl. Phys. 12, 151-164. Elsworth, D. (1989). "Thermal permeability enhancement of blocky rocks: one dimensional flows." Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 26(3/4), 329- 339. Goodman, R. E. (1974). "The mechanical properties of joints." Proc. of the Third Congress of Int. Soc. for Rock Mech., 127-140. Huyakorn, P. S., Lester, B. H., and Faust, C. R. (1983). "Finite element techniques for modeling groundwater flow in fractured aquifers." Water Resour. Res., 19, 1019-1035. Iwai, K. (1976). "Fundamental studies of fluid flow through a single fracture," thesis presented to the University of California, at Berkeley, California, in partial ful- fillment of the requirements of the degree of Doctor of Philosophy. Kazemi, H. (1969). "Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution." Soc. Pet. Engrg. J., 9, 451-462. Kazemi, PL, Merrill, L. S., Jr., Porterfield, K. L., and Zeman, P. R. (1976). "Nu- merical simulation of water-oil flow in naturally fractured reservoirs." Soc. Pet. Engrg. J., 16, 317-326. Kazemi, H., and Merrill, L. S. Jr. (1979). "Numerical simulation of water imbibition in fractured cores." Soc. Pet. Engrg. J., 19, 175-182. Khaled, M. Y., Beskos, D. E., and Aifantis, E. C. (1984). "On the theory of consolidation with double porosity—III. A finite element formulation." Int. I. Numer. Anal. Meth. Geomech., 8, 101-123. Mandel, J. (1953). "Consolidation des sols (etude mathematique)." Geotechnique, 3, 287-299. Nur, A., and Byerlee, J. D. (1971). "An exact effective stress law for elastic deformation of rock with fluids." J. Geophys. Res., 76(26), 6414-6419. Odeh, A. S. (1965). "Unsteady-state behavior of naturally fractured reservoirs." Soc. Pet. Engrg. I., 5, 60-66. Pruess, K., and Narasimhan, T. N. (1985). "A practical method for modelling fluid and heat flow in fractured porous media." Soc. Pet. Engrg. J., 25, 14-26. Rice, J. R., and Cleary, M. P. (1976). "Some basic stress diffusion solutions for fluid saturated elastic media with compressible constituents." Rev. Geophys. Space Phys., 14(2), 227-241. Ryan, T. M., Farmer, I., and Kimbrell, A. F. (1977). "Laboratory determination of fracture permeability." Proc. of the 18th U.S. Symp. on Rock Mechanics, Skempton, A. W. (1960). "Effective stress in soils, concrete and rock." Proc. of the Symp. on Pore Pressure and Suction in Soils, Butterworths, London, U.K., 1. Terzaghi, K. (1923). "Die Berechnung der Durchasigkeitsziffer des Tones aus dem Verlauf der hydrodynamishen Spannungserscheinungen, Sitzungsber." Acad. Wiss. Wien Math Naturwiss. Kl. Alot. 2A, 132, 105 (in German). Terzaghi, K. (1943). Theoretical soil mechanics, John Wiley & Sons, New York, N.Y. Thomas, L. K., Dixon, N. T., and Pierson, G. R. (1983). "Fractured reservoir simulation." Soc. Pet. Engrg. J., 23, 42-54.

Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. Touloukian, Y. S., Judd, W. R., and Roy, R. F. (1989). Physical properties of rocks and minerals. 11(2), McGraw-Hill, New York, N.Y. Warren, J. E., and Root, P. J. (1963). "The behavior of naturally fractured reservoirs." Soc. Pet. Engrg. I., 3, 245-255.

123

J. Geotech. Engrg., 1992, 118(1): 107-124 Wilson, R. K., and Aifantis, E. C. (1982). "On the theory of consolidation with double porosity." Int. J. Engrg. Set, 20(9), 1009-1035. Witherspoon, P. A., Wang, J. S. Y., Iwai, K., and Gale, J. E. (1980). "Validity of cubic law for fluid flow in a deformable structure." Water Resour. Res., 16(16), 1016-1024. Yamamoto, R. H., Padgett, J. B., Ford, W. T., and Boubeguira, A. (1971). "Com- positional reservoir simulator for fissured systems—the single block model." Soc. Pet. Engrg. J., 11, 113-128.

APPENDIX III. NOTATION

The following symbols are used in this paper:

A = derivatives of shape function matrix N; B = strain displacement matrix; 5, = Skempton's fluid pressure coefficients in true dual-porosity system; B = Skempton's fluid pressure coefficients in pseudodual-porosity system; c, = compliance matrices; D, = elasticity matrices; _1 D12 = elasticity matrix (C± + C2 ) ; E = Young's modulus; f = vector of boundary tractions; K = fluid transfer coefficient; K = permeabilities; K = joint normal stiffness; Kt = fluid bulk modulus; Ks = solid grain bulk modulus; m = one dimensional vector; for three-dimensional problem, mT = (1 1 1 0 0 0); for two-dimensional problem, mT = (110); m, = modified intergranular stress relationship; m„ = (m - l/3i&D„m); n, = porosities; Pi = fluid pressure; s = fracture spacing; t = time; to = dimensionless time; u = displacement vector; V,' = flow velocity vector; Z = elevation of control volume; d = partial differential operator; «/ = hydrostatic compressibility; 7 = unit weight of fluid; d = partial differential operator; e — strain vector; in three-dimensional Cartesian coordinates, eT =

(*• = dynamic viscosity; V = Poisson ratio; and = r

Downloaded from ascelibrary.org by Pennsylvania State University on 03/25/16. Copyright ASCE. For personal use only; all rights reserved. a stress vector; in three-dimensional Cartesian coordinates, rr =

Subscripts i = 1, 2, denoting pore phase and fracture phase, respectively. 124

J. Geotech. Engrg., 1992, 118(1): 107-124 4:4 Mechanical deformation - 1D and 2D Elements [5:1][5:2] http://youtu.be/WQbG588gOg4 [5:1]!Solid!Mechanics! Principle!of!virtual!work! 1D!element! 2D!element! ! ! ! ! ! ! ! ! ! ! ! !

[5:2]!Solid!Mechanics! Constitutive!Relations! ! ! ! ! ! ! ! ! ! ! ! ! ! !

4:5 Coupled Hydro-Mechanical Models [6:2] https://youtu.be/ve1EGEOT97k [6:2]!Linked!Mechanisms! HM!–!Poromechanics! Effective!stresses! FE!equations! Summary!equations!(Biot,!1941)! EGEEfem! Comsol! ! ! !

Fru tiT eL6ry6a.)T HFfTk' D

Brat e1anA*'< hk ' Co'nshtLT'u'' /c-""'< e?VtJ'4nL

l,4n^"a'^ qfl'w*, t _9 )y- & ej : b 3eV^-.t. (/-2t) Dx Ex

t:.'"""f ofat .Du,l. \ 'L /,P qx'' u!''L* I ,].* 2/ /.a q4 = b ts (rf") ilJ /ft a* /fv

fr"- e1uz1-'^

,/'-i, +la6t Wv'? =gA rl* I ' ,-* f" &At

Huln'*

r7P\ l: +f+:j{:f'=f ( Lo 6JL7l:1'Q," Lc DJ/fJ T