Computers and Geotechnics

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1 Computer and Geotechnic 35 (2008) Content lit available at ScienceDirect Computer and Geotechnic journal homepage: Unaturated oil: From contitutive modelling to numerical algithm Daichao Sheng a, *, Antonio Gen b, Delwyn G. Fredlund c, Scott W. Sloan a a Centre f Geotechnical and Material Modelling, The Univerity of Newcatle, Autralia b Department of Geotechnical Engineering and Geocience, Technical Univerity of Catalonia, Barcelona, Spain c Golder Aociate Ltd., Sakatoon, SK, Canada article info abtract Article hity: Available online 30 September 2008 Keywd: Unaturated oil Contitutive modelling Numerical algithm Thi paper preent an overview of contitutive modelling of unaturated oil and the numerical algithm f olving the aociated boundary value problem. It firt dicue alternative tre and train variable that can be ued in contitutive model f unaturated oil. The paper then dicue the key iue in unaturated oil modelling and how thee iue can be incpated into an exiting model f aturated oil. Thee key iue include (1) volumetric behaviour aociated with aturation uction change; (2) trength behaviour aociated with aturation and uction change, and (3) hydraulic behaviour aociated with aturation uction change. The paper alo how how hyterei in oil water characteritic can be incpated into the elato-platic framewk, leading to coupled hydro-mechanical model. Finally, the paper demontrate the derivation of the incremental tre train relation f unaturated oil and dicue briefly the new challenge in implementing thee relation into the finite element method. Ó 2008 Elevier Ltd. All right reerved. 1. Introduction All oil can be unaturated with repect to water. In thi regard, unaturated oil are nothing pecial. However, the firt fifty year of oil mechanic hity have been primarily concerned with oil aturated with water and mot oil mechanic principle developed in that period apply to aturated oil only. Thi htcoming i actually one of the main driving fce f the emerging ubject of unaturated oil mechanic. Another imptant driving fce i due to the ditinct volume, trength and flow characteritic of certain oil when they become unaturated with water. Some oil can experience a ignificant volume change upon a change of the degree of aturation. Soil that expand upon wetting are known a expanive oil, whilt oil that compre upon wetting are known a collapible oil. Due to the very large volume change that they may undergo, both of thee oil type can everely damage foundation and the tructure that they uppt. The hear trength of a oil can alo change ignificantly a it degree of aturation change, and a related engineering problem i lope failure caued by rainfall. Unaturated oil alo have ditinct hydraulic propertie which have ignificant implication in the perfmance of oil cover ytem in wate containment. Therefe, the key iue in unaturated oil mechanic are (1) the volumetric behaviour aociated with aturation uction change; (2) the trength behaviour aociated with aturation * Creponding auth. Fax: addre: daichao.heng@newcatle.edu.au (D. Sheng). and uction change, and (3) the hydraulic behaviour aociated with aturation uction change. In term of contitutive modelling, the key quetion i how thee iue can be incpated into a aturated oil model o that it can alo be ued f unaturated tate a well. In the firt part of thi paper alternative method f addreing thee key iue in unaturated oil model are dicued. In particular, the focu i put on the comparion between a recent model that the auth are aociated with and the other common model. A uch, the coverage of the paper i inevitably elective, and can not erve a a complete tate-of-the-art review of the ubject of contitutive modelling of unaturated oil. The econd part of the paper outline the challenge and olution f implementing unaturated oil model into the finite element method. 2. Contitutive modelling: an overview 2.1. Stre and train variable Contitutive relation ued to repreent the mechanical behaviour of material are uually decribed in a tre pace. The choice of the tre pace i thu a fundamental iue in contitutive modelling. Ideally the definition of tree hould be independent of the behaviour the tate of the material, o that the tre pace doe not change with the material tate. There i little argument that total tree hould be ued f ingle phae material uch a metal and dry and. It i alo widely accepted by the oil mechanic community that effective tree (the difference X/$ - ee front matter Ó 2008 Elevier Ltd. All right reerved. doi: /j.compgeo

2 D. Sheng et al. / Computer and Geotechnic 35 (2008) between the total tree and pe water preure) can be ued f aturated oil. The definition of the total tree f dry oil and the effective tree f aturated oil are naturally independent of the oil behaviour oil tate. The tre pace in thee cae are thu eparated from the material tate. We alo note that the o-called total tre in a dry and i actually the difference between the abolute total tre and the atmopheric pe air preure. F oil that are partially aturated with one pe fluid, the choice of the tre pace become me complicated and the tre pace may become dependent on the material tate. In 1960, great efft were made to identify a ingle effective tre that can be ued to decribe the defmation and trength characteritic of unaturated oil [1]. Bihop [8] uggeted the following effective tre concept r 0 ij ¼ r ij u a d ij þ vðu a u w Þd ij ¼ r ij þ vd ij ð1þ where r ij i the total tre, r 0 ij i the Bihop effective tre, r ij i the net tre, u a i the pe air preure, u w i the pe water preure, i the oil (matric) uction, and v i a parameter that may depend on the degree of aturation on the uction. The oil uction in thi paper refer to the matric uction which conit of the capillary and adptive potential. When the pe water exit a capillary water at relatively high degree of aturation, the capillary potential (W c ) i dominant in the matric uction W c = u a u w. When the pe water exit a adbed water film in the oil, the adptive potential (W a ) become dominant in the matric uction. In thi cae the true water preure i not well defined ince it i not unique at one material point and i dependent on the proximity to the particle urface. An apparent water preure can be introduced to quantify the adptive potential: u w = u a W a, i.e. the apparent water preure repreent the negative adptive potential meaured in exce of air preure. When the air preure i atmopheric (zero), the apparent water preure i then the negative adptive potential. Such an apparent water preure i then unique at one material point. With uch a definition of u w, the matric uction can be expreed a = u a u w and can be ued continuouly f a relatively large range of aturation, from fully aturated to very dry tate. Even though the new definition of the effective tre in 1960 ha led to ome ucce in decribing the hear trength of unaturated oil, it ha not led to great ucce in modelling the general mechanical behaviour of unaturated oil, not at leat until the lat decade o. Some limitation of the ingle Bihop effective tre in explaining volume collape during wetting of unaturated oil were repted by Jenning and Burland [36]. Me imptantly, becaue the parameter v uually depend material tate (e.g. the degree of aturation) and even on tre path (e.g. the tranition uction between aturated and unaturated tate), the tre pace defined by Eq. (1) depend on the material behaviour and change with material tate. Therefe, the contitutive behaviour of the material i embodied in both the contitutive relation and the tre pace where the contitutive relation i defined. A pointed out by Mgentern [53], we nmally link equilibrium conideration to defmation through contitutive behavi and hould not introduce contitutive behavi into the tre tate. In 1960 and 1970, it wa realied that it wa poible to ue two independent et of tre variable to model unaturated oil behaviour rather than combining them into one ingle effective tre. F example, Coleman [14] uggeted the ue of the net axial and radial tree and the net pe water preure to repreent triaxial tre tate. Bihop and Blight [9] ued the concept of independent tre tate variable when plotting volume change in an unaturated oil. [51] ued the independent tre variable (called tate parameter ) to decribe the volumetric behaviour of unaturated oil. Numerou other reearcher have ubequently preented the volume change behaviour a urface defined by independent tre tate variable [2,18,5]. Fredlund and Mgentern [19] further provided a theetical bai and jutification f the ue of two independent tre tate variable. The jutification wa baed on the uperpoition of coincident equilibrium tre field f each of the phae of a multiphae ytem, within the context of continuum mechanic. Three poible combination of independent tre tate variable were hown to be jutifiable from the theetical continuum mechanic analyi. However, it wa the net tre and the matric uction combination that proved to be the eaiet to apply in engineering practice: r ij u a d ij ¼ u a u w r ij The net nmal tre primarily account f the external application and removal of total tre (e.g., by excavation, fill and applied load). The matric uction primarily account f the impact of the climatic environment above the ground urface. Fredlund et al. [20] alo preented a hear trength equation uing the independent tre variable. In the context of contitutive modelling, Alono et al. [4] were the firt to provide a complete elato-platic framewk f modelling unaturated oil behaviour. Thi model ue the net tre and uction a the tre variable and became known a the Barcelona Baic Model. A large number of other elato-platic model oon followed (e.g., [58,27,42,52,88,15,10,6,49,78,13,64,62,56]. All thee model deal with tre train relation only. Me recent model have incpated uction aturation relationhip with hyterei into tre train relationhip ([86,16,82,87,24,69,77,79,76, 41,63,81]). Review of contitutive modeling of unaturated oil can be found in e.g. [26,86,37,29,30,65]. A common feature of thee model i that the uction i conidered a an additional tre variable, at leat a an additional hardening parameter [10,49]. However, there i little conenu on whether an independent tre (e.g. net tre total tre) tre variable uch a Bihop effective tre hould be ued. A different argument wa put fth by Houlby [34] uing the wk-conjugate variable in the wk input f a oil element. In term of wk input, it i poible to define a et of train variable that are wk-conjugate to the choen et of tre variable. F example, the wk-conjugate train variable f the two et of independent tre variable, defined by the net tree and the uction, are the oil keleton train and the volumetric water content: r ij u a d ij u a u w () e ij h where e ij i the oil keleton train and h i the volumetric water content. We note that the definition of the tre variable in expreion (3) are independent of each other and independent of material tate. However, their wk-conjugate train variable are not independent, i.e. the oil keleton train and the volumetric water content are dependent on each other. It i alo noted that the net tre become the total tre when the oil i fully aturated and the pe air i under atmopheric preure. When the air preure i kept at the atmopheric value (which i approximately true f mot in itu condition), the matric uction i equivalent to the negative pe water preure. Early model uing the net tre and uction thu uffer a maj htcoming in that they are not continuou at the tranition between aturated tate and unaturated tate, becaue the tre variable ued f unaturated tate (total tre) do not change to the tre variable f aturated tate (effective tre). However, it ha recently been demontrated in the SFG model by Sheng et al. [66] that thi et ð2þ ð3þ

3 812 D. Sheng et al. / Computer and Geotechnic 35 (2008) of tre variable can alo lead to a continuou decription of oil behaviour. Me dicuion of the SFG model will be given in later ection. On the other hand, the wk-conjugate tre variable to the two et of independent train variable, defined by the oil keleton train (e ij ) and the degree of aturation (S r ), are the average tree and a modified uction (n). Thee train variable are conidered to be independent becaue a change in one of them doe not necearily reult in a change in the other. The poity n play a role in caling the wk input (due to a change in aturation) per unit void volume to the wk input per unit volume of the oil matrix. Therefe, the econd et of alternative tre variable take the following fm [69]: r ij u a d ij þ S r ðu a u w Þd ij () e ij ðu a u w Þ S r where S r i the degree of aturation. The tre variable defined in expreion (4) are dependent on one another, a well a on the material tate (S r ). The tre (r ij u a d ij + S r (u a u w )d ij ) become the Terzaghi effective tre (r ij u w d ij ) when the oil become aturated. Therefe, the above tre variable can be ued both f aturated and unaturated tate. The tre (r ij u a d ij + S r (u a u w )d ij ) i alo called Bihop tre and ha been found to repreent the average tre acting on the olid phae by Haanizadeh and Gray [32] uing the entropy inequality exploited via the Colemann-Noll procedure, by Lewi and Schrefler [45] uing volume averaging, and by Hutter et al. [35] on the bai of mixture they. The et of tre variable given in expreion (4) have been ued in recent unaturated oil model uch a thoe of Sheng et al. [69], Sun et al. [76], and Santagiuliana and Schrefler [63]. Me recently Nuth and Laloui [54] and Laloui and Nuth [44] refer to thi et of tree a the generalied tree f unaturated oil and have provided further experimental evidence to ende it ue. It hould be noted that the tre the train variable in expreion (3) and (4) may not have the ame phyical meaning. F example, uction i a phyically different quantity than tre, and aturation i a phyically different quantity than train. They are grouped together to fm pace f etablihing contitutive relation. Thi i very imilar to chooing a codinat ytem (e.g. x, y, z, t) to decribe a function. We alo note that tre and uction have the ame unit, which i why the two variable can ometime be added together through dimenionle multiplie (v, S r 1). In thi context, the argument that uction i phyically not a tre variable [50] i not particularly pertinent. It i alo generally true that the complex tre variable defined by expreion (4) tend to lead to impler contitutive equation, wherea the impler tre variable defined by expreion (3) tend to lead to me complex contitutive equation. The complex tre variable in expreion (4) depend on material tate and are not eaily controllable in labaty teting. Therefe, it i not poible to develop a completely new contitutive relationhip in term of thee variable, unle an exiting framewk i ued. However, it i poible to tranfm an exiting contitutive relationhip fmed in term of the impler tre variable (expreion (3)) to the complex tre pace. Such a tranfmation can often overcome the dicontinuity problem at the tranition between aturated and unaturated tate, a wa done by Sheng et al. [70,72] f the Barcelona Baic Model, me recently by Kohler and Hoftetter [43] f the cap model. In thi regard, it i probably preferable to call the complex tre in expreion (4) the contitutive tre rather than the effective tre, meaning that they are pecific variable ued f contitutive modelling [69,29]. However, we have kept the terminology Bihop effective tre in thi paper f conitency with other publication in the area. The choice of the tre variable alo ð4þ ha a ignificant influence on the yield and failure urface, which i later dicued in thi paper Contitutive model f unaturated oil The firt complete elato-platic model deigned explicitly f unaturated oil wa preented in [4] (and in a me ummary fm in [28]. Thi model wa fmulated in term of net tree and uction. With ome light modification, it came to be known a the Barcelona Baic Model (BBM) and can perhap be ummaried by Fig. 1, where a three dimenional yield urface in p q pace i depicted. Here, p i the mean net tre and q i the deviat tre. Under aturated condition, the yield urface crepond to the Modified Cam Clay (MCC) ellipe [61] and the ize of the elatic domain increae a the uction increae. The rate of increae, repreented by the Loading Collape (LC) curve, i one of the fundamental characteritic of the model. One of the main objective of the development of the BBM wa to try to inert unaturated oil mechanic into the maintream of current and pat development in aturated oil mechanic. Thi aim guided many of the choice adopted in the definition of the model and explain the rough implicity of many of it feature. It wa intended that the model could be ued to make qualitative prediction by imple hand manipulation in the ame way that the conceptual critical tate framewk i often ued. Thi implied the adoption of net tree a one of the baic tre variable. If other tre variable are ued, it i quite difficult to follow conventional labaty tre path in an effective manner. Indeed, the firt ue of the concept underlying the BBM wa preented in [3], befe the mathematical fmulation wa fully developed. The need f a clear connection with aturated oil mechanic led to the adoption of the MCC model a the reference model in uch a way that the BBM contitutive law become the claic MCC model when become zero (i.e. on reaching aturation). In fact, many other elato-platic aturated model could have been ued, a the unaturated fmulation i quite general (ee, f example [58,42,52,10,13,64,62]. Further example of the implifying aumption adopted in the model are the ue of traight line f the void ratio ln p relationhip (implying a continuou increae of the collape train upon wetting) and the linear increae of apparent coheion with uction. A direct benefit of thi implicity i that only a limited number of additional parameter are required. Meover, in pite of it lack of complexity, the model can decribe a large number of typical feature of the mechanical behaviour of unaturated oil in a natural unfced way [3,4]. Some example are: the variation of wetting-induced welling collape train depending on the magnitude of applied tree, the reveral of volumetric train oberved ometime during wetting-induced collape, the increae q Fig. 1. Three dimenional yield urface of Barcelona baic model. LC p

4 D. Sheng et al. / Computer and Geotechnic 35 (2008) of hear trength with uction, the tre path independency aociated with wetting path and the oppoite when the tre path involve drying the apparent increae of preconolidation tre with uction. Following the introduction of the BBM, other model were quickly developed that ought to limit ome of it htcoming, while at the ame time keeping the ame ce of baic aumption. Thu [38] ued non linear relationhip f the variation of void ratio with ln p o that the collape train did not increae indefinitely but went through a maximum befe reducing to zero at high tree. Gen and Alono [27] and Sánchez et al. [64] extended the BBM f highly expanive oil by conidering the interaction between macrotructure and microtructure and generalied platicity they. Wheeler and Sivakumar [88] ued model function me cloely baed on experimental reult. Cui et al. (1995) adopted a aturated yield function typical of aniotropically conolidated oil. Me recently, the mechanical part of the hydromechanical model of Vaunat et al. [82], Wheeler et al. [87], Gallipoli et al. [24], Sheng et al. [69] and Sun et al. [76] all follow a framewk imilar to the BBM, with the effect of aturation being conidered by incpating oil water characteritic curve and hydraulic hyterei. Intead of going through the different feature of variou model in detail, we will dicu how key iue can be incpated into a aturated oil model o that it can alo be ued f unaturated tate a well. Thee key iue include (1) the volumetric behaviour aociated with aturation uction change; (2) the trength behaviour aociated with aturation and uction change, and (3) the hydraulic behaviour aociated with aturation uction change. Our attention here i given to alternative poibilitie f conidering thee key iue in a contitutive model and their implication. In particular, we will focu on the comparion between a recent model that the auth are aociated with (namely the SFG model by Sheng et al. [66] and the other common model uch a the BBM). 3. Key ingredient in unaturated oil model 3.1. Volumetric tre train model Suction affect the volumetric behaviour, yield tre and hear trength of an unaturated oil. It generate capillary inter-particle fce nmal to contact, while pe preure generate iotropic tree around oil particle. A uch, it play a me complex role than the preure mean tre. A pointed out by Li [46], ome meaure of oil fabric hould be incpated when conidering the effect of uction. However, any uch meaure i difficult to define and ha not yet been ued in contitutive modelling. Intead, uction i uually treated a a imilar quantity to the mean tre. Under uch a framewk, the only extra contitutive law that i required to extend a aturated oil model to unaturated oil i the volume-tre uction relationhip. The effect of uction on the yield tre and hear trength can be incpated into the model baed on the volume-tre uction relationhip. The change in the pecific volume (v) of an unaturated oil in repone to uction () mean tre (p) change i typically modelled in one of the following way: dv ¼ dnðþ d k vp ðþ dp d p ln p dk vpðþ d d dv ¼ dnðþ d d k vp ðþ dp0 p 0 dp dv ¼ k vp p k d v ln p 0 dk vpðþ d d ð5þ ð6þ ð7þ where k vp i a material parameter repreenting the tre compreibility under contant uction, p i the net mean tre, p 0 i Bihop effective mean tre, k v i a material parameter repreenting the uction compreibility under contant mean tre, and N() i the pecific volume of the oil when the mean tre i 1 (unit of tre). The parameter N and k vp in Eq. (5) and (6) and the parameter k v in Eq. (7) are uually conidered a function of uction. Over a certain tre range, the parameter k vp i uually approximated by one two contant, depending on the preconolidation preure. Eq. (5) and (6) are uually preented in the literature in total fm. All thee equation imply linear relationhip in the v ln p ln pace. It i alo common to aume linear relationhip in the ln v ln p ln pace. In the later cae, the differential of the pecific volume (dv) can be replaced by the negative differential of the volumetric train ( de v ). Eq. (5) ue the net tre p and i ued in the Barcelona Baic Model of Alono et al. [4] and many other model. Eq. (6) ue Bihop effective tre and i ued in the fmulation of Kohgo et al. [42], Bolzon et al. [10] and Let and Khalili [49]. Eq. (7) ue the net tre p [21] and eparate the compreibility due to a tre change (k vp ) from that due to a uction change (k v ). All thee equation are of coure baed on Cam clay elato-platicity f aturated oil: dv ¼ k vp dp 0 p 0 dp ¼ k vp p þð u w Þ k dð u w Þ vp p þð u w Þ ðsaturated oilþ Eq. (5) (8) are confuingly imilar, but they bear different implication. F example, Eq. (5) and (7) do not recover Eq. (8) f aturated tate. Thi can be verified imply by conidering the cae where the pe air preure (u a ) remain atmopheric. Under thi condition, Eq. (5) and (7) are valid only f zero pe water preure when the oil become aturated. If the tranition uction between aturated and unaturated tate i not zero, the tre-induced volume change i undefined at thi uction in Eq. (5) and (7). The reaon f thi dicontinuity i that the net tre doe not cover the effective tre and that the zero uction doe not ufficiently repreent all aturated tate. Indeed, a confuing concept that i often cited in the literature i to treat zero uction a the equivalent to aturated tate. Thi concept ha two htcoming. Firt, it doe not conider the different uction level that crepond to full aturation (S r =1) during drying and wetting repectively. Second, it doe not provide a continuou treatment between poitive and negative pe water preure. When the pe air preure remain contantly atmopheric, a better alternative would be to treat the atmopheric air preure a zero and the uction a a negative pe water preure f all aturated tate. In thi cae, the net tre imply become the total tre and the uction become the negative pe water preure. Such an alternative will provide a continuou tranition between aturated and unaturated tate. Eq. (6) fully recover the tandard model f aturated oil, but contain Bihop effective tre parameter v. Thi parameter often depend on the material a well a the material tate, leading to the quetionable outcome that the tre pace where the material i modelled change with the material behaviour and even the material tate. Eq. (7) eparate the compreibility due to a tre change (k vp ) from that due to a uction change (k v ). A uch, the parameter k vp i not necearily a function of the uction, which i an advantage over the other two equation. A chematic view of the prediction accding to Eq. (5) and (6) i hown in Fig. 2. Thee two model both have difficulty in explaining the curvature of the nmal compreion line at poitive uction f oil dried from lurry. Let u conider the cae where a lurry oil i firt dried to a pecified uction and then iotropically compreed ð8þ

5 814 D. Sheng et al. / Computer and Geotechnic 35 (2008) e a κ a < 1 < 2 < 3 e lne = 0 0 < 1 < 2 < 3 λ ln p ln p Fig. 2. Qualitative prediction of void ratio veru mean tre under contant uction accding to Equation (5), (6). ln p Fig. 4. Qualitative prediction of the nmal compreion curve under contant uction accding to Equation (9), (11). at thi uction. The iotropic compreion line f thi oil i uually curved in the e lnp pace, a hown by Jenning and Burland [36]. To approximate thi repone uing Eq. (5) and (6), we would have to ue the overconolidation concept, o that the curve i approximated by two traight line, an initial elatic line with a lope of j followed by an elato-platic line with a lope of k, a illutrated in Fig. 2. However, the tre and the uction have never been decreaed during the drying and loading procee and the lurry oil ha never been overconolidated. It i thu very difficult to jutify where the overconolidation effect come from. Me recently, Sheng et al. [66] propoed the following model f the volumetric behaviour of unaturated oil: dp dv ¼ k vp p þ k vðþ d p þ ðnet tre; SFG modelþ where the lope k vp can be independent of uction, and the lope k v varie between k vp f aturated tate and zero f uction above the reidual uction (Fig. 3). Thi model 1. Recover the equation f aturated tate, i.e. Eq. (8). 2. Separate the compreibility due to tre and uction change. 3. Can predict the mooth curvature of the nmal compreion line under contant uction f oil dried from lurry, without the ue of the overconolidation concept (ee Fig. 4). Eq. (9) i very imilar to Eq. (8), but with the negative pe water preure replaced by the uction and the total mean tre ð9þ replaced by the net mean tre. It i alo reaonable to tate that a change in uction doe not necearily have the ame effect a a change in mean tre once the oil become unaturated. Sheng et al. [66] howed that Eq. (9) can capture a number of imptant feature in unaturated oil behaviour and can repreent experimental data very well. Another iue with all the above volumetric model concern the uction range and oil type where they can be applied. Firtly, all thee model apply only to a continuum and become invalid once deiccation occur. Secondly, f dry granular oil where the water phae become dicontinuou, the concept of uction i le meaningful. Indeed, a dry and behave in a imilar way to the aturated and under fully drained condition. Such a phenomenon can not be predicted by the concept of uction. In addition, the bae model f aturated oil, i.e. Eq. (8), i known to be me applicable to clay than to granular oil. Neverthele, other contitutive model ued f aturated oil can be generalied to unaturated oil in a imilar way. F example, Sheng et al. [73] howed that the following equation predict very well the volume change behaviour of aturated dry and: de e ¼ k dp 0 vp ðsaturated andþ ð10þ p 0 þ p re where e i the void ratio, and p re i a hifting tre which depend on the initial void ratio of the oil a well a k vp, and it can be interpreted a the tre level where ignificant particle cruhing occur. Note that the parameter k vp in Eq. (10) i the lope in the double 0.84 Data of Jenning & Burland (1962) e e λ vp λ v Prediction uing equation (9) and 1+ a λv = λvp 1+ λ = 0.01, = 30kPa, p = 0 vp a Shrinkage limit ( re ) a (kpa) water content Fig. 3. Void ratio veru uction under zero net mean tre ( a : uction creponding to full aturation, re : reidual uction, right: a claic hrinkage tet reult).

6 D. Sheng et al. / Computer and Geotechnic 35 (2008) logarithmic lne ln(p 0 + p re ) pace. If Eq. (10) i ued f a aturated and, Eq. (9) can be modified a follow 8 de < e ¼ k dp vp pþp k d re þ vðþ pþp re 6 þ re ð11þ : dp k vp > re pþp re þ re where re i the reidual uction (ee Fig. 3). A threhold uction ( re ) i introduced in Eq. (11) and above thi value uction ha no effect on the volume change. Becaue the reidual uction f and i relatively mall (<100 kpa) compared to preure needed f particle cruhing (1 100 MPa), the effect of uction on the volume change i relatively limited. Setting p re = 0 in Eq. (10) recover Eq. (8). Therefe, Eq. (11) can alo be ued f clay (with p re = 0) Yield tre and yield urface Becaue uction i an additional tre variable, it i neceary to determine the variation of the yield tre with uction, the extenion of the yield urface in the tre uction pace. The yield urface f an iotropic hardening oil uually repreent the contour of the platic volumetric train (i.e. the hardening parameter). A uch, the variation of the yield tre with uction can be derived from the volumetric model. F example, f the volumetric model defined by Eq. (5), it i poible to how that the following function repreent the contour of platic volumetric train in the p pace (ee [66]): 8 < p c0 6 a p c ¼ kvp j k p vp0 j ð12þ : r > a p c0 a p r where p c i the yield tre at the uction, p c0 i the yield tre at zero uction, p r i a reference mean tre and p r ¼ 1 if the oil pecific volume at p r i given by parameter N(), ee Eq. (5), k vp0 i the lope of the nmal compreion line f aturated tate, k vp i the lope of the nmal compreion line f unaturated tate (at uction ), and j i the lope of the unloading reloading line f aturated tate. Eq. (12) i of coure the o-called loading collape (LC) yield urface in the BBM of Alono et al. [4], though only a part of the LC curve f > 0 (with a = 0) wa defined in Alono et al. [4]. A chematic view of the loading collape yield urface defined by Eq. (12) i hown in Fig. 5a. A number of obervation can be made here. Firtly, thi yield urface i uually hown in the literature f uction above the aturation uction only. Becaue the net tre become the total tre f aturated tate, the yield urface actually follow the 45 line f < a. Secondly, the yield tre p c increae with increaing uction only if (1) k vp < k vp0 and p c0 > p r, (2) k vp > k vp0 and p c0 > p r. Thee two alternative condition are a prerequiite to modelling wetting-induced collape. Thirdly, an additional yield tre, p 0, repreenting the p 0 p c p 0 p c apparent tenile trength f > 0, ha to be defined (ee Fig. 5a). F the BBM, p 0 take the fm: p 0 ¼ < a ð13þ a P a On the other hand, if the volumetric model i baed on Bihop effective tre, i.e. Eq. (6), the creponding yield tre become: 8 < p 0 c0 6 a p 0 c ¼ kvp j ð14þ p 0 p 0 k : c0 vp0 j r > p 0 a r p 0 0 ¼ 0 ð15þ In Eq. (14), p 0 r i a reference mean tre and p0 r ¼ 1 if the oil pecific volume at p 0 r i given by parameter N(), ee Eq. (6). The yield urface p 0 c i hown chematically in Fig. 5b. Becaue of the ue of Bihop effective tre, it i uually aumed that an apparent coheion i zero. In addition, the loading collape yield urface extend to the aturated zone following a vertical line. Since the effective mean tre i not contant under contant net mean tre but varying uction, there are certain contraint on the effective tre definition which have not been well dicued in the literature. F example, the effective mean tre mut decreae lower than the yield tre a uction decreae under contant net mean tre, in der to model wetting-induced collape. On the other hand, the effective mean tre mut increae fater than the yield tre a uction increae under contant net mean, in der to imulate drying-induced yielding of a lurry oil. The loading collape yield urface in Fig. 5 cannot be ued f a oil dried from lurry. Thi i becaue drying a lurry oil i imilar to compreing the oil o that the tre tate hould alway be on the current yield urface, which i clearly not poible in Fig. 5a. Indeed, Fig. 5a would predict a purely elatic repone f drying a lurry oil under contant tre. In Fig. 5b, the tre tate could be on the current yield urface only if the effective mean tre increae at a fater rate than the yield tre a uction increae. The SFG model provide a mooth tranition between aturated and unaturated tate. The yield tree, p 0 and p c, can be derived from the volumetric model, i.e. Eq. (9) and (11), provided that the platic volumetric train i taken a the hardening parameter and that an explicit function f the parameter k v i given. F example, the following yield tre wa derived from Eq. (9) f a oil conolidated from a lurry tate: ( p c ¼ p c0 6 a p c0 a ð a þ 1Þ ln þ1 ð16þ aþ1 > a Thi yield tre decreae with increaing uction. Therefe, drying a lurry oil will alway caue the tre point on the current yield urface (ee Fig. 6). F a lurry oil that ha never been conolidated, p c0 i zero. The yield tre then become ( p 0 ¼ < a a ð a þ 1Þ ln þ1 ð17þ aþ1 P a a Elatic zone p c0 45 o 45 o p a Elatic zone p c0 p The above yield tre p 0 alo define the apparent tenile trength of the oil a a function of uction. F an unaturated oil that i compreed compacted at a uction above the aturation uction, the yield tre p c then change to [66]: ( p cn0 < a p cn ¼ ð18þ p c0 þ a ð a þ 1Þ ln þ1 aþ1 P a p cn0 p c0 (a) net mean tre uction pace (b) effective mean tre - uction Fig. 5. Schematic view of loading collape yield urface in mean tre uction pace. where p c0 i the initial preconolidation preure at zero uction, p cn0 i the new preconolidation preure at zero uction (ee Fig. 6).

7 816 D. Sheng et al. / Computer and Geotechnic 35 (2008) Initial elatic zone 600 (kpa) 500 p0 p c pcn p c0 0 p cn p (kpa) Fig. 6. Initial yield urface f a oil that wa conolidated to 300 kpa at zero uction and it evolution when the oil i then compreed at different uction level ( a = 100 kpa). The yield tree given by Eq. (16) (18) are illutrated in Fig. 6. The curve p 0 repreent the apparent tenile trength of the oil caued by uction, while the curve p c depict the yield tre if the oil i air-dried. F example, f a lurry oil that wa firt conolidated to 300 kpa and then air-dried at zero mean tre, the uction that caue platic yielding i 730 kpa. If the air-dried oil i compreed under contant uction, the new yield tree are then repreented by the curve p cn. Therefe, p c repreent the yield tre f an air-dried lurry oil and p cn repreent the yield tre f a compacted oil. The yield tre increae with increaing uction along the curve p cn, not p c. It can alo be noted that (1) the tranition between aturated and unaturated tate i continuou and mooth along all the three yield tree; (2) like the model in Fig. 5, the yield urface p c and p cn are non-convex in the p pace; (3) the model i tre-path dependent and different tre path may reult in different yield urface. If the Modified Cam Clay (MCC) model i ued a the bae model f the aturated oil, the elliptic yield urface can be extended to the uction axi accding to Eq. (16) (18): f ¼ q 2 M 2 ðp p 0 Þðp c pþ ¼0 ð19þ where f i the yield function, q i the deviat tre, and M i the hear trength parameter that define the lope of the critical tate line in q p pace. The yield urface accding to Eq. (19) are hown in Fig. 7 f two type of unaturated oil Shear trength with uction The hear trength of an unaturated oil i uually a function of uction. Fredlund et al. [20] propoed the following relationhip which conveniently eparate the hear trength due to tre from that due to uction: h i ¼ c 0 þðr n u a Þ tan / 0 Šþ½ðu a u w Þ tan / b ¼ c þðr n u a Þ tan / 0 ð20þ where i the hear trength, c 0 i the effective coheion and i uually zero unle the oil i cemented, r n i the nmal tre on the failure plane, / 0 i the effective friction angle of the oil, / b i the frictional angle due to uction, and c i the apparent coheion which include the friction due to uction. Obviouly, if / b i et to / 0 in Eq. (20), the effective tre principle f aturated oil i recovered. Thi hear trength equation wa iginally publihed in a linear fm, but experimental reult how that the econd term i in fact nonlinear (ee, f example, [17]. The hear trength due to uction tart to deviate from the effective angle of internal friction at a uction which i approximately equal to the air entry value of the oil. The oil uction veru hear trength relationhip then appear to have a gradual curvature until reidual uction condition are reached. Once thee condition are reached the hear trength remain approximately contant a the uction are further increaed. However, it i alo poible f the hear trength to decreae f and and increae f clay a the uction are increaed beyond reidual condition. There are a number of model available in the literature f determining the friction angle / b [84,23,55,7]. In elato-platic model, the hear trength of an unaturated oil i uually embodied in the apparent tenile trength function p 0. F example, the apparent coheion in the Barcelona Baic Model i given a c ¼ p 0 tan / 0 tan /0 < a; c 0 ¼ 0 ¼ ð21þ a tan / 0 P a ; c 0 ¼ 0 where a i a material contant. The friction angle / b i then given by tan / b ¼ c tan /0 < a ¼ ð22þ a tan / 0 P a In thi cae, the friction angle / b i independent of uction. In the SFG model, the apparent coheion due to uction i ( tan / 0 c ¼ p 0 tan / 0 ¼ < a; c 0 ¼ 0 tan / 0 a þð a þ 1Þ ln þ1 aþ1 P a ; c 0 ¼ 0 ð23þ

8 D. Sheng et al. / Computer and Geotechnic 35 (2008) Fig. 7. Modified Cam Clay yield urface extended to uction axi ( a = 100 kpa). Therefe, the friction angle / b i given by: ( tan / 0 tan / b ¼ < a tan / 0 P a a þ aþ1 ln þ1 aþ1 ð24þ In thi cae, the friction angle / b i a function of uction a well a the aturation uction. The predicted hear trength variation with uction i hown in Fig. 8 and compared with experimental data f Guadalix Red ilty Clay in Fig. 9. The prediction of Eq. (24) appear to be reaonable, at leat qualitatively. On the other hand, if Bihop effective tre i ued, the hear trength i uually aumed to be unique in the effective tre pace: ¼ c 0 þ r 0 n tan /0 ¼ c 0 þðr n u a Þ tan / 0 ¼ vðu a u w Þ tan / 0 ð25þ The above equation alo implie that tan / b = v tan/ 0. Recently Nuth and Laloui [54] provided ome experimental evidence f the uniquene of c 0 and / 0 in Bihop effective tre pace with v = S r Hyterei of oil water characteritic Extenive reearch ha been done on oil water characteritic curve f unaturated oil, firt in the field of oil phyic and later within geotechnical engineering (ee, e.g., [33]). The oil water Shear trength (kpa) Soil uction (kpa) Fig. 8. Shear trength veru oil uction up to 10,000 kpa f oil with air entry value of 10, 100, and 1000 kpa, predicted from equation (24). characteritic curve (SWCC) i uually preented in the pace of the volumetric water content (h) veru oil uction in the pace of the degree of aturation (S r ) veru uction. A number of empirical h relation exit in the literature, and the one that are com-

9 818 D. Sheng et al. / Computer and Geotechnic 35 (2008) Shear Strength (kpa) 1000 monly ued include thoe of Gardner [31], van Guenuchten [85] and Fredland and Xing [22]. Thee relation are uually written a continuou function and do not explicitly conider the hyteretic behaviour during a drying wetting loop. However, in an elato-platic modelling framewk which mut predict the repone f all poible wetting and drying path, an incremental fm (between d and dh between d and ds r ) i preferred. Recently Li [47] preented an incremental oil water characteritic relationhip between d and ds r. Thi incremental SWCC model include mooth hyteretic repone to arbitrary wetting/drying path, and can be incpated into elato-platic model f unaturated oil. However, the model by Li [47] follow the bounding urface framewk which i omewhat different from the claical elatoplaticity framewk dicued in thi paper. Lin et al. [48] preented a model f hyteretic SWCC from which an incremental fm can alo be etablihed. Me recently, Pedroo et al. [6,59,60] have developed an incremental aturation uction relationhip that incpate hyteretic behaviour. Thi model i fmulated in the ame framewk a elato-platicity and can be conveniently incpated into an elato-platic tre train relation. In thi paper, a very imple model preented by Sheng et al. [66] i decribed. Thi imple model doe not conider the hyteretic behaviour within the main drying and main wetting curve (ee Fig. 10). A a imple approximation, a piece-wie linear relationhip between the degree of aturation S r and logarithmic oil uction can be aumed: ds r ¼ k w d S r λ w κ w Main wetting curve a Guadalix Red Silty Clay (σ - u a ) = 120 kpa ae Experimental Reult Predicted Shear Strength Suction (kpa) Fig. 9. Predicted hear trength value f Guadalix Red ilty Clay [17] by equation (24). Main drying curve we re Scanning curve κ w ln ð26þ Fig. 10. Degree of aturation veru uction (dahed line repreent implification). where the lope k w may change with uction. F oil uction below the aturation uction, the oil i aturated and the degree of aturation remain eentially contant. F oil uction larger than the reidual uction, the water content gradually decreae to zero at a uction of 10 6 kpa [21]. The lope i aumed to be contant between the air entry and the reidual uction f a drying oil [87]. Therefe, we have, f increaing uction a hown in Fig. 10: 8 0 < a >< j w a < a k w ¼ ð27þ k w ae < re >: j w P re where ae i the air entry value, and re i the reidual uction (ee Fig. 11). The above equation i only valid f the main drying curve. F the main wetting curve and the canning curve, the lope mut be adjuted accdingly (ee Fig. 10). The oil uction veru water content relationhip i affected by the mean net tre primarily through it influence on the air entry uction and the rate of deaturation (ee, e.g., [83]), and i not conidered here. Hyterei in oil water characteritic i uually conidered to be too imptant to igne. Therefe, a wetting curve mut be added to the drying curve. The wetting curve i characteried by the water entry value we and ha a imilar lope to the drying curve, k w (ee Fig. 10). A erie of parallel line having a lope j w, are ued to repreent recoverable change in S r between the drying (deption) and the wetting (adption) curve. Thee curve are called canning curve. F the purpoe of thi tudy, the lope of the canning curve i aumed to be identical to the lope of the drying curve f uction below the air entry value and f uction above the reidual value. The lope of the wetting curve f uction above the water entry value i alo aumed to be j w (ee Fig. 10). The implification adopted here are imilar to thoe in the model by Wheeler et al. [87]. In the implified model, the maximum uction that crepond to full aturation i the aturation uction ( a ), not the air entry value ( ae ). Hyterei of oil water characteritic can alo be explained within the ame framewk of elato-platicity [69]. Under uch a framewk, an unaturated tate alway lie within the main drying and wetting curve. Drying wetting from within the hyterei loop will only caue recoverable water content change until the uction reache the main drying wetting curve. Once the oil uction reache the main drying wetting curve, further drying wetting will caue irrecoverable water content change. Therefe, the drying and wetting curve define the boundarie of recoverable water content change and are imilar to the nmal compreion line. The canning curve define the recoverable water content change and are imilar to the unloading reloading line. On the p plane, two additional boundarie can be added, repreenting the main drying and wetting curve, repectively (Fig. 11). 4. Finite element implementation 4.1. Incremental tre train relation One of the ultimate goal of contitutive modeling i to develop an incremental tre train relation o that it can be implemented in a numerical method to olve boundary value problem. F unaturated oil, thee incremental relation can either be written a in [66] dr ¼ Dep W ep de ð28þ d R G dh a in [69]

10 D. Sheng et al. / Computer and Geotechnic 35 (2008) p 0 SI: Drying p c 600 (kpa) ELASTIC ZONE SD: Wetting p Fig. 11. Elatic zone encloed by the yield urface and the drying and wetting urface. dr 0 ¼ Dep W ep de d R G ds r ð29þ depending on the tre variable choen. In the diplacement finite element method, the pe preure and diplacement are firt olved from the equilibrium and continuity equation. Therefe, the train and uction increment are known, and the tre and water content increment are to be found from the contitutive equation. In uch a context, Eq. (28) and (29) have to be refmulated o that all known increment are kept on the right-hand ide. Here the SFG model i ued to demontrate the derivation of the incremental tre train equation. The Modified Cam Clay model i ued a the bae model f aturated oil. The yield function then take the fm of F ¼ q 2 M 2 ðp p 0 Þðp c pþ ¼0 The conitency condition become: df ¼ of T dr þ of op 0 of op c of op c d þ d þ op 0 o op c o op c oe p v de p v ¼ 0 The train decompoition and the flow rule can be written a de ¼ de e þ de p ¼ de e þ _ K og ð30þ ð31þ ð32þ where g i the platic potential function, and _ K i the platic multiplier to be olved from the conitency condition. The elatic tre uction train relation can be written a de e ¼ðD e Þ 1 dr þðw e Þ 1 d; dr ¼ D e de e D e ðw e Þ 1 d ¼ D e de K _ og W e d ð33þ where D e i the elatic tre train tiffne matrix, W e i the elatic uction train vect, and W e ¼ D e ðw e Þ 1. The platic multiplier can be found from Eq. (31): T of D e de þ of o p 0 o p 0 o þ of o p c o p c o _K ¼ T of D e og of o p c o p c oe p v of! T W e og op And the tre train relation i thu: dr ¼ D e de T D e og of D e de þ D e og of T D e og d of o p 0 o p 0 o þ of o p c o p c o of o p c og o p c oe p v op ð34þ of! T W e d ð35þ The uction-water content relation i given by d dh ¼ nk w þ S rde v ¼ k w n d þ S rm T de ð36þ where m T = (1,1,1,0,0,0) and n i the poity of the oil. Note that in the equation above it i aumed that the volumetric train of oil keleton i due to the change to void volume only (i.e. the liquid water and olid particle are not compreible). The lope k w hould be replaced by j w f uction change along the canning curve. Therefe, uing the following notation T D e og of D e D ep ¼ D e T of D e og of o p c og o p c op oe p v H ¼ k w n= H ¼ j w n= D e og of op 0 op W ep 0 o þ of op c op c o of! T W e ¼ T of D e og of op c og op c oe p v op T ¼ S r m

11 820 D. Sheng et al. / Computer and Geotechnic 35 (2008) the final incremental tre train relationhip can be written a dr ¼ Dep W ep de ð37þ dh T H d where D ep i a 6 6 matrix, T i a row vect of 6 element, W ep i a column vect of 6 element, and H i a calar. The of train rate and oil uction rate are both on the right-hand ide, o the fmulation i conitent with the diplacement finite element method where pe preure and diplacement are firt olved from equilibrium and continuity equation. The incremental tre train relationhip defined by Eq. (37) can be implemented into the finite element method to olve boundary value problem Global governing equation Contitutive model have to be implemented into numerical method uch a the finite element method to olve boundary value problem. The governing equation f unaturated oil uually involve the equilibrium of momentum, the balance of ma and the balance of energy [57,25,80,40,72]. Under certain condition, thee equation can be implified. F example, under iothermal condition, the heat tranfer and the temperature field can be neglected. Meover, under atmopheric air preure, the flow of pe air can be neglected. Therefe, in the implet fm, the governing equation contain the equilibrium equation of momentum and the continuity equation of water flow. The equilibrium equation can be written in a weak fm a Z Z Z ðoe T r t ÞdV ðou T t t ÞdS ðou T b t ÞdV ¼ 0 ð38þ V t S t V t where du i a virtual diplacement field atifying the diplacement boundary condition, de denote the variation of the train ten, r i the tre ten, b i the body fce vect, t i the ditributed fce acting on the boundary S of the volume V, and the upercript t tand f quantitie that are meaured at time t. The continuity equation follow from the ma conervation of pe water and Darcy law: div k g ðru w b w Þ þ o ot ðq w hþ¼0 ð39þ where div i the divergence operat, k i the permeability ten, g i the gravity acceleration, u w i the pe water preure, b w i the body fce vect of pe water, and q w i the water denity. After appropriate patial dicretization the diplacement and pe preure field can be approximated a u ¼ NU U w ¼ N w U w ð40þ ð41þ where N i the diplacement hape function, U i the nodal diplacement vect, N w i the pe preure hape function and U w i the nodal pe preure vect. The tre i updated incrementally: r t ¼ r t ot þ ð42þ where r t ot i the tre at the lat equilibrium tate. Now the contitutive equation mut be incpated into Eq. (38) and (39). The finite element fmulation may vary lightly, depending on the tre variable ued in the contitutive equation. F example, if Eq. (37) i ued, we have f zero air preure: dr ¼ Dep W ep de ð43þ dh T H du w Eq. (43) can then be ubtituted into Eq. (38) and (39). Due to the nonlinearity of the material behaviour, the governing equation are uually olved incrementally and the olution at time t i ought with the known olution at time t Dt. Therefe, the dicretied finite element equation are uually written in rate fm: K ep U _ þ L Uw _ ¼ F _ ext L 0 U _ þ S Uw _ þ HU _ w ¼ Q _ ext where K ep ¼ X Z V e ðb T D ep BÞdV e L ¼ X Z ðb T W ep N w ÞdV e V e _F ext ¼ X Z Z ðn T bþdv _ e þ ðn T _tþds e V e S e L 0 ¼ X Z V e q w ðn T w TBÞdV e S ¼ X Z q w ðn T V e w HN wþdv e _H ¼ X Z B T k V e w g B w dv e _Q ext ¼ X Z ðn T S e w qt ÞdS e X Z B T k V e w g b w dv e ð44þ ð45þ If Bihop effective tre i ued in the contitutive equation, i.e. Eq. (28), the definition of the above matrice and vect will change omewhat, but the governing equation will have the ame fm a (44) and (45). The dicretied governing equation can alo be written in a compact fm of dinary differential equation: C _ X þ KX ¼ _ W ð46þ where X contain the global unknown nodal diplacement and pe water preure, C and K are coefficient matrice, W contain the external fce and flow vect, and the uperi dot tand f the rate with repect to time. Due to the material nonlinearity, Eq. (46) ha to be olved numerically and appropriate algithm have to be ued at both the global and local level. The global unknown X i uually olved tep by tep uing a time integration cheme. F elato-platic conolidation problem it i uually recommended to ue an implicit time tepping cheme, where the coefficient matrice are etimated at the time level where the unknown i ought and iteration are needed. Automatic time tepping cheme can be deigned to control the integration err. Me detailed dicuion of time tepping cheme f conolidation problem can be found in [89,74,68]. Once the diplacement and pe preure are olved, the train and uction can be computed at Gau point. The tree and the volumetric water content ( the degree of aturation) are then olved from the contitutive relation, again numerically. A number of tre integration cheme are available in the literature, e.g. thoe by Sheng et al. [71], Sloan et al. [75], Sheng et al. [70] and Bja [11]. The following ection dicue a pecific challenge aociated with Gau point tre integration of unaturated oil model, i.e. the problem of non-convex yield urface Gau point tre integration One of the main challenge in implementing an unaturated oil model into finite element code arie from the non-convexity of the yield urface around the tranition between aturated and unaturated tate. The non-convexity exit irrepective of the tre variable ued in the model and i demontrated in Fig. 12. F given train and uction increment, the current tre tate and internal variable mut be updated in accdance with Eq. (28), (29). Thi update i generally carried out uing numerical