Analysis of the Extension of the Elastic Parameters for Modelling Highly Expansive Unsaturated Soils with the Barcelona Basic Model

,


Introduction
Deep geological disposal is considered a viable option for the long-term management of the spent nuclear fuel generated in nuclear power plants. In November 2015, the Finnish government granted Posiva Oy [1][2][3][4] a licence for the construction of an encapsulation and disposal facility for spent nuclear fuel in Olkiluoto (municipality of Eurajoki, Finland). In January 2022, the Swedish government approved the Svensk Kärnbränslehantering AB (SKB) [5] application for the fnal repository for spent nuclear fuel in Forsmark (municipality ofÖsthammar, Sweden), and an encapsulation plant in the municipality of Oskarshamn (Sweden). In Finland, Posiva Oy is in charge of the disposal of the spent nuclear fuel, while in Sweden, SKB is responsible. In Finland and Sweden, the spent nuclear fuel is intended for disposal in a geological repository following the KBS-3 method, based on the so-called multibarrier principle, where several redundant engineered barriers ensure long-term safety. Posiva and SKB are studying the KBS-3V design alternative [5,6]. In this design (Figure 1), disposal tunnels at a depth of 400-500 m are excavated in crystalline rock, along which vertical disposal holes are excavated. Te copper/iron canisters ( Figure 1, barrier 1) containing spent nuclear fuel are surrounded by a clay barrier (referred to as the "bufer") consisting of compacted bentonite (Figure 1, barrier 2), a highly expansive clay, and emplaced individually in the disposal holes as shown in Figure 1. Te disposal tunnels are flled with additional compacted bentonite (referred to as the "backfll," Figure 1, barrier 3), and all remaining connections with the surface are sealed with closure structures. Te copper/iron canister, the bufer, the backfll, and other closure structures form the EBS, while the natural barrier is the host rock ( Figure 1, barrier 4).
Groundwater fows from the host rock to the disposal tunnels and holes and hydrates the bentonite, which expands, flling the remaining gaps and sealing the repository. Once all the gaps have been flled, the bentonite starts to develop swelling pressure. Swelling pressure is also important because the pressure prevents or limits the growth of microbes. Te main reason for avoiding microbial activity in the bufer is to prevent the formation of corrosive agents such as sulphide. On the other hand, the bufer should allow gases forming in the corrosion process to escape. Te spent nuclear fuel generates heat, so the clay barrier will also heat up. For the safety case of a deep geological repository, the evolution of the clay barriers over time needs to be described. Tis analysis can be performed by simulating the behaviour of the engineered and natural barriers, including the compacted bentonite. Tis simulation requires the use of complex constitutive models. Te Barcelona Basic Model (BBM, [7]) is one of the constitutive models used for the simulation of the compacted bentonite in spent nuclear fuel repositories [8][9][10][11][12]. Tis model allows the simulation of the thermo-hydro-mechanical (THM) processes which take place in clay barriers due to heating, water uptake, and swelling pressure development. In particular, the maximum temperature near the canister containing the spent nuclear fuel, time needed for bufer saturation, bufer stress state due to bentonite swelling, density distribution after bufer saturation, and possible canister movements. Such simulations have been carried out for the construction licence application in Finland [6,13,14] and Sweden [9,10] and in the operational licence application in Finland [15].
Te BBM was originally intended for partially saturated soils which are slightly or moderately expansive. Because experimental evidence was still limited at the time of model development, the model was kept simple in order to provide a basic framework from which extensions for other applications were possible. Results from extensive experimental programmes are used to calibrate models such as the BBM, and new experimental results have led to modifcations or extensions of the models. For partially saturated and highly expansive materials, the parameter κ (logarithmic compliance with respect to changes in net pressure) of the BBM model was extended to a function κ i (s) of suction s and the parameter κ s (logarithmic compliance with respect to changes in suction) of the BBM model was extended to a function κ s (p, s) of net pressure p and suction s. Tese functions were implemented in CODE_BRIGHT computer code [16,17], and the resulting extension of the BBM model is the Barcelona Basic Model for Expansive soils (BBMEx). Other reasons for modifcations or extensions of the models include being able to simulate parts of the clay barriers which will have special conditions such as flling cavities and voids, where free swelling with large volumetric changes is expected.
Te extension of the elastic part of the BBM model has some advantages. It allows the BBM model to be able to simulate the behaviour of expansive clays without the implementation of new yield surfaces such as in models presented by Gens and Alonso [18], Navarro et al. [19], or Tachibana et al. [20]. Te functions proposed for κ and κ s can also be implemented in these models and also in other models related to unsaturated soils, such as the models presented by Gallipoli et al. [21] and Wenjing and Sun [22] where the efect of the degree of saturation on mechanical behaviour is considered. In these two models, the water retention curve must be measured. Tis measurement presents large scatter when it is measured in tests keeping constant volume [23].
Using the BBMEx has the disadvantage of increasing the number of parameters. Te main limitation of the BBMEx model is that the capability for the simulation of swelling clays is lost when there are cycles of loading-unloading and wetting-drying because there are plastic strains that cannot be considered, but this limitation disappears in spent nuclear fuel repositories' simulation because the main paths are wetting due to the groundwater fow and loading due to the swelling pressure development during the hydration of the bentonite, so using the BBMEx model can be justifed in the analysis of the spent nuclear fuel repositories since the assembly of the clay barriers till fully saturation is reached, considering the initial path of drying-wetting close to the canisters and the unloading due to temperature reduction after the swelling as minor efects that will not change the results signifcantly. Te BBM model is also implemented in other computer codes such as COMSOL Multiphysics [19,24], PLAXIS [25,26], Lagamine [27], and FLAC, where the BBM is implemented in combination with TOUGH [28]. Te extension of the elastic parameters presented in this article can also be implemented in these computer codes (it is already considered in Lagamine) to provide more fexibility in the simulation of highly expansive clays, albeit assigning parameter values with caution.
Te BBMEx should not be confused with the Barcelona Expansive Model (BExM, [18]), developed for addressing double structure expansive soils. Te BBM model has been used for the simulation of the evolution of the bufer in the Finnish and Swedish safety assessments for the KBS-3V design ( Figure 1) [29][30][31][32].
Te calibration of the BBM model may be difcult because individual aspects of the isotropic virgin behaviour are being controlled by multiple parameters, while, at the same time, a single parameter controls more than one aspect of the soil behaviour. Gallipoli et al. [33] propose a method for streamlining the parameter selection carrying out a sequential procedure. Wheeler et al. [34] assessed the BBM and presented a practical method for the evaluation of BBM plastic parameters. Diferent methods for the BBM parameterisation were also presented by D'Onza et al. [35] in a benchmark where some laboratory tests in compacted samples were used for the parameterisation of the BBM. Zhang et al. [36] presented the parameterisation of the BBM from pressuremeter tests using the particle swarm optimization method. Tese methods may be used for the parameterisation of the BBMEx taking into account the limitations presented in this article.
Te elastic parameters of the BBM and BBMEx are presented in Table 1. Te paper considers a set of conditions on the values of the elastic functions of the BBMEx, and, for each one, determines the following: (1) for all positive net pressures and all non-negative suctions, the range of parameter values is such that the condition is satisfed and (2) for given values of the parameters, the maximal positive net pressure and the maximal non-negative suction such that the condition is satisfed. In this way, it is possible to prevent those conditions on the values of the elastic functions from being inadvertently not satisfed during calculations.
Te plastic parameters of the BBM and BBMEx are the same and are presented in Table 2. Table 3 presents the variables.

Barcelona Basic Model for Expansive Soils
Te BBMEx model (Barcelona Basic Model for Expansive soils) is an extension of the BBM model (Barcelona Basic Model) for expansive soils which is available in CODE_-BRIGHT [16,17]. Tis extension concerns only the elastic response, and in that, it generalizes the parameter κ of the BBM model to a function κ i (s) of suction s and it generalizes the parameter κ s of the BBM model to a function κ s (p, s) of net pressure p and suction s. Te functions κ i (s) and κ s (p, s) will be referred to as the elastic functions of the BBMEx model.
Te volumetric compressive behaviour of the BBMEx model is defned by Load and Collapse (LC) yield locus p − p 0 s, p * 0 � 0, (3) LC yield locus hardening rule dp * 0 � (5) and the functions where p * 0 is the hardening variable, dμ ≥ 0 is the plastic multiplier, e is the void ratio, p at is the atmospheric pressure, and κ i0 , κ s0 , λ(0), β, r, p c , α i , α il , α sp , α ss , and p ref are the parameters of the BBMEx model. By (3) and (9), the LC yield loci of the BBMEx model and of the BBM model are the same. We note that if α i � α il � α sp � α ss � 0, then the BBMEx model reduces to the BBM model with parameters κ � κ i0 , κ s � κ s0 , λ(0), β, r, and p c . We also note that, at s � 0, both the BBM model and the BBMEx model reduce to the modifed Cam clay model [37] with parameters κ � κ i0 and λ � λ(0) and preconsolidation pressure p 0 � p * 0 . From (2) and (5), the variation of the volumetric deformation reads (i) In the elastic regime, dε v � dε e v so that (10) reduces to (2).
which at constant suction s reduces to and at constant net pressure p reduces to From (12), it follows that, in the BBMEx model, unloading-reloading at constant suction s is represented in the (1 + e) − ln p plane by a line with slope κ i (s).
Integrability of the variation of volumetric elastic deformation (11) (independence from the Advances in Materials Science and Engineering 3 integration path) on a simply connected domain of the p − s plane requires the following: for all points (p, s) of that domain. (ii) In the elastoplastic regime (in compression), the consistency condition required by (3) is and by using p � p 0 (10), (10) becomes so, the variation of the volumetric deformation for normal loading at constant suction s is From (17), it follows that, in the BBMEx model, normal loading at constant suction s is represented in the (1 + e) − ln p plane by a line with slope λ(s) From (17) and (12), it follows that the variation of the volumetric plastic deformation at constant suction s of the BBMEx model reads as follows: Volumetric elastic compliance to changes of net pressure Volumetric elastic compliance to changes of suction  which is the same as the variation of volumetric plastic deformation at constant suction s of the BBM model. In order to prevent unrealistic predictions, it is of interest to fnd the set of values of the parameters of the BBMEx model such that certain conditions are satisfed at all points (p, s) of the subset(0, ∞) × [0, ∞) of the space p − s. Te following conditions will be considered: (1) Te volumetric elastic compliance to changes of net pressure is positive and fnite, which (since p > 0) by (12) is equivalent to 0 < κ i (s) < ∞ for all 0 ≤ s < ∞.
(2) Te volumetric plastic compliance to changes of net pressure is positive and fnite, which (since p > 0) by (3) Te volumetric elastic compliance to changes of suction is positive and fnite, which (since s + p at > 0) by (13) is equivalent to 0 < κ s (p, s) < ∞ for all 0 < p < ∞ and all 0 ≤ s < ∞.
(4) At constant net pressure p, the volumetric elastic compliance does not increase with suction s, which by (12) is equivalent to κ i ′ (s) ≤ 0 for all 0 ≤ s < ∞, which means that the stifness of unsaturated soils increases with the suction increase.
(5) At constant net pressure p, the volumetric compliance for normal loading does not increase with suction s, which by (17) is equivalent to In saturated soils (zero suction), consolidated soils are softer than overconsolidated soils.
(6) Te variation of the volumetric elastic deformation is integrable (independent from the integration path), which by (14) is equivalent to z s (κ i (s)/p) � z p (κ s (p, s)/(s + p at )) for all 0 < p < ∞ and all 0 ≤ s < ∞. Te compliance of this condition is important in cyclic loading-unloading processes and less important in monotonic loading or unloading or in loading-unloading or unloading-loading processes.
We note that all these conditions are satisfed by the BBM model, provided that its parameters κ, κ s , λ(0), β, r, and p c satisfy the conditions as follows: Consequently, it is assumed that the corresponding parameters of the BBMEx model satisfy these conditions with κ � κ i0 and κ s � κ s0 . Furthermore, it is assumed that 0 < p ref < ∞ so that the function κ s (p, s) is well defned.
As noted earlier, the extension of the BBM model to the BBMEx model only concerns the elastic response. With the assumptions made, the function λ(s) satisfes the condition λ ′ (s) ≤ 0 for all 0 ≤ s < ∞. Terefore, if condition 4 is satisfed, then condition 5 is also satisfed. Discarding the weaker condition 5 (in which the function λ(s) is also involved), the remaining conditions only involve the elastic functions κ i (s) and κ s (s, p).
Te problem set earlier seeks to fnd the set of values of a set of material parameters such that a certain condition is satisfed on the subset(0, ∞) × [0, ∞) of the space p − s. A related problem seeks to fnd the maximal subset (0, p * ) × [0, s * ) or (0, p * ] × [0, s * ] of the space p − s on which a certain condition is satisfed for given values of a set of material parameters. We note that s ≥ 0, but p > 0, so that formulas (2) and (7) are well defned. If the condition holds for all p > 0 (all s ≥ 0), it will be indicated by setting p * � ∞ (s * � ∞) so that the corresponding end of interval symbol "]" should in fact be ")." Sketches of the sets to be found in each of these problems are shown in Figure 2. We note that in the frst problem, the restrictions are on the material parameters, whereas in the second problem, the restrictions are on the subset of the space p − s. Te values of the acceptable parameters and the range of p and s are summarised in diferent tables. We note that the tensile values of the net pressure are not considered (soil is always in compression) and suction values are always positive.

Calculation of the Conditions
Given the equations (6) and (7) depending on the param-

Determine
(1) Te set of values of the parameters α i , α il , α sp , α ss , and p ref > 0 such that the condition holds on the subset (2) If the subset of the space p − s to be determined for a condition is connected and contains points arbitrarily close to (0, 0), then from any point of that subset, an unloading towards the limit "natural" state of zero net pressure and zero suction is possible without violating the condition. We note that s ≥ 0, but p > 0 so that formulas (2) and (7) are well defned.
Advances in Materials Science and Engineering Since  (19). We note that f(0) � 1, and (with In   Advances in Materials Science and Engineering Since α i > 0, from the second equation, it follows that which when substituted into the frst equation yields the following: With x � − α il /(α i p at ), this condition is equivalent to and s > 0 is equivalent to x > 1. Since the function g(x) is such that g(1) � 0, lim x⟶∞ g(x) � ∞ and g ′ (x) � ln x > 0 for all x > 1, it is a one-to-one map from the interval [1, ∞) onto the interval [0, ∞). Terefore, for any α i > 0, the , f(s) � 0 has no positive solutions; and if − α il > α i p at g − 1 (1/(α i p at )), f(s) � 0 has two positive solutions separated by s � p at (g − 1 (1/(α i p at )) − 1). Geometrically, α i > 0 is a parameter of the (straight with positive slope) graph of f 1 (s) and − α il > 0 is a parameter of the (strictly increasing, strictly convex upward) graph of f 2 (s). If the parameter α i > 0 is fxed, then the graph of f 1 (s) is fxed. For sufciently small values of − α il > 0, the graph of f 2 (s) is below the graph of f 1 (s) for all 0 < s < ∞ so that f(s) � 0 has no positive solutions; as − α il increases, the graph of f 2 (s) grows until eventually is a tangent to the graph of f 1 (s) so that f(s) � 0 has only one positive solution; as − α i > 0 further increases, the graph of f 2 (s) further grows and intersects the graph of f 1 (s) at two points so that f(s) � 0 has two positive solutions.
Te condition κ i (s) > 0 for all (p, s) ∈ (0, p * ] × [0, s * ] is summarised in Table 4. We note that g − 1 (1/α l p at ) is the solution x of the From these results, it follows
We note that, since it can be shown that lim x ⟶ 0 + xg − 1 (1/x) � 0 + , the limit of the frst condition as α i ⟶ 0 + requires α il to be greater than an arbitrary close to zero negative number so that α il cannot be negative. Tus, the second condition is the limit of the frst condition as α i ⟶ 0 + .

3.2.
Condition κ s (p, s) > 0 . From defnition (7) of the function κ s (p, s), it follows that, since κ s0 > 0 and exp(α ss s) > 0 for any α ss and all s ≥ 0, the sign of κ s (p, s) is Advances in Materials Science and Engineering the same as the sign of f(p) � κ s (p, s)/(κ s0 exp(α ss s)) so that the problem is reduced to the analysis of the function.
We note that case 2 is the limit of case 1 as α sp ⟶ 0 − . Since f ′ (p) � α sp /p, in case 1, f(p) is a strictly decreasing function.
We note that the fulflment of the condition does not depend on the values of α ss and p ref > 0.

Condition κ i
′ (s) ≤ 0. From defnition (6) of the function κ i (s), its derivative reads as follows: since κ i0 > 0, the sign of κ i ′ (s) is the same as the sign of f(s) � κ i ′ (s)/κ i0 so that the problem is reduced to the analysis of the function.
. We note that f(0) � α i + α il /p at ≤ 0, which is equivalent to α il ≤ − α i p at , is necessary for the condition to hold.
Te s * < p at (g − 1 (1/α i p at ) − 1) such that f(s * ) � 0 Table 5: Condition κ i (s) > 0. Ranges of values of the parameters α i and α il such that the condition is satisfed for all non-negative suctions.
α i α i and α il Condition holds

Advances in Materials Science and Engineering
Te summary of the results obtained is

then the condition does not hold
Since κ i ′ (s) does not depend on p, it follows that p * � ∞.
We note that if α i > 0 and α il � − α i p at , then the condition holds only for saturated states so that in fact, it does not hold for the BBMEx model, which is intended to model unsaturated states. Te condition κ i ′ (s) ≤ 0 for all (p, s) ∈ (0, p * ] × [0, s * ] is summarised in Table 8.
From these results, it follows.
If α i ≤ 0 and α il ≤ − α i p at , then the condition holds. If α i > 0 or α il > − α i p at , then the condition does not hold. Table 9.
Te condition z s (κ i (s)/p) � z p (κ s (p, s)/(s + p at )) for all (p, s) ∈ (0, ∞) × [0, ∞) is summarised in Table 11. Table 6: Condition κ s (p, s) > 0. Maximal positive net pressure p * and maximal non-negative suction s * for given values of the parameters α sp and α ss such that the condition is satisfed ("none" means that the condition cannot hold).
α ss is any real None None

Laboratory Test Model Validation
Te BBMEx has been extensively used in [16,38,39] for the simulation of clay barriers made of FEBEX bentonite [8,40] and in [9,11,12] for the simulation of clay barriers made of MX-80 bentonite [41,42]. In this section, it is presented as an example of two laboratory test simulations carried out with the BBM and the BBMEx and the verifcation that the parameters fulfl the conditions described in Section 3. Te test set-up is presented in Figure 3. Water was pumped at the bottom of the samples keeping constant pressure (200 kPa) and measuring the water infow. More details concerning this test can be found in [43]. Te tests were performed in crushed pellets made of two bentonites, one from Italy and the second from Bulgaria. Tese materials are options as barrier 3 (Figure 1) in the Finnish spent nuclear fuel repository. Long-term thermo-hydromechanical response of crushed pellets of bentonite in the Finnish spent nuclear fuel repository concept can be found in [15]. More information about crushed pellets of bentonite can be found in [38,[39][40][41][42][43][44] and information about the Italian and Bulgarian bentonites in [45]. Figures 4 and 5 show the evolution of the axial and radial stresses measured in the infltration test performed in crushed pellets made of Italian bentonite and Bulgarian bentonite, respectively, and the results of the simulations of the same tests carried out using the BBM and BBMEx models. At the end of the test, radial stresses measured and simulated were around 2000-2400 kPa in Italian bentonite and around 2500-3500 kPa in Bulgarian bentonite, depending on the distance to the injection point. Figure 6 presents the evolution of the suction-net mean stress during the tests. Te computed suction-net mean stress paths for crushed pellets' material (BBM and BBMEx) for Italian bentonite (a) and Bulgarian bentonite (b) are shown in Figure 6.
Te long-term evolution was simulated better by the BBMEx model when the test was performed in Italian bentonite, but it was not possible to conclude the same in the case of the test performed in Bulgarian bentonite. Te shortterm evolution was also simulated better in the Italian bentonite test by the BBMEx model, but it was not possible to appreciate an improvement in the simulation when the test was performed in Bulgarian bentonite. Te swelling pressure development was slower in the Bulgarian bentonite test than in the Italian bentonite test, and this could be the reason for why the improvement of the BBMEx was not clear in the Bulgarian bentonite. Te parameters of both models are presented in Table 12. Tese parameters were used to simulate long-term hydromechanical response of backfll ( Figure 1-Barrier 3) in a single deposition tunnel [15].
Te BBMEx model was used for the calculation of κ s (p, s). From Table 6, it is possible to see that the compliance of the condition κ s (p, s) > 0 was fulflled because, since for α sp < 0, α ss can reach any real value and p * is 19.3 MPa (Italian bentonite) and 6691 MPa (Bulgarian bentonite), larger than the mean swelling pressure measured in both tests.
Te compliance of the condition z s (κ i (s)/p) � z p (κ s (p, s)/(s + p at )) was not fulflled because it is never fulflled if α il � 0 or α sp � 0 and α ss ≠ 0. Due to the path being monotonic (loading), the compliance of this condition is not important and the BBMEx can be used. Table 8: Condition κ i ′ (s) ≤ 0. Maximal non-negative suction s * for given values of the parameters α i and α il such that the condition is satisfed ("none" means that the condition cannot hold). Table 9: Condition κ i ′ (s) ≤ 0. Ranges of values of the parameters α i and α il such that the condition is satisfed for all non-negative Table 10: Condition z s (κ i (s)/p) � z p (κ s (p, s)/(s + p at )). Maximal positive net pressure p * and maximal non-negative suction s * for given values of the parameters α i , α il , α sp , and α ss such that the condition is satisfed ("none" means that the condition cannot hold).
None None α i ≠ 0 α il and α sp are any reals α ss is any real None None Table 11: Condition z s (κ i (s)/p) � z p (κ s (p, s)/(s + p at )). Ranges of values of the parameters α i , α il , α sp , and α ss such that the condition is satisfed for all positive net pressures and all non-negative suctions. α i α il and α sp α ss Condition holds

Conclusions
Te BBM model is a geomechanical constitutive model which is used to simulate the elastoplastic behaviour of unsaturated soils which are slightly or moderately expansive. Te BBMEx model is an extension of the BBM model to highly expansive unsaturated soils that can be used for the design of the clay barriers in spent nuclear fuel repositories and for assessing the evolution of the clay barriers needed in the safety case of such repositories. Tis extension only concerns the elastic response, and in that, it generalizes the constants κ and κ s of the BBM model to functions κ i (s) and κ s (p, s) of net pressure p and suction s that depend on a set of parameters.
Although calibration of the parameters of the functions κ i (s) and κ s (p, s) could allow the modelling of the results of   Table 13: Maximal positive net pressure p * and maximal non-negative suction s * for which the considered conditions hold for the parameters of the Italian bentonite and of the Bulgarian bentonite in Table 12. Reference is made to the tables used to obtain these results. 12 Advances in Materials Science and Engineering experimental tests over certain ranges of net pressures and suctions, some conditions on these functions may eventually not be satisfed over the ranges of net pressures and suctions considered in a modelling problem. Tis could lead to unrealistic predictions. Four conditions on the functions κ i (s) and κ s (p, s) have been considered: (1) κ i (s) > 0; (2) κ s (p, s) > 0; (3) κ i ′ (s) ≤ 0; and (4) z s (κ i (s)/p) � z p (κ s (p, s)/(s + p at )). For each of these conditions, two problems have been solved: (1) we fnd the ranges of the values of the parameters of the functions κ i (s) and κ s (p, s) such that the condition is satisfed on the subset (0, ∞) × [0, ∞) of the space p − s and (2) given a set of values of the parameters of the functions κ i (s) and κ s (p, s), we fnd the maximal subset (0, p * ) × [0, s * ) or (0, p * ) × [0, s * ] of the space p − s such that the condition is satisfed.

Conditions
Te results of the frst problem could be useful during the calibration of the parameters of the functions κ i (s) and κ s (p, s) so that the conditions are satisfed for all net pressures and suctions. Te results of the second problem could be useful for checking if the conditions are satisfed over the ranges of net pressures and suctions used in a modelling problem with a given set of values of the parameters of the functions κ i (s) and κ s (p, s).
Overall, the BBMEx may help develop internally consistent THM models to simulate the evolution of the clay barriers in spent nuclear fuel disposal facilities [15]. Te parameter conditions presented in this article are useful when implemented in computer codes that use the BBMEx in order to avoid nonphysical values or for warning the user that inconsistent values are used in the simulation.

Data Availability
Te data used to support the fndings of this study will be made available on request from the corresponding author.

Conflicts of Interest
Te authors declare that they have no conficts of interest.