Pore pressure coefficient in frozen soils Open Access

Frozen saturated soils generally consist of solid minerals, ice and water. This poses a question: what is a good effective stress measure for frozen soils? This question reflects the difference in stiffness between water and ice and the influence on how the ice forms in the soil – that is, mainly in the centre of the pores or on the surface of the grains indicated as the typical microstructure of frozen saturated soil in Fig. 1(a). For investigating a relevant effective stress measure, the Skempton pore pressure coefficient, B, defined as the variation in pore pressure u with the unit change in confining pressure p under undrained condition (Skempton, 1954), is essential

As one of the parameters in Biot theory (Berryman & Milton, 1991; Berryman, 1992), the B value plays an important role in many aspects of permafrost or frozen soil behaviour. It is essential for estimating seismic compressional wave velocity (Dou et al., 2017), thermo-hydro-mechanical behaviours (Ghoreishian Amiri et al., 2016) and permafrost landslide risk (Dramis et al., 1995; Patton et al., 2019). There is a limited number of studies available in the literature on B value measurements and relevant theoretical studies for frozen soils. However, some measurements and theoretical developments have been carried out for unfrozen soils and rocks (Hart & Wang, 2010; Makhnenko & Labuz, 2016). To the present authors’ knowledge, only Kia (2012) has addressed B value measurement for frozen saline sand, with significant effort. A key reason is the challenge in keeping a hydraulic connectivity between the unfrozen water phase in the sample and the pore pressure sensor. A common standard for pore pressure measurement of frozen soils has not been established. Another reason is the requirement to apply a relatively limited range of hydraulic pressure for the B value measurement of frozen soils. Frozen soil in geotechnical and permafrost engineering is typically subject to a low stress range (order of 100 kPa), unlike in rock engineering, where much higher stress levels are involved (up to the order of MPa or even tens of MPa), for example as encountered in mountain tunnels and mine shafts. Besides, a high overall stress level can be further magnified at local grain-to-grain/ice contacts, and result in significant ice melting and further changes in the soil skeleton for frozen soils. As a result, relatively low pressure (up to 100 kPa) is favoured to ensure that the unfrozen water content and skeleton remain roughly unchanged.

In order to study the B value of frozen soils both experimentally and theoretically, the present work firstly followed the pore pressure measurement proposed by Arenson & Springman (2005), who conducted B value measurement in a specially designed triaxial cell similar to that shown in Fig. 2. P-wave velocity for frozen Onsøy clay was measured using the experimental method suggested by Yun et al. (2007) in order to estimate the unfrozen water saturation (UWS) defined as S_w = V_w/(V_w + V_i) in Fig. 1(b) and the skeleton stiffness. Regarding the theoretical analysis of B value, this study adopted three effective medium theories: namely, the modified Kuster and Toksӧz's model (KT model) (Kuster & Toksöz, 1974; Zimmerman & King, 1986), matrix cementing (Ecker et al., 1998) and grain coating (Dvorkin et al., 1999) to estimate the change in B value with decrease in the UWS based on different ice crystallisation modes. The paper finally discusses the effect of pore bulk modulus and loading history on the B value of frozen soils.

Onsøy clay, which is found in south-eastern Norway near Fredrikstad, has been the subject of research for 50 years by the Norwegian Geotechnical Institute (NGI) (Oslo, Norway) and others, and has recently been chosen as one of five Norwegian geotest sites (NGTs) (Gundersen et al., 2019). It is homogeneous, fully saturated, soft marine clay with typical salinity 20–30 g/l and water content 50–70%. The soil is slightly overconsolidated and contains a significant amount of glacial rock flour, clay minerals (mostly illite and chlorite), quartz and feldspars, with typically 50% clay and 50% silt content. The plasticity limit and liquid limit are of the order 20% and 60%, respectively, and the permeability coefficient at in situ effective stress is 10⁻⁹m/s.

Natural Onsøy clay was used to prepare artificial frozen samples with a ‘slow’ freezing method, which minimises the thermal gradient and further induced water redistribution during freezing. A cylindrical soil sample 150 mm long and 74 mm in diameter was sealed and immersed inside anti-freezing liquid in a container with much larger volume (12 l). The sample and container were placed inside a low-freezing-rate fridge. The temperature was lowered down to 1°C from room temperature until the whole system reached thermal equilibrium. Further freezing continued down to the testing temperatures (−3, −5 and −10°C in this study). The whole freezing process took at least 48 h. A triaxial set-up (shown in Fig. 2) was developed with precise temperature control and the capability of B value measurement. The apparatus allowed both constant strain rate and constant stress loading. The feedback temperature control was achieved by continuously adjusting the temperature of circulating fluid through a spiral copper coil inside the cell to maintain the temperature, as measured by a thermocouple 30 mm away from the sample, equal to the testing temperature.

Particular care is necessary for B value measurement in frozen soils. It is achieved by ensuring a low system compressibility and minimum amount of extra fluid volume by adopting both rigid pore pressure sensors and short high-pressure stainless steel tubing (Bishop, 1976; Mesri et al., 1976; Makhnenko & Labuz, 2016). The pressure connectivity between the unfrozen water phase and the pore pressure sensor requires flow path connectivity and fluid phase continuity inside frozen soil. This study adopted anti-freeze liquid (a mixture of ethylene glycol and water) to keep the drainage system (including porous stone and pipelines) unfrozen for flow path connectivity in the subzero temperature. The ethylene glycol concentration should be relatively low to minimise any thawing effect on frozen soils. In clay the water phase can become discontinuous with the degree of saturation below 30–40% (Romero et al., 1999; Romero & Simms, 2008). Pore pressure measurement for T = −10°C testing might not secure fluid phase continuity and could result in measurements that are only representative of locality. The system saturation should also be maintained, although it is quite difficult to check this. The B value of unfrozen Onsøy clay, tested at the same triaxial set-up, is equal to 0·98 with 0–100 kPa back-pressure. This indirectly suggests that the natural soil was fully saturated, and proper pipeline flushing and ‘low’ back-pressure can de-air the system properly. X-ray scanning was also conducted and potentially freezing-induced air bubbles or voids were not found. Besides 300 kPa cell pressure and 50–100 kPa back-pressure were applied to the post-freezing consolidation to secure the pore pressure equilibrium before the B test. In the B tests, p was increased by Δp = 20 kPa in each step until a maximum value of 400 kPa. Fig. 3 shows the u plotted against p response marked as ‘pre-loading’ for natural Onsøy clay with different salinity and temperature values. The cell pressure was reduced back to 300 kPa following the same path (no hysteresis observed). Next, a 1%/h constant strain rate loading test was conducted in the undrained condition until 20% vertical strain was reached. After the constant strain rate loading test, the load was gradually relaxed to zero. Furthermore, the pore pressure reached equilibrium again. In order to test possible B value changes due to the loading test, the same B test procedure was followed again and termed the post-loading–unloading B value measurement (results marked as ‘post-loading’ in Fig. 3). The B value results show a significant increase after loading–unloading and decrease with the decrease of temperature (−3 to −10°C) and salinity (no. 2, 30 g/l as opposed to no. 3, 20 g/l).

In this study, P-wave velocity measurements were conducted and estimates of UWS were made based on the KT model for natural Onsøy clay at the tested temperatures. The piezo disc elements, which have around 100 kHz central frequency, were coated with silver paint and grounded to eliminate electrical cross-talk (see details in the paper by Lee & Santamarina (2005)). These prepared piezo disc crystals (12 mm dia. and 1·5 mm thick) were coated with epoxy for water proofing and attached to the inside wall of the 76 mm dia. acrylic cylinder in which the sample was placed. Two PT-100 sensors (a type of resistance thermistor) with 0·1°C accuracy and good repeatability were inserted to measure the actual temperature and its spatial variation inside the frozen soil sample (test set-up details can be found in Fig. 4). This set-up has been verified based on the accurate measurement of water acoustic velocity. The whole system was placed inside a temperature-controlled freezer to maintain the temperature during the testing. The waveform and measured velocities are presented as a function of temperature in Fig. 5(a) for the tested Onsøy clay with 20 g/l salinity and 48% water content. The KT model (see equations (10) and (11) in the Appendix) was adopted to calculate the undrained moduli of frozen soils and P-wave velocity as a function of UWS ranging from 0 (fully frozen state) to 100% (fully unfrozen state). The UWS value that corresponds to the measurement result of P-wave velocity was taken as the estimate at a given temperature. The UWS estimate for frozen Onsøy clay is presented in Fig. 5(b) as a function of temperature.

To qualitatively verify the estimate, Fig. 5(b) also shows theoretically calculated UWS plotted against temperature curves for S = 20–40 g/l frozen brine according to the thermodynamic equilibrium of water–ice phases. The UWS is expected to remain quite significant, even below −10°C for saline ice, because ions are expelled from freezing ice to unfrozen water and the electrolyte concentration in the remaining unfrozen water increases with ice crystallisation. Therefore, it leads to further freezing point depression of the remaining unfrozen water. In addition, surface effects, such as electrical double layers, are induced by fine-grained particles in frozen soils, and also have a significant contribution to UWS in frozen fine-grained soils. The UWS of Onsøy clay with the combined effects of ions in pore fluid and fine-grained particles should be higher than UWS in saline ice at a given temperature and salinity. As Fig. 5(b) also indicates, the UWS (S_w plotted against T) curve of Onsøy clay with 20 g/l salinity is between the S = 30 g/l and 40 g/l frozen brine at a given temperature, which indirectly shows that the interpreted UWS is reasonable. The estimated UWS together with the measured B value for Onsøy clay is presented in Figs 6(a) and 6(b). The results suggest that the B value decreases with a decrease in UWS (50–100%).

Kia (2012) also developed a fibre optic rigid piezometer and conducted a B value test for frozen saline Ottawa sand with 30 g/l salinity at the range of 100 kPa hydraulic pressure. Combining UWS results measured by Hivon & Sego (1995), the B value against S_w results for frozen saline Ottawa sand are marked as triangles in Figs 6(c) and 6(d). Similarly to the Onsøy clay, the B value decreased with decreasing UWS, but an opposite trend was observed at UWS below 50%.

The pore pressure coefficient can be written as a function of the fluid bulk modulus K_f, the pore bulk modulus K_n (defined in equation (3)), the skeleton modulus K, the porosity n and the Biot coefficient α = 1 − K/K_s (K_s: solid bulk modulus defined in equation (4)) for porous media (Gassmann, 1951; Berge et al., 1993; Berge & Berryman, 1995)

where V is the sample volume and p′ is the Terzaghi mean effective stress defined as p′ = p − u. Here, p is the mean total stress and is defined as the average of three principal stresses σ₁, σ₂ and σ₃

Alternatively, B is expressed in terms of K_u (undrained bulk modulus), K_s, and K (Brown & Korringa, 1975)

The B value approaches 1 for unconsolidated, unfrozen porous materials, typically like soils, indicated by equation (6) when K/K_s, K/K_u and K/K_f are sufficiently small, such as < 0·01. Gundersen et al. (2019) concluded that the oedometer modulus for reloading–unloading is around 1–2 MPa for the natural Onsøy clay above 10 m depth. The skeleton bulk modulus K is in the range of 0·7–1·5 MPa (estimated from the oedometer modulus) and negligible compared with K_s (20–44 GPa) and K_w (2·25 GPa). In addition, the value can be closer to 1 for low-porosity porous media with n→0 in equation (2) and K_u→K_s in equation (6) (Berge et al., 1993) – for example, B = 0·95–1 for Vermont marble and Barre granite with around 2–3% porosity at p′ = 200 kPa (Mesri et al., 1976).

The B value is expected to correlate with porosity n, since a decrease in porosity generally leads to an increase of K and K_u, and a decrease of α (Lee, 2002), although porosity distribution, microcracks and pore shapes complicate the physical properties of porous media (Mesri et al., 1976; Berge et al., 1993). Some laboratory test results for rocks, as one of the typical porous materials, also indicate this tendency: B = 0·93–0·98 for n = 0·4 Nevada Tuff at p′ = 20–50 MPa (Fredrich et al., 1995), B = 0·68 for n = 0·19 for Berea Sandstone at p′ = 7 MPa (Hart & Wang, 1995) and B = 0·53–0·69 at n = 0·13 for Indiana Limestone at p′ = 2–10 MPa (Hart & Wang, 1995). In summary, both theoretical and experimental evidence shows the relationship between porosity and B value, although in addition p′ plays a significant role. Therefore, ice crystallisation following the decrease in UWS can play an important role in B value variation during freezing.

Gassmann's equation assumes the invariability of n with u to secure K_n = K_s as indicated in equations (3) and (4) for a homogeneous porous medium composed of a monotonic solid phase when p′ remains constant (Berge, 1998; Hart & Wang, 2010). However, K_n is not bound by K_s when more than one solid constituent is present (like coexistence of soil particles and ice in frozen soils). This discrepancy becomes significant, and K_n value can theoretically even become negative. It can take place when compressible solid grains (e.g. ice with 8·4 GPa) fill the pore spaces such as the pore-filling crystallisation mode as seen in Fig. 1, and are covered by a five times ‘stiffer’ solid medium (like quartz with 44 GPa bulk modulus) (Berge & Berryman, 1995; Helgerud et al., 1999; Hart & Wang, 2010). Although difficult to measure directly, the back-calculated K_n indicates negative value for rocks composed of quartz and clays with a large stiffness mismatch (Berge, 1998). Similarly, frozen soil can possibly show a negative K_n value because of the large stiffness mismatch and comparable proportion for soil and ice according to the theoretical analysis of Berge & Berryman (1995).

It is noted that ice bulk stiffness shows slight temperature-dependency and increases by around 1·6% when the temperature decreases from a freezing temperature down to −10°C (Vogt et al., 2008; Vaughan et al., 2016). For simplicity, K_i is assumed to remain unchanged in the range of the testing temperature (−3 to −10°C) in the current paper. The other issue is the wide stiffness range of clay minerals. Reported theoretical values of clay bulk modulus range from 20 to 50 GPa (Vanorio et al., 2003). The laboratory test results show chemical composition, mineral type and testing method influence the stiffness values of clay minerals. For example, Aleksandrov (1961) and Katahara (1996) suggested 52–56 GPa bulk modulus for illite, kaolinite and chlorite. Han et al. (1986) and Berge & Berryman (1995) proposed 21·4 GPa bulk modulus for the situation of mixed clay minerals. Wang et al. (2001) measured 60·1 GPa for illite and up to 164·3 GPa for chlorite (ripidolite) and 127 GPa for Mg-rich chlorite. In this study, the content of different clay minerals and chemical compositions in each mineral remain unknown. For simplicity of analysis, a typical clay bulk stiffness value, 20·9 GPa, was adopted to calculate the solid bulk stiffness of Onsøy clay with mixed clay minerals. However, a higher value is expected because laboratory test results suggest much higher stiffness for illite and chlorite minerals, which are the two major clay composites in Onsøy clay. The assumption of K_n = K_s in Gassmann's equation can further deviate from the reality if the stiffness of clay minerals is of the order 100 GPa. In the discussion section of this paper (entitled ‘Discussion’), the effect of the variation in clay bulk stiffness on the B value of frozen soils will be analysed.

The KT model has been proven to achieve accurate UWS estimates for saline sand and clay (Lyu et al., 2020). The model was used earlier to estimate UWS according to measured P-wave velocities, and here it is used to investigate the likely change in B value with the decrease of UWS and increase in K during freezing. The KT model outputs a K_u estimate (see Appendix), and K as the function of K_u can be back-calculated from the extended Gassmann's equation (Brown & Korringa, 1975; Berge, 1998)

where effective porosity n₁ = nS_w ranging from 0 (completely frozen) to n (completely unfrozen). The solid bulk and shear moduli, K_s and G_s, can be calculated based on the Voigt–Reuss average suggested by Hill (1952) 

where m is the number of solid components (ice, clay and quartz); f_j, K_j and G_j are the volumetric fraction, bulk and shear moduli corresponding to the jth component in the solid phase. The ice fraction f_i increases and, correspondingly, quartz and clay volumetric fractions f_q and f_c decrease with freezing, which can be calculated as follows

where C_q = V_q/(V_c + V_q) and C_c = V_c/(V_c + V_q). The quartz and clay volumetric content, V_q and V_c, are defined in Fig. 1(b).

Frozen soils can typically consist of various microstructures such as ice growing in the pore space, coating soil particles, or ice enveloped by soil particles. Soil particles can maintain contact with each other, or ice can invade, and split particles apart, as suggested in Fig. 1(a). In order to simplify the complexity, this work considers two monotonic ice crystallisation modes: the pore filling and grain coating models, as ‘very’ weak and strong K evolution modes with freezing, suggest upper and lower bounds of B value with soil freezing, respectively. It should be noted that there is always a thin layer of water molecules between ice and fine-grained soil particles as seen in Fig. 1(a). This indicates that the grain coating model essentially deviates from reality, especially when it is applied to Onsøy clay. However, it can be suggested as a scenario to provide the upper boundary of skeleton stiffness K. The detailed calculations regarding the B value as a function of UWS are presented here. Similarly to the KT model, both equations (8) and (9) were used again to first update K_s with the variation of S_w. A set of equations (14)–(17) and (19)–(22) (in the Appendix) was then adopted to calculate skeleton bulk modulus K at a given ice crystallisation mode (pore filling or grain coating). Finally, the extended Gassmann equation (18) was applied to calculate K_u from the K value, and relevant bulk moduli were introduced into either equation (2) or (6) to obtain the B value at a given S_w. It should be noted that the original studies (Dvorkin & Nur, 1996; Helgerud et al., 1999) applied Gassmann's equation (assuming K_n = K_s). Instead, this paper highlights the extended Gassmann equation with the relaxation of K_n = K_s assumption to calculate K_u from K in order to consider the effect of K_n on the B value of frozen soils in the discussion (Brown & Korringa, 1975). The key equations of the KT, pore filling and grain coating models can be found in the Appendix.

Table 1 presents the input parameters for the three models and other properties for Ottawa sand and Onsøy clay. In addition, two more parameters in the grain coating and pore filling models are also determined as coordination number (CN = 9) and critical porosity (n_c = 0·37), for an assumed case of p′ = 100 kPa, according to Helgerud et al. (1999). The lines in Figs 6(a) and 6(c) present diverse B value plotted against S_w variation by assuming K_n = K_s, and Figs 6(b) and 6(d) present the results with relaxation of the K_n = K_s assumption.

Generally, both theoretical and experimental results suggest a decrease in B value with lowering UWS from 100% to 40–60%. The measured B value of frozen Ottawa sand in Fig. 6(c) seems to follow the lower bound suggested by the grain coating mode when K_n = K_s is assumed. However, this ice crystallisation mode only has been shown to be unlikely to occur (Dou et al., 2017; Lyu et al., 2020). Furthermore, the measured pre-loading B value for frozen Onsøy clay is even below this lower bound suggested by the grain coating mode as indicated in Fig. 6(a). Although more pore pressure coefficient measurements should be conducted for frozen soils, the fact that both B value results deviate from the KT model indicates the possibility of K_n≠K_s, and even a negative K_n value for frozen soils. With the relaxation of the K_n = K_s assumption, the proper K_n values were sought to make the B value variation estimated by the KT model consistent with the measurement results. Fig. 6(d) suggests K_n = −10 GPa to reach the general consistency between experimental results and the KT model for frozen Ottawa sand. Various negative values of K_n with the range from −1 to −5 GPa are indicated for frozen Onsøy clay in Fig. 6(b). However, this consistency would not be taken as the definitive support for negative K_n values. Here, the experimental data are explained by a K_n that is variable with mostly negative values according to the UWS. Further measurement for parameter sets (e.g. K_u, K and B value) is recommended to minimise the error influence of a single parameter measurement and to reach a more accurate estimate of K_n (Hart & Wang, 2010).

In addition to UWS (or temperature) dependency, the present study also found the significant increase in B value after loading–unloading, which is an important finding in the paper. Two possible mechanisms can contribute to this phenomenon. First, K can become significantly smaller because of ice crystal crushing during the loading–unloading phase. Dai & Santamarina (2017) also reported stiffness loss as a consequence of creep and loading–unloading based on the evolution of P-wave velocity. The other possible reason is the loading history induced K_n evolution. As Fig. 6(a) suggests, the post-loading B value of Onsøy clay is enveloped by the upper and lower bounds suggested by pore filling and grain coating, respectively, when K_n = K_s is assumed. The negative value of K_n occurs in some specific soil–ice microstructures as mentioned above, and this structure can be highly degraded during different loading processes. However, more evidence is needed to confirm these explanations.

As suggested in the earlier section entitled ‘Pore pressure coefficient and Gassmann's equation’, a wide range of bulk stiffness values for clay minerals has been reported in the literature, and the solid bulk stiffness for Onsøy clay is likely to be higher than the 20·9 GPa adopted in this work. Therefore, it is meaningful to study the effect of variation in clay mineral stiffness on the B value for frozen soils. In this discussion, four scenarios are proposed with different clay mineral stiffness: (a) K_c = 10 GPa, G_c = 5 GPa (lower bound of clay stiffness); (b) K_c = 20·9 GPa, G_c = 6·85 GPa (typical value for mixed clay minerals); (c) K_c = 60·1 GPa, G_c = 25·3 GPa (illite stiffness measured by Wang et al. (2001)); (d) K_c = 127 GPa, G_c = 82 GPa (Mg-rich chlorite stiffness measured by Wang et al. (2001)). The calculated B values are presented in Fig. 7 for four different clay stiffness scenarios based on KT when K_n = K_s and K_n = −5 GPa. As the results indicate, the clay stiffness shows some influence on the B value of frozen soil, and a maximum variation of ≈ 0·15 in the B value is suggested with the wide variation of clay stiffness. Besides, the B value at low UWS (less than 30%) was more influenced by the variation in clay stiffness compared with at high UWS (above 60%).

The KT model was developed to simulate the elasticity of unconsolidated permafrost (King et al., 1988), and it was used for UWS estimates in this paper because it requires a very limited number of parameters and has been widely adopted (Kneisel et al., 2008; Hauck et al., 2011). However, this model only takes spherical shapes of water and soil phases into consideration, which deviates from the geometry of natural clay particles. In order to study the capability of the KT model for UWS estimation, Lyu et al. (2020) conducted a systematic comparison between the KT model estimate and UWS measurement results for a wide range of soil types. They found that the KT model can achieve quite an accurate UWS estimate for saline unconsolidated sand and saline Boom clay based on P-wave velocity measurement. This indicates an insignificant shape effect of ‘hard inclusions’ (soil particles in frozen soils) on the UWS estimate. To further investigate the accuracy of the UWS estimate, the UWS plotted against temperature for different types of frozen saline clay with various salinity is also summarised and presented together with Onsøy clay data in Fig. 8. The UWS for S = 20 g/l Onsøy clay as a function of temperature is comparable with silty clay with S = 30 g/l (Patterson & Smith, 1985) and Morin clay (S = 29 g/l) (Zhou et al., 2020) in the range from −3 to −10°C. Although Onsøy clay has lower salinity compared with the other two types of soils, there is a much higher clay content in Onsøy clay compared with Morin clay (50% against 12·5%). In addition, the UWS of lean clay with S = 19 g/l is slightly lower than that of Onsøy clay (S = 20 g/l), because the clay content of the former soil type is also lower than that of the latter soil type. To sum up, Fig. 8 proves the coupled effects of ion concentration in pore fluid and clay or fine content on UWS. The UWS estimate of Onsøy clay appears to have reasonable accuracy.

It should also be noted that the Gassmann equation has been adopted throughout this study. However, it assumes well-connected pore spaces, which can be violated for frozen soil with low UWS, although the adopted generalised Gassmann equation in this study already relaxes the assumption of isotropy, monomineral solid phase and homogeneity (Smith et al., 2003). As well as the research on the B value of frozen soil, there are quite a few studies on the validation of the Gassmann equation on low-UWS frozen soils, or even on other types of porous material – for example, rocks. Han et al. (1986) and Han & Batzle (2004) suggested the validity of the Gassmann equation for sandstone samples above n = 0·15, and estimated that errors of bulk modulus below n = 0·15 become more significant with the decrease in porosity. This also indicates that the relation between B value and UWS shown in Figs 6 and 7 can deviate from reality when UWS is below 20–30%. However, the B value approaches 1 when effective porosity n₁ becomes closer to 0 based on the previous analysis and reported experimental results. This suggests that the tendency towards B value variation in UWS below 20–30% could be reasonable even when it is analysed based on the Gassmann equation. Further study regarding new model development is necessary to have a better understanding of the B value for frozen soils.

This study presents a systematic work regarding the B value testing of frozen soils. It includes a ‘slow-freezing’ preparation of artificial frozen samples and an experimental framework for pore pressure measurement in frozen soils. In addition, P-wave velocity was successfully measured for frozen Onsøy clay, and the KT model was applied to estimate UWS as a function of temperature. The UWS comparison for different frozen saline clays indirectly proves the reasonable accuracy of the estimate for Onsøy clay. In addition, this study draws several other original conclusions, as outlined below.

The B value of frozen soil shows temperature dependency. Both experimental results and theoretical analysis suggest a decrease in B value with lower UWS from 100% (completely unfrozen) to 40–50%, as the temperature drops.

The deviation of the B value between the measurement results presented and theoretical analysis suggests K_n ≠ K_s, and that K_n can vary with UWS and even take a negative value.

The B value of frozen soils also shows a loading history dependency. This is likely to be induced by a loss in skeleton stiffness as the pore stiffness approaches the solid stiffness due to microstructure evolution.

A wide range of clay stiffness can contribute limited impact on the B value of frozen Onsøy clays, and this influence is more significant at low UWS (less than 30%) compared with high UWS (above 60%).

This publication is part of the Nunataryuk project. The project has received funding under the European Union's Horizon 2020 Research and Innovation Programme under grant agreement no. 773421. The first author would like to acknowledge the Norwegian Geotechnical Institute (NGI) for providing soil samples. The authors also appreciate paper proofing and comments provided by Michael Long, Associated Professor at University College Dublin.

The ‘KT model’ suggests the undrained bulk and shear moduli, K_u and G_u, for a two-phase medium composed of spherical inclusions (with moduli K_in and G_in) embedded in a continuous matrix medium (with moduli K_ma and G_ma)

where c is the volumetric fraction of spherical inclusions.

As Zimmerman & King (1986) and King (1984) suggested, equations (10) and (11) are adopted iteratively to first estimate the effective moduli of the ice–water mixture by embedding spherical water inclusions into a continuous ice phase and further to calculate the moduli of frozen soil composed of spherical soil grains built into the ice–water mixture.

The frozen soil density ρ_m and estimated P-wave velocity V_p as a function of S_w are further calculated as follows

where ρ_c, ρ_q, ρ_w and ρ_i are clay, quartz, water and ice densities. At given inputs of C_c and n, the S_w value can be determined when the corresponding estimated V_p is consistent with the measurement result. The skeleton bulk modulus K can be back-calculated using equation (7), as discussed in the main text of the paper.

The ‘pore filling model’ was developed from Hertz–Mindlin contact theory to estimate skeleton bulk modulus K with the consideration of soil particle state, consolidation or suspension, which is distinguished by a critical porosity n_c (Dvorkin & Nur, 1996; Helgerud et al., 1999)

where K_HM and G_HM are calculated as per the following equations (15) and (16) (CN, coordinator number; P, effective stress; ν_s, solid Poisson's ratio)

In the pore filling model, the ice phase is assigned as a fluid phase, and the fluid bulk modulus, K_f, is a weighted reciprocal average of water and ice bulk moduli (K_w and K_i) according to UWS

Instead of the original Gassmann's equation, the extended Gassmann equation (18) was adopted by the relaxation of the K_n = K_s assumption to calculate the undrained bulk modulus K_u from skeleton modulus K (Brown & Korringa, 1975; Berge, 1998; Hart & Wang, 2010)

Regarding the ‘grain coating model’, K can be calculated following equation (19)

where the ice constraint modulus M_i and the parameter S(ψ) can be obtained as follows

where the ν_i are ν_g are ice and grain (sand and clay) Poisson's ratio.

B

pore pressure coefficient

C_c

clay volumetric content

C_q

quartz volumetric content

f_i, f_q, f_c

ice, quartz and clay volumetric fraction

f_j, K_j, G_j

volumetric fraction, bulk and shear moduli corresponding to the jth component in the solid phase

G_w, G_i, G_c, G_q

water, ice, clay and quartz shear modulus

K

skeleton modulus

K_u, K_f, K_n, K_s

undrained, fluid, pore and solid bulk modulus

K_w, K_i, K_c, K_q

water, ice, clay and quartz bulk modulus

M_i

ice constraint modulus

n

porosity

n_c

critical porosity

P

effective stress

p

confinement pressure

p′

Terzaghi mean effective stress

S

salinity

S_w

unfrozen water saturation (UWS)

u

pore pressure

V

sample volume

V_p

estimated P-wave velocity

α

Biot coefficient

ν_g

grain (sand and clay) Poisson's ratio

ν_i

ice Poisson's ratio

ν_s

solid Poisson's ratio

ρ_w, ρ_i, ρ_c, ρ_q

water, ice, clay and quartz density