Relative density estimation from shallow depth CPTs in siliceous sand: an updated approach

a

empirical fitting parameter (Equations 7 and 10)

C_0.deep

empirical fitting parameter (Equations 2 and 3)

C_1.deep

empirical fitting parameter (Equations (2) and 3)

C_2.deep

empirical fitting parameter (Equations 2 and 3)

C_0.shallow

empirical fitting parameter (Equation 6)

C_1.shallow

empirical fitting parameter (Equation 6)

C_2.shallow

empirical fitting parameter (Equation 6)

D_r

relative density

D_r.deep

predicted relative density considering deep failure penetration

D_r.shallow

predicted relative density considering shallow failure penetration

d_c

cone diameter

e_max

maximum void ratio

e_min

minimum void ratio

f_transition

transition function (Equation 9)

K

lateral stress coefficient (Equation 4)

K₀

coefficient of earth pressure at rest

m′

stress exponent (Figure 7)

NC

normally consolidated

N_q

bearing capacity factor (Equation 4)

n₁

fitting parameter (Equation 9)

n₂

fitting parameter (Equation 9)

OC

overconsolidated

p_a

reference stress (=100 kPa)

p′

mean effective stress

q_c

cone resistance

q_c.deep

predicted cone resistance considering deep failure penetration

q_c.global

predicted cone resistance considering both shallow and deep failure penetration

q_c.shallow

predicted cone resistance considering shallow failure penetration

z

depth of penetration

q_t

corrected cone resistance (Figure 7)

z_cr

critical depth

z/d_c

normalised penetration depth

γ′

effective unit weight

σ′

normalising effective stress

σ′_h

horizontal effective stress

σ′_p

preconsolidation stress (Figure 7)

σ_v

vertical total stress

σ′_v

vertical effective stress

φ′

peak friction angle

φ′_cs

critical state friction angle

When sufficient embedment is achieved during cone penetration in sand, a deep failure mechanism develops, characterised by a localised deformation zone near the cone tip (Arshad et al., 2014). Deep penetration has been widely analysed using the analogy between cone penetration and spherical cavity expansion (e.g., Salgado and Prezzi, 2007). In contrast, the upper ≈0.2–1.0 m of standard field cone penetration tests (CPTs) is typically governed by a shallow failure mechanism, associated with upward displacement of surface sand (Biarez and Gresillon, 1972). A transition zone separates the shallow and deep mechanisms, marking a shift in the governing soil deformation pattern (Puech and Foray, 2002).

Industry-standard methods for estimating the relative density (D_r) of sand from CPT cone resistance (q_c) are largely empirical, based on CPTs in calibration chambers under deep failure conditions (e.g. Baldi et al., 1986; Jamiolkowski et al., 2003). Applying these methods to near-surface sands, where shallow failure governs, leads to erroneous D_r estimates. Accurate estimation of relative density at shallow depths is of practical importance in various geotechnical applications. Near-surface sands often control the performance of shallow foundations, pipelines, cables, and earthworks. Erroneous classification of shallow sand as loose, when it is in fact dense, may lead to overly conservative designs, unnecessary ground improvement, or incorrect assessment of serviceability performance. Conversely, overestimation of D_r may result in unconservative predictions of foundation performance.

Despite its practical importance, guidance for interpreting CPT data at shallow depths remains limited. Although several researchers have advanced understanding of shallow failure penetration and the transition zone (Emerson et al., 2008; Jensen, 2024; Kim et al., 2015; Lehane et al., 2022; Puech and Foray, 2002), these developments have not yet been widely incorporated into routine engineering. This paper provides a critical review of existing methods for estimating D_r at shallow depths and introduces an updated global model that extends the formulation of Jensen (2024).

While Jensen (2024) introduced a unified framework for combining shallow and deep penetration mechanisms, the preliminary calibration of the model exhibited limitations in the very near-surface zone and was based on a very small data set. The present study addresses these limitations through recalibration using comparisons with alternative theoretical and empirical interpretation methods, previously published controlled laboratory tests, field CPTs with measured D_r, and a new database of 132 onshore and offshore CPTs. This approach enables both quantitative validation and broader qualitative evaluation. Refinement of the shallow penetration coefficient and transition function improves predictive performance in the shallow failure zone and reduces bias in near-surface D_r estimation, thereby strengthening the model’s practical applicability.

As illustrated in Figure 1, cone penetration in uniform sand deposits generally exhibits a transition in the measured cone resistance profile, q_c(z), with depth (z), from a shallow failure mode to a deep failure mode. In the very near-surface zone, shallow failure dominates, producing an upward-concave q_c(z) profile characterised by an increasing gradient (dq_c/dz). The mechanics of shallow failure penetration are governed primarily by the mobilised friction angle and soil dilatancy (Durgunoglu and Mitchell, 1973). This penetration mode has been the subject of several studies (e.g., Balachowski, 2007; Kim et al., 2015; Puech and Foray, 2002).

Experimental observations by Kim et al. (2015), consistent with the general findings of Lehane et al. (2022), indicate that the transition from shallow to deep failure occurs at a depth of penetration of ≈3–6 cone diameters in loose, clean silica sand, and about 15–20 cone diameters in dense silica sand. Based on their tests, Kim et al. (2015) proposed the following expression with a form that describes the upward-concave q_c(z) profile representative of shallow failure penetration (q_c.shallow) in clean silica sand:

where p_a is a reference stress (=100 kPa), σ′_v is vertical effective stress, and d_c is the cone diameter. Following Kim et al. (2015), both q_c.shallow and σ′_v are expressed in MPa in Equation 1. Although this convention may appear dimensionally inconsistent, it is retained to ensure comparability with their formulation. Equation 1, along with other approaches, is discussed further later in this paper.

At greater depth, the penetration response transitions to a deep failure mode with decreasing dq_c/dz, as shown in Figure 1. Deep failure penetration is primarily governed by sand compressibility characteristics (Konrad, 1998; Yu and Mitchell, 1998). Widely used CPT-D_r correlations for deep failure are based on Schmertmann’s (1976) formulation:

where C_0.deep, C_1.deep, and C_2.deep are empirical fitting parameters, and σ′ is the normalising effective stress.

In normally consolidated (NC) sands, calibration is often performed using the vertical effective stress σ′_v (e.g., Baldi et al., 1986; Schmertmann, 1978). However, Houlsby and Hitchman (1988) demonstrated that the horizontal effective stress (σ′_h) has a stronger influence on q_c. Consequently, many methods adopt the mean effective stress, p′ = (σ′_v + 2σ′_h)/3, improving generality, particularly in overconsolidated (OC) sands where the coefficient of earth pressure at rest (K₀), on which σ′_h depends, is significantly higher (e.g., Jamiolkowski et al., 2003; Krogh et al., 2022).

In the upper 3–5 m of OC sands, where K₀ and the overconsolidation ratio (OCR) vary significantly with depth, stress normalisation using p′ becomes particularly important to avoid overestimation of D_r, as may occur when normalising solely with σ′_v. Depth-dependent K₀(z) can be evaluated using the approach of Krogh et al. (2022) (procedure given in the flowchart in Figure 7 presented later in the paper), where preconsolidation stress is estimated following Agaiby and Mayne (2019). Importantly, Emerson et al. (2008) and Jensen (2024) demonstrated that better interpretation of shallow CPT data requires integrating the K₀(z)-based stress normalisation with shallow failure considerations, as discussed in the following.

Puech and Foray (2002) developed a limit equilibrium model for shallow failure penetration in siliceous sand based on bearing capacity theory, incorporating friction angle, cone diameter, and lateral friction:

where γ′ is effective unit weight of sand, ϕ′ the peak friction angle, L is the dimension of a soil cylinder considered around the cone (L = d_c·exp(tan(ϕ′·(π/2)))·tan(π/4 + ϕ′/2)), N_q = 1.058·exp(6.168·tanϕ′), and K is a lateral stress coefficient, different from K₀ and ranging from ≈1–3 (Emerson et al., 2008). Equation 4 is to be applied with curve fitting while relating D_r via the following correlation with ϕ′:

Distinguishing between shallow and deep failure penetration is critical, since deep failure penetration in loose sand can produce q_c-profiles that resemble those from shallow penetration in dense sand, leading to misinterpretation of relative density. To address this issue, Emerson et al. (2008) proposed a global q_c model (q_c.global) that via a transition function combines Puech and Foray’s (2002) model for q_c.shallow with the widely applied calibration of deep failure penetration (q_c.deep; Equation 3 by Jamiolkowski et al. (2003)); see Table 1. In their global model, the transition between shallow and deep failure penetration is defined by a critical depth (z_cr), primarily dependent on D_r; lower D_r causes shallower onset of deep failure.

Figures 2(a) and 2(b) show Emerson et al.’s (2008) global model applied to two CPTs at a German test site with homogeneous, clean, uniformly graded, fine to medium OC siliceous sand, using a standard 10 cm² cone (d_c = 3.57 cm) (data from Krogh et al., 2022). Nuclear densometer (ND) tests performed at the site indicate D_r = 60% ± 10% (CPT-02; Figure 2(a)) and D_r = 80% ±10% (CPT-03; Figure 2(b)). A weakness in Emerson et al.’s model lies in the parameter K (Equation 4), which is somewhat arbitrary. Although K strongly influences the predicted q_c.shallow profile in the model, no robust guidance exists for selecting an appropriate value in practice. For comparison within the shallow failure zone, Emerson et al.’s (2008) global model is evaluated against the experimentally derived expression for q_c.shallow proposed by Kim et al. (2015) (Equation 1). The two approaches show good agreement in the shallow zone when Emerson et al.’s model is applied with K = 2.5 (Figures 2(a) and 2(b)). In these comparisons, representative relative density ranges of D_r = 0.6–0.8 (Figure 2(a)) and 0.65–0.85 (Figure 2(b)) were used, corresponding to the ND test results.

Although the curves now show reasonable agreement with the direct ND measurements of D_r in the shallow zone, Emerson et al.’s model does not accurately capture the transition from shallow to deep failure, predicting the shift at a considerably greater penetration depth than indicated by the CPT data (Figure 2). Furthermore, the abrupt shift from increasing dq_c/dz (shallow zone) to decreasing dq_c/dz (deep zone) does not fit the general observation of a gradually changing gradient (Figure 1). If applying a lower value of K, the transition to the deep failure penetration could have been captured better, but that would have then compromised the accuracy significantly in the shallow failure zone. Similar discrepancies were noted by the author comparing with the extensive internal CPT database, presented later in this paper.

Centrifuge CPTs, due to limited penetration, are well suited for studying shallow failure penetration (e.g., Balachowski, 2007; Bolton et al., 1999). Lehane et al. (2022) used a large centrifuge database to describe the shallow-to-deep transition with a hyperbolic tangent function:

where C_0.shallow, C_1.shallow, and C_2.shallow are empirical parameters calibrated by Lehane et al. (2022) for clean, uniformly graded, fine to medium, freshly deposited NC silica sands using σ′ = σ′_v (Table 1), and a is an empirical curve fitting parameter, given by Lehane et al. (2022) as follows:

Equation 7 (Lehane et al., 2022) is shown in Figure 3 together with the a values derived from their centrifuge database, and supplemented with values inferred by the author from data presented by Puech and Foray (2002) and Emerson et al. (2008) from CPTs performed with a standard 10 cm² cone. Although designated ‘shallow’, Equation 6 may also apply for deep failure penetration in freshly deposited NC sands, since it essentially transitions into a deep failure expression (Equation 2) when the tangent hyperbolic function approaches unity with depth, as controlled by the a parameter. In this study, however, it is primarily used to model the shallow zone and the transition.

When applied using D_r = 0.6–0.8 (Figure 4(a)) and D_r = 0.65–0.85 (Figure 4(b)), predicted q_c values fall below the field data, with the discrepancy increasing beyond ≈0.3–0.4 m, as expected due to overconsolidation effects. Jensen (2024) noted that Equations 6 and 7 effectively captures the shape of field q_c profiles in moderately compressible siliceous sands. However, overconsolidation effects influences deep q_c significantly and must be accounted for.

For shallow failure penetration, Jensen (2024) argued that overconsolidation has little, if any, effect on shallow failure penetration resistance. This conclusion was supported by two main pieces of evidence. Firstly, numerical simulations by Krogh et al. (2022) (Figure 5) demonstrated comparable q_c.shallow for relative densities of D_r = 0.65 (Figure 5(a)) and D_r = 0.85 (Figure 5(b)), under both constant K₀ = 0.414 (representative of NC sand) and depth-dependent K₀(z) (representative of OC sand). Secondly, similar trends can be inferred from the initial penetration curves reported in experimental centrifuge CPT studies by El-Sekelly et al. (2015), Roy (2020), and Richards et al. (2021). These experiments, conducted at varying g-levels, effectively simulated a range of OCR values, including NC conditions.

Building on these observations, Jensen (2024) proposed a global q_c model that combines Equations 2 and 6, with the transition governed by OCR (or K₀). An assumption in this approach is therefore that very shallow q_c values are equivalent for both normally consolidated and OC sands, with OCR effects becoming more pronounced with increasing embedment approaching deep failure penetration.

To capture both failure regimes and the transition, Jensen (2024) proposed:

where

with n₁ = 0.5 and n₂ = 2 suggested as initial proof of concept values, subject to further calibration. Jensen (2024) also proposed that the calibration of Equation 6 could be extended to deep failure (Equation 2) by making small adjustments to C_0.shallow, thereby enabling a shift from σ′_v to p′ in stress normalisation (Table 1).

The global model, defined by Equations 8 and 9 and incorporating Jensen’s (2024) preliminary calibration, is presented in Figures 6(a) and 6(b). The predicted q_c.global profiles closely match the shape of the measured CPT data and align well with the ND measurements. However, a current key limitation is evident in the upper ≈0.4 m: the profiles converge in the initial penetration for the applied D_r range, preventing objective differentiation of D_r. Although the model successfully captures the overall profile shape, this shortcoming highlights the need for refinement, which is addressed in the following section.

To support refinement of the global model, an internal database of 132 shallow CPTs from onshore and offshore industry projects was compiled and analysed in conjunction with previously published controlled laboratory tests. The internal database comprises onshore tests conducted in Denmark and Germany, and offshore tests performed in the North Sea and Irish Sea; details are summarised in Table 2. The onshore CPTs were mainly carried out in recently placed, clean siliceous sand fill with varying degrees of compaction, whereas the offshore CPTs were conducted in young, surficial, clean siliceous deposits. As noted in the descriptions for each site (Table 2), the tested sands generally consist of fine to coarse, uniformly graded sand, as confirmed by sampling. Of the total tests, 96 CPTs were performed with a 10 cm² cone and 36 CPTs using a 15 cm² cone, all from the same site (Site B(2)). Each CPT was advanced to a depth constraining both shallow and deep failure penetration.

A qualitative review of the internal CPT database indicated that the transition to deep failure penetration generally occurs at a slightly shallower depth, particularly for loose to medium dense sands, than predicted by the current model governed by the coefficient a defined in Equation 7. That is, a controls the hyperbolic tangent function in Equation 6, which drives the transition of the q_c profile to deep failure penetration as the function approaches unity. To account for this, a minor modification to Equation 7 is proposed, resulting in an improved fit to the internal database and a closer overall correspondence with the data shown in Figure 3:

The review of the internal database also indicates that Lehane et al. (2022) profiles (Equation 6), which forms the basis for the global model by Jensen (2024) in the very shallow zone, consistently underestimate q_c.shallow in the near-surface range for both loose and dense natural sand deposits. This trend is further supported by the distinct discrepancy observed in Figure 4, where comparisons between Lehane et al. (2022) (Equation 6) profiles and q_c.shallow profiles from Kim et al. (2015) reveal systematic large deviations in the shallow failure zone. As was demonstrated in Figure 2, the profiles by Kim et al. (2015) show good agreement in the shallow failure zone with both the theoretically derived model of Puech and Foray (2002) and the measured data by Krogh et al. (2022), further confirming the observed underestimation of q_c.shallow in natural deposits with Lehane et al. (2022).

To improve agreement, an increase in C_2.shallow from 3.2 to 3.8 is proposed (Table 1), and a slight adjustment to f_transition (Equation 9) is similarly made; specifically, increasing n₁ to 0.6, while retaining n₂ = 2. As shown in Figure 4 (profiles named ‘this study’), these adjustments enable Equation 6 to follow the upward concave portion of the Kim et al. (2015) profiles more closely in the shallow failure penetration zone. The complete updated approach is summarised in the flow chart in Figure 7, which is easily implemented in a spreadsheet. In the proposed approach, D_r is treated as a representative constant value, whereas curve fitting may be performed over selected depth intervals depending on the desired level of detail.

As shown in Figure 8, curve fitting with the updated global model indicates D_r ≈ 0.5–0.7 for CPT-02 (Figure 8(a)) and D_r ≈ 0.6–0.8 for CPT-03 (Figure 8(b)), both in excellent agreement with the ND measurements (D_r = 60% ± 10% and D_r = 80% ± 10%, respectively). The updated model achieves two key improvements: (1) it prevents convergence of the q_c.global profiles in the surficial sand, allowing objective estimation of D_r; and (2) it yields strong agreement with the internal database, as demonstrated with numerous representative examples from the database later in this paper. Furthermore, the following section discusses cone size effects, in which the updated global model similarly show great performance with controlled tests by Iqbal (2002) and Larsen and Ibsen (2006).

The variability observed in the internal database is reflected in Table 3, which summarises relative density interpretations for the upper 1 m of CPT data, using the updated global model. The descriptive categories from ‘loose’ to ‘very dense’ that are linked to the D_r values (see Table 3; descriptions correspond to interpretations from top to bottom of the upper 1 m profile) indicate substantial variation in the database.

As embedded in Equation 1 (Kim et al., 2015) and Equation 4 (Puech and Foray, 2002), the cone diameter (d_c) plays an important role in the q_c.shallow profiles. Specifically, an increase in d_c leads to a reduction in cone resistance at shallow failure penetration. By analysing CPT data obtained with varying cone diameters in uniform sand samples, Lehane et al. (2022) observed that data fall into a consistent trend when presented in a normalised chart, where the stress normalised cone resistance is plotted against the normalised penetration depth (z/d_c). This observation formed the basis for their q_c.shallow profile expression (Equation 6), upon which the updated global model is an extension.

Essentially, Equation 6 expresses a strong influence of cone diameter during shallow failure penetration, which diminishes with large penetration well into the deep failure penetration zone where profiles of varying d_c will eventually converge. This is consistent with experimental observations (e.g., Balachowski, 2007).

The updated global model is evaluated in Figure 9 using CPTs conducted under controlled laboratory conditions by Iqbal (2002). The tests were performed with two significantly different cone diameters, d_c = 1.13 cm and d_c = 3.57 cm (standard field size cone), in fine to medium, uniformly graded silica sand. Three separate sand bodies with D_r ranging 0.62–0.84 were tested, with densities confirmed by direct measurements following uniform compaction using a vibration screen unit. As shown in Figure 9, the updated global model exhibits excellent agreement with the measurements, remaining within about 5%–10% (in absolute terms) of the measured values for both d_c = 1.13 cm and d_c = 3.57 cm in all three sand bodies.

Because shallow failure penetration is more pronounced in denser sand, the cone size effect is correspondingly greater. In contrast, shallow failure effects are minimal in loose sand. In Figure 10, the global model is evaluated against a CPT with d_c = 1.50 cm performed in a loose, fine to medium, uniformly graded silica sand by Larsen and Ibsen (2006). Direct measurements indicated D_r ranging about 0.3–0.45. The sand specimen was prepared by water pluviation followed by minimal external vibration. Accordingly, a representative value of K₀ = 0.5, consistent with NC conditions, was assumed in the global model rather than using the recommended K₀(z)-procedure. Under these controlled conditions with minimal compaction disturbance, overconsolidation effects are expected to be negligible. As shown in Figure 10, the global model again performs well, with curve fitting ranging about D_r = 0.30–0.45.

At a Danish onshore site, three CPTs (CPT-A1 to A3) using a standard 10 cm² cone and 2 ND tests were conducted within a 15 m radius to assess compaction of a sand fill, see Figure 11. Limiting void ratios of e_min = 0.45 and e_max = 0.79 were measured for the sand fill, typical for clean, subrounded to subangular silica sand (Youd, 1973). The ND tests are considered representative within the shallow failure zone, approximately the upper 0.45 m (Figure 11(a)).

In this zone, D_r values with Equation 3 (D_r.deep) are unreliable; therefore, ND data are used as the primary reference (Figure 11(b)). The ND measurements yielded D_r = 0.81 and 0.87, indicating dense to very dense sand (D_r ≈ 0.84) in the near-surface layer. Below this, D_r decreases to ≈0.6 at a depth of 2 m, except in CPT-A3, where the sand fill ends at about 1.4 m.

Applying D_r = 0.84 in the updated global model results in q_c.global profiles closely matching all three CPTs in the shallow zone (Figure 11(a)). At greater depths, the measured CPT data diverge from q_c.global, consistent with the generally decreasing D_r.deep profile (Figure 11(b)), indicating that the compaction activities produced a denser sand deposit near the surface, which is not unusual of surface vibration compaction.

At a second Danish site (Site B), CPTs were performed using 10 and 15 cm² cones (Table 2) to evaluate compaction of a medium to coarse sand fill. Particle size distribution tests showed a uniformity coefficient (UC) ranging from 3 to 6, indicating minor local variations in gradation. These characteristics differ slightly from those of the uniformly graded sands (UC ≈ 2) used in the development of the reference model.

Two example CPTs at Site B, conducted with a 10 cm² cone, are presented in Figure 12 (CPT-B1, Figure 12(a); CPT-B2, Figure 12(b)). As illustrated in Figure 12(c), the D_r.deep profiles indicate that at CPT-B1, the relative density remains nearly constant with depth (D_r ≈ 0.65–0.70), except in the shallow zone. In contrast, CPT-B2 shows a slight gradual increase in relative density with depth, from D_r ≈ 0.50 to D_r ≈ 0.60 at greater depths. Application of the updated global model (Figures 12(a) and 12(b)) yields excellent agreement in the shallow zone, with estimated D_r values of 0.68 and 0.48, respectively. These results are consistent with the trends indicated by the D_r.deep profiles in Figure 12(c), where indicative ‘corrected’ values are provided for the shallow zone.

At Site B, 36 CPTs were carried out using a 15 cm² cone. These are the only tests in the internal database not performed with a 10 cm² cone and thus provide a valuable case for examining the applicability of the global model, although further validation is required to gain more confidence with this cone size. Two representative CPTs are shown in Figure 13 (CPT-B3, Figure 13(a); CPT-B4, Figure 13(b)). The D_r.deep profiles (Figure 13(c)) indicate that CPT-B3 is dominated by very dense sand (D_r ≈ 0.90–0.95), except in the shallow zone. In contrast, CPT-B4 reveals two distinct layers: a medium-dense near-surface sand (D_r ≈ 0.60) overlying a very dense layer (D_r ≈ 0.88). Application of the updated global model (Figures 13(a) and 13(b)) shows close agreement in the shallow zone, with estimated D_r values of 0.93 and 0.63, respectively, consistent with the D_r.deep trends. The indicative ‘corrected’ values shown in Figure 13(c) illustrate that the shallow zone increases with increasing relative density.

Due to hydrostatic pore pressures, stress conditions during offshore CPTs differ from those in onshore tests, with the effective stresses typically being much lower. Figure 14 presents four offshore CPTs from the North Sea (Sites D and E, Table 2), where the upper layer consists of clean, uniformly graded siliceous sands (confirmed by sampling), illustrating a range of conditions. For clarity, the x-axis is plotted with a consistent scale across all charts in Figure 14.

In Figure 14(a), the updated global model is applied to a CPT in loose sand within the top 1 m. Using D_r = 0.26, the q_c.global profile reproduces the observed shape well and indicates only limited shallow failure (≈0.1–0.2 m). By contrast, methods that assume deeper shallow failure (e.g., Emerson et al., 2008) would significantly overestimate D_r in this case.

Figure 14(b) shows a CPT in medium dense sand, where q_c.global reflects D_r = 0.60. Figure 14(c) presents a near-uniform dense top layer (≈1 m thick), with q_c.global indicating D_r ≈ 0.77 at the surface. In both cases, D_r.deep would misclassify the deposits as loose to very loose in the near-surface.

In Figure 14(d), q_c.global captures slight variations from dense to very dense sand within the upper ≈1.5 m, with fitted D_r values ranging from ≈0.7 to 0.9. Here again, D_r.deep would misclassify the surficial sand as loose to very loose.

The updated model builds on Jensen (2024), which was presented with an initial proof of concept calibration and emphasised the need for further calibration. Nevertheless, Jensen (2024) demonstrated an improvement over existing approaches in capturing the realistic shape of shallow CPT profiles in sand. The shortcomings associated with the initial calibration are addressed in the present study through a recalibration based on previously published controlled laboratory tests and a large internal database of 132 CPTs compiled from industry projects.

As was demonstrated in Figure 6, the initial Jensen (2024) model results in convergence of q_c profiles in the very shallow zone for certain ranges of D_r, preventing objective differentiation of D_r; this limitation is resolved with the updated model. Furthermore, the initial calibration tends to underestimate q_c in the shallow zone, with the discrepancy becoming increasingly pronounced with increasing D_r. This behaviour is illustrated in Figure 15 through comparison of the initial calibration and the updated model using tests that were presented in Figures 12 and 14. Overall, the updated model captures the shape of the q_c profiles more accurately, in agreement with the full internal database, and the difference between the initial and updated formulations increases with D_r.

Figure 16 presents a CPT in a sand deposit visually described as slightly silty to silty and possibly heavily OC due to past glacial activity. In this case, the updated global model underperforms near the surface, probably due to increased sand compressibility not represented in the model calibration. The shallow failure zone appears smaller than predicted by the global model, with deep failure initiating at ≈0.3–0.4 m. This behaviour is likely attributable to the fines content, although in other cases it may arise from angular particles, which increase compressibility while also enhancing dilatancy during shallow failure.

Figure 17 illustrates these effects: centrifuge CPTs in highly compressible, angular carbonate sand (Kwinana sand) and in moderately compressible silica sand (UWA sand), both at similar relative densities (D_r = 0.70 and 0.78), show that Kwinana sand develops higher shallow failure q_c (q_c.shallow) due to greater dilatancy from particle angularity, but lower q_c at deep failure penetration due to higher compressibility.

Discrepancies such as those in Figure 16 could also reflect ageing, bonding, or cementation – factors not captured by the model, which was developed for young, unaged, clean sands. Similarly, intense man-made compaction activities can modify fabric and yield very high near-surface q_c values that are not reproduced by the model, necessitating careful engineering judgement when applying the global model at shallow depth.

The global model is more reliable for fully saturated or dry siliceous sands, as partially saturated conditions can compromise effective stress estimation due to pore pressure effects. Near the surface, particularly within the top 2–4 cone diameters, model reliability is increasingly affected by second-order effects and measurement sensitivity. In this shallow zone, factors such as precise mudline definition and minor cone disturbances can influence the estimated D_r. Improved methods for mudline identification are therefore recommended to enhance interpretation in the near-surface region.

The model is derived from clean, unaged, uncemented, moderately compressible siliceous sands. Application to sands with differing characteristics may introduce bias. A simple comparison of D_r.deep estimates performed by the author, using the calibrations of Jamiolkowski et al. (2003) and Jensen (2024) (Table 1), applied to a database of 480 calibration chamber tests on 11 siliceous sands (compiled by Jefferies and Been, 2015), yielded standard deviations of 0.15 and 0.14, respectively, between measured and predicted D_r. These results support the general applicability of Jensen’s (2024) calibration for q_c.deep while also underscoring the uncertainties associated with D_r estimation, partly due to the inherent difficulty of defining limiting void ratios (Lunne et al., 2019).

This paper has critically reviewed existing approaches for estimating relative density (D_r) from shallow depth CPTs in sands and presented an updated global model that unifies shallow and deep failure mechanisms, calibrated against CPTs performed under controlled laboratory testing and a comprehensive internal database of 132 onshore and offshore CPTs. The main conclusions are as follows:

1. Advancement beyond Jensen (2024):

• The updated calibration refines the shallow penetration formulation and transition function proposed in Jensen (2024), enabling more objective differentiation of D_r in the shallow zone.

• The revised model reduces shallow-zone bias and provides improved agreement with measured q_c profiles across a substantially expanded data set.

1. Validation:

• The model has been evaluated against numerous controlled laboratory tests, by comparison with alternative theoretical and empirical approaches, and an extensive internal database of 132 suitable CPTs from onshore and offshore sites, covering a broad range of conditions.

• Validation is primarily based on tests with the most widely used 10 cm² cone, which represents the current basis of calibration. While controlled laboratory testing as well as the limited number of tests with 15 cm² cones show promising agreement, further validation is required before general application across cone sizes.

• Case studies confirm strong agreement with independent direct relative density measurements.

1. Applicability and limitations:

• The model is most reliable for clean, young, uncemented, uniformly graded siliceous sands under fully saturated or dry conditions.

• Shortcomings are illustrated for sands with higher compressibility where factors such as fines content and very angular particle shape may alter the penetration response beyond the calibrated range.

• Within the upper 2–4 cone diameters, reliability decreases due to mudline definition and cone disturbance effects, reinforcing the need for improved mudline definition method.

1. Recommendations for practice:

• As natural soils rarely exhibit idealised behaviour, site-specific validation of the model against direct D_r measurements is required, even when soils appear to fall within the calibrated range of the model.

• Engineering judgement is required to assess potential effects of ageing, bonding, cementation, or intense man-made compaction, where soil fabric effects can produce very high near-surface q_c values not reproduced by the model.

• Future work could focus on extending calibration to a wider variety of sands (including angular and highly compressible types), further validating across cone sizes, and improving mudline identification methods for shallow penetration analysis.

The author gratefully acknowledges Geo for granting permission to compile and use the database drawn from internal projects for this paper, and for covering the article processing charge.