Microphysics of Inelastic Deformation in Reservoir Sandstones from the Seismogenic Center of the Groningen Gas Field

When either one of the above criteria for grain splitting is reached, transgranular cracks are produced. In the deviatoric case, these cracks propagate diametrically from B to B (Fig. 4g, h), in reality opening preferentially towards open pore space (cf. Fig. 2h—red arrow). In the hydrostatic case, all grain contact types (i.e., A, B, C and D) are equally stressed, while transgranular cracks will form normal to any one pair of grain contacts of the unit cell, i.e., normal to A, B, C or D, assuming that one of these fails slightly earlier than the others (Fig. 4f). Here, we explore the magnitude of crack opening (2c) for the deviatoric case, the strain increment that may develop because of it, and the effect it has on the subsequent loading behavior and intragranular crack evolution. The hydrostatic scenario can be evaluated in a similar way, as will be briefly shown at the end of this section.

The assumed geometry of crack opening, and the geometrical factors considered in our analysis are defined in Fig. 5. On the one hand, crack opening by an amount 2c (c = half opening) will cause vertical expansion \((u_{ \exp } )\) normal to the failing contact by opening space between the contact and the overlying clay film (Fig. 5b). This expansion is given by: \(u_{ \exp }\) = − sin(β)r = − cr/(2R_c) (Fig. 5b). On the other hand, vertical contraction occurs, at the same site, due to indentation of the split grain halves into the overlying clay films, as seen in micrographs of lab-deformed, Slochteren sandstone (Fig. 2j). At the opened end of the crack (Fig. 5c), this contraction is given by: \(u_{\text{cont}}^{o}\) = sin(β)(r − m). At the opposite (closed) end of the crack (Fig. 5d), indentation of the tilted contact margin into the contact clay film yields a contraction: \(u_{\text{cont}}^{c}\) = sin(β)n. The net contraction \((u_{\text{net}} )\) parallel to a single meridional crack, due to its opening, is then the sum of these contraction and expansion effects, given: \(u_{\text{net}} = u_{\text{cont}}^{\text{o}} + u_{\text{cont}}^{\text{c}} + u_{ \exp } ,\) or: \(u_{\text{net}} = \frac{c}{{2R_{\text{c}} }}\left( {n - m} \right)\). In the limit of small c, the resultant vertical strain increment of the unit cell due to intragranular splitting \(({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{s}}}} )\) can hence be written as:

Because intragranular splitting is considered for laterally unconfined grains (see Sect. 7.1), opening will not induce any lateral strains, so that \({\text{d}}\varepsilon_{3}^{{{\text{cr}},{\text{s}}}} = 0\).

The values of c, m and n in Eq. 22 are found by balancing the forces supported by the grain contacts at both ends of the crack prior to opening, versus those supported after opening, i.e., by the portions of the clay contact films that are indented by the split grain halves. The locally increased clay film consolidation strains implied by the indentations shown at both the opened and closed cracks ends in Fig. 5c, d are additive to the reference consolidation strain accumulated in the clay films at the B contacts up to the point of intragranular splitting. This reference strain is given: \(\tilde{\varepsilon }^{*} = B\ln (\tilde{\sigma }^{*} /\tilde{\sigma }_{\text{h}} ),\) while the corresponding reference clay film thickness is: \(z^{*} = z(1{-}\tilde{\varepsilon }^{*} )\). Assuming crack opening is symmetric on both sides of each failing grain contact, only half of the clay film thickness is indented on each side (i.e., z*/2—see Fig. 5c, d). At the opening end of the crack, the contraction of the half clay film at a distance x from the contact margin is: Δl_o(x) = (x − m) sin(β) (see Fig. 5c). This implies that at this end, the increase in normal strain \((\hat{\varepsilon }_{o} (x))\) within the half film at location x is given:

Similarly, at the closed end of the crack, the increase in normal strain in the half clay film at distance x from the contact margin is:

Combining Eqs. 23a and 23b with Eq. 7, and inserting \(\tilde{\sigma }^{*} \approx \frac{{4R_{\text{c}} \sqrt {E_{\text{q}} YR} }}{{\sqrt 3 \pi r^{2} \left( {1 - \frac{r}{R}} \right)}}\) (cf. Eq. 20) as the reference normal contact stress \(\tilde{\sigma }_{0}\), we obtain the grain contact normal stresses supported by the clay films present at the opened \((\hat{\sigma }_{o} (x))\) and closed ends \((\hat{\sigma }_{\text{c}} (x))\) of the crack, as a function of x:

In the clay film present at the opened end of the crack, the normal force acting on a constant element dx, at position x is given: \({\text{d}}\hat{F}_{\text{o}} (x) = S(x) \hat{\sigma }_{\text{o}} (x){\text{d}}x\). Here, S(x) is the crack-parallel width of the circular grain contact, at x (see Fig. 5e). At the closed end, the normal force contribution at x is \({\text{d}}\hat{F}_{\text{c}} (x) = S(x) \hat{\sigma }_{\text{c}} (x){\text{d}}x\). The normal force \((\tilde{\sigma }^{*} \pi r^{2} )\) supported on grain contact B upon splitting must be equal to the force supported after opening, given the assumed microstructure. At the opening end of the crack (Fig. 5c), this force balance means:while at the closed end (Fig. 5d), we obtain:

For easy integration, we approximate the circular grain contact area as being hexagonally shaped (see Fig. 5e), such that for x ≤ r/2, then S(x) = 4x, while for r/2 < x ≤ r, S(x) = 2r. The hexagonal area is similar to the circular solution within 5%. The description of S(x) in Eqs. 25a and 25b thus depends on whether m and n are larger or smaller than r/2, e.g., if 0 < m < r/2, then S(x) is equal to 4x within the bounds of m ≤ x < r/2, and to 2r when r/2 ≤ x < r. If m ≥ r/2, then S(x) = 2r. For the opened end of the crack, we will assume, for now, that 0 < m ≤ r/2 (see Fig. 5e). Later calculations demonstrate that this is indeed the case. The force balance for the grain contact at the open end of the crack is then obtained by inserting Eq. 24a along with the two simplified descriptions given above for S(x), into Eq. 25a, which gives:where the first integral term represents the force contribution within the bounds of m ≤ x < r/2, and the second integral term represents the force contribution along r/2 ≤ x < r. After integration, we obtain:

Similarly, assuming that at the closed end of the crack, r/2 ≤ n < r, and inserting Eq. 24b and the above two simplified descriptions of S(x) into Eq. 25b leads to:which on integration gives:

Equations 26b and 27b can be numerically solved to yield the values of m and n, corresponding to an arbitrary half crack opening c. An additional constraint is provided by the fact that c, and the corresponding values of m and n, must be such that the external work increment per unit volume (dW_ext) done through deforming the unit cell by intragranular splitting is approximately equal to the internal work increment per unit volume (dW_int) done (dissipated) by indenting all B-type clay films within the cell upon crack opening, i.e., assuming that the energy required to create new crack surface is provided by elastic energy released as described by the Kendall criterion (Kendall 1978). For the presently considered, deviatoric case with zero \({\text{d}}\varepsilon_{3}^{{{\text{cr}},{\text{s}}}}\), dW_ext is given:

At the opened end of the crack (Fig. 5c), the work done to achieve clay indentation over a clay film strip of width dx (length S(x)), at location x is given by the product of the local force acting on this strip at x (i.e., \(S(x) \hat{\sigma }_{\text{o}} (x){\text{d}}x\)), and the corresponding indentation displacement \((\Delta l_{\text{o}} (x) = \hat{\varepsilon }_{\text{o}} (x)(z^{*} /2)),\) hence by the product: \({\text{d}}T_{\text{o}} = \left[ {\hat{\varepsilon }_{\text{o}} (x)\left( {\frac{{z^{*} }}{2}} \right)S(x)\hat{\sigma }_{\text{o}} (x){\text{d}}x} \right]\). Similarly, at the closed end of the crack (Fig. 5d), the work done by clay indentation at location x is given: \({\text{d}}T_{\text{c}} = \left[ {\hat{\varepsilon }_{\text{c}} (x)\left( {\frac{{z^{*} }}{2}} \right)S(x) \hat{\sigma }_{\text{c}} (x){\text{d}}x} \right]\). Recalling that two intragranular cracks are formed in each unit cell (Fig. 5a), each indenting the clay on both sides, and at both ends, the corresponding internal work per unit volume dW_int is given:

Here 8√3R_c³ is the volume of the unit cell, while the first and second integral terms reflect the internal work contributions due to indentation of the clay films at the opening and closed ends of the two cracks, respectively. Inserting Eqs. 23a, 23b, 24a and 24b and the relations given above for \(S(x)\), in Eq. 29a, and integrating over x gives:

For the hydrostatic case, transgranular cracks may form normal to any pair of the contact types A, B, C or D, all of which are equally likely to fail (Fig. 4f). The external and internal work increments per unit volume associated with crack opening plus indentation at A, C and D will be the same to the corresponding work increments associated with failure of B described above, while the orientation of the strain increments developing in the unit cell due to cracking will depend on which of the contacts fail. Assuming that at the sample scale, all contacts fail at roughly equal proportions, the average, principal, inelastic strain increments \({\text{d}}\varepsilon_{i}^{{{\text{cr}},{\text{s}}}}\) (i = 1,2,3), developing during hydrostatic compression are approximately uniform, and each roughly equal to a third of the vertical strain increment given for the deviatoric case by Eq. 22.

We now briefly apply the above equations to the case of deviatoric loading at an effective confining pressure of 40 MPa, to explore whether our analysis implies realistic magnitudes of crack opening (2c), and what increments in vertical strain \(({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{s}}}} )\) should be associated with such opening. We first employ Eqs. 26b, 27b, 28 and 29b, to numerically yield the values of m, n, dW_ext and dW_int corresponding to a range of c values between 10 nm and 10 µm. We use the input parameters corresponding to initial porosities of 13.4, 21.5 and 26.4%, listed in Table 2. For these initial porosities, the values obtained for half opening c, at which the external and internal work increments per unit volume are balanced (dW_ext = dW_int) yield 0.1, 0.3 and 0.5 µm, respectively. These values of c imply full crack openings (2c) between 0.2 and 1.0 µm, which are similar, although perhaps on the low side, of the crack openings typically seen in micrographs (0.1–10 µm; Fig. 2g–j). The corresponding, splitting induced, vertical strain increments \(({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{s}}}} )\) ranged between 0.01 and 0.03%, being again higher with increasing porosity. We further test whether the values calculated for the indentation limits m and n indeed fall within the ranges of 0 < m < r/2, and r/2 ≤ n < r, as previously assumed in deriving Eqs. 26a, 26b, 27a and 27b. For the full range in stress changes considered here, i.e., for deviatoric compression at P_c = 5–80 MPa and for hydrostatic compression up to \(P_{\text{c}}^{*}\), the obtained magnitude of m always fell in between 0 and r/2, while the resultant values of n ranged between r/2 and r. This validates the previously assumed bounds on m and n. Based on these results, we infer that our analysis of the inelastic strain associated with grain splitting, as outlined in Sect. 7.4.1, leads to representative magnitudes of crack opening, and hence of the strain increments developing because of it.

For given contact stress (e.g., at B in the θ = 0° configuration), the normal stresses acting around a newly formed, meridional crack will be elevated where the clay film is indented the most, notably around the opened crack end (Fig. 5c), and near the grain contact periphery at the closed end (Fig. 5d). Near the grain contact periphery at the closed end, the elevated contact stress leads to elastic compression within the bulky volumes constituting the sides of the split grain, and are otherwise likely of little effect. However, at the opened end, the normal stress acting on the most uplifted edges within one flaw spacing width d (3 µm; Table 2) from both sides of the initial meridional crack will be strongly enhanced and will tend to extend the first neighboring contact flaws to produce edge cracks running parallel to the initial crack surface (Lawn 1993; Tada et al. 2000). An estimate of the enhanced edge stress \((\tilde{\sigma }_{\text{edge}} )\) acting over an edge width d is obtained by subtracting the normal force acting on one split grain contact half between m < x < r − d, from the nominal force supported by the full grain contact half (i.e., 1/2 \(\tilde{\sigma }_{\text{B}} \pi r^{2}\)). Using the approximate relations for S(x) outlined in Sect. 7.4.1, an expression for the normal stress acting on the most uplifted edge \((\tilde{\sigma }_{\text{edge}} )\) alongside the opened end of the meridional crack can hence be written as:which after integration yields:

Here, c and m can be obtained at given stress conditions, grain size and porosity by following the approach outlined in Sect. 7.4.1. For the presently considered range in porosity (13.4–26.4%) and confining pressures (5–80 MPa), these enhanced normal stresses are thus estimated to range between 480 and 928 MPa. The critical normal stress \((\sigma_{\text{edge}}^{*} ),\) required for mode I edge crack propagation has been described by Thouless et al. (1987) to be equal to \(\sigma_{\text{edge}}^{*} = 0.87 \frac{{K_{\text{Ic}} }}{\sqrt d } = 1.23\sqrt {\frac{{YE_{\text{q}} }}{d}} ,\) where K_Ic is the Griffith critical stress intensity factor for mode I crack propagation. For the E_q, Y and d values listed in Table 2, this critical normal stress equals 489 MPa. Hence, for virtually all presently considered porosities and stresses, the enhanced normal stress acting on the edge adjacent the initial meridional crack \(\tilde{\sigma }_{\text{edge}} \ge \sigma_{\text{edge}}^{*} ,\) implying that initial meridional splitting and crack opening is spontaneously followed by edge cracking, at assumed constant stress.

Upon edge cracking, the two grain slices located on both sides of the initial split between r − d < x < r (Fig. 5c) will break off, and be displaced into the gaps opened by the initial, intragranular split (Fig. 5a). If the broken-off slices do not sufficiently fill this gap, they will not support load. The contact normal stress will now be supported on the remaining contact area, where it is largest at the now most-uplifted edge adjacent to the first set of edge cracks (i.e., between r − 2d < x < r − d; Fig. 5c). The normal stress acting on this edge will again exceed \(\sigma_{\text{edge}}^{*}\), resulting in edge crack propagation from of the next pair of flaws. Thus, at constant stress, an instable sequence of multiple stages of edge cracking is expected, which will progress until the gap opened by the initial split is filled, such that the broken-off slices will start to support load. The strain increments (\({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{m}}}}\) and \({\text{d}}\varepsilon_{3}^{{{\text{cr}},{\text{m}}}}\)) developing up to this point due to multi-edge cracking of grain contacts B are estimated by considering the gap filling process in the 2D plane, where the 2D gap area is given by: cR_c + c(r − n)²/R_c (see Supplementary Figure S2). The 2D porosity fraction φ_cr characterizing this 2D gap cross section before the cracked grain starts to support load is taken to be 0.5, i.e., equivalent to a typical porosity fraction of loosely packed, angular sand (0.4–0.5; Mavko et al. 2009). During deviatoric loading, \({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{m}}}}\) is then given by:while \({\text{d}}\varepsilon_{3}^{{{\text{cr}},{\text{m}}}}\) is again zero. For the values and porosities listed in Table 2, the value of \({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{m}}}}\) obtained for deviatoric compression at P_c = 5–80 MPa ranges between 0.08 and 0.40%. During hydrostatic compression, multi-edge cracking occurs at whichever grain contacts A, B, C or D an initial meridional crack was formed. As it was assumed that the unit cells including initial meridional cracks from any one of the contact types A, B, C or D, are equally present throughout the microstructure at the sample scale (cf. Sect. 7.4.1), the average, principal inelastic strain increments (\({\text{d}}\varepsilon_{i}^{{{\text{cr}},{\text{m}}}}\), i = 1, 2, 3) developing due to multi-edge cracking at the sample scale are then each equivalent to a third of the deviatoric, vertical strain increment \({\text{d}}\varepsilon_{1}^{{{\text{cr}},{\text{m}}}}\), described by Eq. 31. Note that the sum of the strain increments due to initial splitting, and those resulting from subsequent multi-edge cracking developing at constant stress, constitutes the total inelastic strain increment developing at the vertical stress \((\sigma_{1}^{*} )\) corresponding to initial intragranular splitting (Eqs. 21a and 21b).

The triaxial stress–cycling experiments reported by Pijnenburg et al. (2019a) were performed on Slochteren sandstone samples with initial porosities (φ₀) of 13.4, 21.5 and 26.4%. These samples were either deviatorically compressed at a constant effective confining pressure (P_c) of 5 MPa from the outset, or else first hydrostatically compressed from an initial 5 MPa, to 20, 40 or 80 MPa, followed by deviatoric compression at constant P_c until stage 3d or stage 3c behavior was seen. The stress conditions used in the model were chosen to match. The modeled mean effective stress (P = [σ₁ + σ₂ + σ₃]/3 = [σ₁ + 2σ₃]/3) versus inelastic porosity reduction (Δφ_i ≈ ε₁^inel  +2ε₃^inel ) behavior and the P versus total porosity reduction (Δφ_t ≈ ε₁^el  + 2ε₃^el  + Δφ_i) behavior are shown in Fig. 6 (lines), where they are compared to the corresponding experimental data (circles).

At each of the initial porosities explored, the modeled and experimentally obtained P versus Δφ_t and Δφ_i curves obtained are typically similar within 0.1–0.2% porosity reduction (Fig. 6). An exception is seen at the lowest P_c of 5 MPa tested, where the model significantly overestimates the P–Δφ_t behavior by a factor of 1.5–2, particularly for φ₀ ≥ 21.5% (Fig. 6b, c). The experimental behavior likely reflects the anisotropic elastic behavior shown to occur at these low values of P_c (Pijnenburg et al. 2019a), which is at present not included in our model (Sect. 3). Still, during hydrostatic compression up to P_c = 20, 40 or 80 MPa, the modeled and experimental P–Δφ_t and P–Δφ_i are closely similar. During subsequent deviatoric compression, the model initially shows no inelasticity, reflecting the inability of the model to compact by serially coupled clay consolidation and slip, when slip is not yet activated (i.e., when \(P_{\text{c}} < \sigma_{1} < \left. {\sigma_{1}^{\text{slip}} } \right|_{{\left( {\theta = 0^\circ } \right)}}\); Eq. 12a). In our experiments, Δφ_i is never fully absent, albeit small at low deviatoric stresses. In our model, the onset of inelastic deformation during deviatoric compression (i.e., when \(\sigma_{1} = \left. {\sigma_{1}^{\text{slip}} } \right|_{(\theta = 0^\circ )}\)) is accompanied by an instantaneous increase in Δφ_i of about 0.02–0.05% (Fig. 6). This reflects the re-equilibration of the clay films by clay consolidation and slip to the contact stresses prevalent at this point in loading (behavior cf. Eqs. 15a and 15b). During subsequent compression, the model shows more gradual P–Δφ_t and P–Δφ_i behavior, yielding similar, roughly linear (stage 2) hardening rates to those shown by the experimental data at the same P. Stages 1 and 2 are then followed by stage 3d or stage 3c behavior. The dilatant (suffix d) or compactive (suffix c) nature of these stages, as seen in our experiments (solid arrows in Fig. 6), is in most cases well-predicted by the model (open arrows), except for φ₀ = 13.4%, at P_c = 40 MPa and for φ₀ = 26.4%, at P_c = 20 MPa. For these two cases, stage 3c is predicted, while stage 3d was observed. Where described correctly, the predicted mean effective stress marking the onset of stage 3d or stage 3c falls within 10 MPa from the experimentally obtained value. In our experiments, stage 3c is associated with concave-down P–Δφ_i behavior, while our model implies an instantaneous increase in Δφ_i at constant stress. On the other hand, the inelastic porosity reduction seen in our experiments as stage 3c starts to develop (indicated by c-arrows), is closely similar to the modeled inelastic porosity reduction after initiating stage 3c, i.e., within a relative difference of 5%. Despite the noted, modest discrepancies, the modelled and experimental P–Δφ_t and P–Δφ_i data show a general agreement, implying that the P–Δφ_t and P–Δφ_i behavior can be largely accounted for by the mechanisms assumed.

Yield envelopes describing the differential stress (Q = σ₁ − σ₃) versus mean effective stress (P = (σ₁ + σ₂ + σ₃)/3) conditions required to activate each of the assumed, inelastic processes at modelled initial porosities of 13.4, 21.5 and 26.4% are shown in Fig. 7.

The yield envelopes obtained for dilatant intergranular slip (i.e., stage 3d) and for serially coupled clay slip plus clay consolidation are insensitive to φ₀, while the envelopes obtained for intragranular cracking (stage 3c) are markedly φ₀-sensitive (Fig. 7). Furthermore, the combined outline of the predicted stages 3d and 3c envelopes describes a tri-linear, or broadly elliptical shape in Q–P space, transecting the P axis at the origin and at the hydrostatic confining pressure for isotropic grain crushing \((P_{\text{c}}^{*} )\). This shape and the predicted φ₀-(in)sensitivities characterizing stages 3d and 3c in our model are in qualitative agreement with the behavior seen in experiments performed on Slochteren sandstones (Pijnenburg et al. 2019a) and on many other sandstones (Wong and Baud 2012—see also our Introduction). Hence, at least for these high strain stages 3d and 3c (ε > 1%), the agreement between predicted and observed stress conditions is found to be satisfactory. This implies that the dilatant behavior is well-captured by considering the onset of tension at grain boundaries developing during intergranular slip (see also: Guéguen and Fortin 2013), while stage 3c initiation can be accounted for by considering intragranular cracking, which at Q > 0.6P is made easier by accompanying intergranular slip. We now go on to discuss whether the model captures the inelastic yield envelope expansion implied for stages 1 and 2 of the experiments performed by Pijnenburg et al. (2019a). In that way, we evaluate to what extent the associated hardening behavior due to inelastic densification is accounted for by consolidation of- and slip on the intergranular clay films.

To constrain the model hardening behavior (i.e., the yield envelope expansion) due to inelastic porosity reduction, we show contour lines delineating fixed values of inelastic porosity reduction in Fig. 7 (thin green lines; magnitude indicated in %). These data are obtained by interpolating Q–P–Δφ_i data rendered in multiple model runs, in which hydrostatic- and subsequent deviatoric compression is simulated at a maximum P_c in the range of 5–140 MPa and at a φ₀ value of 21.5%. Similar Δφ_i-contour lines were obtained for initial porosities of 13.4 and 26.4%. These are not shown, as the behavior was highly similar to that seen for φ₀ = 21.5%, albeit less- and more compliant, respectively.

Overall, the modeled inelastic porosity reduction contours obtained for a φ₀ value of 21.5% at the Q and P conditions enveloped by stage 3d (dilation) and stage 3c (intragranular cracking) are positively and steeply inclined (Fig. 7), indicating that inelastic compaction is primarily sensitive to P, not to Q. The interspacing between the lines decreases with increasing P, reflecting stiffening of the clay films as these consolidate. Recall that throughout hydrostatic compression, the unit cell deforms by inelastic clay consolidation (cf. Eq. 9). However, at deviatoric stress conditions below those required to activate intergranular slip (Q < 0.6P) and intragranular cracking, the unit cell cannot deform inelastically. Therefore, at these deviatoric stresses, corresponding to inelastically rigid behavior, the slope (dQ/dP) of the inelastic porosity reduction contours is equal to 3, i.e., equivalent to the slope dQ/dP implied during deviatoric loading at constant confining pressure. Upon activating intergranular slip plus clay consolidation (Q = 0.6P), all contours shift along the corresponding envelope to lower Q and P, reflecting the instantaneous increase in Δφ_i accompanying the onset of serially coupled, intergranular slip and clay film consolidation, as noted earlier (Sect. 8.1.1). At Q > 0.6P, contour lines are steeper (dQ/dP = 5 ± 1), due to inelastic compaction by active clay consolidation plus slip, at these conditions. Where Δφ_i contours intersect the intragranular cracking envelopes (Fig. 7), their orientation changes to follow these envelopes towards lower P values, implying negative slopes (dQ/dP) of − 0.5 (if accompanied by intergranular slip), or − 1.5 (no intergranular slip). Hence, the onset of intragranular cracking in our model marks a sharp change from primarily P-sensitive- (due to clay film deformation) to P- and Q-sensitive inelastic porosity reduction behavior (due to intragranular cracking).

In the experiments performed by Pijnenburg et al. (2019a), a similar transition was noted, notably from P-sensitive Δφ_i behavior during stage 1 and the initial portions of stage 2, to P- and Q-sensitive behavior, towards the end of stage 2, and into stage 3c. The initial P-sensitive Δφ_i behavior of stages 1 and 2 was then speculated to be caused by a yet unidentified, intergranular deformation process. The increased Q-sensitivity towards the end of stage 2 and into stage 3c was tentatively attributed to a gradually increasing role played by intragranular cracking, in accordance with accompanying crack density data and earlier inferences (Curran and Carroll 1979; Wong et al. 1997). The present microphysical model provides a physical basis to underpin these earlier inferences. It confirms that while predominantly P-sensitive hardening behavior can be explained by intergranular clay consolidation and accompanying intergranular slip, Q-sensitive hardening behavior is to be attributed to a role played by intragranular cracking (Shah and Wong 1997).