Journal of Earth Science  2018, Vol. 29 Issue (6): 1319-1334 PDF     0
Development and Evolution of the Size of Polygonal Fracture Systems during Fluid-Solid Separation in Clay-Rich Deposits
Teodolina Lopez1,2, Raphaël Antoine3, José Darrozes4, Michel Rabinowicz4, David Baratoux4,5
1. CESBIO, Université de Toulouse, CNES/CNRS/IRD/UPS, Toulouse, France;
2. International Space Science Institute(ISSI), Bern, Switzerland;
3. CEREMA, Laboratoire Régional de Rouen, Groupe Sciences de la Terre, CS 90245, F-76121 Le Grand Quevilly, France;
4. GET, Université de Toulouse, UPS/CNRS/IRD/CNES, Toulouse, France;
5. Institut Fondamental d'Afrique Noire(IFAN), Cheikh Anta Dio, Dakar, Senegal
ABSTRACT: In continental and oceanic conditions, clay-rich deposits are characterised by the development of polygonal fracture systems (PFS). PFS can increase the vertical permeability of clay-rich deposits (mean permeability ≤ 10-16 m2) and are pathways for fluids. On continents, the width of PFS ranges from centimeters to hundreds of meters, while in oceanic contexts they are up to a few kilometers large. These structures are linked to water-solid separation during deposition, consolidation and complete fluid squeeze of the clay horizon. During the last few decades, modeling of melt migration in partially molten plastic rocks led to rigorous quantifications of two-phase flows with a particular emphasis on 2D and 3D induced flow structures. The numerical modeling shows that the melt migrates on distances almost equal to a few times the compaction length L that depends on permeability and viscosity. Consequently, polygonal structures in partially molten plastic rocks are resulted from the melt-rock separation and their sizes are proportional to L. Applying these results to fluid-solid separation in clay-rich horizons, we show that (1) centimetric to kilometric PFS are resulted from the dramatic increase of L during compaction and (2), this process involves agglomerates with 100 μm to 1 mm size.
KEY WORDS: compaction    clay deposit    agglomerates    polygonal fractures    desiccation cracks

0 INTRODUCTION

Polygonal-shaped fractures have been described in fine-grained sediments (e.g., Cartwright and Dewhurst, 1998) and are designated as polygonal fracture systems (hereinafter named PFS). PFS have been described in various lithologies such as evaporites (e.g., De Paola et al., 2007; Alsharhan and Kendall, 2003; Lowenstein and Hardie, 1985), sandstones/conglomerates (e.g., Sweet and Soreghan, 2008; Kocurek and Hunter, 1986), chalks (Tewksbury et al., 2014; Hansen et al., 2004; Hibsch et al., 2003; Cartwright and Dewhurst, 1998) and clay-rich deposits. On continents, polygonal fractures, also referred as desiccation cracks, are common in clay-rich deposits (e.g., Li and Zhang, 2011; Baer et al., 2009) and formed during layer contraction and dewatering. The width of desiccation cracks ranges from a few centimeters from experiments in slurry (e.g., Tang et al., 2011, 2010; Rayhani et al., 2008, 2007) to ~30 cm in natural deposits (e.g., Li and Zhang, 2011; Weinberger, 1999). Exceptionally, large desiccation cracks up to ~50–300 m were reported. They are referred as giant desiccation polygons (e.g., Harris, 2004; Neal et al., 1968).

In sub-marine clay-rich deposits, PFS have been described in the North Sea Basin (Cartwright et al., 2003; Dewhurst et al., 1999; Cartwright and Dewhurst, 1998; Lonergan et al., 1998; Cartwright, 1994), the mid-Norway margin (Davies and Ireland, 2011; Hustoft et al., 2007; Stuevold et al., 2003; Davies et al., 1999), the lower Congo Basin (Andresen and Huuse, 2011; Gay et al., 2004), the Lake Hope region in Australia (Watterson et al., 2000) and the Qiongdongnan Basin in the South China Sea (Sun et al., 2010, 2009). The width of the polygons varies from ~100 to ~1 000 m with some exceptions that can reach ~3 000 m as in the Qiongdongnan (Sun et al., 2010, 2009) and the lower Congo basins (Gay et al., 2004). Since these polygons generally develop in the absence of tectonic forces, the fracturing phenomenon is attributed to stresses due to horizontal density variations, generated during the subsidence of the basin. Most of the hypotheses proposed to account for these density variations have been reviewed and evaluated by Goulty (2008) and Cartwright et al. (2003). A first hypothesis links the development of polygonal fractures to a downslope sliding of deposits (Higgs and McClay, 1993). Since some polygonal fields are developed in flat basins, such process cannot be universally applied (Cartwright et al., 2003). Another model considers that fluid preferentially escapes during compaction from the deposit borders leading to their sealing (Sun et al., 2010, 2009; Watterson et al., 2000; Henriet et al., 1991). That vertical density inversion allows the fluid-rich clay horizon to fold by diapirism. However, the lack of correlation between folds and fractures in most giant polygonal fields does not support this hypothesis (Goulty, 2008; Cartwright et al., 2003). Goulty(2002, 2001) proposed that because of the low coefficient of friction of some types of sediments (Bishop et al., 1971; Skempton, 1964), the normal gravitational loading is sufficient to generate a failure, without invoking additional stresses or overpressures. Pratt (1998) suggested that syn-sedimentary earthquakes can lead to the development of PFS in clay-rich deposits, but there are no observations that substantiate such process (Goulty, 2008). Finally, syneresis has been invoked to generate giant polygons (e.g., Cartwright et al., 2003). It is described as a spontaneous horizontal contraction (shrinkage) of the solid network, without evaporation, that occurs in gels: i.e., when inter-grains attractive forces are greater than repulsive ones by ionic concentrations or salinity change (Brinker and Schere, 1990). However, Goulty (2008) noted that this hypothesis fails to explain the development of PFS across hundreds of meters thick horizons, the shot-time scale of the process as well as the occurrence of the fractures in different materials. More recently, a new hypothesis has been proposed by Davies and Ireland (2011) and Davies et al. (2009). The re-precipitation of microcrystalline opal-CT (cristobalite and tridymite) from amorphous biogenic silica leads to localised differential compaction and causes faults propagation. However, this hypothesis can only be applied to siliceous successions (Davies and Ireland, 2011).

Whatever the source of stress or excess pressure that may drive fluid redistribution in clay horizons is, the development of polygonal fractures are resulted from the interstitial fluid squeeze of the deformable solid matrix. The clay rheology depends on the interstitial fluid volume designated here as the matrix porosity ϕ. In the studies considered above, the clay horizon is assumed to be consolidated: solid grains are in contact, form a compact system (Fig. 1) and cannot glide along other grains (e.g., Mondol et al., 2007; Vasseur et al., 1995; Karig and Hou, 1992; Engelhardt and Gaida, 1963). The porosity ϕ at which the two-phase system is consolidated is 26% for equal spheres in compact hexagonal packing (McGeary, 1961). In practice, consolidation occurs at ϕ=36% for random hexagonal packing or even up to 47% for cubic packing (McGeary, 1961). The porosity ϕcons is hereafter considered as the consolidation threshold of the deposit. As we can see later, the major problem concerns the evolution of the clay horizon when ϕ decreases from ~90% to the consolidation threshold ϕ (Fig. 1). In all cases, the permeability K of the connected interstitial fluid network depends on ϕ on the size d of the clay grains (Fig. 2). It may be deduced from the Kozeny-Carman's Law which writes (Carman, 1961)

 Download: larger image Figure 1. Schematic representing the clay size evolution from microstructures to macroscopic one. (a) Image obtained from an electronic microscope of kaolinite plates and stacks (modified from Murray, 1991); (b) example of natural aggregate obtained by scanning electron microscope (SEM) in secondary electron imaging (SEI) (modified from Paszkowski, 2013); (c) photomicrography of smectites agglomerates (modified from Quaicoe et al., 2013); (d) photograph of desiccation cracks obtained with a water content of ~5% (modified from Tang et al., 2011); (e) Google Earth extract from Red Lake playa, showing giant desiccation polygons; and (f) seismic map showing the position of well-defined polygonal fractures in North Sea Basin (modified from Dewhurst et al., 1999); (g) plot of the clay viscosity versus the porosity ϕ, simulating the evolution of the rheology as a function of water content and viscosity. Schematics display in a deconsolidated domain, the evolution of the clays grains from individual grains to agglomerates under free compaction. Then, under forced compaction, the agglomerates impact the macroscopic scale of clay-rich deposit by the development of desiccation cracks and giant polygons.
 Download: larger image Figure 2. Evolution of the permeability versus porosity from Eq. 1 when the grain size d is 10-6 m (dashed line) and when d is 10-7 m (solid line).
 $K \approx \frac{{{\phi ^3}{d^2}}}{{{{\left({1 - \phi } \right)}^2}172.8}}$ (1)

When the driving term of the two-phase flow is essentially due to the pressure difference between solid and liquid, the fluid-solid separation is designed as a free compaction. When the movement is essentially driven by density variations of the fluid-solid mixture, the process is called diapirism. Alternatively, when the stress is essentially due to volume changes during dehydration, freezing or even syneresis, it is designed as a desiccation-like process. Finally, if the fluid movement is driven by compression or stretching along borders of the clay horizon, the process may be called forced compaction.

In the present study, we propose a new concept to explain the formation of PFS in clays that can be compared to the Russian doll system. This Russian doll system enlighten the variety of size and shape of PFS. Firstly, the modeling of compaction in an unconsolidated and consolidated media (based on two different formalisms) is reviewed. Secondly, we show that during consolidation (i.e., when the porosity ϕ decreases from 90% to ϕcons), the interstitial forces acting on sub-micrometric clay grains lead to the formation of ~10–100 μm agglomerates. Then, we describe (1) the process occurring when the interstitial fluid concentration is close to the consolidation threshold ϕcons and (2) how generation of centimeter size terrestrial desiccation crack occurs at that stage. Finally, we concentrate on the evolution of the clay deposit rheology when the porosity ϕ is several percent lower than ϕcons and on the formation of ~100– 1 000 m size PFS under marine conditions.

1 MODELING OF COMPACTION OF UNCONSOLIDATED AND CONSOLIDATED MEDIA 1.1 Equations of the Compaction

Currently, the modeling of fluid-rock separation in sedimentary horizons has essentially been concerned with the vertical evolution of porosity during subsidence (Yarushina and Podladchikov, 2015; Connolly and Podladchikov, 2014, 2013, 2000; Bernaud et al., 2006; Suetnova and Vasseur, 2000; Schneider et al., 1996). On one hand, the understanding of the 2D/3D distribution of aggregates and fractures is only emerging (Okamoto and Shimizu, 2015; Räss et al., 2014) but the parameters controlling the size and shape of the PFS are not clearly determined. On the other hand, modeling of melt migration in partially molten plastic rocks (e.g., Bercovici et al., 2001; McKenzie, 1984) led to rigorous quantifications of the involved two-phase flow with a particular emphasize on the 2D and 3D induced flow structures (Wiggins and Spiegelman, 1995). For both formalisms developed by Bercovici et al. (2001) and McKenzie (1984), two velocity field must be calculated, the centre of mass velocity $\overrightarrow C$ and the segregation velocity $\overrightarrow S$ (Rabinowicz et al., 2001)

 $\overrightarrow C = \phi {\overrightarrow V _f} + \left({1 - \phi } \right){\overrightarrow V _s}$ (2a)
 $\overrightarrow S = \phi \left({{{\overrightarrow V }_f} - {{\overrightarrow V }_s}} \right)$ (2b)

and

 ${\overrightarrow V _s} = \overrightarrow C - \overrightarrow S$ (2c)

where ${\overrightarrow V _f}$ and ${\overrightarrow V _s}$ designate the fluid and solid velocity, respectively. Then, the mass conservation of the suspension and that of the fluid writes

 $\vec \nabla \cdot \vec C = 0$ (3a)

and

 $\frac{{\partial \phi }}{{\partial t}} + \vec \nabla \cdot \left({\left({1 - \phi } \right)\vec S} \right) + \vec C \cdot \vec \nabla \left(\phi \right) = 0$ (3b)

McKenzie (1984)'s formalism is adapted to describe the fluid-grain separation in an unconsolidated suspension, as he supposes that solid grains soak in the fluid. As the solid grains are extremely small in comparison to the height of the compacting domain, the solid pressure ps is undistinguishable from that of the fluid pf. Consequently, the segregation velocity $\overrightarrow S$ is directly linked to the dynamic pressure pdyn=pfphydros, where phydros=ρfgz, g is the gravity constant, the depth z in the compacting layer positively oriented toward depth and ρf is the density of the fluid. More precisely, according to the Darcy Law, $\overrightarrow S$ depends on the permeability K of the suspension and on the fluid viscosity μ as follows

 $\vec S = - \frac{{K(\phi)}}{\mu }\nabla {p_{dyn}}$ (4)

The suspension is modeled as a compressive Newtonian fluid with a shear viscosity η representing the stress/strain relationship (due to the shear of the suspension) and a bulk viscosity λ (related to the stress required to compact or dilate the suspension). Finally, the constitutive equation for stress σ writes (McKenzie, 1984)

 $\begin{array}{l} \sigma = \left({1 - \phi } \right)\\ \left({\left({\lambda + \frac{2}{3}\eta } \right)\vec \nabla \cdot {{\vec V}_s} \cdot {\rm I} + \eta \left({\nabla {{\vec V}_s} + {{\left({\nabla {{\vec V}_s}} \right)}^T} - \frac{2}{3}\vec \nabla {{\vec V}_s} \cdot {\rm I}} \right)} \right) \end{array}$ (5)

and the equation of motion of the suspension writes (McKenzie, 1984)

 $\nabla \cdot \sigma - \vec \nabla \left({{p_{dyn}}} \right) + \left({1 - \phi } \right)\delta \rho g = 0$ (6)

In particular, $\nabla \cdot \sigma$ verify

 $\begin{array}{l} \nabla \cdot \sigma = - \nabla \left({\left({1 - \phi } \right)\left({\lambda + \frac{1}{3}\eta } \right)\vec \nabla \cdot \vec S} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\nabla \times \left({\left({1 - \phi } \right)\eta \nabla \left({\vec S - \vec C} \right)} \right) \end{array}$ (7)

where I and δρ are the identity matrix and the density difference between the density of the solid ρs and one of the fluid ρf. The first term of the constitutive Eq. 5 represents the stresses generated by compaction and the last one represents the shear. In most compaction studies, shear is neglected along with the movement of the mass centre ${\vec C}$, being in most cases negligible in comparison with the compaction stress. In that case, equation 6 can be approximated as in Wiggins and Spiegelman (1995)

 $\mu \vec S - K\left(\phi \right)\left({\vec \nabla \left({\left({1 - \phi } \right)\left({\lambda + \frac{1}{3}\eta } \right)\vec \nabla \cdot \vec S} \right) - \left({1 - \phi } \right)\delta \rho \vec g} \right) = 0$ (8)

The divergence of above vector field leads to the following scalar one

 $\begin{array}{l} \mu \vec \nabla S - \vec \nabla K\left(\phi \right)\left({\vec \nabla \left({\left({1 - \phi } \right)\left({\lambda + \frac{1}{3}\eta } \right)\vec \nabla \cdot \vec S} \right)} \right)\\ = - \delta \rho g\frac{{\partial \left({\left({1 - \phi } \right)K\left(\phi \right)} \right)}}{{\partial z}} \end{array}$ (9)

Introducing η*= $\left({1 - \phi } \right)\left({\lambda + {1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}\eta } \right)$ and $D = \vec \nabla \cdot \vec S,$ thereafter designed as the effective compaction viscosity and the compaction rate, respectively, we derive the following scalar equation (easily resolved analytically or numerically in 2D and 3D, Wiggins and Spiegelman, 1995)

 $\mu D - \vec \nabla K\left(\phi \right)\left({\vec \nabla \left({\eta *D} \right)} \right) = - \delta \rho g\frac{{\partial \left({\left({1 - \phi } \right)K\left(\phi \right)} \right)}}{{\partial z}}$ (10)

Bercovici et al. (2001) formalism strictly applies to the compaction of a consolidated suspension. In that case, the solid grains composing the suspension are in contact with each other, and the solid pressure ps remains close to the lithostatic one (defined as plithsgz that notably differs from phydros). Indeed, they proposed that the pressure jump ΔP (considered hereafter as an effective pressure) between the solid and the fluid phases ps and pf verifies

 $\Delta P = \frac{{O\left(1 \right)\eta }}{{\phi \left({1 - \phi } \right)}}\frac{{D\phi }}{{dt}}$ (11a)

where O(1) is a factor close to 1 and /dt= $\partial \phi /\partial t + \vec C \cdot \vec \nabla \phi .$ Finally, η represents the shear viscosity of the solid fraction of the suspension which is also assumed to be equal to the bulk one. Then, assuming that (ⅰ) ${\vec C}=0$, (ⅱ) O(1)=1 and (ⅲ) ϕ is small (i.e., (1–ϕ)=1 in Eq. 11a), we derive that (Rabinowicz et al., 2002)

 $\Delta P = - \frac{\eta }{\phi }\frac{{\partial \phi }}{{\partial t}}$ (11b)

This equation was found to be exact in the case of water flow inside a partially molten ice cap (Fowler, 1984). The last equation of state is very important, because it associates the fluid and the solid flow, via the viscosity η, which dramatically depends on the fluid concentration, temperature, stress, rock softening/hardening and eventually effective pressure (Connolly and Podladchikov, 2013). Consequently, because of the variations of η, those expected from the O(1) and (1–ϕ) terms in Eq. (11a) are negligible. In Bercovici et al. (2001)' formalism, both the Darcy and solid flow equations take into account the balance of stresses between fluid and solid and thus writes

 $- \nabla ({p_f} - {p_{hydros}}) - \frac{\mu }{{K(\phi)}}S + \Delta P\vec \nabla \phi = 0$ (12a)

and

 $\begin{array}{l} - \left({1 - \phi } \right)\nabla \left({{p_s} - {p_{lith}}} \right) + \nabla \cdot \sigma - \frac{\mu }{{K\left(\phi \right)}}\\ \phi S + \left({1 - \phi } \right)\Delta P\vec \nabla \phi = 0 \end{array}$ (12b)

When we manipulate the Eqs. 12a and 12b, we get the following set of equations resulting to be appropriate to model the fluid-solid separation of a consolidated suspension (Rabinowicz et al., 2002)

 $\frac{{\partial \phi }}{{\partial t}} = - \frac{\eta }{\phi }\Delta P$ (13a)
 $- \frac{\mu }{{K(\phi)}}\vec S = - \nabla \left({{p_{dyn}}} \right) + \Delta P\vec \nabla \phi = 0$ (13b)
 $\frac{{\phi \Delta P}}{\eta } - \vec \nabla \frac{{K(\phi)}}{\mu }\vec \nabla \left({\Delta P} \right) = - \frac{{\delta \rho g}}{\mu }\frac{{\partial \left({K\left(\phi \right)} \right)}}{{\partial z}}$ (13c)
 $\vec \nabla \left({K\left(\phi \right)\vec \nabla {p_{dyn}}} \right) = - \frac{{\mu \phi \Delta P}}{\eta } + \vec \nabla K\left(\phi \right)\Delta P\vec \nabla \left(\phi \right)$ (13d)

Note that the 3D compaction flow depends only on the two non-linearly coupled scalar variables: the fluid concentration ϕ and effective pressure ΔP. If we write ${\eta ^*} = \eta /\phi$ Eq. 13c writes

 $\mu D - \vec \nabla K\left(\phi \right)\vec \nabla {\eta ^*}D = \delta \rho g\frac{{\partial \left({K\left(\phi \right)} \right)}}{{\partial z}}$ (14)

To conclude, the review of Bercovici et al. (2001)' and McKenzie (1984)'s formalisms can be summarised by Eqs. 11 and 14, respectively. When we compare both equations assuming that η*=(1–ϕ)(λ+1/3η)=η/ϕ, we see that both equations are identical. This means that either the suspension is consolidated or not, there is no difference in using the formalism of McKenzie (1984) or Bercovici et al (2001) to describe compaction. Actually, the parameters governing this process are the permeability K(ϕ), the effective viscosity laws η* as well as the initial distribution of porosity ϕ.

1.2 Solution of the Compaction Model

In a compaction experiment, the major parameter to consider is the compaction length L defined as (Bercovici et al., 2001)

 $L = \sqrt {\frac{{{K_0}{\eta _0}}}{{{\phi _0}\mu }}}$ (15)

where K0, ϕ0, and η0 designate the averaged value of the initial permeability, porosity, and viscosity of the solid matrix, respectively. If we assume that (ⅰ) D=sin(kz/L) where k is the wavenumber of the instabilities and (ⅱ) all the parameters that appears in the left side of Eq. 14 are constants, it results that this left side is equal to $\mu \left({1 + {k^2}} \right)$ sin(kz/L) That relation shows that when the instabilities have a large wavenumber k, they cannot grow with time. It implies that during compaction, fluid concentrates in structures whose sizes are proportional to the compaction length L (Ribe, 1985). The 3D numerical experiments run by Wiggins and Spiegelman (1995) demonstrate that when (ⅰ) the effective viscosity η* is constant, (ⅱ) the height of the compactive layer is extremely large in comparison to L and (ⅲ) the initial porosity is essentially constant, but just perturbed by a small-amplitude random field, spherical structures are formed by closely packed solid interbedded between tubes rich in fluid. The 2D experiments by Rabinowicz et al. (2001) shows that when (ⅰ) porosity is essentially constant but just perturbed by a small-amplitude large wavelength field and that (ⅱ) the effective viscosity η* dramatically drops for above a fluid concentration ranging from 5% to 20%, the suspension initially consolidated form spherical pockets of deconsolidated material separated by consolidated walls. Rabinowicz et al. (2002)'s 1D numerical experiments demonstrate that when the permeability K(ϕ) at the top of the compacting domain dramatically drops and that η*=η/ϕ, where η is a constant, the compaction waves are proportional to a few times the compaction length L. Finally, Grégoire et al. (2006) and Rabinowicz and Ceuleneer (2005) studied 2D compaction in a consolidated layer using the same conditions that those in Rabinowicz et al. (2002). The experiments show that an initial planar wave of thickness ~2L develops just below the top interface, which eventually splits into several spheres rich in fluid with a radius ~2L (Fig. 3). Their numerical models also show that the effective pressure ΔP at the top of the spherical spheres reaches at minimum a value of where δρ represents the density difference between the liquid and the solid. Then, a second planar wave develops at a depth ~2L below the first one. This wave progressively splits into ~L radius spheres. Finally, other planar compaction waves successively develop at increasing depth but their sizes and growth rates become gradually weaker. When the variations of permeability K(ϕ) and viscosity η are small, the growth rate of the compaction waves is weak. On the contrary, when the instabilities are close to an impermeable layer (Rabinowicz et al., 2001) and/or when the viscosity variations are strong, for example near the consolidation threshold ϕcons (Rabinowicz et al., 2002), the growth of the compaction waves becomes dramatically strong.

 Download: larger image Figure 3. Snapshots of a dimensionless water concentration and effective pressure fields during compaction experiment in a computing box of 450 km (modified from Grégoire et al., 2006). An initial planar wave front develops with a thickness of ~2L. This wave progressively splits into ~L radii spheres that contain a water concentration of ~30%–35%. As the compaction process proceeds, others waves successively develop. The effective pressure at the top of the spheres is sufficient to fracture the overlying material.
 $\Delta P = - 2\delta \rho L$ (16)

The formalism of Bercovici et al. (2001) also applies to the fluid migration in a consolidated solid network deformed by pure shear (Rabinowicz et al., 2010; Rabinowicz and Toplis, 2009; Rabinowicz and Vigneresse, 2004). In that case, the fluid concentrates along lines of maximum compressive stress ${{\vec \sigma }_1}$. In the zone where the fluid is squeezed, the effective pressure ΔP becomes positive. Finally, the solid framework breaks along the two conjugated directions of maximum shear and the fluid, which initially concentrated in ${{\vec \sigma }_1}$ lines, now concentrates inside shear fractures (Rabinowicz et al., 2010). A basic fact deduced from this modeling is that contrary to the case of the compaction waves, the wavelength of the most instable waves triggered during shear deformation is zero: this means that the fluid inside two successive fluid films is separated by parallel solid ribbons (consisting of a row of individual solid grains stuck together) (Gardien et al., 2016). Consequently, shear deformation modifies the microscopic topology of the fluid-solid interface, but not the macroscopic one.

2 FORMATION OF AGGLOMERATES BY COMPACTION IN A CLAY MUD

Structural bonds in clays are determined by several forces: chemical (valence bond), ion-electrostatic, molecular, electrostatic (Coulomb) and magnetic. Based on theoretical calculations of Osipov (1975), an assessment is made on the role of each type of force in forming the structural bonds at the contact point between two grains. Proceeding from the character and the energy of contact interactions, three types of contacts between clay grains are defined (Fig. 4): coagulation, atomic (or point) and phase (or developed) contacts. The coagulation contact is resulted from the cohesion of grains due to molecular, electrostatic and magnetic forces acting at large distance. The characteristic feature of such contact is the presence of a thermodynamically stable hydrate film with a thickness from several Å to hundreds of Å (Osipov, 1975). Coagulation contact generates small forces from 10-12 to 10-8 N. When the hydrate film is ruptured as during consolidation and/or drying, atomic contact occurs. This contact is due to ion-electrostatic or chemical forces. For ion-electrostatics, the contact occurs for cations with sizes between 4 Å and 10 Å and substantially exceeds the molecular forces (Osipov and Sokolov, 1978). For contact formation due to chemical forces, the grain distance ranges between 0.5 Å and 3.5 Å. Grain contacts are associated with forces stronger than coagulation ones, ranging from 10-9 to 10-7 N than coagulation contacts. Phase contacts are due to even stronger chemical forces. They occur at higher pressures and temperatures and result to plastic deformation. Their associated forces range from 10-7 to 10-5 N.

 Download: larger image Figure 4. Schematic representation of the contact interactions between grains. (a) coagulation with a thermodynamically stable hydrate film with 1) and 3) short contacts, 2) long contact, (b) weak contact forces due to atomic contact 1') weak chemical forces, 2') weak mechanical forces and (c) stronger contact forces due to stronger chemical forces 1") and 3") short contacts, 2") long contacts.

In a clay suspension, these structural bonds are responsible for the flocculation of grains into larger aggregates. In presence of cations, as in seawater, the van der Waals forces, the crystallization of salts and/or the adhesion processes lead to the formation of aggregates floating in the liquid (Pansu and Gautheyrou, 2006; Holdich, 2002; Lifshitz, 1956; Hamaker, 1937). In seawater, the resulting size of the aggregates is micrometric (Pansu and Gautheyrou, 2006; van Olphen, 1977), while in fresh water, the aggregation of grains is less efficient and their size is about a fraction of a micrometer (Neumann et al., 2002). However, a recent laboratory analysis reveals that muddy episodic fluvial suspensions released during flood events contain about 60% in volume of clay aggregates with a size between 50 and 100 μm (Haberlah and McTainsh, 2011). These types of aggregates are seen as the result of an "aggregation of aggregates" (Fig. 4), sometimes designated as agglomerates (Lee-Desautels, 2005). Sergeyev et al. (1980) have defined five main structure of clay aggregates and agglomerates of various ages, origin and degree of lithification.

(1) The honeycomb un-oriented microstructure (Casagrande, 1932) is known as a syngenetic one; i.e., the microstructure is formed during the sedimentation both in fresh or salt water. It is characterised by isometric illite/montmorillonite cells of 2 to 12 μm agglomerates in coarser grains. In this case, aggregates can "coagulate" (Osipov and Sokolov, 1978) and thus the aggregates boundaries disappear. The agglomerated clay layer has a high porosity (60%–90%) and a pseudo-liquid consistency. This kind of clay deposit is highly compressible even under small loads.

(2) The skeletal un-oriented microstructure is thixotropic and does not create a continuous matrix. It is usually found with a porosity of 30%–50%.

(3) The matrix un-oriented microstructure is characterised by continuous un-oriented clay mass known as matrix. Pores sizes, depending on the compaction level, range from 1 to 8 μm. This structure is made up of illite or mixed clay and the porosity does not exceed 60%. Its rheology is viscous-elastic. This structure can be found in various contexts as marine, lacustrine or alluvial deposits.

(4) The turbulent oriented microstructure is described as clay aggregate oriented along the bedding plane, disturbed by local heterogeneity like sand grains. This microstructure is formed during the clay compaction of other types of microstructures (matrix, honeycomb), and is mainly found in marine environment. The porosity can decrease to 10%–30% and the sizes of pores and agglomerates are ~20 and 40–50 μm, respectively.

(5) The laminar structure is characteristic of clay deposits of diverse mineral composition, but with a high content in clay minerals (> 40%) and can be syngenetic (in fresh water) or post-diagenetic. In this case, the compaction is pronounced and the initial microstructure is replaced by laminar one. This structure is observed in varve (glacial lacustrine clays), as well as argillites (compacted clay with very low permeability ~10-20 m2). Their unconfined compression strength varies from less than 11 Pa (varve) to MPa (argillites).

In continental areas, straw incorporation may raise soil organic matter levels (Hooker et al., 1982; Rasmussen et al., 1980), increasing the formation and stability of the agglomerates, whose widths rise up to more than 1 mm (Christidis et al., 2005; Lynch and Elliott, 1983; Kaila, 1952; Martin, 1942). With temperature increase (hydrothermal clay) or weathering effect on clay-rich soil, it is possible to generate larger agglomerates that contain, in most cases, relict microstructures inherited from the sedimentary state. These relict microstructures have been classified by Sergeyev et al. (1980): (1) coarse aggregates leading to agglomerates made up of axially-oriented kaolinite. (2) Pseudo-globular or agglomerate coarser spherical aggregates. Their size can reach 20 μm. (3) The sponge microstructure found in hydrothermal systems with montmorillonite clays. It consists of larger agglomerates (up to 80 μm) in diameter, with predominantly edge-to-face or face-to-face contacts (Fig. 4).

The size of the agglomerates dramatically depends on the mineralogy: for kaolinite (K), it is less than several 10 μm, for illite (Ill), it is less than 50 μm and for smectite (Sm), it reaches 100 μm (Sergeyev et al., 1980). In some cases, for mixed agglomerates (e.g., organic matter/Sm or sand/Sm), it can reach a millimetric scale (Sergeyev et al., 1980). The agglomeration mechanisms have a higher probability to develop when approaching the consolidation threshold (Fig. 1). Indeed, the extremely small distance on which the attracting forces act (< ~1 μm) requires that grains in a suspension must become extremely close to paste together. This statement is independent of the causes of grains coalescence in a suspension. Actually, combination of surface tension and Stokes velocity immobilise the fluid film around the grains, with a thickness proportional to the grain size. Therefore, once sub-micrometric grains become so close that the fluid films are in contact, the grains are immediately pasted by chemical forces. This explains why the aggregation of grains are resulted from collisions and pasting of grains dragged by a turbulent fluid during flocculation. However, the coalescence of grains in agglomerates is more complex. Indeed, it requires the squeeze of the agglomerate fluid films at the contact point with the grain. To squeeze the fluid film, the stress transmitted through the agglomerate to the grain is equivalent to the agglomerate relative weight $\tilde w$ defined as

 $\tilde w = \delta \rho gd$ (17)

When the agglomerate has a size d of 100 μm or 1 mm, $\tilde w$ is ~1 to 10 Pa, respectively. Consequently, above a size d of several μm, the agglomeration is determined by the collision of grains in a viscous fluid, depending on their differences of Stokes velocity. It suggests that the agglomeration of clay grains are resulted from the fluid-solid separation in an unconsolidated suspension, at the conditions adopted by McKenzie (1984). Consequently, we propose that in clay deposits, the agglomeration process is triggered by compaction waves. It is expected that at time t the size of the agglomerate d(t) verifies

 $d\left(t \right) = \sqrt {\frac{{K\left({\phi \left(t \right)} \right)\eta \left(t \right)}}{{\phi \left(t \right)\mu }}}$ (18)

where η(t), K(t), ϕ(t) represents the transient values of the viscosity, permeability and porosity of the suspension, respectively. Initially at t=0, the porosity ϕ(t) ranges between 90% and 50% and η(t) is described by the Roscoe-Einstein rule (Roscoe, 1952): η=(1–ϕ)-2.5, which leads to values of one order of magnitude greater than that of water (μ=10-3 Pa·s). Alternatively, when the porosity of the suspension ϕ drops to a value close to the consolidated threshold ϕcons, the effective viscosity of the mud η(t) may reach a value up to 106 Pa·s (Kopf et al., 2005). Besides, for clay suspensions essentially composed of aggregates (Fig. 1) with a size d of ~10-7 and 10-6 m, the application of the Kozeny-Carman Law (Eq. 1) reasonably explains plug measurements that reveal a permeability K(t) of 10-17 and 10-15 m2 (Fig. 2), respectively (Schwinka and Moertel, 1999). Consequently, the compaction length L reaches ~100 μm and 1 mm, respectively (Eqs. 15 and 18). This corresponds to the characteristic sizes of agglomerates discovered in smectite and mixed agglomerates (Sergeyev et al., 1980). Finally, when the fluid concentration reaches ϕcons, agglomerates form a connected solid framework tied at their contact by chemical forces. Such process increases the strength of this framework. When its yield strength exceeds the relative weight ${\tilde w}$ of the agglomerates (~1 or 10 Pa), it becomes rigid. At that time, the ~100 μm to 1 mm size agglomerate network has a gel-like structure (van Olphen, 1977), implying that the free compaction process stops.

3 FORCED COMPACTION AND GENERATION OF TERRESTRIAL DESICCATION CRACKS

The width of desiccation cracks ranges from a few centimeters from experiments in a slurry (e.g., Tang et al. 2011, 2010; Rayhani et al., 2008, 2007) to ~30 cm in natural deposits (e.g., Li and Zhang, 2011; Weinberger, 1999). According to the understanding of the compaction process previously described, we propose that the development of desiccation cracks can be linked to the forced compaction of a consolidated clay layer. As an illustration of this hypothesis, we consider the desiccation experiments made by Tang et al. (2011). The used clay slurry is essentially composed of an illite-smectite unconsolidated mixture with an initial porosity of 70%. It results from a 3 day compaction of a 8 mm thick clay layer (Hi) mixed with 77% of interstitial water (ϕi) and maintained isolated to prevent evaporation. The reduction of porosity from 77% to 70% during 3 days is consistent with a separation velocity ${\vec S}$ between fluid and solid of ~2×10-9 m·s-1 (Eq. 4). Besides, at the initiation of the compaction process the Darcy Equation (Eq. 4) can be approximated by

 $\vec S = - \frac{{K\left(\phi \right)}}{\mu }\nabla {p_{dyn}} \approx - \frac{{K\left(\phi \right)}}{\mu }\delta \rho g$ (19)

where we assume that the dynamical pressure $\nabla {p_{dyn}}$~δρg is equal to the maximum of the difference between the fluid and the hydrostatic pressure. From Eq. 19, it can be deduced that the permeability of the slurry K is ~10-16 m2, which corresponds to the permeability of a suspension with a grain size d of ~10-7 m and a porosity of 0.7 (Eq. 1): i.e., the typical size of clay aggregates which had grown in fresh water (Haberlah and McTainsh, 2011).

When evaporation of the water in the slurry starts, the porosity decreases progressively at its top, while in its bulk, the reduction is more sluggish. When the porosity of the top clay horizon is close to ϕcons, it becomes plastic with a yield strength related to the strength due to capillary suction. Using demonstration in section 1, we see that the compaction process triggered by the top dehydration leads to the formation of ~100 μm size agglomerates composed of 10-7 m clay aggregates, eventually increasing the slurry permeability by six orders of magnitude, from 2×10-16 to 2×10-10 m2 (Eq. 1). The desiccation cracks are triggered when the slurry thickness Hf is 3 mm, suggesting a reduced final porosity ϕf of ((Hi–Hf)ϕi/Hf ~39%. Because of the buoyancy of the bottom water-rich horizon, the overlying horizon is submitted to a vertical compression and thus to horizontal tension. Assuming that the bulk of the bottom horizon still contains a volume of 70% of water and the overlying one a volume close to ϕcons (here we assume that ϕcons=35%), the porosity difference Δϕ between both horizons induces a density contrast Δρ of Δϕ(ρs–ρf) ~380 kg·m3. Integrated along the 3 mm thick Hf compacted layer, we deduce that a maximal vertical compression ΔρgHf of 8.4 Pa is induced by the underlying fluid-rich and buoyant horizon. That stress greatly exceeds the 1 Pa fluctuation of pressure ${\tilde w}$ due to the relative weight of the ~100 μm size agglomerates. Thus, the horizontal extension induced by the lower buoyant horizon overcomes the capillary suction of the overlying one, and permits the generation of the desiccation cracks in Tang et al. (2011) experiments. Finally, the water contained in the bottom horizon escapes through the desiccation cracks. Such process generates a compaction wave in the horizontal direction, which triggers cylindrical waves with a radius proportional to the compaction length L (Section 1). Assuming that the agglomerates which developed in the bottom horizon have a size of ~50 μm and that the slurry has an effective viscosity of 104 Pa·s, we find that L ~1 cm (Eqs. 1 and 15). This compaction length L is identical to the centimeter size of the desiccation cracks obtained by Tang et al. (2011). This result confirms that on terrestrial context, the forced compaction generated by a lower buoyant horizon generates a compaction length L that controls the width of the desiccation cracks. We propose to designate this forced compaction has a diapiric compaction process.

4 EVOLUTION OF THE RHEOLOGY OF A CLAY DEPOSIT AFTER ITS CONSOLIDATION

We have seen in sections 1 and 3 that the clay suspension develops a yield strength that exceeds several Pa when the porosity approaches the consolidation threshold ϕcons. A very complicated problem is to decipher how the yield strength and the effective viscosity evolve when the porosity drops substantially below ϕcons. Indeed, for clay samples from mud volcanoes, Kopf and Behrmann (2000) measured a strength of 35 MPa for samples having a fluid concentration of 24% and submitted to a strain rate ${\dot \varepsilon }$ of 3×10-3 s-1. Experiments using cemented oceanic clay display strengths of ~0.1 MPa that increases with the confining pressure (Moses et al., 2003). Such experiments are extremely difficult to run and to interpret because they strongly depend on (1) the porosity of the clay, (2) on its nature (Rutter and Wanten, 2000) and (3) on the shape and size of the grains and solid framework, which evolve during deformation. Finally, beyond a deformation greater than ~0.3, interstitial water concentrates into microscopic veins where the grains, aggregates and agglomerates are free to slide (Rabinowicz et al., 2010). Such process dramatically softens the clay rheology, explaining why the strain-stress law of clay-rich landslides dramatically depends on the time scale of the observations and rainfall (Rutter and Green, 2011; Rutter et al., 2003). These are also the main reasons why precise determinations of the yield strength of clays and the relation between the deviatoric stress σ versus the infinitesimal strain ${\dot \varepsilon }$ for consolidated clays are rare. To avoid these sliding effects, Li et al. (2004) have investigated the rheology of saturated clay-rich soils below the freezing temperature. In the experiments, 90% of soil volume consists of agglomerates with sizes ranging from 100 μm to 1 mm. The samples were initially moistened and then variably compacted until a "dry density" ρd of 1 380 kg·m-3 is obtained. Then, a volume of water of 34% was added to saturate the sample and thus represents its porosity. Finally, the sample was cooled and then deformed at temperatures T ranging between 258 and 271 K.

One of the very interesting aspects we can highlight from Li et al. (2004) experiments is that due to these low temperatures, no pressure-dissolution process contributes to the deformation of the clay/silt-rich soil, as described by Rutter and Wanten (2000). Moreover, it is worth to be noted that during the experiments, the interstitial ice does not exceed a volume of 30%. Thus it cannot significantly modify the bulk rheology of the samples, with respect to the one containing the same volume of interstitial fluid (Ślizowski and Lankof, 2003). As clays are hydrophiles, the freezing temperature of water in the porosity channels decreases as a function of their sizes (Alba-Simionesco et al., 2006). This means that the experiments run with lowest temperatures represent cases where there is less interstitial fluid; i.e., capillary and gravitational waters. Then, the deformation essentially results from the translation and rotation of individual micrometic grains, aggregates and/or agglomerates (Buessem and Nagy, 1954). Such movements are possible thanks to the presence of fluid films partially filling the micro-porosity between grains and agglomerates. Accordingly, the evolution of the rheological law with temperature essentially represents an evolution of the clay rheology, as the volume of interstitial fluids decreases 1) by freezing large pores (between agglomerates) and then 2) by micro-pores (between grains and aggregates). Thus, it mimics what may happen when the fluid in a consolidated clay deposit is progressively squeezed during forced compaction.

Li et al. (2004) experiments show that the samples present yield strength even when the applied stress is relatively low (~0.1 MPa). During the experiments, the deviatoric stress σ and the strain velocity ${\dot \varepsilon }$ are measured close to overcome the yield strength of the samples. The experiments are fitted to a power law (as in So and Yuen, 2014), with an interstitial fluid concentration ϕ of 34%, relating the effective viscosity ηcons (Pa·s) of a clay to the temperature T (K) and the stress σ (MPa) (Li et al., 2004)

 ${\eta _{cons}} = 1.51 \times {10^6}\frac{{{{\left({273.15 - T} \right)}^{5.636}}}}{{{\sigma ^{4.636}}}}$ (20)

This formula is valid for T < 273.15 K. When T=271 K and σ=1, 0.3 and 0.1 MPa, the effective viscosity ηcons is 108, 2.5×1010 and 4.5×1012 Pa·s, respectively. At temperatures < 258 K, the entire volume of fluid permitting the sliding of grains or aggregates is likely frozen. The clay rheology becomes similar to that of silicate rocks (Mangold et al., 2002), which requires a deviatoric stress σ of several hundred MPa to fail. When T approaches 273.15 K, ηcons tends to zero. According to the discussion presented in Section 1 and to Kopf (2002), it can be deduced that at T=273.15 K: (ⅰ) the fluid concentration is very close to the consolidation threshold ϕcons, (ⅱ) samples require a strength of several Pa to creep and (ⅲ) ηcons reaches a value as low as 106 Pa·s. Besides, it is likely that at T=271 K, Eq. 20 represents the rheological law of a consolidated clay with a porosity ϕ several percent lower than ϕcons. Considering a stress σ ranging between 0.1 and 1 MPa, the effective viscosity of the clay may range between 108 and 1012 Pa·s; i.e., 2 to 6 orders of magnitude greater than the maximum viscosity close to the consolidated threshold (Kopf et al., 2005).

5 PFS COMPACTION AND GENERATION OF MARINE GIANT PFS 5.1 PFS Development by the Deposition of an Impermeable Horizon

Polygonal fractures have been discovered in the Qiongdongnan Basin, in the South China Sea using high-resolution 3D seismic data (Sun et al., 2010, 2009). The size of the polygons ranges from 50 to 3 000 m. The major part of the study area is under pelagic conditions since the Early Miocene. Since then, the sedimentary sequences consist of fine-grained sediments deposited in the Meishan (15.5 to 10.5 Ma), the Huangliu (10.5 to 5.5 Ma), the Yinggehai (5.5 to 1.9 Ma) and the Ledong formations (1.9 Ma to present). The estimated average sedimentation rate of the Meishan and the Huangliu formations is very low (0.1 and 0.12 mm·yr-1, respectively), while the deposition rate increases to 0.5 and 0.7 mm·yr-1 for the Yinggehai and the Ledong formations, respectively (Sun et al., 2009). Although analysis of the proportions of the clay minerals is lacking, the first two horizons are qualified of "coarse" and the last two of fine (Sun et al., 2009). The reduction of the grain size of the Yinggehai and Ledong formations may simply result in an illite-rich deposit (Li et al., 2008; Wei et al., 2001), leading to finer agglomerates (d ~10 μm) (Sergeyev et al., 1980). For the "coarse" horizons, a high content of smectite, sand and organic matter (Li et al., 2008; Wei et al., 2001) is observed, leading to millimetric agglomerates (Sergeyev et al., 1980). The total thickness of the Meishan and the Yinggehai formations, in which PFS have been identified, is ~252 m and in these areas the mean extension induced by the faults represents ~13%. The hypothesis proposed by Sun et al. (2009) to explain the development of the polygonal fractures can be resumed as follow. After the deposition of the Meishan Formation, the fluid is expelled during burial and leads to its consolidation. When the Huangliu Formation settles, the Meishan Formation becomes isolated and the progressive increase of the effective pressure triggered the failure of the sediment and the development of polygonal fractures (Luo and Vasseur, 2002; Cartwright and Lonergan, 1996). Finally, the deposition of the Yinggehai Formation impeded the expulsion of the fluid contained in the Huangliu one and, as for the Meishan Formation, lead to the formation of a new set of polygonal fractures.

This reasoning is essentially supported by our analysis of the compaction processes. We hypothesise that the 4 orders of magnitude permeability drop, resulting from the 100 μm to 1 mm drop of the agglomerates size, produces the top drastic obstruction (see Section 1) necessary to generate the compaction waves in the Meishan and the Huangliu consolidated formations (Rabinowicz et al., 2002). The ~13% horizontal extension indicates that the compaction waves permit the reduction of the porosity at the consolidation threshold ϕcons to ~22%. Finally, the dynamic pressure at the top of the Meishan and the Huangliu formations represents at most pdyn=δρgH≈1 MPa. The compaction length L of a clay with a fluid concentration several percent lower than ϕcons and submitted to a stress ranging between 0.1 to 1 MPa (Section 4) ranges between 3 to 3 000 m (Eqs. 15 and 20). It is due to (1) the 108 and 1012 Pa·s effective viscosity range, respectively and (2) the 10-10 and 10-8 m2 (Eq. 1) permeability range of clay agglomerates with a size d ranging between 100 μm and 1 mm, respectively. Due to the concentration of water inside the compaction spheres, the clay may be deconsolidated. Thus, the clay suspension may have an effective viscosity similar to that of water. Conversely, as the walls of the spheres are consolidated, they must break under the 0.1 to 1 MPa stresses due to the buoyancy of their fluid-rich cores. Thus, sets of faults are generated inside the walls of the spheres which have a polygonal planform. We suggest that the PFS described in Sun et al.(2010, 2009) are resulted from this compaction process.

5.2 PFS Development by Vertically Superposed Spherical Compaction Spheres

In the Lower Cretaceous sequence of Lake Hope region of South Australia, ~1 km wide PFS are observed in three horizons: the oldest Cadna-Owie Formation (the deepest), then the Coorikiana Formation and the youngest Mackunda Formation (Watterson et al., 2000). Sonic velocity logging in a well reveals that the interfaces of these formations (Fig. 5a) are located between hundred meters thick low- and high-velocity layers (i.e., a low-velocity layer is associated to a high-porosity layer and a high-velocity one to a low-porosity layer). The Top Cadna-Owie interface presents the most drastic velocity drop, referred in Fig. 5a as the low-density layer. This high-porosity layer is maintained overpressured by a top seal. A 3D map of the Cadna-Owie/ Coorikiana interface (Fig. 5b) shows that it is clearly polyhedral, where polygons have a bowl-shape and borders, an ellipsoid one. Watterson et al. (2000) suggested that such structures are resulted from a radial extension of a brittle material by a process comparable with "boudinage", in a system characterised by an oblate strain ellipsoid. The vertical cross-section of the basin (Fig. 5c) highlights the existence of conjugated faults that seems to stop propagating close to the Top Cadna-Owie interface. These conjugated faults may result from the stress induced by the Rayleigh-Taylor instabilities generated by the buoyant low-density layer (Watterson et al., 2000).

 Download: larger image Figure 5. (a) Sonic velocity logging for a well drilled in the Eromanga Basin (Australia). High velocity zone are interpreted as a low porosity zone, while a low velocity layer is associated to a high porosity zone. The sonic velocity log displays various velocities, but is characterised by the presence of a very low velocity zone, referred as the low density layer above the Cadna-Owie interval. (b) Isopach map of the Cadna-Owie interval, located below the low density layer. The colour bar represents the thickness variation leading to the formation of polyhedral structures, highlighted by the dashed line. (c) Seismic cross-section of Eromanga Basin where faults are interpreted in yellow and the different interfaces Cadna-Owie, Coorikiana and Mackunda in red, green and blue respectively.

According to these observations, we suggest that the Top Cadna-Owie structures are resulted from the development of a horizontal compaction wave, which thereafter splits into ellipsoid shape fluid-rich zones (Grégoire et al., 2006; Rabinowicz and Ceuleneur, 2005). This hypothesis explains why the conjugated faults stop there, as they cannot propagate in the overpressured fluid-rich and very ductile underlying clay horizon. Finally, the conservation of the overpressured ellipsoid shape fluid-rich zones may be explained by the absence of connections of this structure with the overlying system of conjugated faults. Concerning the Top Mackunda and Top Coorikiana interfaces, the sonic velocity sounding shows a reduction of the velocity contrast, in comparison with the one of the Top Cadna-Owie interface. We propose that this reduction of the velocity contrast is associated to a water migration through the conjugated faults. Moreover, the faults overlying the Mackunda/Coorikiana interface indicate that these horizons were overpressured when the horizontal compaction waves in these horizons developed. In Section 1, we mentioned that a drastic porosity reduction at the top of a very thick consolidated layer triggers successive horizontal compaction waves that eventually split into fluid-rich spherical structures (Fig. 3). Consequently, the giant PFS observed at these three interfaces result from the superposition of ellipsoidal compaction waves.

5.3 PFS Development in the North Sea Basin

In the North Sea Basin, a precise 3D seismic map of the fault system in a compacting horizon has been described (Dewhurst et al., 1999). From Late Paleocene to Middle Miocene, smectite, sand and organic matter-rich mudrocks developed from sub-aqueous alteration of ash falls. The basin is divided from bottom (~2 km deep) to top (~1 km deep) by four distinct seismic horizons: layer A (~150 m thick), layer B-C (~500 m) and layer D (~150 km thick). Fault networks were inferred from the seismic reflectors discontinuities (Fig. 6a). A network of conjugated steep fractures is uniformly distributed inside the two A and D layers providing an extension rate of ~13%.

 Download: larger image Figure 6. (a) Seismic cross-section survey from the Central North Sea Basin. Four distinct layers (A to D) are visible. They highlight three layers presenting numerous faults compared to the undisturbed strata above and below. White circles represent zones devoid of fractures, interpreted later as cores of the compaction spheres. Black arrows illustrate the ${{\vec \sigma }_1}$ direction (modified from Dewhurst et al., 1999). (b) Map showing the position of well-defined faults in layer B, which is located at ~1.5 km depth (modified from Dewhurst et al., 1999). Note that the traces of the faults delimit polygonal networks with characteristic lengths of ~500 m and ~1 km, respectively. Two to six fault planes (in white) are emitted from a hub (white dot) corresponding to a vertical axis. This observation can be explained by the development of Rayleigh-Taylor instabilities (Talbot et al., 1991).

Several steep hundred meters long fractures roughly orthogonal to the reflector developed in the B-C layer, inducing an extension rate of only 6%–7%. This layer is also characterised by ~1 km wide regions with tight fractures spacing separated by ~1 km wide regions more or less devoid of fractures. Moreover, the fractures displacement in this layer is notably smaller than the one estimated in layers A and D. Fractures intersections with the mid-horizontal plane of the basin show polygons-like features with characteristic sizes of ~500 m and ~1 km, respectively (Fig. 6b). The basin deposits, investigated from a drill core, reveals a content of 85% and 60% of smectites in layers A and D, respectively (Dewhurst et al., 1999). In the B-C layer, a grain mixture of 50% silt and 50% smectite is recovered. Actually, in the B-C horizons (Fig. 6a), the rather circular regions (white circles) devoid of fractures may represent paleo-cores of compaction spheres with a diameter h of ~500 m. As the compaction wave moves up, the fluid content at the top of the wave increases, while the one present in the bottom of the wave decreases. It results that inside the core of the compaction sphere, fluid concentration exceeds by few percent the consolidated threshold ϕcons, while in the exteriors of the spheres it becomes few percent lower than ϕcons. During compaction, the solid grains move downwards, while the water moves upwards. Then, because of the ~1 400 kg·m-3 density contrast between silt and clay grains, the Stokes velocity of silt grains, aggregates and/or agglomerates are expected to be faster than the clay-rich ones. This fact may explain why the concentration of smectites in the top of horizon D, and in the horizon A is greater than in the B-C horizon (Dewhurst et al., 1999). Considering the 6%–7% to 13% rates of extension induced by the development of the fault arrays, we deduce a porosity step Δϕ of ~10% between consolidated walls and the deconsolidated core of the spheres and the density contrast Δρ when the compaction spheres develop is ~55 kg·m-3. The resulting difference of weight integrated along the diameter h=500 m of the compaction sphere core represents a vertical stress σcomp of ~0.27 MPa. It implies that, at the top of the spheres, the consolidated walls are submitted to strong stresses with maximal compressive stresses ${{\vec \sigma }_1}$ oriented vertically. Initially, interstitial fluids orient parallel to the ${{\vec \sigma }_1}$, but rapidly the walls crack and generate conjugated fractures (Rabinowicz et al., 2010, black arrows in Fig. 6a). On the contrary, at the spheres bottom, the maximal compressive stresses ${{\vec \sigma }_1}$ is horizontal and weaker. This hypothesis can explain the lack of conjugated fractures in horizon A. In zones separating two spheres, the fluid concentration remains relatively high and the deformation of the clay relatively weak. It implies that the generated fractures rather follow the ${{\vec \sigma }_1}$; i.e. essentially parallels to the plastic flow lines of the Rayleigh-Taylor instability resulting from the inhomogeneous distribution of fluid during compaction. Figure 6b displays the fractures polygonal geometry of the horizon B. Two to six fractures intersect at one hub (white dots). The development of these hubs is specific either to thermal convection or Rayleigh-Taylor instability (Talbot et al., 1991). The only plausible explanation for the development of the hubs observed in Fig. 6b is to assume that the compaction spheres grow while the Rayleigh-Taylor instabilities develop (Fig. 7).

 Download: larger image Figure 7. Schematic representing the development of Rayleigh-Taylor instabilities and compaction spheres. Consolidated clays sunk beneath the polygonal fractures (solid lines). The ascending plans are highlighted by the dashed lines. Between both convective currents, compaction spheres develop (grey circles) (modified from Talbot et al., 1991).
6 CONCLUSION

Different studies demonstrated that polygonal fracture systems (PFS) are resulted from the fluid-solid separation during compaction of consolidated clay deposits. However, the processes and parameters controlling the size of the PFS in terrestrial or marine contexts have never been addressed. In this study, we propose that the PFS size is resulted from the interaction between the microphysics of the agglomeration process with the non-linear two-phase flow at the macroscopic scale.

The development of agglomerates is of high importance in our present study. Their formation can be easily explained by the squeeze of the fluid film surrounding the clay grains during their pasting. In the early stages of the agglomerate formation, the grains weight is sufficient to allow the squeeze of water films. But, when the consolidation progresses (i.e., during the decrease of the water content), the yield strength required to squeeze the fluid rises and thus overcomes the relative weight of the agglomerates. The agglomeration process stops when the clay deposit is (1) closed to the consolidated threshold, (2) has a porosity of ~35%, (3) a yield strength of several Pa and (4) when agglomerates have a size ranging from 100 μm to 1 mm. At that time, the clay deposit has a permeability 4 to 6 orders of magnitude greater than the one prior to the development of the agglomerates.

Just below the consolidation threshold, the solid framework constituted by agglomerates is purely plastic. Using deformation experiments on frozen clays, we derive possible rules determining the evolution of the yield strength and effective viscosity during compaction, when the fluid concentration decreases. Then, processes like desiccation, syneresis, local extensional tectonics, Rayleigh-Taylor instability and hydrothermal and/or plastic convections can generate stresses exceeding the yield strength of the solid framework of the clay deposit. In such cases, new compaction stages are triggered. Between viscosities of 104 to 108 Pa·s, compaction leads to macroscopic structures ranging from a few centimeters to hectometers corresponding to the compaction length L of the slurry. This explains the development of desiccation polygonal cracks on continents. Giant PFS are resulted from the drastic reduction of the permeability at the top of a few hundred to a few kilometers thick consolidated clay deposits. This implies that during compaction, the fluid squeeze is so important that the yield strength of the overpressured horizons ranges between ~0.1 to 10 MPa and have a compaction length L of a few hundred meters to a few kilometers; i.e., equivalent to the size of PFS described by Sun et al.(2010, 2009). When the consolidated horizon is much thicker, several superimposed horizontal compaction waves are generated, as those found in Australia (Watterson et al., 2000). Finally, the buoyancy of the fluid-rich horizons generates Rayleigh-Taylor instabilities. During that time, fault arrays accommodating the volume of fluid reduction during compaction are generated (Watterson et al., 2000; Dewhurst et al., 1999).

At the end of the compaction process, the reduction of the porosity is accommodated by a variation of shape of the grains, aggregates and agglomerates composing the media. Such process likely modifies the fluid connectivity within the deposit, which in turn drastically reduces the effective permeability. The system is rapidly locked and the permeability irremediably drops to values in the 10-16 to 10-22 m2 range, typical of mudstones (e.g., Kopf, 2002). Such a process is irreversible and overprints the features associated to the successive compaction steps. It is thus inferred that the development of large-scale PFS with their associated compaction lengths, evaluated above, occurs just at the moment when the compaction waves are generated.

ACKNOWLEDGMENTS

This research has benefited from the support by the French Space Agency CNES, PNP (Programme National de Planétologie) and TOSCA (Terre, Océan, Surfaces Continentales, Atmosphère). It has also benefited from the support of Commissariat Général au Développement Durable (CGDD) from the French Ministry of Environment, as part of the CEREMA internal research project HYDROGEO. We thank two anonymous reviewers for their constructive criticisms which significantly improved the paper. We also want to thank David A. Yuen for his support and scientific discussions. The final publication is available at Springer via https://doi.org/10.1007/s12583-017-0814-9.

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