Compaction of siliciclastic sediments is of interest for the study of numerous transport processes occurring in sedimentary basins. Mechanical compaction of sand/clay mixtures depends on the clay content, the effective stress history, and both the mechanical compaction coefficients and the depositional porosities of the two end-members (clean sand and pure shale). The porosity/depth profiles of siliciclastic sediments result from the superposition of two kinds of spatial variations. The first component corresponds to compaction of the mixture with depth of burial. The second component corresponds to small-scale variations of the clay content during deposition and clay infiltration processes. The porosity/depth data are bounded by a minimum porosity/depth envelope corresponding to clay content at the limit between the shaly sand domain and the sandy shale domain. There are two possible upper bounds corresponding to small or large clay contents. The porosity of the clean sand end-member decreases with the depth of burial until a critical porosity of 0.25–0.40 is reached. This critical porosity corresponds to the porosity of random assemblages of more or less spherical grains and depends on the grain-size distribution of the sand grain assemblage. The critical porosity of the shale end-member is much smaller because of the high aspect ratio of the clay particles. The model developed here is applied to downhole measurements made in a borehole that penetrates 3 km of Mio-Pleistocene shaly sand series. The compaction model agrees well with the observed compaction profile.
Athy  introduced the first compaction law relating porosity to the depth of burial. Athy's law is a decreasing exponential law independent of the pore fluid pressure [e.g., Hoholick et al., 1984]. Because of its simplicity, this law is still very popular and was used in a large number of studies [e.g., Korvin, 1984; Hoholick et al., 1984; Baldwin and Butler, 1985; Bethke, 1985, 1986; Henry et al., 1990; Le Pichon et al., 1990; Mello et al., 1994]. However, Athy's law is purely empirical and does not include the influence of disequilibrium compaction in overpressured sections of sedimentary basins. It also does not account for lithological variations in the sediment column. It can thus be the source of large errors when evaluating pressure regimes in sedimentary basins. Dickinson  and Bredehoeft and Hanshaw  showed that porosity is primarily a function of effective stress rather than depth [see also Shi and Wang, 1986, 1988]. Again, lithogical variations were ignored in their works. However, clay minerals do mix with sand grains during sedimentation. This was observed in various sedimentary basins around the world, particularly in deltaic environments [e.g., Matlack et al., 1988]. Therefore there is a need for a compaction model that accounts for this mixture. A first step in this direction was undertaken in this work.
2. Compaction Model
2.1. Mechanical Compaction of Sands and Shales
 Two main phenomena are responsible for bulk compaction in sedimentary basins, pressure solution (pervasive or stylolitic) and mechanical compaction. Pervasive pressure solution occurs by dissolution of stressed grain-to-grain contacts and precipitation on unstressed faces. This mechanism has been recognized as the major mechanism responsible for irreversible deformation of “clean” (i.e., clay-free) sandstones in sedimentary basins [e.g., Weyl, 1959; Yang, 1997; Fowler and Yang, 1999]. The influence of clays on pervasive pressure solution depends on the clay content. At very low clay content (say <5–10% weight), the presence of thin films of clay (illite, kaolinite, smectite) coating sand grains enhances the kinetics of pressure solution [e.g., Heald, 1956; Thomson, 1959; Weyl, 1959; Hickman and Evans, 1995; Bjørkum, 1996]. In such a case, the clay coating probably acts as a physical catalyst by increasing the efficiency of the diffusion pathways at the grain-to-grain contacts of sandstones [Weyl, 1959; De Boer, 1977; Hickman and Evans, 1995]. The most extensive quartz cementations resulting from pervasive pressure solution are found in practically clay-free sands and sandstones or in sands and sandstones with less than 5–10% clay content [e.g., Siever, 1959]. Field studies and laboratory experiments show that the presence of a clay content higher than 5–10% completely inhibits pressure solution of sand grains [e.g., Siever, 1959; Weyl, 1959]. Pettijohn  noted an inverse relationship between the clay content and the amount of quartz cement resulting from pressure solution. Here, we are only interested in mechanical deformation of sand/shale mixtures with clay content higher than 5–10%. Therefore pressure solution can be neglected even for the fictitious “clean” sand end-member. Pervasive pressure solution of clean (i.e., clay-free) sandstones is investigated by Revil .
 Mechanical deformation has been studied mainly by soil engineers [e.g., Terzaghi and Peck, 1948]. In a well-sorted uncemented granular porous material, the cohesion of grains is low. Mechanical compaction results from slippage and rotation of the grains, which change their position and orientation but not their shape (Figure 1) [e.g., Rieke and Chilingarian, 1974]. In a well-sorted clean sand, the porosity decreases from the depositional porosity ϕ0 (in the range 0.50–0.70) to a residual porosity ϕc ≈ 0.25–0.40. The range of ϕc is independent of the grain size for unimodal and narrow grain-size distributions but it depends strongly on the grain shape and grain-size distribution for multimodal grain-size distributions. A residual porosity of 0.36–0.40 corresponds to the porosity of a random packing of identical spherical grains [Terzaghi and Peck, 1948; Scott, 1960; McGeary, 1961]; ϕc = 0.36 is the porosity of a close random packing of identical spheres. Mechanical compaction is very efficient in shales and mudstones due to the plate-like shapes of the grains [e.g., Terzaghi and Peck, 1948; Hubbert and Rubey, 1959] (see Figure 1). In shales and mudstones, the porosity decreases from the depositional porosity ϕ0 (≈0.60–0.70) to a residual porosity ϕc (≈0.05–0.10). The residual porosity of shales is much lower than the residual porosity of sands due to the high aspect ratio of their grains, which allows closer packing of the particles (Figure 1).
 In this study, we assume that, during mechanical compaction, porosity changes are proportional to the change in effective stress, dσeff, and to the difference between the existing porosity ϕ and the residual porosity ϕc, i.e., dϕ = −βm(ϕ − ϕc) dσeff. The proportionality constant βm is a pseudopore mechanical compaction coefficient (the term “compressibility” is usually reserved for poroelastic, reversible, deformation). The compaction coefficient βm is assumed to be independent of temperature, porosity, and effective stress. The pore compaction coefficient associated with mechanical compaction is usually defined by
where ϕ0 is the porosity in a reference state with no stress applied. Hereinafter ϕ0 corresponds to the depositional porosity at the seafloor, i.e., the reference state is taken at the top of the sedimentary column. According to equation (1), the pore compaction coefficient decreases with porosity and is equal to zero when the porosity reaches the residual porosity ϕc. In other words, the smaller the porosity, the less compressible is the porous aggregate by mechanical compaction. This is qualitatively in agreement with the experimental observations made by Hamilton  concerning mechanical consolidation of marine shales and by Terzaghi and Peck  for sands and shales.
 Because mechanical deformation is an inelastic process, the porosity change depends on the stress path. Consequently, the constitutive relationship accounting for poroelastic and inelastic mechanical compaction can be written as
where βe is the poroelastic (reversible) compressibility. From observations of the compaction curves of shales and mudstones observed in various sedimentary basins and in mechanical compaction experiments, the mechanical compaction coefficient βm is in the range (3–7) × 10−8 Pa−1 as shown in Figure 2. Equations (2) and (3) do not account for inelastic mechanisms of deformation other than mechanical compaction. For example clay swelling, which is an important process at very low pore fluid ionic strength in shale, is ignored. We can neglect the poroelastic contribution in equation (2) because βe (∼10−11 Pa−1) ≪ βm (∼5 × 10−8 Pa−1).
 We now integrate equation (2) to obtain the end-member porosity-depth relationships in the hydrostatically pressured formations. The differential of the effective stress is given by dσeff = dσ − dp ≈ dPH − dpH, where PH and pH represent the lithostatic pressure and the pore fluid pressure, respectively, in hydrostatic pore fluid pressure conditions (the subscript “H” refers below to hydrostatic conditions). The changes in lithostatic and pore fluid pressures under hydrostatic pore fluid pressure conditions over a depth increment dz are given by
where ϕH is the “hydrostatic porosity,” i.e., the porosity under hydrostatic pore fluid pressure conditions, ρf is the density of the fluid in the connected pore space (assumed to be constant), ρg is the grain density, and g is the acceleration of gravity. Equations (4) and (5) can be combined with equation (2), yielding
Integration of equation (6) from the top surface of the basin (z = 0, ϕ = ϕ0) to depth z yields
where zm is a characteristic depth of the compaction profile, which is inversely proportional to the compaction coefficient of the sediment. Equation (7) is valid for clean sands and pure shales, i.e., equation (7) describes the porosity trends of the two end-members of the sand/clay mixture problem under hydrostatic pore fluid pressure conditions. Assuming that ϕc (shale) ≈0, the porosity profile in hydrostatic fluid pressure conditions for a clean sand and a pure shale are given from equation (7) by
respectively, where zmSd = 1/[(1 − ϕcSd)gβmSd(ρg − ρf)] and zmSh = 1/[gβmSd(ρg − ρf)]. The parameters entering into equations (9) and (10) are defined in Table 1.
Table 1. Definition of the Main Model Parameters
Value in the Field Case
compaction coefficient of the shale end-member
4 × 10−8 Pa−1
compaction coefficient of the sand end-member
6 × 10−8 Pa−1
porosity of a sand/clay mixture
function of depth
permeability of a sand/clay mixture
function of depth
bulk density of the grains (clay and sand)
2650 kg m−3
bulk density of the pore water
1050 kg m−3
porosity (permeability) of the shale end-member
function of depth
porosity (permeability) of the sand end-member
function of depth
depositional porosity (permeability) of the shale end-member
0.65 ± 0.07
depositional porosity(permeability) of the sand end-member
0.54 ± 0.05
permeability of a clay-filled sand
function of depth
residual porosity of the sand end-member
0.45 ± 0.11
characteristic length defined by zmSh = 1/[gβmSh (ρg − ρf)]
characteristic length defined by zmSd = 1/[(1 − ϕcSd)gβmsh(ρg − ρf)]
minimum porosity of a sand clay mixture
function of depth
clay fraction (in volume)
function of depth
clay fraction (in weight)
function of depth
critical clay fraction (in weight)
function of depth
2.2. Mechanical Compaction of Sand/Clay Mixtures
 A question arises first about the formation of sand clay mixtures in sedimentary basins. Indeed, spherical sand-sized grains of quartz and platey clay minerals are not depositionally compatible due to very different grain sizes. Argillaceous sediments may be deposited simultaneously with sand through clay flocculation in deltaic environments. Sandy shales in which sand grains “float” in a clay matrix are observed in sedimentary basins and could result from such a process. Clay minerals can also be mixed with sand grains during infiltration of clay minerals into a sand matrix during pedogenesis or during shallow burial [e.g., Brewer, 1964; Walker et al., 1978; Matlack et al., 1988]. Howard and Reineck  showed that extensive homogenization of muds and sands is quite common in sedimentary basins.
 We now develop compaction equations valid for sand/clay mixtures. We use the scheme of Marion et al.  originally developed to model acoustic velocities in sand/shale mixtures. The purpose of the mixture equations is to derive the properties of a mixture from the properties of the end-members and their respective volumetric contents. Parameters describing the sand/shale mixture are illustrated in Figures 3 and 4 The porosity of an ideal sand/clay mixture is related to the clay content and the porosity of the two end-members by
where φV is the clay volume fraction. The condition φV ≤ ϕHSd corresponds to the “clayey or shaly sand” domain, whereas the condition ϕHSd ≤ φV ≤ 1 corresponds to the “sandy shale” domain (Figures 3 and 4). In the clayey sand domain, there is not enough clay to fill the sand pore space so some of the sand porosity remains open. In this domain, the sand pore texture is preserved, i.e., the critical porosity of the sand is not affected by filling the space with clay minerals. In the sandy shale, the sand grains float in the shale matrix and the pore space is entirely in the shale component. In shaly sands, porosity decreases with clay content while in sandy shales it increases with clay content (Figure 5).
 In equations (11) and (12), we have assumed ideal packing of sand clay mixtures. This means that the infiltration of the clay grains does not perturb the packing of the coarse grains. This condition arises when the ratio between the grain size of the sand grains and that of the clay grains is very large (≫1). Yu and Standish  and Koltermann and Gorelick  provide more realistic packing models for binary granular mixtures. Differences between these models and the ideal packing model occur mainly at the limit between the sandy shale and the clayey sand domains. At this limit, the perfect packing model underestimates the porosity of the binary mixture [e.g., Koltermann and Gorelick, 1995]. Yu and Standish  show that the difference between real and ideal packings is only a function of the ratio of the grain diameters of the two end-members. Because clay particles have diameters much smaller than the mean grain diameter of sands (<5 μm for clay particles and >50 μm for sands), we assume that ideal packing is a good first order approximation (keeping in mind of course that more complex mixtures can occur in some sedimentary sequences).
 The bulk density in the clayey sand domain is given by (see Figure 4)
Here ρgSd, ρgSh, and ρf are the bulk density of the sand grains, of the clay mineral grains without their bound water (ρgSd ≈ ρgSh ≈ 2650 kg m−3 [Ellis, 1987, p. 203, Table 10-1]), and the pore fluid, respectively. In the sandy shale domain, the bulk density can be calculated from the volume fractions shown in Figure 4:
 We note φW, the clay weight fraction, which represents the ratio between the mass of dry clay to the total mass of the sediment. In the clayey sand domain, and the relationship between φV and φW is given by
In the sandy shale domain, the relationship between φW and φV is given by
Hereafter we assume ρgSd ≈ ρgSh ≈ 2650 kg m−3 = ρg [Ellis, 1987, p. 203, Table 10-1]. This assumption is a good approximation if we consider the clay minerals without their bound water (the inclusion of the bound water in the grain density of clays decreases the values of the grain density depending on the specific surface area of the grains). From equations (15) and (16), we obtain
where φWcrit is the critical clay content characterizing the boundary between the clayey sand domain and the sandy shale domain and defined by the condition φV = ϕHSd (see Figure 3).
2.3. Overpressured Formations
 In the overpressured portion of the sedimentary column, all the mixture equations remain valid (The difference between the overpressured case and the hyrostatic case is that in the former equations (9) and (10) are not any more valid.) So equations (11), (12), (15), (16), and (19) remain valid in these formations with the subscript H dropped. Note that the variable φW does not depend on the state of compaction (it is a weight fraction) whereas φV does as it refers to a volume fraction. Combining equations (15), (16), (17), and (18) yields
Consequently, equation (23) can be used to estimate the porosity of the clean sand end-member in the clayey sand domain from the porosity of the sediment and the clay weight fraction (both easily determined from downhole measurements analysis, see section 3). Using equation (24), we can determine the porosity of the shale end-member from the porosity and the clay weight fraction in the sandy shale domain.
2.4. Permeability of Sand/Clay Mixtures
 A complete determination of the occurrence of pore fluid overpressure (above hydrostatic) requires the determination of the permeability of the sediment in addition to the compaction laws. Laboratory measurements of permeability evolution during compaction in siliciclastic sediments [e.g., Dewhurst et al., 1996a, 1996b, 1998] show that permeability depends strongly on the clay mineralogy and clay content. Dewhurst et al. [1998, p. 660] conclude by stating “the permeability of argillaceous sediments is unlikely to be simply related to the total porosity of the sediment … These results show that the use of generic mudstone porosity-permeability curves in basin models will yield significant errors in predictions of fluid flow and fluid pressure. One pragmatic approach to this problem would be to recognize that permeability might be reasonably inferred if both the porosity and the lithology of the mudstones were known.” In this section, we follow this idea. We include in the compaction model derived in section 2 the means to determine permeability as a function of effective stress, clay content, and clay mineralogy. The permeability model for a sand/clay mixture is assumed to follow (modified from Revil and Cathles )
where d represents the mean grain size of the sand grains and the permeabilities kSd and kSh represent the permeabilities of the clean sand and pure shale end-members, respectively (see definitions in Table 1). The permeability kCfs represents the permeability of the clay-filled sand at the boundary between the two domains (clayey sands and sandy shales). Here ϕSd and ϕSh are given by equations (23) and (24), depending on the domain (clayey sand or sandy shale) or by explicit function of depth in the hydrostatic section (equations (9) and (10)). Equations (25) to (28) represent very well the permeability of a sand/clay mixture as a function of the volumetric clay content (Figure 6). To go further, we need a relationship to determine the permeability of the shale end-member. For granular porous materials, Revil and Cathles  obtained a relationship between the permeability kSh, the mean grain diameter dSh, and a grain shape parameter called the electrical cementation exponent mSh.
The grain size is rarely reported for clay minerals. However, it can be related to a chemical measurement, the cation exchange capacity (CEC) [e.g., Patchett, 1975], which is traditionally determined and used in the interpretation of electrical resistivity logs [e.g., Revil et al., 1998, and references therein]. The cation exchange capacity (usually expressed in meq g−1, where 1 meq g−1 = 96320 C kg−1 in SI units) represents the amount of titrated surface charge per unit weight of mineral. Its range for various clay minerals has been summarized in a number of papers [see Patchett, 1975; Revil et al., 1998, and references therein]. Using the relationship between the grain size and the CEC, we obtain
According to equation (31) the permeability of the clay matrix at a given porosity (say 0.50) depends directly from the square of the cation exchange capacity of the fine fraction. This property is checked in Figure 7 in which permeability data from the literature (at a reference porosity of 0.50) are shown for the three main shale end-member families (kaolinite, illite, and smectite). These data are plotted versus the cation exchange capacity of the respective shale end-members. According to equation (31), there is a direct relationship between the permeability and the cation exchange capacity at a given porosity as suggested by the model. This relationship can be described by
where CECSh is here expressed in meq g−1 and kSh0 is in millidarcies (1 mdarcy ≈ 10−15 m2) (the subscript zero indicates that the permeability is determined at the reference porosity 0.50), a = −5.79 ± 0.07 and b = 1.73 ± 0.04. Once the permeability has been determined at the reference porosity of 0.50, the variation of the permeability with porosity can be determined by combining equations (31) and (32). This yields
We need now to test the model for the case of a mixture of clays. This is done in Figure 8 using the experimental data from Dewhurst et al. [1996a]. In this example, the cation exchange capacity of the silt/clay mixture is related to the CEC of each clay end-member by
where χi represents the relative (weight) fraction of the clay mineral “i” and CECi represents the cation exchange capacity of the clay mineral “i” (see values of Patchett  and Revil et al. ). Using this relationship and the proportions of clays, we obtain a fairly good agreement between our model and the experimental data reported in Figure 8.
3. Case Study
 In this section, we apply the compaction model to a case study corresponding to a borehole drilled in a basin located on an active Southeast Asia margin in a deltaic-type depositional environment (Figure 9) (the location of the area and a description of the tectonic context are given by Grauls and Cassignol  and Grauls and Baleix  and will not be repeated here). The depths are referred to sea level (meters below sea level, mbsl) or to the seafloor (meters below the seafloor, mbsf).
3.1. Sedimentary Context and Downhole Measurements
 Three main units are identified in the drilled section shown in Figure 9. The upper unit (217 to 1800 mbsl) is formed by Pleistocene to late Pliocene silty and sandy argillaceous sediment. Early Pliocene clay, silty clay, and very fine argillaceous sandstones comprise the middle unit (1800 to 2341 mbsl). The lower unit (2341 to 3406 mbsl) is late Miocene (possible middle Miocene from 3170 mbsl) shaly and silty shale sediment. No diagenetic features associated with pressure solution are observed in the cored sand and sandstone samples from this section.
 The available downhole measurements include electrical resistivity, gamma ray, litho-density, neutron porosity, P wave travel time, and geochemical logs. Ellis  provided a complete description of the tools used to obtain these downhole measurements (see also Revil et al.  for a description of a method of interpretation of these logs). The intersections between the borehole and both stratigraphic and tectonic features are indicated in Figure 9. The pore fluid pressure is hydrostatic between the seafloor (146 mbsl) and 1800 mbsl. The sediments below 1800 mbsl are overpressured and the borehole intersects a set of growth faults between 1800 and 2221 mbsl observed in the seismic section [see Grauls and Cassignol, 1992].
 From 2270 to 3185 mbsl, the borehole penetrates a strong acoustic anomaly observed previously in interval velocity determinations obtained from analyze of seismic profiles. This seismic velocity anomaly was interpreted by as corresponding to the presence of an undercompacted massively hydrofractured sediment volume. Hydrofractures were observed at 2482, 2905, 3305, and 3110 mbsl. While drilling, the occurrence of successive drilling mud gains and losses below 2400 mbsl confirms the presence of highly permeable fractures in formations characterized by a low matrix permeability (<1 mdarcy). In this zone, the fluid pressure is close to the lithostatic pressure (see Figure 10 where the excess pore fluid pressure corresponds to the pore water pressure in excess of hydrostatic). Geochemical analyses indicate that the gas-saturated water within this undercompacted zone has migrated along the network of hydrofractures from the downthrown high-pressure zone. Hydrocarbons recovered from the shallow sand reservoirs are more mature than in the surrounding source rocks and have probably migrated from greater depths. Below 2970 m, the porosity decreases again with depth as shown by in situ electrical, acoustic and density measurements. The gas and the pore fluid pressure gradient observed during drilling also decrease below 2970 mbsl.
 Other data include in situ temperature measurements (corrected temperature from BHT measurements using the classical Horner method, see Ellis ) and the observation of the presence of gas. The seafloor temperature is 25°C and the geothermal gradient in the hydrostatically pressured formations is 28.2°C km−1 (Figure 10). Transitions between hydrostatically pressured formations and deeper overpressured zones are often accompanied by significant increases in the geothermal gradient [e.g., Kharaka et al., 1980]. A major temperature anomaly appears between 2231 and 2710 mbsl. This depth interval corresponds to the top of the strong acoustic anomaly at 2450 mbsl, Figure 9 [see Grauls and Cassignol, 1992]. The presence of gas (mainly methane) observed below 2050 mbsl during drilling (the maximum amount of gas was found in the cuttings between 2500 and 3000 mbsl) is probably responsible for the strong temperature anomaly below this depth. Indeed, the presence of gas in the connected pore space of a sediment decreases dramatically its thermal conductivity [Somerton, 1992; Revil, 2000]. This temperature anomaly could also result from upward flow of hot gas and gas-rich water in the network of open hydrofractures present in the Miocene sediments below 2270 mbsl.
 The fluid pressure is hydrostatic between the seafloor and 1800 mbsl. Below this depth the fluid pressure increases above hydrostatic levels (Figure 10). Between 1800 and 2800 mbsl, the fluid pressure gradient is 37 MPa km−1 (the hydrostatic and lithostatic fluid pressure gradients are 10.2 and 22.2 MPa km−1, respectively). At the bottom of the growth fault zone (at ∼2220 mbsl), the fluid pressure is 80% of the lithostatic pressure (using a fluid hydrostatic pressure equal to 22.5 MPa). At 2850 mbsf, the fluid pressure reaches a quasi-lithostatic level (the fluid pressure at this depth equals 96% lithostatic, Figure 10). Such high levels of pore fluid pressure explain the presence of hydrofractures observed between 2480 and 3300 mbsl. Below 2900 mbsl the fluid pressure gradient decreases to near the hydrostatic fluid pressure gradient (Figure 10).
3.2. Compaction Trend
 We focus on the hydrostatically pressured interval (seafloor to 1796 mbsl) in order to test the hydrostatic compaction equations introduced in section 2. The interpretation of downhole measurements follows from the methodology developed by Revil et al. . The gamma ray log is used to estimate the shaliness of the formations [Ellis, 1987]. The clay weight fraction is related to the gamma ray level, γ, by [e.g., Revil et al., 1998]
where γSd is the gamma ray level of a pure sand assumed to be equal to 10 gamma ray units [Ellis, 1987], whereas γSh is the gamma ray level of a pure shale which is calculated by
Here χi are the clay fractions of each clay mineral relative to the total shale fraction, and γi is the associated gamma ray level. The clay mineralogy observed in the recovered sediments changes little along the entire well. The clay fractions are approximately χ(I) = 1/3, χ(C) = 1/3, and χ(K) = 1/3 where I, C, and K represent illite, chlorite, and kaolinite, respectively. This clay mineralogy is representative of a terrestrial source. The associated gamma ray levels are γ(I) = 230, γ(C) = 150, and γ(K) = 70 [Ellis, 1987]. With the previous values for the clay mineralogy and the gamma ray levels for each clay component, we calculate from equation (40) γSh = 150. The porosity and the clay weight fraction in the hydrostatic zone are shown in Figure 11. The porosity shown in Figure 11 was obtained directly from the density log assuming water-saturated formations [see Ellis, 1987; Revil et al., 1998].
 As in the area investigated, there is very little evidence of pressure solution in the first 3 kilometers, we assume that compaction is mainly the result of mechanical deformation of sands mixed with clays. In the hydrostatic section, the porosity curve is a composite of two phenomena. The first phenomenon corresponds to a monotonic decrease of the porosity with the depth of burial. This trend corresponds to compaction of the sediments. The second signal corresponds to high frequency variations due to variations of the clay content with depth (see Figure 9). Considering small depth intervals, the porosity actually reaches the upper bounds for clean sand and pure shale, and a minimum value when the shale fraction fills exactly the porosity of the sand (Figure 5). These minimum porosity values were extracted from the data set to generate a minimum porosity/depth envelope, which is shown in Figure 12. According to equations (11) and (12), the minimum porosity of the sand clay mixture at a given depth is given by
where the unknown parameters βmSh, βmSd, ϕcSd, ϕ0Sh and ϕ0Sd are defined in Table 1 (we use ρg = 2650 kg m−3 and ρf = 1050 kg m-3). As shown by Figure 2, the compaction coefficients βmSh and βmSd are likely to be in the range 4–6 × 10−8 Pa−1. The values of the different parameters used in the compaction model are fixed and given by βmSh = 4 × 10−8 Pa−1 and βmSd= 6 × 10−8 Pa−1 as very reasonable values for the mechanical compaction coefficients of pure shale and clean sands, respectively (see section 2 and Figure 2).
 Using the Simplex algorithm [e.g., Caceci and Cacheris, 1984], we inverted the depositional porosities for pure shale and clean sand, ϕ0Sh and ϕ0Sd, respectively and the residual porosity ϕcSd using the expression for ϕm(z), the previous values for the compaction coefficients and densities, and the porosity data shown in Figure 12. We used as a priori (initial guess) values ϕ0Sh = 0.60 (from Figures 2a and 2c), ϕ0Sd = 0.60, and ϕ0Sd = 0.40. After optimization of these three parameters, we obtain ϕ0Sh = 0.65 ± 0.07, ϕ0Sd = 0.54 ± 0.05, and ϕ0Sd = 0.45 ± 0.11. Using these values, we are now in the position to compute the porosity trends of the end-members in the hydrostatic section, ϕSh(z) and ϕSd(z), using equations (9) and (10). The critical clay weight fraction, φWcrit is determined then using equation (22) and plotted in Figure 11. The critical clay weight fraction allows us to discriminate at each depth if sediments are clayey sand or sandy shales.
 In the upper hydrostatic interval, we observe in Figure 11 only two major sandy units located between 600 and 670 mbsl and between 730 and 780 mbsl, respectively. These two formations are likely to be very good aquifers or reservoirs. Most of the other formations are sandy/silty shales with clay content never exceeding 70% by weight.
 We selected three small depth intervals containing large variations in clay content. They correspond to 1075 to 1150 mbsl, 1425 to 1500 mbsl, and 1550 to 1600 mbsl, respectively. In Figure 13, we plot their porosity as a function of their clay weight content. As in Figure 5 (which shows laboratory data), in situ data exhibit the same characteristic trends and porosity minimum corresponding to the limit between the clayey sand and the sandy shale domains. We can compute the porosity from our model using equations (20) to (22) assuming ρgSd ≈ ρgSh ≈ 2650 kg m−3 = ρg. The agreement between the field data and the model (the solid lines in Figure 13) is satisfactory.
 We now know all the parameters used to determine the compaction trends of the two end-members, the shale weight fraction, and the critical shale weight fraction. We can then reconstruct the porosity curve taking into account the clay content variations. Figures 14 to 1516 compare the predicted and observed porosity profiles for some sandy and shaly lithological units. The porosity is predicted by substituting equations (9) and (10) into equations (11) and (12). The agreement between the model and the porosity data indicates that the combination of clay content variations and mechanical compaction of sand clay mixtures explains relatively well the small-length scale porosity variations. The underprediction of the porosity near the critical level at the limit between the clayey sand and sandy shale domains can be explained from the fact we used an ideal packing model. As shown in Figure 17, the discrepancy between ideal and nonideal packings is maximum near the critical limit between the two domains. We believe that our compaction scheme could be improved with the help of a better mineralogical composition inversion and the use of a nonideal packing model.
4. Concluding Statements
 We have developed a model describing the plastic mechanical compaction of sand clay mixtures. The model takes into account the influence of the clay content, effective stress, and the mechanical compaction coefficients and depositional porosities of the two end-members (clay-free sand and pure shale). At each depth interval the porosity is minimum for clay contents at the limit between the shaly sand and the sandy shale domains, and is maximum for small or large clay contents. The porosity of the fictitious free-clay sand end-member decreases until a critical porosity of 0.25–0.40 is reached (“fictitious” because pressure solution would decrease the porosity below this level). This critical porosity corresponds to the porosity of a random assemblage of spheroidal grains and it depends on the grain shape. This result is valid only if the clay content is larger than 5–10% in the sedimentary formations analyzed. This is because the presence of clays inhibits deformation of the sand grains by pressure solution above this clay content value.
 To test the validity of our approach, the compaction model was applied to a field case concerning a borehole crossing 3 kilometers of Mio-Pleistocene shaly and sandy formations. The model agrees well with the observed compaction profile including the small-scale porosity variations caused by variations in the sand shale ratio with the depth. Possible next steps include (1) further testing the predictions of the compaction model developed in this paper using a more extensive database concerning downhole measurements in deltaic-type sedimentary environments (using, for example, the Ocean Drilling Program database) and (2) including the influence of the deviatoric component of the stress tensor upon the porosity of sand/shale mixtures and transport properties like permeability.
 This work is supported by the GBRN (Global Basin Research Network) at Cornell University, the companies sponsoring the GBRN, and the CNRS (Centre National de la Recherche Scientifique). We thank Elf-Aquitaine for initial authorization to publish this work. Larry Cathles is thanked for very fruitful discussions during various stages of this work. We appreciate very much the helpful comments made by Richard L. Carlson and Yves Bernabé.