The geometrical distribution of various components in a composite sandstone is decisive for its overall stiffness and seismic velocities. Information about which constituents, for example, are load bearing, dispersed in pore fluid or present as contact cement is, therefore, necessary if the seismic properties are to be modelled reliably. A distribution scheme for quartz cement, K-feldspar and some of the most common clay minerals in sandstones (illite, kaolinite, smectite and chlorite) is suggested on the basis of thin-section observations made by a number of authors. This classification scheme facilitates rock physics modelling as a function of mineral concentrations. A composite rock physics model has also been developed to account for simultaneous combinations of mineral distributions. Well-known mineral reactions are used to make simple models of mineralogy versus temperature (depth) from different starting scenarios, as various minerals tend to follow different and predictable paths during burial and increasing temperature. The mineralogical trends are then entered into the composite rock physics model to produce the diagenetic evolution of seismic rock properties, and the procedure is used to estimate the effective rock properties of sandstones in a well log. The modelling allows deductions to be made about possible mineralogies and their distributions from seismic parameters. Finally, reflection coefficients resulting from sandstones subjected to various diagenetic processes are modelled and analysed. The results show that it is possible to discriminate between reflections emanating from the interfaces of a selection of common diagenetic scenarios.
Sandstones are rarely composed entirely of quartz and often contain admixtures of various clay minerals. Worden & Morad (2003) recently described several ways in which clay minerals can be incorporated in sandstones, while Anstey (1991), Dvorkin & Nur (1996), Gal et al. (1999) and Sams & Andrea (2001), among others, stated that the elastic moduli of shaly sandstones strongly depend not only on the volume of clay but also its position. Thus, two sandstones with the same clay volume may exhibit significant differences in stiffness. Sams & Andrea (2001) used four clay distributions: clay-rich framework grains (structural); grain boundary clay (structural, interstitial); pore-filling and pore-lining clay in the pore spaces (dispersed); and clay laminations. They also presented four distinct models for estimating their effects on sandstones. They did not, however, explicitly model contact cement or study the effects of more than one distribution of clay at the same time (e.g. structural and dispersed). A rock physics model aiming to mimic a real sandstone containing clay minerals should be able to accommodate varieties of simultaneously occurring clay distributions. A model for estimating effective anisotropic rock properties in shales during diagenesis was presented by Dræge et al. (2006).
In this paper any mineral precipitated in the rock is termed ‘cement’. The precipitation of a mineral is a chemical reaction that can be predicted given the constituents and environment, and mineral reactions follow certain chemical/physical laws and adhere to certain trends. Geologists and geochemists have recognized different temperature intervals in which certain detrital minerals dissolve and re-precipitate as authigenic cements (e.g. Bjørlykke & Aagaard 1992; Bjørkum 1996; Ketzer et al. 2003; Worden & Morad 2003). These studies provide the constituents and their concentrations, but that alone is insufficient for rock physics approaches. Rock physicists need to seek mineral distribution patterns because of the considerable impact they have on the overall stiffness.
A clay distribution scheme for the most common clays in sandstones is therefore suggested, based on thin-section observations made by a number of authors. This classification scheme is combined with geological knowledge of some of the most essential mineral reactions, so that theoretical predictions can be made of their evolving composition and distribution during diagenesis. Furthermore, a rock physics model is created to incorporate the composite effect of the different types of mineral distribution.
In some cases micromechanical models, such as the ones applied here, may have limited validity. However, the essential part of this research is the integration of geological processes and constraints with rock physics modelling, not the rock physics model itself. In other words, the rock physics model described below allows predictions to be made of the diagenetic evolution of seismic rock properties during diagenesis, given the mineralogy and porosity.
The quality of the mineralogical and porosity estimations obviously influences the quality of the rock physics modelling. It should also be emphasized that this paper presents a rock physics model and modelling strategy, not a study of advanced geological modelling. Hence, a simplified and somewhat unrealistic porosity model has been used for the introductory examples, where the porosity–depth model is kept constant for various mineralogy–depth scenarios, and porosity differences are thus excluded. Nevertheless, well-log porosities are applied to ensure realistic input to the rock physics model when comparing it with real data.
Distribution types and rock physics models
The distributions are based on the solid constituents, except for framework quartz, which is used as a ‘reference medium’.
Four types of distribution are defined: (1) cement that lies at grain contacts (but not between grains) and thus stabilizes the contacts between the framework grains; (2) pore-filling cement, which contributes little to the overall rock stiffness until it becomes pore-bridging, when approximately 40% or more is present in the pores; (3) grain-coating and pore-lining cement, which envelops the framework grains but does not carry load in the framework grain–grain contacts; (4a) replacive clay or clay clasts, which act as part of the load-bearing framework and (4b) grain-coating clay cement that prevents contact between framework grains and is thus load bearing. The mineral distribution types are illustrated in Figure 1, with abbreviations above each stage (explained below) indicating the rock physics model used in the modelling procedure. It is assumed that when the cement content exceeds 50% of the intergranular volume (i.e. the volume unoccupied by framework grains), the rock starts to become increasingly seismically impermeable until the cement content exceeds 75%, at which stage it is considered to have lost all of its seismic permeability.
Seismic permeability denotes the ability of fluid to move into neighbouring pores during the transient passage of a seismic wave through the rock, not to its permeability over geological time-scales. Thus, seismically isolated pores (seismic permeability = 0) should not be confused with completely isolated pores in a geological sense. The limits for seismic permeability changes can be adjusted according to local differences in pore sizes. Small pores with narrow pore throats are more prone to seismic isolation due to pore-filling cements than more spacious pores. Decreasing fluid mobility leads to enhanced rock rigidity, as demonstrated by Wang (2000).
The rock physics models were chosen to reflect the increasing seismic isolation of pores when pore space is reduced. In transitions from uncemented to cemented rock, and from seismically permeable to seismically impermeable rock, the Hill (1952) average is applied between the various rock physics models to obtain respectively smooth transitions in the 0–5% and 50–75% cement intervals of the intergranular volume (Fig. 1).
The weighting function used when applying the Hill average in those cases is explained in Appendix A.
The Differential Effective Medium (DEM) approach (Nishizawa 1982) can be formulated as a set of differential equations whose physical analogue is that of inclusions gradually being embedded in or removed from a background medium. The procedure is as follows (Johansen et al. 2002): (1) start with a background medium with known properties, in this case a quartz framework; (2) embed or remove inclusions (e.g. pores or cement) to form a new composition; and (3) compute the effective properties characterizing the new background medium. Steps (2) and (3) are repeated until the desired material composition is reached, at which stage the required elastic properties are obtained. New constituents added to the composite media using the DEM approach will be added as unconnected, seismically isolated inclusions. When using DEM for mixing framework quartz with isolated clay clasts and pores, quartz and clay are first mixed together, whereupon the pores are added.
The Combined Effective Medium Theory (CEMT) model is applied to approximately reproduce connected phases in a composite medium. The theory consists of two steps, the first making use of the Self Consistent Approach (SCA) (Willis 1977). The SCA model accounts for interactions between inclusions when a wave travels through the medium by treating all constituents of the composite as embedded in a ‘virtual’ medium having the required effective medium properties. The SCA solution occurs when the net (scattering) effect of all of the inclusions is zero. The term ‘self-consistent’ means that the results do not depend on selection of host medium to embed the remaining constituents, just their volume fraction. If the concentrations of the constituents in a two-phase medium are approximately between 40% and 60%, the SCA solution will correspond to the effective properties of a two-phase medium in which both phases constitute connected phases (Hornby et al. 1994; Jakobsen et al. 2000).
The second step of the CEMT model is to use the DEM approach to estimate the effective properties of the two-phase, composite medium at the desired concentrations. The DEM model has the ability to preserve the connectivity status when the concentrations of the constituents change. Thus, connected phases will remain connected when the concentrations change (and vice versa for unconnected phases). The CEMT is used to model connected cement/inclusion phases when the concentration of Type (3) cement exceeds 50% of the intergranular volume.
Permeable rock with distribution Types (2), (4a) and (4b), and when uncemented, is modelled using the combined theories of Hertz–Mindlin (HM) (Mindlin 1949) and Hashin & Shtrikman (1963) (HS lower bound), as described by Dvorkin & Nur (1996). This combined model (HMHS) connects two end-members: one with zero porosity and the modulus of the solid phase; the other with critical porosity and a pressure-dependent modulus, as given by the Hertz–Mindlin theory.
The Contact Cementation Theory (CCT) of Dvorkin et al. (1999) is applied to model the first stages of Type (1) cement. The CCT provides the effective elastic properties of an aggregate of spheres, where the spheres are in direct point contact and cement fills the space around the contacts. The CCT cannot be used to estimate the elastic constants of an aggregate where cement fills the entire pore space or large portions of it (Dvorkin et al. 1999) and is therefore combined with the DEM approach when the cement fills more than 50% of the intergranular pore volume.
The Coated Sphere model (CS) corresponds to the ‘scheme 2’ cement of Dvorkin & Nur (1996). This model treats the cement as being evenly deposited on grain surfaces, leading to coated framework grains. The CS model is combined with the CEMT to approximate connected phases of cement and framework when cement concentrations become high.
If there are more than two types of load-bearing framework minerals, the isolated (Type (4a)) minerals are first added by mixing mineral 1 with framework quartz (DEM). The result is then used as the framework for the next process when mineral 2 is mixed in. Finally, if there is any grain-enveloping cement present, it is mixed with the framework using the CEMT in order to obtain connectivity. Again, the Hill average is applied to the pore-filling cements to obtain the cement properties of an effective pore-filling cement if more than one type is present. If there is more than one type of coating sphere cement or contact cement, the effective rock properties are calculated for one cement type at a time and the Hill average is applied.
If more than one distribution is present, the effective pore fluid stiffness is calculated using the SCA by mixing fluid and Type (2) minerals. Minerals with distribution Type (4a) are considered as isolated framework grains, which contribute to the effective load-bearing framework. If distribution models (1) and (3) are both present, the Hill (1952) average is applied between the modelling results for each inclusion type (Fig. 2). Type (4b) cement is grain enveloping and thus prevents the precipitation of Types (1) and (3) cements in the areas in which it occurs. Thus, in the case of Types (1), (3) and (4b) all being present, the Hill average is applied between the effective stiffness calculated for each distribution type to obtain the overall effective rock moduli (Fig. 2).
The fluid effects for seismically permeable sandstones without pore-bridging cements are modelled using the Bound Averaging Model (BAM) (Marion & Nur 1991) for all types of cement distributions. This model is an approximation of the Gassmann (1951) model for viscous fluids and is applied because the pore-filling cement is suspended in the fluid, making it more viscous.
The BAM relies on the assumption that the bulk modulus is a weighted average of the lower and upper limits for the given combination of grain modulus and fluid modulus, and that the weighting function is independent of the fluid. The weighting function can be found for the dry rock and then used to determine the modulus for the wet rock. Sams & Andrea (2001) are followed and the Reuss (1929) and Voigt (1928) averages applied for the lower and upper limits. Ideally the BAM and Gassmann models should coincide at the end points with no pore-filling cement and complete cementation; although Sams & Andrea (2001) found that the BAM model predicts slightly higher pressure (P-) wave velocities for rocks with pure fluids.
In the case of pore-bridging cement (the effective fluid having positive shear stiffness), the CEMT is used to model the mix of framework and effective fluid in the seismically permeable rock. The DEM is applied for modelling the fluid effects in non-seismically permeable rocks for all distribution types. Brine is used as the pore fluid in all of the modelling work and is mixed with pore-filling cement when present.
Figure 3 shows the bulk and shear moduli for different distributions in a rock with 30% intergranular volume. The rock exclusively loses its porosity by cementation. The figure demonstrates that the effective stiffness can vary significantly for equal cement concentrations due to different distributions. The amount of load-bearing cement is decisive for the effective stiffness of a rock containing soft cement and a rigid framework. Thus, some of the curves do not converge when the porosity approaches zero. The shear modulus of the Type (2) distribution slightly increases during the first stage of cementation. This increase, however, is not due to the cement but to an imposed linear porosity–pressure relationship that increases the moduli from the HMHS model. The bulk modulus will also increase as a function of a stiffer pore fluid as cementation proceeds. The rock physics models applied to isolated and connected load-bearing cements (Types (4a) and (4b)) do not produce large differences, except for the shear modulus for seismically isolated pores. Considerable changes in all of the stiffness gradients are observed when the pores starts to become seismically isolated at 15% porosity, and when all of the pores are seismically isolated at 7.5% porosity.
The K/μ plot reveals three main trends with decreasing porosity: an initial, sudden drop in the ratio (Types (1) and (2)); a smooth decrease (Types (4a) and (4b)); and an initially little-changed ratio, followed by a sudden drop (Type (3)).
There are five dominant groups of clay minerals in sandstones (Worden & Morad 2003): kaolinite; illite; chlorite; smectite and mixed-layer varieties (e.g. illite–smectite (I/S); and chlorite–smectite (C/S)).
Some important sandstone minerals are now considered, having been classified according to the four distribution classes in order to make the rock physics modelling more convenient. Normally a cemented sandstone contains mixtures of these distributions. Furthermore, when cement concentrations reach certain levels, contact cement becomes pore filling and pore-filling cement eventually becomes pore bridging.
Commonly there are more minerals present in sandstones than those considered here, and these will lead to additional mineral reactions. This paper, however, focuses on a selection of essential and frequent clay mineral reactions and neglects minerals that commonly play inferior roles.
Kaolin minerals generally display two growth habits: vermicular booklet-like kaolinite which, in open systems with good communication between pores, is progressively replaced by well-developed, blocky crystals of dickite at temperatures between 90°C and 130°C (Cassagnabere 1998; Worden & Morad 2003) (Fig. 4).
The conversion of kaolinite to mixed-layer illite–smectite (I/S) and illite via diagenetic dissolution/precipitation reactions can occur at temperatures greater than 120°C (Ehrenberg & Nadeau 1989). Mixtures of detrital kaolin and illite can also be present as load-bearing clay clasts (e.g. inclusion Type (4)). On the other hand, authigenic kaolin typically occurs as pore-filling and pore-lining phases (Van Der Gaag 1997; Jolicoeur et al. 2000; Jones et al. 2000; Khidir & Catuneanu 2002; Thomas et al. 2002; Marfil et al. 2003; Shaw & Conybeare 2003). Pore-filling kaolinite is not expected to generate continuous networks throughout the rock, and will not contribute very much to overall rock stiffness until the concentration exceeds about 40% of the total pore space. Then the kaolinite starts to become pore bridging. Authigenic and infiltrated kaolinite is a typical clay mineral of distribution Type (2).
The chlorite group consists of several minerals with some variations in the aluminium, iron, lithium, magnesium, manganese, nickel, zinc and silicon contents. The chlorite composition suggested by Worden & Morad (2003) is chosen for the various reactions that involve chlorite described in this paper. The most common chlorite morphology is a grain-coating boxwork, with the chlorite crystals attached perpendicular to grain surfaces (Worden & Morad 2003). Early authigenic chlorite coatings are commonly the products of chloritization of other less stable grain-coating minerals.
Ehrenberg (1993) and Grigsby (2001) concluded that syn-sedimentary Fe-rich clay is an important precursor for grain-coating chlorite, which can prevent or inhibit the precipitation of quartz on grain surfaces, thereby preserving relatively high porosities even at considerable depths (Ehrenberg 1993; Anjos et al. 2003; Rossi et al. 2003). Chlorite is generally grain coating and acquires distribution Type (3).
The smectite group contains a variety of minerals, which principally display variations in calcium, sodium, aluminium, magnesium, iron, zinc and silicon contents. The chemical composition of smectite is adopted from Worden & Morad (2003). Smectite usually occurs as flakes curling up from an attachment zone on detrital sand grain surfaces (Worden & Morad 2003). The grains are small and usually vary between 0.4 μm and 0.9 μm in length.
In this study smectite is divided into detrital and authigenic varieties. Detrital smectite envelops framework grains before burial, thus preventing direct contact between framework grains. Grain-coating authigenic smectite is deposited after the grain contacts in the framework have been established and is thus generally absent from the contacts. During diagenesis, smectite can be converted to chlorite or illite (Fig. 4). The conversion is gradual and mixed illite–smectite (I/S) and chlorite–smectite (C/S) layers are transitional forms that become increasingly smectite poor with increasing depth/temperature. Authigenic smectite is a typical representative of a Type (3) distribution, like chlorite. Detrital smectite belongs to the Type (4b) inclusions, as it is load bearing. However, in contrast to load-bearing clasts, detrital smectite is considered as a connected phase, because it covers grain surfaces and prevents framework grain contacts. The intermediate forms of smectite, S/I and S/C, commonly occur as pore-lining or grain-replacive clays (Ketzer et al. 2003).
Illite can occur as flakes, filaments or hair-like crystals (Worden & Morad 2003). The thin, elongated crystals vary in length from <1 μm to 7 μm, whereas the flakes are approximately 2 μm in diameter (Lemon & Cubitt 2003). Illite morphology may cause a dramatic drop in permeability because the presence of illite significantly increases the specific surface areas of pore walls. When the illite concentration increases, the illite lattice bridges the pores, which reduces permeability, establishes communication between grains and strengthens overall rock stiffness (e.g. Chuhan et al. 2001; Net 2003) (Fig. 5). Illite can precipitate from pore fluid and as a product of a number of precursor minerals (Fig. 4). Grain-coating and pore-filling illite, which is absent on grain contacts, has been observed and described by Ketzer (2002), Storvoll et al. (2002) and Patrier et al. (2003). Hence, illite is considered to stiffen the pore fluid only until it becomes pore bridging. These properties coincide with inclusion Type (2) criteria.
Bjørkum (1996) argued that the most important quartz dissolution process in sandstones occurs when quartz grains are in contact with mica or illitic clay surfaces. This study suggested that pressure plays a minor role in quartz dissolution; the quartz grains depend only on a certain minimum of effective pressure to stay in contact with the mica grains as dissolution proceeds. The quartz cementation process consists of three sub-processes: the dissolution of quartz; the transport of aqueous silica; and the precipitation of quartz cement. The step that controls the rate of initial quartz cementation is precipitation (Walderhaug 1996).
From reservoir temperatures of 80°C, the precipitation rate increases exponentially with further temperature elevation and quartz cementation becomes effective. The mineralogy of the sandstone can influence the cementation rate; for example, pure quartz sandstone experiences more rapid cementation than arkosic sandstone. In other words, porosity reduction tends to be less severe if feldspar grains are present (Walderhaug 1996). The surface area of quartz grains is decisive for the extent of cementation. Small quartz grains have larger surface areas than large grains and are thus likely to produce more cement under otherwise similar conditions. Clay minerals or calcite cement covering quartz surfaces reduce the available surface area and makes the quartz grains less susceptible to dissolution. Quartz cement commonly grows as a continuation of the original quartz grain. Together, the grain and cement form a single crystal, although the two parts have different ages (Boggs 1995). A continued increase in grain contact area due to quartz precipitation during chemical compaction has been observed and described by Storvoll & Bjørlykke (2004). Contact-cementing quartz has also been observed and studied by Avseth et al. (1998) and Florez-Niño & Mavko (2004). It is assumed that quartz cement preferentially precipitates near grain contacts until the cement volume exceeds 30% of the intergranular volume, after which the cement starts to fill up the pore space. This corresponds to the definition of distribution Type (1).
The last mineral considered here is K-feldspar. Such grains commonly form part of the load-bearing framework (e.g. McKinley et al. 2003) and thus acquire a Type (4a) distribution. However, if the mineral precipitates in situ, it is expected to form a coating sphere cement (e.g. Bjørlykke & Brendsdal 1986), which qualifies it as a Type (3) distribution.
The mineral transitions considered in this paper are illustrated roughly in Figure 4, although additional reactants and by-products are omitted. Considerable changes in mineral distributions occur according to the various mineralogical processes shown. The relative volumes are calculated from stoichiometrically balanced (balanced due to number of atoms) mineral reactions by estimating the relative amount of moles for the different constituents and then calculating their relative masses. Finally, the masses, together with densities from Table 1, provide the relative volumes of the various constituents. Table 2 shows the volumetrically balanced mineral reactions. Aqueous anions and cations are omitted. More complete stoichiometrically balanced reactions can be seen in Worden & Morad (2003).
Diagenetic illite–smectite can also form from kaolinite and K-feldspar in Norwegian Continental Shelf shales (Nadeau et al. 2002). This pathway, however, is neglected in Figure 4, as well as in the simplified mineralogical modelling described in this paper.
Even if quartz is a product of reactions at temperatures lower than 80°C, it is not expected to precipitate until the temperature exceeds this value. Thus, since the pore water is always supersaturated with respect to silica, the excess silica enters other diagenetic mineral reactions, such as the precipitation of feldspars. This phenomenon is also excluded in the modelling. The influence of chemical compaction on total porosity due to mineral dissolution during transitions is also disregarded.
The mineral transitions shown in Table 2 have a significant effect on overall rock stiffness, because they not only entail new constituents with new properties but changes in the distributions and volumes of cement. In Figure 6, effective stiffness changes are modelled during transitions progressing from a starting mineral to a product mineral. The porosity is held constant at 20% throughout all reactions – hence, differences between reactant and product volumes are compensated by varying the amounts of the framework minerals. In all of the examples, 25% of the solid framework consists of clay clasts, while the remainder is detrital quartz. The pore-filling cements are all dispersed in the fluid, which results in the total amount of solids becoming higher when the cement is present.
Effective moduli increase when authigenic smectite is replaced by illite and quartz cement. The increment is mainly due to quartz cement at grain contacts, but the bulk modulus also increases due to stiffer pore fluid. Even if the volume of the products is less than the volume of the reactants (Table 2), the contact-cementing quartz, together with increased cement stiffness, increase the moduli. When modelling with detrital rather than authigenic smectite, the moduli have lower starting points because the cement is load bearing and prevents grain to grain contacts. Otherwise the trends are similar to authigenic smectite and the curves converge when all of the reactants are consumed.
When authigenic smectite converts to chlorite and quartz cement, both the effective bulk modulus and the shear modulus increase smoothly, although the total cement volume decreases, because of the significantly greater stiffness of the chlorite. Detrital smectite follows the same pattern but, as before, the moduli increase more because the starting point is lower.
The kaolinite to illite transition entails a slight increase in effective bulk modulus due to the greater stiffness of illite. The cement bridges the pores, which leads to a positive shear modulus for the effective fluid. This also affects the rock shear modulus. It first increases due to increased effective cement stiffness as illite is stiffer than kaolinite; however, when the reaction moves towards completion, the volume of pore-filling cement is reduced so much that the shear modulus decreases slightly. The bulk modulus is less dependent on the volume of pore-filling cements and hence increases throughout the entire reaction.
Illite and some quartz cement are produced when kaolinite and K-feldspar react. The quartz causes an initial increase in stiffness due to stiffer framework contacts. The illite is not present in large enough concentrations to bridge the pores and, therefore, plays a subordinate role in this example.
When moving from K-feldspar to illite and quartz cement, the contacts become stiffer because of the quartz cementation, and the fluid bulk modulus increases. The rock framework properties increase if the K-feldspar is softer than the other minerals in the framework. The sum of the products is less than the reactant volume, but the contact cement nevertheless dominates the evolution of the moduli and increases the bulk modulus and shear modulus as the reaction proceeds.
When dispersed kaolinite reacts to produce pore-lining chlorite, the bulk modulus increases significantly because the chlorite contributes greatly to strengthening the grain contacts. The reaction also consumes water and ions that might originate from other minerals, and produces a larger volume than the original kaolinite volume. During the first stage of the reaction, two strong but opposing processes determine the effective rock stiffness. The kaolinite is pore bridging but the volume decreases as the reaction proceeds; the pore bridging thus ceases and the moduli decrease. The cementing chlorite, however, counteracts this and leads relatively quickly to an increase in net stiffness.
The K/μ plot (Fig. 6, lower) reveals large differences between the reactions. Except for the kaolinite to illite reaction, all reactions lead to a net decrease in the K/μ ratio. Figure 6 also illustrates relative ratios of the influence of stiffness between various individual reactions when porosity changes are neglected. If several reactions proceed simultaneously, some are more likely to dominate the stiffness evolution than others. This kind of analysis can be performed easily for all kinds of mineralogical reactions in rocks, as long as the stoichiometrically balanced equations and densities are known, and the minerals are classified according to distribution.
In real rocks the porosity is unlikely to remain constant as in these examples. On the contrary, the rocks will be subjected to chemical compaction, which means that when minerals dissolve the framework collapses and starts to occupy liberated space. To perform realistic modelling of real rocks, porosity should either be measured (e.g. from well logs) or modelled consistently with mineral reactions and chemical compaction versus depth. Such considerations are not taken into account for the results shown in Figure 6.
Coupled geological and rock physics modelling
Three scenarios are defined and the diagenetic evolution of mineralogy and rock properties is modelled for each of them. When estimating geochemical reaction rates, a ‘rule of thumb’ is adopted from Worden & Burley (2003), who assert that the rate doubles for every 10°C increase in temperature.
The linear effect of time is not accounted for in the modelled mineral reactions. Slow burial rates would result in higher degrees of mineral transformations at a given temperature than at high burial rates, because the latter has had less time to reach equilibrium. At a depth of 200 m, 10% of the solids consist of framework clay clasts, 70% of framework quartz and the remaining 20% of other minerals. The apportionment of the minerals at 200 m is shown in Table 3, together with the mineral reactions involved in each case.
The porosity–depth curve used in all of the modelling is shown in the upper left of Figure 7. It is taken from the model of Ramm & Bjørlykke (1994) and yields an exponential decrease in porosity with depth. Mineralogy-dependent porosity evolution with depth is not dealt with in this paper; thus the effect of specific minerals on porosity (e.g. the possible porosity-preserving effect of chlorite) is disregarded when obtaining the porosity curve used in the following examples. By using the same porosity curve in all cases, the velocities versus depth curves are unrealistic but allow the rock physics modelling of the three cases to be compared solely on the basis of mineralogy and distribution differences. The concentrations of all minerals not involved in reactions increase with depth, due to an increased concentration of solids as porosity decreases.
The temperatures are stippled in all of the curves shown in Figure 7. Case 1 produces no chlorite coating but less quartz cement than Case 2. Case 2 results in extensive quartz cementation, while the chlorite coating in Case 3 inhibits the precipitation of quartz cement, which results in only a small amount of quartz cement being produced. This is approximated by considering that quartz cementation for Cases 1 to 3 amounts to 25%, 50% and 12.5% of the total porosity loss after 80°C, respectively. All of the pore-filling cement in Cases 1 and 2 becomes pore bridging at depths below about 3.9 km and 3.3 km, respectively, and the cement of distribution Type (2) dominates at depth. In Case 3, chlorite coating of distribution Type (3) dominates.
According to these models, when the total amount of cement shown in Figure 7 exceeds 0.5 (50% of the intergranular volume), the rock starts to lose its seismic permeability. This takes place at different stages in the three cases: at 4 km, 3.5 km and 3.8 km, respectively. When calculating total cement content due to the intergranular volume, Type (4a) minerals are not included as a cement because they are part of the effective load-bearing framework and do not fill the space between framework grains.
As the porosity, framework clay and initial framework quartz concentrations are the same in all cases, the velocity differences shown in Figure 8 are not very large; nevertheless, they are large enough for the curves to be distinguished easily after incipient cementation. The circles around the curves reflect mineralogical and physical changes in the rock.
Five depth-related changes characterize the Case 1 velocity curves:
the initiation of quartz cementation at 2.5 km;
incipient pore bridging at 3.96 km;
the entire consumption of smectite at 4.0 km (125°C);
incipient seismic isolation of pores at 4.1 km;
the entire consumption of K-feldspar at 4.2 km.
Not all of these changes have the same impact on the velocity curves. Figures 3 and 6⇑ provide good indications as to which mineralogical transformations have the dominant influence on the overall stiffness in the case of simultaneous transformations. The first figure shows those distributions that yield the highest stiffness, while the latter illustrates stiffness changes during various mineral transformations.
Case 1 has the highest smectite content and equal amounts of detrital (Type (4b)) and authigenic (Type (3)) smectite. It is clear that the decreasing smectite content has a considerable influence on the velocities. The second factor (together with smectite) exerting the greatest influence on velocities is the bridging of pores, which yields effective pore fluid with positive shear stiffness. Cases 1 to 3 are all influenced moderately by the incipient seismic isolation of pores, while the other diagenetic changes play inferior roles in Case 1.
A similar analysis of Case 2 leads to the conclusion that, in addition to the two factors that dominated Case 1, quartz cementation is more extensive here and should be included as an important factor in the velocity evolution. Reactions (3) and (4) (Table 2) only have a minor influence on the velocities in this case.
Case 3 is dominated by the chlorite reactions (reactions (2) and (7)). Figure 6 shows that a considerable increase in effective stiffness can be expected from both reactions. Pronounced velocity gradient changes are also observed in Figure 8 when all of the smectite and kaolinite are consumed and chlorite production ceases at 4.35 km and 4.5 km.
The velocity differences between Cases 1 and 3 would have been greater if porosity had also been modelled as a function of volumetric differences arising from the mineral reactions. However, porosity evolution as a function of mineralogy and mineral transformations is beyond the scope of this paper. The volumes of the mineral reactions in Table 2 were used to obtain the correct relative ratio between the various minerals at given porosities.
Real well example
Figure 9 compares modelled velocities with well-log velocities. The black dots in the three lower figures are well-log velocities, which are the same in all of the figures. The modelled velocities vary from Case 1 to 3. The mineralogical concentrations for Case 1 are shown in the upper plots. The temperature–depth gradient is 36.5°C km−1 in the studied interval, and some of the mineral reactions are not completed by 4.4 km (c. 120°C) (see Fig. 9). Thus, the concentrations of the precipitated cements are too small to induce the same variations between the different models as observed in Figure 8. The relative concentrations of the minerals consumed in the reactions do not necessarily decrease exclusively with depth, because porosity loss increases the relative concentrations of the solids and thereby the concentrations of each mineral. The density, porosity, clay content, temperature and fluid properties derived from the well log are used in the velocity modelling. When it comes to estimating mineral concentrations, the relative ratios between the minerals are calculated for Cases 1 to 3 and then normalized to the clay content from the log.
At all depths, 10% of the clay is represented by clay clasts in the framework. The reactions involved are shown in Table 3. In the shallow section of the log (depth <2.5 km), it is assumed that no Type (1) and Type (3) cements are present, because the log velocities are so low. However, authigenic smectite is introduced at depths greater than 2.5 km (T≈50°C). The mean deviations for Vp and Vs are: ±151.9 m s−1 and ±132.5 m s−1 in Case 1; ±177.8 m s−1 and ±174.4 m s−1 in Case 2; and ±147.3 m s−1 and ±115.5 m s−1 in Case 3. All of the modelled velocities in the deepest part of the well are too high and, although the deviations are rather close to each other, some deductions about the mineralogy can be made. For example, Case 1 would have shown a better fit if load-bearing soft minerals had been present. In Case 2, the quartz cement is responsible for the high velocities; lower concentrations would have decreased the gap between the modelled and real velocities. Case 3 is marginally closest to the real data; however, although the amount of quartz cement is low and chlorite is yet to become extensive, the velocities are too high for the deep data. Greater amounts of load-bearing or pore-filling clay, rather than contact-cementing clay, would have lowered the velocities.
These considerations are obviously limited by the simplified mineralogical modelling. Nevertheless, the methodology as it stands is suitable for more advanced geological input and the conclusions would be more nuanced.
Reflectivity of diagenetic processes
A hypothetical downgoing P-wave reaching an interface between two different layers (rock types) is now considered. The reflected wave is split into P- and S-waves due to impedance differences between the upper and lower layer (see Table 4 for the layer properties). Figure 10 shows the reflection coefficients versus the angles of the reflected P- and S-waves, designated Rpp and Rps, respectively. The peaks in the curves represent the critical angles at which no P-waves are transmitted to the underlying layers. At higher angles, the reflection coefficients consist of real and imaginary parts. Only the real part is plotted, but commonly angles lower than the critical angles are in focus for conventional seismic data. Differences in reflection coefficients down to 0.02 should be identifiable from good quality seismic data.
In the two plots above (Fig. 10), the upper layer is an isotropic cemented shale with 3% porosity, which has been modelled using the shale cementation theories of Dræge et al. (2006). The lower layer has been modelled using the theories advanced in this paper for sandstone with the following variable characteristics:
quartz cemented with seismically impermeable pores;
quartz cemented and seismically permeable;
containing pore-bridging cement;
The porosity of the sandstone varieties was held constant at 10%, with the exception of the chlorite-cemented example in which the porosity was set at 22.5%. The reason for this is that Ehrenberg (1993) observed that porosities are 10–15% higher in chlorite-cemented sandstones, because the chlorite inhibits the development of quartz cement.
The figure shows that the quartz-cemented sandstone containing isolated pores stands out by reaching the critical angle before the others. In addition, this sandstone clearly elicits the highest P-wave reflectivity of the normal incident wave. The quartz-cemented and pore-bridged sandstones differ most at low angles for Rpp, and at medium angles for Rps, but are otherwise hard to distinguish from each other. A normal incident P-wave never returns reflected S-waves, so there must be a non-zero incident angle to separate the coefficients of the S-wave reflections. The chlorite-cemented sandstone has the largest critical angle, while the uncemented sandstone has an almost negligible S-wave reflection and a lower P-wave reflection than the others for large incident angles.
The two following plots show the evolution of the reflection coefficients versus angle for an overlying uncemented sandstone. The same patterns are seen, but the positive Rpp coefficients are somewhat higher in this example. All scenarios differ enough at sub-critical angles to be separated seismically.
The last reflection example is for a high porosity, chlorite-cemented sandstone overlying quartz-cemented sandstone. The coefficients were estimated when the underlying cemented sandstone was seismically permeable and seismically isolated. In the first case, the S-wave reflections are practically absent for all angles, while the P-wave reflections are pronounced but considerably lower than in the case with seismically isolated pores at low angles. Hence, the two scenarios are separated easily by P-wave reflections alone. In addition, the S-wave reflections differ considerably, with the critical angle varying by about 20°. The modelling indicates that a high-porosity chlorite-cemented layer may constitute a good reflector when overlying quartz-cemented sandstone with lower porosity. Furthermore, seismically impermeable sandstones have different signatures than their seismically permeable counterparts, which should be recognizable from seismic data.
Although the reflection coefficients indicate the potential presence of good reflections in some places, there are limits to how thin a layer can be and still be seismically detectable. This limit depends on the wavelength, which further depends on the frequency and velocity. A reflector that is thinner than 1/4 of the wavelength is generally considered to be beyond seismic resolution. For the chlorite-cemented sandstone the wavelength of a 50 Hz wave will be 4343 m s−1/50 s−1 = 86.9 m. The minimum thickness for this layer to be seismically detectable should thus be c. 22 m. Slower layers have shorter wavelengths for a given frequency and, hence, better resolution.
The model and strategy presented here depend on knowledge of the mineralogical properties of the minerals involved. Effective elastic properties of clay minerals have been derived from theoretical computations (Katahara 1996), combined theoretical and experimental investigations on clay–epoxy mixtures (Wang et al. 2001) and direct measurements (Vanorio et al. 2003). In Figure 11, the mineral transitions from Figure 6 are reproduced with altered mineral properties. The alternative properties are listed in Table 5. The modelling shows that alternative kaolinite properties yield the greatest changes; however, even if the kaolinite bulk modulus and the shear modulus increase by 433.6% and 328.3%, respectively, the ‘new’ kaolinite only alters the corresponding effective rock moduli by 18.5% and 31.8%. Alterations of the properties of the pore-filling minerals induce smaller changes unless they are pore bridging. This is very well illustrated for the shear modulus with altered illite properties (blue curve) when kaolinite reacts to illite. The difference is large in the first part of the reaction where there still is pore-bridging cement; but, when pore bridging ceases, the curve coincides with that showing the original illite values (stippled brown).
An increase in the properties of authigenic smectite results in a larger increase in overall rock stiffness than for detrital smectite, because the latter is load bearing and still softens the framework significantly. New chlorite properties reduce the effective shear modulus slightly.
Figure 11 shows that the distribution type assigned to each mineral is more important for overall rock stiffness than the mineral properties themselves. This also agrees with the conclusion of Dvorkin & Nur (1996), who suggested that reducing the stiffness of contact cement does not significantly reduce the stiffness of the cemented aggregate.
The chemical formulae used to calculate mineral molar masses and relative mineral volumes are adopted from Worden & Morad (2003), but they are not all unique. For example, the composition of some clay minerals, such as illite, chlorite and smectite, may vary within certain limits due to the chemical environments in which they are generated. The chemical formulae applied here can be viewed as average compositions. The new mineral densities shown in Table 5 will also change the relative volumes of the reactions shown in Table 2. By normalizing the reactions to the reactants, the alternative illite density in reaction (1) entails an illite volume decrease of 2.8%, and the new smectite density results in a volume increase of 4.7% for quartz and illite in the reaction. Similarly, the new chlorite density in reaction (7) leads to a reduction in chlorite volume by 5.5%, which is the same result as obtained when applying the kaolinite density from Table 5. Figure 11 does not account for changes in volumetric relations between the constituents, only stiffness changes. But the volumetric changes also influence the overall rock stiffness because of porosity and increased/decreased concentrations of compliant or stiff minerals.
Only a selection of diagenetic transformations has been used in this paper, while there exist numerous other more or less inferior transformations of local importance. Another aspect not considered here is the possibility of calcite cementation. Calcite is assumed to behave in a similar fashion to quartz; i.e. it will be distributed near the contacts (as in Type (1) cement). Calcite will thus occupy the same space as quartz cement. Calcite often precipitates at lower temperatures than quartz (Bjørkum & Walderhaug 1990) and might therefore prevent or reduce the extent of quartz precipitation.
It is also assumed that all elongated components in the rock are orientated randomly, which will lead to isotropic sandstones. Sandstones containing thin shale laminations are not considered.
When comparing these models with real well-log data, it is assumed that the detrital composition of the sandstone minerals remains unchanged during deposition. This is probably a dubious assumption, but the problem can be approached by decomposing the sandstone log into smaller intervals that were deposited in similar depositional environments fed by similar sources, and performing separate modelling on each of them.
A way of bringing this strategy further is to introduce holistic and advanced geological modelling that includes the complicated interplay between various mineral reactions and related porosity evolution.
This paper presents some innovative ways of incorporating geological and geochemical processes in rock physics modelling. The idea of classifying minerals according to their distribution is the key to implementing mineralogical reactions into rock physics modelling; and the development of a rock physics model capable of accommodating multiple simultaneous distributions is the enabler. When combined, these two aspects constitute a new interdisciplinary workflow, in which the first task involves the advanced geological modelling of mineralogy and porosity evolution, followed by the rock physics modelling of seismic properties that have been affected by diagenesis.
The rock physics modelling reveals significant differences between diagenetic scenarios when reflection coefficients are plotted versus angles of incidence. It should, therefore, be possible to discriminate between pore-bridging quartz-cemented sandstones, high porosity chlorite-cemented sandstones and uncemented sandstones using high quality seismic data, when they occur in sufficiently thick intervals. The strategy presented here is thus considered to be a new and valuable rock physics tool for exploration purposes, including previously little used information. Furthermore, when combined with high quality geological input, the procedure can be inverted to predict subsurface saturation, lithology, porosity and microstructures.
A. D. would like to thank The Norwegian Academy of Science and Letters (VISTA) for financial support. The support from Statoil and Norsar is also acknowledged.
- Received September 29, 2005.
- Accepted March 10, 2006.
- 2006 EAGE/Geological Society of London