Manuscript received by the Editor 18 December 2007; revised manuscript received 11April 2008; published online 18 November 2008.

Transcription

1 GEOPHYSICS, VOL. 73, NO. 6 NOVEMBER-DECEMBER 2008; P. E197 E209, 9 FIGS / Rock physics modeling of unconsolidated sands: Accounting for nonuniform contacts and heterogeneous stress fields in the effective media approximation with applications to hydrocarbon exploration Ran Bachrach 1 and Per Avseth 2 ABSTRACT By treating contact stiffness as a variable, one can extend the effective medium approximation used to obtain elastic stiffness of a random pack of spherical grains. More specifically, we suggest calibrating effective media approximation based on contact mechanics by incorporating nonuniform contact models. The simple extension of the theory provides a better fit for many laboratory and field experiments and can provide insight into the micromechanical bonds associated with unconsolidated sediments. This approach is motivated by repeated observations of shear-wave measurements in unconsolidated sands where observed shear-wave velocities are lower than predicted by the Hertz-Mindlin contact theory. We present the calibration process for well-log data from a North Sea well penetrating a shallow-gas discovery and a deepwater well in the Gulf of Mexico. Finally, we demonstrate the benefit of using this model for amplitude variation with angle AVA analysis of shallow sand targets in exploration and reservoir studies. INTRODUCTION The effective properties of sphere packs have been used as an analog for the behavior of unconsolidated sands for many years. Contact between two spheres has been well characterized for different loading paths, boundary conditions, and grain radii Johnson, 1985, and the forces acting on a two-particle arrangement have been characterized using normal and tangential stiffness Winkler, 1983; Norris and Johnson, 1997; Mavko et al., The effective medium approximation EMA associated with granular media attempts to average two-grain contacts into the effective behavior of a pack of aggregates with many contacts. Gassmann 1951 calculates the effective elastic properties of a dense hexagonal pack of spheres and compares the results to those of seismic velocity measurements in unconsolidated sands. Digby 1981 and Walton 1987 develop effective-medium-averaging techniques to estimate the effective properties of a random sphere pack while considering contact laws for adhesive contacts, rough contacts, and smooth contacts. Winkler 1983 shows that for Hertz-Mindlin contact stiffness, the simple volumetric averaging of Digby can be presented in terms of normal to tangential stiffness ratios. Muhlhaus and Oka 1996 analyze dispersion and wave propagation in granular media using homogenization of discrete equations of motion. Norris and Johnson 1997 rederive the EMA of Walton and Digby using energy density functions for different contact models and show that, in general, because the contact force between the grain is path dependent i.e., relates to the history of loading, the EMAwill relate to the loading path. With the advancement of computer power, the use of granular dynamic models has improved the understanding of the elastic behavior of a random pack of spheres and associated force distributions Makse et al., 1999, These studies show that although bulk modulus is predicted well using EMA theory, shear-modulus predictions do not follow conventional EMA predictions because the grains tend to relax from the affine, macroscopic deformation i.e., each grain translates according to the direction of the macroscopic strain or rotate. The effect of rotation of grain is also studied by Pasternak et al The assumption of affinity given by the EMA theory is approximately valid for the bulk modulus but seriously flawed for the shear modulus. Thus, the uniform strain assumption breaks down, causing the EMA approximation for granular pack to differ considerably from observed values. Several experimental studies demonstrate significant differences between the shear modu- Manuscript received by the Editor 18 December 2007; revised manuscript received 11April 2008; published online 18 November WesternGeco/Schlumberger, Houston, Texas, U.S.A. rbachrach@slb.com; ran.bachrach@gmail.com. 2 Rock Physics TechnologyAS, Bergen, Norway. per@rpt.info Society of Exploration Geophysicists. All rights reserved. E197

2 E198 Bachrach and Avseth li predicted by Hertz-Mindlin effective medium models and empirical results, including Winkler 1983, Goddard 1990, and Zimmer et al One consequence of the inadequacies of using EMA to describe shear behavior is in the prediction of Poisson s ratio and V P /V S ratio in unconsolidated sands e.g., Avseth and Bachrach, 2005; Sava and Hardage, Manificat and Guéguen 1998 show that a contact roughness model can explain the higher-than-expected Poisson s ratio observed in sands. Bachrach et al show that while analyzing P- and S-wave velocities in unconsolidated sands, different contact curvatures of grains can be accounted for and will not change the V P /V S ratio. In near-surface sediments, the observed Poisson s ratio can be used to determine the fraction of slipping contacts to nonslipping contacts by simple averaging of two representative media: one with and one without tangential contact stiffness. We address two dependent problems that often appear when using rock-physics models in granular media. The first problem is the expansion of the EMA theory of Norris and Johnson 1997 to account for variable contact models in granular systems. Specifically, we show that choosing a binary model where tangential stiffness of a contact can be zero, or following Hertz-Mindlin, enables us to derive an EMA for granular pack where all contacts are not the same. The second problem we address is applying the theory to real data, where we show the calibration process depends on additional granular properties such as effective contact ratio, not just the coordination number. The paper is organized as follows: We first provide a short review of the basic theory associated with EMAof granular media. Next, we introduce the binary contact model that accounts for nonuniform contacts in EMA. We follow with a detailed discussion of the steps associated with model calibration for real data. Finally, we present examples of how the theory is applied to well-log data and rockphysics template analysis Ødegaard andavseth, 2004; Avseth et al., THEORETICAL BACKGROUND Contact stiffness In this section, we closely follow the derivation of Norris and Johnson All detailed derivation is given in their paper. Here, we only repeat relevant equations associated with nonuniform graincontact forces. Asingle contact between two spheres can be characterized by normal and tangential stiffness, defined as Winkler, 1983; Mavko et al., 1998 S n F n, S F t, where F n,f t are the normal and tangential components of the force acting on the contact and where and are the normal and tangential displacements, respectively, resulting from such a force. In general, the behavior of a single contact between spheres depends on the loading path and additional parameters such as friction, cement, and 1 adhesions Norris and Johnson, The actual normal stiffness between two elastic spheres has been modeled and experimentally verified Johnson, 1985 using the Hertzian contact model: where S n 4aG 1, 2 a R 3 3F n R 1 /8G is the radius of contact area between two spheres, G is the shear modulus, and is Poisson s ratio of the sphere material, which is assumed to be isotropic. The effective contact radius R is defined as R R 1 R 2, 4 where R 1 and R 2 are the radii of the two grains in contact, as shown first by Hertz in his 1882 seminal paper, On the Contacts of Elastic Solids Love, 1927; Johnson, Note that R is related to the curvature of the actual contact surface. To derive the tangential stiffness, one needs to assume a specific loading path and boundary conditions. One choice of the tangential model is the perfectly smooth case Walton, 1987, where the tangential stiffness is zero. Another typical choice associated with Hertzian contact and friction is the Hertz-Mindlin model, where the tangential stiffness is given by Mindlin, S t 8aG 2. 5 As discussed by Norris and Johnson 1997, the tangential stiffness depends on the boundary conditions and loading path. In practice, for most poorly consolidated sediments, these are often not well known. This subject will be discussed further. Derivation of effective bulk modulus for a dry pack of spherical grains under hydrostatic loading To illustrate how the effective bulk modulus of a random pack of identical spheres can be obtained, we present the following simple derivation. The pressure is defined as force/area. If we consider the solid fraction of a spherical volume of radius R with porosity through which the forces are transmitted and we consider n points of contacts per spheres also known as the coordination number, we can write the hydrostatic pressure as P F n1 n 4R 2. 6 This equation is identical to equation 71 in Norris and Johnson Note that in equation 6 the normal contact force, which is a

3 Modeling unconsolidated sands E199 local point, is averaged along a spherical radius R, which is associated with the representative volume of the grain. This point will be addressed later. The effective volumetric increment is defined as equation 20 in Norris and Johnson 1997, where V 0 is the volume of a single grain. Note that the EMAaveraging rule is also used in the simple form of equation 8. e dv V 4R2 dr 3dR 4R 3 /3 R. Bulk modulus relates the pressure increment to the volume increment; therefore, K eff dp e 7 1 n df n 1 n 12R dr 12R S n, 8 Motivation EXTENSION OF EMA FOR NONUNIFORM CONTACTS As discussed in the introduction, the stress distribution on granular packs is nonuniform. The tangential contact stiffness associated with two grains is path dependent, so it is clear that equation 10 will not capture such heterogeneities. In the next section, we discuss one way to accommodate stress-field heterogeneities. which is identical to the formulas derived by Norris and Johnson 1997 and Walton In the simple derivation above, equation 6 is the EMA for hydrostatic state of stress. EMAassumes that because the forces are not interacting with each other, on average the pressure can be represented as the force times the contact number that acts on the solid part of a reference sphere. Equation 7 is the EMA for strain, where it is assumed that the volumetric deformation can be related to the normal displacement dr. We also distinguish between the effective contact radius R and the volumetric averaging radius R. Although R affects the contact stiffness of a two-grain configuration, the volumetric averaging radius R is the volume over which EMA is performed. This distinction is discussed later. Binary scheme We now consider a case where all contacts are not the same.asimple way to introduce different contacts is to assume a binary mix where some contacts may behave as smooth contacts with zero tangential stiffness and other contacts may have finite tangential stiffness, as given by equation 5. This assumption tries to account for the heterogeneities in stress chains as observed in laboratory measurements and numerical simulations Ammi et al., 1987; Mueth et al., 1998; Geng et al., 2001; Makse et al., The binary model, simplified from the true grain pack, can be viewed as a specific probability distribution of nonuniform contacts. Given the binary model assumption, and because the strain energy is linear, we can rewrite equation 10 as Effective elastic modulus with normal and tangential stiffness The derivation of effective bulk modulus is very simple, but the derivation of effective shear modulus for hydrostatic loading and the derivation of elastic modulus for nonhydrostatic loading are more complicated. An elegant derivation of the effective media associated with granular packs is given by Norris and Johnson 1997, who derive the effective elastic moduli by differentiating the strain energy density per unit volume U, defined as U 1 V contacts F du, where F du F n d F t d with respect to strain and V is the volume associated with EMA averaging. The assumption made by Digby 1981, Walton 1987, and Norris and Johnson 1997 is that all contacts are the same and that the volumetric average associated with equation 9 is U 1 n1 F du V contacts V 0 F du 10 9 U U smooth U no-slip F du smooth contacts n1 F du f no-slip contacts V 0 s F du f t F du, 11 where f s is the fraction of smooth contacts in the system and f t 1 f s is the fraction of no-slip contacts. Equation 11 says that the strain energy density for unit volume can be related to more than a specific type of boundary condition. The macroscopic stress-strain relations are derived by differentiating equation 11 with respect to the strain Norris and Johnson, 1997, their equation 30: ij U U smooth U no-slip. 12 e ij e ij e ij The effective elastic modulus is then given by the combination of smooth and no-slip contacts, which is a simple average of elastic moduli formulas derived by Walton 1987 and Norris and Johnson

4 E200 Bachrach and Avseth Similar conclusions can be reached using the forces acting on the contacts as derived by Walton For a homogeneous system i.e., a granular pack where all contacts are the same, one can write the stress-strain relations associated with equation 12 as e.g., Muhlhaus and Oka, 1996; Pasternak et al., 2006 ij C * ijkl S n,s t e kl, 13 which is the effective medium approximation of the elastic modulus derived from equation 12 for identical contacts with given normal and tangential stiffness. Then, from the linearity of equation 12, we can write the macroscopic stress-strain relations for nonuniform contacts as ij f s C * ijkl S n,s t 0 f t C * ijkl S n,s t 0e kl 14 and the overall effective stiffness as C* ijkl S n,s t. The case of f t 0.5 is similar to the case investigated by Manificat and Guéguen It is also interesting to note that equation 14 suggests the Voigt average, which implies isostrain Mavko et al., 1998, is the one appropriate isostrain condition to the binary contact problem. Recall that the macroscopic stress-strain relationships were derived in this case from a general energy density function. Equation 12 assumes that an averaged macroscopic strain can be defined over the volume. Thus, the isostrain interpretation is consistent with equation 14. For the specific case of hydrostatic loading, the bulk and shear moduli for Hertz-Mindlin contacts are given by K eff n1 12R S n1 n, G eff S n 3 20R 2 S t. 15 The effective shear modulus in terms of volume fraction of no-slip contacts f t is given by Poisson's ratio K eff n1 12R S n1 3 n, G eff S n f t 20R 2 S t Fraction of no-slip contacts Figure 1. Effective dry Poisson s ratio as a function of volume fraction of no-slip contacts. Following Bachrach et al. 2000, using equations 2, 3, and 14, we can write the effective Poisson s ratio in terms of f t as eff S n f t S t 4S n f t S t f t 1 2f t f t 1, 17 which clearly shows that for f t 0, eff 1/4, as first derived by Walton Equation 17 also shows that the effective Poisson s ratio is only a function of the mineral Poisson s ratio and the fraction of no-slip contacts, where an increase in no-slip contacts will reduce Poisson s ratio. In Figure 1, we present the result of applying equation 17 to quartz spheres.asimilar result is produced by Bachrach et al. 2000, which is derived using the Hashin-Shtrikman upper and lower bounds. It is important to bear in mind that the maximum value of eff 1/4 is a dry value. The presence of pore fluid in the system should be addressed separately. One approach is to use Gassman s relationships, following Gassman s 1951 example. The theory does not account for surface processes that may occur in the presence of residual water saturation or for cement. Smooth contacts and slip during wave propagation Seismic waves possess very small strain amplitudes, so one may ask if slippage occurs when a shear wave passes. The mathematical derivation of EMAin the previous section uses a given state of stress. As sediments are compacted, pressure increases and porosity decreases. When mapping elastic moduli versus burial depth, as a proxy to elastic moduli versus increasing stress, it is clear that as porosity reduces in a nonreversible way, grains slip and rotate during the compaction process. However, at a given burial depth with a given state of stress and porosity, when one probes the formation with a low-strain-amplitude seismic wave, does zero tangential stiffness imply that grains must slip? Winkler 1979, 1983 points out that, in general, for seismic waves with small strain amplitudes, frictional losses are not observed, which can be interpreted as either there is no slip at the grain contacts or there is no friction between grain contacts. At first, these two explanations seem to be at odds. However, experimental and numerical simulation of granular packs shows that force distribution is not homogeneous and, in general, not all contacts are stressed. Moreover, analysis of photoelastic images of stress chains Majmudar and Behringer, 2005 has shown that under isotropic loading, the normal forces are larger than the tangential forces in granular packs. In addition, stress anisotropy imposes direction-dependent elastic behavior of grain contacts. Thus, the seismic velocities as described by Hertzian contact theory will likely diverge farther from observed values because of stress anisotropy see Xu, 2002 and Vega, 2003, among others.

5 Modeling unconsolidated sands E201 From equation 16, it is clear that shear waves will propagate even when all grains are smooth or if all tangential forces are zero. Therefore, one way to understand the granular pack is by determining what fraction of the low-strain shear load is supported only by normal forces, relative to the fraction supported by both normal and tangential forces. We also note that, in general, f t may be stress dependent. We revisit this issue in the well-log example and discussion sections. Effective contact curvature, coordination number, and model calibration in reservoir settings From equations 2 6 and 16, we can write the effective bulk and shear moduli for hydrostatic loading explicitly as K eff G eff G 2 1/3n 2R 1/ G 2 1/3n 2 1 R 2R G n 2R 1/3 R P 1/3f t. R P 1/3 1/3 P 1/3 1/3 18 Here, P is the effective pressure at grain contacts. When predicting bulk and shear moduli for sands at given depth using this equation, we normally assume mineralogy, porosity, and overburden stress as known parameters. However, as shown in equation 18, calibration parameters are related to both coordination number n and the ratio R /R, unless the granular aggregate is made out of identical spheres where R /R 1. We note, as shown schematically in Figure 2, that R /R can be larger or smaller than one. The ratio R /R captures some aspects of grain angularity and sorting, and it can be viewed as a mechanical parameter that characterizes the averaged effective contact radii and general grain-size distribution. Another issue with model calibration is that, in general, the coordination number changes with the porosity of the sediment and with stress. Murphy 1982 establishes the relationship between coordination number and porosity from theory and observations for porosity ranges between 0.2 and 0.6, where for 20% porosity n 14 and for 60% porosity n Makse et al shows that in their granular dynamic simulation, for low-effective-stress ranges, the coordination number can change without a porosity change but will change with a pressure change. Their coordination number/pressure dependency is formulated as where P 0 is an empirical-fitting parameter. It is interesting to note that for random pack at low effective stress with porosity of 36%, Murphy s 1982 coordination number is about nine, but Makse et al predict a coordination number close to six. In natural materials, stress and compaction are related to each other. In our opinion, it is beneficial to look at the coordination number as a function of porosity and stress. If we follow Bowers 1995 definition of the virgin compaction curve, it is possible to relate the coordination number to porosity and stress in a fashion similar to the way stress and porosity are related to each other along the virgin compaction curve. Thus, the choice of relating porosity to the coordination number Murphy, 1982 is a statement that the sedimentary pack is following a specific compaction curve, such as the loading curve Bowers, One way to reconcile the numbers of both Murphy 1982 and Makse et al is to consider different compaction/stress history associated with sediment burial. Proposed workflow for parameter calibration From the above discussion, it is clear that to calibrate a model to, for instance, well-log data, we must account for porosity, pressure, and effective contact curvature radius. However, without specific knowledge of either coordination number or R /R, we suggest the following steps to calibrate sedimentary characteristics from P-wave, S-wave, and density data Derive dry Poisson s ratio using the Gassmann theory; see Mavko et al and estimate f t from Poisson s ratio or V P /V S ratio using equation 17. Derive effective stress from Terzaghi s principle, P P over P P, where P over is the overburden stress and P P is the pore pressure. Because we are dealing with unconsolidated sediment, we assume that Biot s coefficient of effective stress is close to unity. Use equation 18 to fit functional dependence for effective dry shear modulus such that a constant c n 2 R /R is estimated when fitting the data. For hydrostatic pore pressure under normal loading situations, we recommend using the coordination number/porosity relation compiled by Murphy 1982 and given in Mavko et al. 1998, and we neglect pressure dependencies. If the reservoir has gone through changes in pore pressure because of production, one must define a different dependency because stress path affects the seismic response and compaction of sediments Sayers, This may be done by the calibrating equation 19 through laboratory experiments. np 6 P P 01/3, 19 Figure 2. Illustration of three contact scenarios where R /R can be larger or smaller than one.

6 E202 Bachrach and Avseth 4 5 Use equation 18 to derive dry bulk modulus for the system. Use Gassmann s equation see Mavko et al., 1998 to correct for in situ saturations. The calibration parameter f t is a characteristic of the sedimentary bed. For a single well log with different sand beds, one may have to calibrate this constant for each individual bed. This calibration procedure does not affect dry Poisson s ratio or V P /V S ratio and microstructure parameter f t because they are independent of c. Microstructure parameter c provides insight into the general texture and packing of the sediment. APPLICATION FOR HYDROCARBON EXPLORATION Well-log observations Figure 3 shows well-log data from Avseth et al a shallowgas reservoir offshore Norway, including sonic velocities V P and V S, density, Poisson s ratio, porosity, gamma-ray data, and saturation. Note the large drop in P-wave velocity and Poisson s ratio in the upper thick sand interval i.e., zone with low gamma-ray values. This unit is filled with gas. A thinner sand unit lies below the gas zone, which is water saturated. Here, the Poisson s ratio is higher than the background Poisson s ratio of the shale. We apply the five-step procedure described above and derive the results shown in Figures 4 and 5. If we use the original Hertz-Mindlin contact theory with only no-slip grain contacts f t 1, weobtain a poor match between predicted and observed shear modulus and Poisson s ratio Figure 4. However, using the estimated value of f t 0.07, we obtain a good match for all the elastic moduli. If we use the coordination number as a function of porosity from Murphy s 1982 approximation, we can derive the averaged ratio of R /R 2/3. The shear modulus is still predicted wrong in the zone between 605 and 615 m. A closer look at the well-log data shows that, in this zone, the depositional environment changes and a laminated sand-shale sequence is present, a geologic characteristic not included in our modeling. a) b) c) d) e) f) g) Depth (m) P-velocity (km/s) S-velocity (km/s) Density (g/cm 3 ) Porosity GR S w Figure 3. Well-log observations of shallow unconsolidated sands embedded in shales: a P-wave velocity, b S-wave velocity, c density, d Poisson s ratio, e porosity, f gamma ray GR, and g water saturation S W. The dashed line embraces the zone of interest. Note the small zone of laminated sands between 605 and 615 m data fromavseth et al., 2007.

7 Modeling unconsolidated sands E203 a) b) c) Depth (m) G (Pa) K (Pa) Figure 4. Model versus well-log observation for a shallow-gas well: a Poisson s ratio, b shear modulus G Pa, and c bulk modulus K Pa. We use f t 1 and obtain a good match for bulk but poor for shear.

8 E204 Bachrach and Avseth a) b) c) Depth (m) G (Pa) K (Pa) Figure 5. Model versus well-log observation for a shallow gas well: a Poisson s ratio, b shear modulus G Pa, and c bulk modulus K Pa. Here we use f t 0.07 and obtain a good match for both bulk and shear. The zone where the shear modulus underpredicts the observation is where the depositional environment has changed and a laminated sand-shale sequence is present in the well see gamma-ray plot between 605 and 615 m in Figure 3.

9 Modeling unconsolidated sands E205 Figure 6 presents well-log data from a vertical, deepwater Gulf of Mexico GOM well. These data are presented by Bachrach et al. 2007, and we use vertical effective stress estimates derived in that study as the confining pressure in equation 18. The vertical effective stress is about 23 MPa for the target zone in this well. The data are coming from a deep, poorly consolidated sand reservoir and include sonic and shear log, density, porosity, saturation, and clay volume. We apply the five-step procedure and derive the results shown in Figures 7 and 8. We are able to fit Poisson s ratio, bulk, and shear modulus using ft Again, if we use Murphy s 1982 coordination number porosity relations, we can derive the averaged ratio R /R of 0.15 from the fit. It is interesting to note that, in the case of the deeper well, the fraction of no-slip grains is larger than in the shallower case. This is expected because, at larger effective stresses, more grains are likely to be loaded and f t should be larger.as mentioned, it is difficult to interpret R /R because we are actually calibrating c n 2 R /R. If the assumption that Murphy s 1982 relations are valid for both reservoir sands, then the difference in R /R may indicate that the sediments in the deeper well are more angular than in the shallower well. Rock-physics template analysis of well data Implication for AVO modeling and hydrocarbon exploration Rock-physics templates RPT were introduced by Ødegaard and Avseth 2004 and further developed by Avseth et al as a tool to evaluate the relationship between local geologic parameters, rock-physics properties, and expected amplitude-variation-withoffset AVO responses. Figure 9 shows rock-physics models superimposed on the well-log data shown in Figure 3. The blue line represents brine-saturated sands at the in situ effective pressure. The red a) b) c) d) e) f) g) Depth (m) P-velocity (km/s) S-velocity (km/s) Density (g/cm 3 ) Porosity Vcl S w Figure 6. Well-log observations of deepwater GOM unconsolidated sands embedded in shales: a P-wave velocity, b S-wave velocity, c density, d Poisson s ratio, e porosity, f volume of clay V cl, and g water saturation S W. The zone of interest is marked by a dashed line.

10 E206 a) Bachrach and Avseth b) c) d) V cl Depth (m) G (Pa) K (Pa) V cl Figure 7. Model versus well-log observation for a deepwater GOM well: a Poisson s ratio, b shear modulus G Pa, c bulk modulus K Pa, and d volume of clay V cl. We have used f t 1 and obtained a good match for bulk, but poor for shear.

11 Modeling unconsolidated sands E207 a) b) c) d) V cl Depth (m) G (Pa) K (Pa) V cl Figure 8. Model versus well-log observation for a shallow gas well: a Poisson s ratio, b shear modulus G Pa, c bulk modulus K Pa, and d volume of clay V cl. Here we have used f t 0.35 and obtained a good match for both bulk and shear.

12 E208 Bachrach and Avseth line represents the 100% gas-saturated sands. The green line represents a best-fit shale model. In Figure 9a, we assume no slip for the unconsolidated sand model, whereas in Figure 9b, we use the estimated value of f t 0.07 and obtain a very good match between the sand models and the sand-rich data points. The RPT analysis indicates that the shallow sands are completely unconsolidated and the sand grains are packed loosely with close to no friction at the contacts. The dry and wet V P /V S ratios are affected drastically by this reduced shear effect; the acoustic impedances are affected relatively weakly by this effect. The improved rock-physics models for unconsolidated sands demonstrated in this study can help us better predict expected AVO signatures at very shallow burial depths. In particular, this reduced shear effect can cause an otherwise no- or low-impedance contrast with negative AVA gradient, also known as class II or class III AVO response, to turn into low impedance with positive AVO gradient class IV AVO response see Avseth and Bachrach, 2005 for details. Hence, this effect may explain why class IV AVO signatures are commonly observed in shallow, unconsolidated sediments. DISCUSSION We have shown how we can describe the seismic properties of unconsolidated sands, taking into account reduced tangential shear a) V P /V S b) 5 V P /V S Acoustic impendance (g/m 2 s) Acoustic impendance (g/m 2 s) Figure 9. RPT analysis of well-log data from a shallow gas reservoir: a no-slip i.e., Hertz-Mindlin formulation and b slip factor f t We obtain an improved match with the well-log data using the reduced shear model. Shale Volume Shale Volume stiffness during wave propagation. The parameter f t, derived from Poisson s ratio, can be interpreted as the accommodation of EMA to nonuniform contact properties. In our two examples, f t for the deeper sands was larger than the value for the shallow well. In general, we expect that as stresses increase, more contacts get locked and thus f t should be stress dependent. When we use two parameters, we will always be able to fit compressional and shear data such as well logs. However, our goal is to find a model that can be used in predicting AVO responses and with which we can gain physical insight into the microstructure of the granular aggregate. We have defined the two important microstructure parameters as f t and c n 2 R /R. We have shown that the coordination number should not be considered independently from the contact radius of curvature. In the literature, expressions for n n or n np are found. However, a more general approach is to have n n,p. Because porosity depends on pressure or compaction process as well as sorting, we do not think that the coordination number by itself is a good parameter for model calibration. The parameter c n 2 R /R can capture information about compaction and provide some idea about the general contact stiffness. For a given unconsolidated sand layer, where pressure is not changing much, it is reasonable to relate the coordination number to porosity and to derive R /R from the fitting process, as demonstrated. This parameter should correlate with the geometric aspects of the contact. Laboratory tests can verify if this is indeed the case. CONCLUSIONS The seismic properties of unconsolidated sands can be described by taking into account reduced tangential shear stiffness during wave propagation. We have extended EMA by introducing a binary model for grain contacts and a parameter f t, which represents the amount of nonslipping contacts in the mixture. The parameter f t can be derived from Poisson s ratio and can be interpreted as the accommodation of EMA to nonuniform contact properties. We apply the updated contact models to a shallow hydrocarbon discovery offshore Norway and a poorly consolidated sand reservoir in the deepwater GOM and obtain an improved match between well-log observations and rock-physics models. This modeling approach can help improve our understanding of AVO signatures in unconsolidated sands. ACKNOWLEDGMENTS The authors thank StatoilHydro for permission to publish the data used in this study. Thanks also to Torbjørn Fristad, Erik Ødegaard, and Aart-Jan van Wijngaarden at StatoilHydro, and Gary Mavko at Stanford University for valuable discussions. We thank Yves Guéguen, Boris Gurevich, and three anonymous reviewers for constructive comments that improved this manuscript. REFERENCES Ammi, B., D. Bideau, and J. P. Troadec, 1987, Geometrical structure of disordered packing of regular polygons, comparison with disc packing structure: Journal of Physics D Applied Physics, 20, Avseth, P., and R. Bachrach, 2005, Seismic properties of unconsolidated sands: Tangential stiffness, V P /V S ratios and diagenesis: 75thAnnual International Meeting, SEG, ExpandedAbstracts, Avseth, P., R. Bachrach, A.-J. van Wijngaarden, T. Fristad, and E. Ødegaard, 2007, Application of contact mechanics to hydrocarbon prediction in shal-

13 Modeling unconsolidated sands E209 low sediments: 69th Conference & Technical Exhibition, EAGE, ExtendedAbstracts, F024. Avseth, P., T. Mukerji, and G. Mavko, 2005, Quantitative seismic interpretation: Applying rock physics tools to reduce interpretation risk: Cambridge University Press. Bachrach, R., J. Dvorkin, and A. Nur, 2000, Seismic velocities and Poisson s ratio of shallow unconsolidated sands: Geophysics, 65, Bachrach, R., S. Noeth, N. Banick, M. Sengupta, G. Bunge, B. Flack, R. Utech, C. Sayers, P. Hooyman, L. den Boer, L. Leu, W. Troyer, and J. Moore, 2007, From pore-pressure prediction to reservoir characterization: A combined geomechanics-seismic inversion workflow using trend-kriging techniques in a deepwater basin: The Leading Edge, 26, Bowers, G. L., 1995, Pore pressure estimation from velocity data: Accounting for overpressure mechanisms beside undercompaction: Society of Petroleum Engineers Drilling & Completion paper Digby, P. J., 1981, The effective elastic moduli of porous granular rocks: Journal ofapplied Mechanics, 48, Gassmann, F., 1951, Elastic wave through a packing of spheres: Geophysics, 16, Geng, J., D. Howell, R. P. Behringer, G. Reydellet, L. Vanel, and E. Clement, 2001, Footprints in the sands: The response of granular material to local perturbations: Physical Review Letters, 87, Goddard, J. D., 1990, Nonlinear elasticity and pressure-dependent wave speeds in granular media: Proceedings of the Royal Society of London, SeriesA, Mathematical and Physical Sciences, 430, Johnson, K. L., 1985, Contact mechanics: Cambridge University Press. Love, A. E. H, 1927, Atreatise on the mathematical theory of elasticity: Cambridge University Press. Majmaudar, T. S., and R. P. Behringer, 2005, Contact force measurements and stress-induced anisotropy in granular materials: Nature, 23, Makse, H. A., N. Gland, D. L. Johnson, and L. Schwartz, 1999, Why effective medium theory fails in granular materials: Physical Review Letters, 83, , 2004, Granular packings: Nonlinear elasticity, sound propagation and collective relaxation dynamics: Physical Review E, 70, Manificat, G., and Y. Guéguen, 1998, What does control V P /V S in granular rocks?: Geophysical Research Letters, 25, no. 3, Mavko, G., T. Mukerji, and J. Dvorkin, 1998, The rock physics handbook: Cambridge University Press. Mindlin, R. D., 1949, Compliance of bodies in contact: Journal of Applied Mechanics, 16, Mueth, M. D., H. M. Jaeger, and S. R. Nagel, 1998, Force distribution in granular medium: Physical Review E, 57, Muhlhaus, H. B., and F. Oka, 1996, Dispersion and wave propagation in discrete and continuous models for granular materials: International Journal of Solids and Structures, 33, Murphy, W. F., 1982, Effect of microstructure and pore fluids on the acoustic properties of granular sedimentary materials: Ph.D. dissertation, Stanford University. Norris, A. N., and D. L. Johnson, 1997, Non-linear elasticity of granular media: Journal ofapplied Mechanics, 64, Ødegaard, E., and P. Avseth, 2004, Well log and seismic data analysis using rock physics templates: First Break, 22, Pasternak, E., H. B. Muhlhaus, and A. V. Dyskin, 2006, Finite deformation model of simple shear of fault with microrotations: Apparent strain localisation and en-echelon fracture pattern: Philosophical Magazine, 86, Sava, D., and B. Hardage, 2006, Rock physics models of gas hydrates from deepwater, unconsolidated sediments: 76thAnnual International Meeting, SEG, ExpandedAbstracts, Sayers, C. M., 2006, Sensitivity of time lapse seismic to reservoir stress path: Geophysical Prospecting, 54, Vega, S., 2003, Intrinsic and stress-induced velocity anisotropy in unconsolidated sands: Ph.D. dissertation, Stanford University. Walton, K., 1987, The effective moduli of a random packing of spheres: Journal of the Mechanics and Physics of Solids, 33, Winkler, K. W., 1983, Contact stiffness in granular and porous materials: Comparison between theory and experiment: Geophysical Research Letters, 10, Xu, S., 2002, Stress-induced anisotropy in unconsolidated sands and its effect on AVO analysis: 72nd Annual International Meeting, SEG, ExpandedAbstracts, Zimmer, M., M. Prasad, G. Mavko, and A. Nur, 2007, Seismic velocities in unconsolidated sands: Part 1 Pressure trends from 0.1 to 20 MPa: Geophysics, 72, no. 6, E1-E13.