ROCK PHYSICS MODELING FOR LITHOLOGY PREDICTION USING HERTZ- MINDLIN THEORY Ida Ayu PURNAMASARI*, Hilfan KHAIRY, Abdelazis Lotfy ABDELDAYEM Geoscience and Petroleum Engineering Department Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Perak, Malaysia *Coressponding author : purnamasari.ida@live.com ABSTRACT Rock physics represents the link between geologic reservoir parameters (e.g.,porosity,clay content,sorting,lithology,saturation) and seismic properties (e.g., V P /V S,density,elastic modulus). Most of the sand rock physics models required contact models to calculate the elastic modulus of the dry frame (loose random grain pack) at critical porosity. The friable sand model of Dvorkin and Nur (1995) is used to describe the velocity porosity relation for a clean, high porosity and unconsolidated sand. It assumes that the porosity decreases from critical porosity due to the addition of a solid mineral in the pore space. The bulk and shear modulus of the solid end member are then used to calculate the properties of the sphere pack at critical porosity. This can be done by using one of contact theory models such as Hertz-Mindlin model (Mindlin,1949). End-member bulk modulus calculated using Hertz-Mindlin theory is 1.98 GPa and shear modulus 2.9 GPa (for initial porosity equal 40%). This is usually rigid for such a compressible rock. The dry bulk modulus in clean sand is greater than in shaly-sand. It will makes the P-wave velocity (V P ) in the dry condition also greater in clean sand than shaly-sand so that the clean sand is tend to more compressible. Keywords rock physics model, friable sand model, Hertz-Mindlin theory, dry bulk and shear modulus, P and S-wave velocities INTRODUCTION Rock physics establishes a link between the elastic properties and the reservoir properties such as porosity, water saturation, clay content, cement volume and degree of sorting. Rock Physics models allow us to link seismic properties and geological properties, and the application of rock physics models can guide and improve on the qualitative seismic interpretation. Moreover, if we understand the link between geological parameters and rock physics properties, we can avoid certain ambiguities in seismic interpretation, particularly fluid/lithology, sand/shale and porosity/saturation. The link between rock physics and various geological parameters, including cement volume, clay volume and degree of sorting, allow us to perform lithology
prediction from rock types observed at a given well location to rock types assumed to be present nearby (Bjørlykke, 2010). The way in which geological trends, in this case depositional trends, in an area can be used to constrain rock physics models is also investigated. If we can predict the expected change in seismic response as a function of depositional environment or burial depth, this will increase our ability to predict hydrocarbons, especially in areas with little or no well log information. Understanding the geological constraints in an area of exploration reduces the range of expected variability in rock properties and hence reduces the uncertainties in seismic reservoir prediction (Avseth et al., 2005). A typical approach to testing rock physics models is to define the elastic properties of the end members, being bulk and shear moduli of the two different minerals, in this case clay and sand (quartz). This assumes that the rock s properties must then fall between the two end members and interpolation between them can be calculated using the modified Hashin-Shtrikman bound. Then the P and S-wave velocities can be calculated. Single mineral models consist of just one mineral type that is commonly clean sand or quartz. Rock physics models can generally be separated into different types. Empirical models are the simplest and only require the solid mineral bulk and shear modulus to calculate the changes in velocity as the porosity increases (Harbert, 2006). The Friable Sand Model The friable-sand model, or the unconsolidated line, describes how the velocity-porosity relation changes as the sorting deteriorates. The well-sorted end member is represented as a well-sorted packing of similar grains whose elastic properties are determined by the elasticity at the grain contacts. The friable-sand model represents poorly sorted sands as the well-sorted end member modified with additional smaller grains deposited in the pore space. These additional grains deteriorate sorting, decrease the porosity, and only slightly increase the stiffness of the rock. The friable sand model of Dvorkin and Nur (1995) is initiated by estimating the solid phase end member elastic moduli. The rock at zero percent porosity in sandstones often has the same value as that of pure quartz. The bulk and shear modulus of solid end member are then used to calculate the properties of the sphere pack at critical porosity. This can be done using one of the contact theory models, Hertz-Mindlin model (Mindlin, 1949). Hertz-Mindlin Theory The elastic moduli at high porosity (usually critical porosity) is modeled as an elastic sphere pack subject to confining pressure, given by the Hertz-Mindlin theory (Dvorkin et al., 1995) as follows:
(1) (2) where, Κ HM, µ HM = dry rock bulk and shear moduli, respectively, at critical porosity φ c P = confining pressure, which is equal to effective pressure (i.e., the difference between the overburden pressure and the pore pressure) µ = shear modulus for solid phase (mineral modulus) v = Poisson s ratio for solid phase n = coordination number (the average number of contatcs per grain) The Poisson s ratio can be expressed in terms of the bulk (Κ) and shear (µ) moduli as follows: (3) Effective pressure versus depth is obtained with the following formula: (4) where, g = gravity constant ρ b and ρ fl = bulk density and fluid density, respectively Z = depth The coordination number, n, depends on porosity. The relationship between coordination number and porosity can be approximated by following empirical equation : n = 20-34φ + 14φ 2 (5) Hence, for a porosity φ = 0.4, n = 9. At porosity φ the concentration of the pure solid phase (added to the sphere pack to decrease porosity) in the rock is 1 = φ/φ c and that of the original sphere-pack phase is φ/φ c. Then the bulk (Κ dry ) and shear (µ dry ) moduli of the dry friable sand mixture are : (6) where, (7) (8) Forward Modeling of The Effective Medium Model
The rock physics model of friable sand is applied to model texture, lithology and pressure changes. The effective medium model (friable sand model) is used to predicting velocities in unconsolidated sands, as a function from pressure and porosity. This model is used to predict the high porosity end-members (bulk and shear moduli) using Hertz-Mindlin theory(avseth et al., 2005). This stage of work applied in two different reservoirs condition, clean sand and shaly-sand in dry condition. Here are the to predict P- and S-wave velocity as a function of porosity, using effective medium model (friable sand model) : 1. Estimation of dry bulk and shear modulus, K dry and µ dry for clean sand pack at critical porosity using Hertz-Mindlin contact theory (equation 1 and 2), then apply modified Hashin- Shtrikman bound (equation 6, 7 and 8) to interpolate high porosity end member and the zero porosity mineral point, for the clean sand and shaly-sand reservoirs. 2. Calculate the P- and S-wave velocity in dry condition at the critical porosity, for each porosity. Velocity Porosity Rock Physics Models Based on previously described methodology, the elastic modulus and velocities for the porosity range from 0 to 40% were calculated. The rock properties data are in Table 1. There are 2 types of sandstones used in this model, clean sand and shaly-sand which contained 70% of quartz and 30% of clay. Paramet er Value Paramet er Value Bulk modulus of clay* * 17.5 GPa Critical porosity 40% Shear modulus of clay* * 7.5 GPa Poisson's rat io 0.08 Bulk modulus of quart z* 36.6 GPa Coordinat ion number 9 Shear modulus of quart z* 45 GPa Pef f 20 MPa Densit y of quart z* 2650 kg/ m3 Porosit y 0-40 % Densit y of clay* * 2300 kg/ m3 Rock mineral composit ion 70 % of quart z and 30 % of clay Table 1. Rock properties data used as input parameter model ( * Mavko et al., 1998, ** Avseth et al., 2000) The results are plotted as dry elastic modulus against porosity, dry velocities (V P and V S ) against porosity in dry conditions (Figure 1). End-member bulk modulus calculated using Hertz- Mindlin theory is 1.98 GPa and shear modulus 2.9 GPa (for initial porosity equal 40 %). This is usually rigid for such a compressible rock. The dry bulk modulus in clean sand is greater than in shaly-sand. It will makes the P-wave velocity (V P ) in this dry condition also greater in clean sand than shaly-sand so that the clean sand is tend to more compresible. In Figure 1 (left) near to the critical porosity, the dry bulk modulus has almost the same value with dry shear modulus both in clean sand and shaly-sand. Otherwise, these elastic modulus are approximate to each other at low porosities, both in clean sand and shaly-sand. On the other hand, dry P-wave velocity (V p ) in
shaly-sand is almost similar to dry S-wave velocity (V S ) in clean sand. If these velocities measured at the same time, it may cause an ambiguities for the interpretation. In Figure 1 (right) shows the decrease in both P and S-wave velocities as the porosity increases. The results are subjected to possible errors in uncertainties in the mineral properties, their composition and the model applied. 5. 10 10 Dry Elastic Modulus Against Porosity 7000 Dry Velocities Against Porosity 6000 4. 10 10 5000 Elastic Modulus (Pa) Kdrymineral () φ 3. 10 10 μdrymineral () φ Kdryφ () μdryφ () 2. 10 10 Velocity (m/s) Vpdrymineral () φ Vsdrymineral () φ 4000 Vpdryφ () Vsdryφ () 3000 2000 1. 10 10 1000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 φ Porosity Dry Bulk Modulus Shaly-Sand Dry Shear Modulus Shaly-Sand Dry Bulk Modulus Clean Sand Dry Shear Modulus Clean Sand 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Vpdry Shaly-Sand Vsdry Shaly-Sand Vpdry Clean Sand Vsdry Clean Sand φ Porositiy Figure 1 Theoretically predicted dry bulk and shear modulus (left), P and S-wave velocities (right) for dry condition as a function of porosity with porosity ranging from 0 to 0.4 (P eff = 20 MPa) CONCLUSIONS Most of the sand rock physics models required contact models to calculate the elastic modulus of the dry frame (loose random grain pack) at critical porosity. The friable sand of Dvorkin and Nur (1995) was used to analyse the P and S-wave velocity versus porosity trends for the clean uncosolidated sands and shaly-sand. REFERRENCES Avseth, P., Mukerji, T., and Mavko, G., [2005], Quantitative Seismic Interpretation Applying Rock Physics Tools to Reduce Interpretation Risk, Cambridge University Press Bjørlykke, K., [2010], Petroleum Geoscience: From Sedimentary Environments to Rock Physics: Springer Dvorkin, J.P., Nur, A., [1995], Elasticity of High-Porosity Sandstones: Theory of two North Sea Datasets: SEG Annual Meeting, Conference Paper, p. 890-893.
Harbert, M.K., [2006], Rock Physics Models for Unconsolidated Sands and Shales, Report No : GPH 9/06, Department of Exploration Geophysics, Curtin University of Technology Mavko, G., Mukerji, T. and Dvorkin, J., [1998], The Rock Physics Hand Book: Tools for Seismic analysis in a porous media, Cambridge University Press, Cambridge, U.K Mindlin, R. D., [1949], Compliance of Elastic Bodies in Contact: Journal of Applied Mechanics, 16, 259-268