Differential effective medium modeling of rock elastic moduli with critical porosity constraints

1995 ◽  
Vol 22 (5) ◽  
pp. 555-558 ◽  
Author(s):  
Tapan Mukerji ◽  
Jim Berryman ◽  
Gary Mavko ◽  
Patricia Berge
Keyword(s):  
Elastic Moduli ◽  
Effective Medium ◽  
Critical Porosity

Sediment with porous grains: Rock-physics model and application to marine carbonate and opal

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. E1-E15 ◽  
Author(s):  
Franklin Ruiz ◽  
Jack Dvorkin
Keyword(s):  
Aspect Ratio ◽  
Elastic Moduli ◽  
Rock Physics ◽  
Effective Medium ◽  
High Porosity ◽  
Ocean Drilling Program ◽  
Critical Porosity ◽  
Effective Medium Model ◽  
Suspension Model ◽  
Porous Grains

We offer an effective-medium model for estimating the elastic properties of high-porosity marine calcareous sediment and diatomite. This model treats sediment as a pack of porous elastic grains. The effective elastic moduli of the porous grains are calculated using the differential effective-medium (DEM) model, whereby the intragranular ellipsoidal inclusions have a fixed aspect ratio and are filled with seawater. Then the elastic moduli of a pack of these spherical grains are calculated using a modified (scaled to the critical porosity) upper Hashin-Shtrikman bound above the critical porosity and modified lower (carbonates) and upper (opal) Hashin-Shtrikman bounds below the critical porosity. The best match between the model-predicted compressional- and shear-wave velocities and Ocean Drilling Program (ODP) data from three wells is achieved when the aspect ratio of intragranular pores is 0.5. This model assigns finite, nonzero values to the shear modulus of high-porosity marine sediment, unlike the suspension model commonly used in such depositional settings. The approach also allows one to obtain a satisfactory match with laboratory diatomite velocity data.

On a Spring-Network Model and Effective Elastic Moduli of Granular Materials

1999 ◽  
Vol 66 (1) ◽  
pp. 172-180 ◽  
Author(s):  
K. Alzebdeh ◽  
M. Ostoja-Starzewaski
Keyword(s):  
Granular Materials ◽  
Disordered Systems ◽  
Granular Media ◽  
Elastic Moduli ◽  
Volume Fraction ◽  
Effective Medium ◽  
Solid State Physics ◽  
Two Phase ◽  
Effective Medium Theories ◽  
The Individual

Two challenges in mechanics of granular media are taken up in this paper: (i) development of adequate numerical discrete element models of topologically disordered granular assemblies, and (ii) calculation of macroscopic elastic moduli of such materials using effective medium theories. Consideration of the first one leads to an adaptation of a spring-network (Kirkwood) model of solid-state physics to disordered systems, which is developed in the context of planar Delaunay networks. The model employs two linear springs: a normal one along an edge connecting two neighboring vertices (grain centers) which accounts for normal interactions between the grains, as well as an angular one which accounts for angle changes between two edges incident onto the same vertex; edges remain straight and grain rotations do not appear. This model is then used to predict elastic moduli of two-phase granular materials—random mixtures of soft and stiff grains —for high coordination numbers. It is found here that an effective Poisson’s ratio, νeff, of such a mixture is a convex function of the volume fraction, so that νeff may become negative when the individual Poisson’s ratios of both phases are both positive. Additionally, the usefulness of three effective medium theories—perfect disks, symmetric ellipses, and asymmetric ellipses—is tested.

Stress sensitivity of sandstones

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 444-455 ◽  
Author(s):  
Jack Dvorkin ◽  
Amos Nur ◽  
Caren Chaika
Keyword(s):  
Elastic Modulus ◽  
Effective Stress ◽  
Elastic Moduli ◽  
Hydrostatic Stress ◽  
Clay Content ◽  
Stress Sensitivity ◽  
High Porosity ◽  
Effective Pressure ◽  
Critical Porosity ◽  
Difference Formula

Our observations made on dry‐sandstone ultrasonic velocity data relate to the variation in velocity (or modulus) with effective stress, and the ability to predict a velocity for a rock under one effective pressure when it is known only under a different effective pressure. We find that the sensitivity of elastic moduli, and velocities, to effective hydrostatic stress increases with decreasing porosity. Specifically, we calculate the difference between an elastic modulus, [Formula: see text], of a sample of porosity ϕ at effective pressure [Formula: see text] and the same modulus, [Formula: see text], at effective pressure [Formula: see text]. If this difference, [Formula: see text], is plotted versus porosity for a suite of samples, then the scatter of ΔM is close to zero as porosity approaches the critical porosity value, and reaches its maximum as porosity approaches zero. The dependence of this scatter on porosity is close to linear. Critical porosity here is the porosity above which rock can exist only as a suspension—between 36% and 40% for sandstones. This stress‐sensitivity pattern of grain‐supported sandstones (clay content below 0.35) practically does not depend on clay content. In practical terms, the uncertainty of determining elastic moduli at a higher effective stress from the measurements at a lower effective stress is small at high porosity and increases with decreasing porosity. We explain this effect by using a combination of two heuristic models—the critical porosity model and the modified solid model. The former is based on the observation that the elastic‐modulus‐versus‐porosity relation can be approximated by a straight line that connects two points in the modulus‐porosity plane: the modulus of the solid phase at zero porosity and zero at critical porosity. The second one reflects the fact that at constant effective stress, low‐porosity sandstones (even with small amounts of clay) exhibit large variability in elastic moduli. We attribute this variability to compliant cracks that hardly affect porosity but strongly affect the stiffness. The above qualitative observation helps to quantitatively constrain P‐ and S‐wave velocities at varying stresses from a single measurement at a fixed stress. We also show that there are distinctive linear relations between Poisson’s ratios (ν) of sandstone measured at two different stresses. For example, in consolidated medium‐porosity sandstones [Formula: see text], where the subscripts indicate hydrostatic stress in MPa. Linear functions can also be used to relate the changes (with hydrostatic stress) in shear moduli to those in compressional moduli. For example, [Formula: see text], where [Formula: see text] is shear modulus and [Formula: see text] is compressional modulus, both in GPa, and the subscripts indicate stress in MPa.

An approximation for the Xu‐White velocity model

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1406-1414 ◽  
Author(s):  
Robert G. Keys ◽  
Shiyu Xu
Keyword(s):  
Clay Content ◽  
Velocity Model ◽  
Compressional Wave ◽  
Effective Medium ◽  
Critical Porosity ◽  
Shear Moduli ◽  
Effective Medium Method ◽  
Computationally Intensive ◽  
Medium Method ◽  
Rock Bulk

In 1995, S. Xu and R. E. White described a method for estimating compressional and shear‐wave velocities of shaley sandstones from porosity and shale content. Their model was able to predict the effect of increasing clay content on compressional‐wave velocity observed in laboratory measurements. A key step in the Xu‐White method estimates dry rock bulk and shear moduli for the sand/shale mixture. This step is performed numerically by applying the differential effective medium method to the Kuster‐Toksöz equations for ellipsoidal pores. This step is computationally intensive. Using reasonable assumptions about dry rock elastic properties, this step can be replaced with a set of approximations for dry rock bulk and shear moduli. Numerical experiments show an extremely close match between velocities obtained with these approximations and velocities computed with the differential effective medium method. These approximations simplify the application of the Xu‐White method, and make the method computationally more efficient. They also provide insight into the Xu‐White method. For example, these approximations show how the Xu‐White model is related to the critical porosity model.

Elastic instabilities in dry, mesoporous minerals and their relevance to geological applications

2010 ◽  
Vol 74 (2) ◽  
pp. 341-350 ◽  
Author(s):  
E. K. H. Salje ◽  
J. Koppensteiner ◽  
W. Schranz ◽  
E. Fritsch
Keyword(s):  
Bulk Modulus ◽  
Porous Materials ◽  
Power Law ◽  
Master Curve ◽  
Elastic Moduli ◽  
External Stress ◽  
Effective Elastic Moduli ◽  
Critical Porosity ◽  
Elastic Instabilities ◽  
Linear Decay

AbstractThe collapse of minerals and mineral assemblies under external stress is modelled using a master curve where the stress failure is related to the relative, effective elastic moduli which are in turn related to the porosity of the sample. While a universal description is known not to be possible, we argue that for most porous materials such as shales, silica, cement phases, hydroxyapatite, zircon and also carbonates in corals and agglomerates we can estimate the critical porosity ϕc at which small stresses will lead to the collapse of the sample. For several samples we find ϕc ~0.5 with an almost linear decay of the bulk moduli with porosity at ϕc <0.5. The second scenario involves the persistence of elasticity for porosities until almost 1 whereby the bulk modulus decreases following a power law κ ~ (1–ϕm, m>2, between ϕ = 0.5 and ϕ = 1.

Estimating dry rock frame moduli of high-resolution 3D digital rock images using the contact-mechanics-based effective medium approach

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. MR235-MR243
Author(s):  
Abdulla Kerimov ◽  
Jennie Cook ◽  
Nathan Lane ◽  
Dmitry Lakshtanov ◽  
Glen Gettemy
Keyword(s):  
High Resolution ◽  
Contact Mechanics ◽  
Prediction Accuracy ◽  
Elastic Moduli ◽  
Effective Medium Theory ◽  
Contact Stiffness ◽  
Effective Medium ◽  
Contact Radius ◽  
Ambient Conditions ◽  
Digital Rock

We have developed a method to estimate the dry frame elastic moduli of high-resolution 3D digital rock images using the contact-mechanics-based effective medium theory (EMT) model. The existing EMT models often are used to predict the effective dry frame elastic moduli of granular aggregates as a function of porosity, average number of contacts per grain, grain radius, contact radius, and contact stiffnesses of an elastic two-grain combination. But, it is almost impossible to measure the number of contacts per grain, contact radius distribution, or contact stiffness distribution in complex rocks. Therefore, explicit assumptions based on simplified microstructural geometries often are made to predict these contact properties in granular aggregates. As a result, the predictions of dry frame elastic moduli using EMT models may fail to match the observed properties because of numerous simplified assumptions, which can be violated in complex rocks. Our method uses the morphological contact properties (i.e., the grain-to-grain contact radius distribution, grain radius distribution, and coordination number distribution) directly extracted from 3D digital rock images to improve the prediction accuracy of dry frame elastic moduli using the EMT models. With integration of digital rocks technology, there is no longer a need to assume the size and shape of the grains, contact size, and number of contacts. The prediction accuracy of our method is validated on high-resolution 3D micro-CT digital rock images of miniplugs extracted from plugs with ultrasonic velocity measurements under dry conditions at different confining pressures. Image-computed dry frame elastic moduli using the EMT model are consistent with laboratory-measured moduli extrapolated to ambient conditions.

Estimating 3D elastic moduli of rock from 2D thin-section images using differential effective medium theory

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. MR211-MR219 ◽  
Author(s):  
Sadegh Karimpouli ◽  
Pejman Tahmasebi ◽  
Erik H. Saenger
Keyword(s):  
Porous Media ◽  
Thin Section ◽  
Elastic Moduli ◽  
Effective Medium ◽  
Physical Parameters ◽  
Reconstruction Method ◽  
Digital Rock ◽  
Aspect Ratios ◽  
Rock Structure ◽  
2D Images

Standard digital rock physics (DRP) has been extensively used to compute rock physical parameters such as permeability and elastic moduli. Digital images are captured using 3D microcomputed tomography scanners that are not widely available and often come with an excessive cost and expensive computation. Alternative DRP methods, however, benefit from the highly available low-cost 2D thin-section images and require a small amount of computer memory use and CPU. We have developed another alternative DRP method to compute 3D elastic parameters based on differential effective medium (DEM) theory. Our investigations indicate that the pore aspect ratio (PAR) is the most crucial factor controlling the elastic moduli of rock. Based on digital rock modeling in a dry calcite sample with 20% porosity, the bulk modulus is reduced by 51%, 80.7%, and 96.8% for aspect ratios of 1, 0.2, and 0.05, respectively. Similarly, the shear modulus is reduced by 52%, 73.8%, and 92.8% for the same PARs. These findings confirm the importance of the PAR in wave propagation through porous media. Such an evaluation, however, can be very expensive for 3D images because one requires using several of them for drawing a reliable conclusion. Therefore, we aim to capture the PAR distribution from 2D images. This distribution is, then, used to estimate 3D elastic moduli of sample by DEM equations. Three orthogonal 2D images were used and results indicated that 2D PARs in orthogonal orientations could address pore shapes more effectively. Moreover, a stochastic porous media reconstruction method was also used to generate more scenarios of rock structure and those of which that are not seen in 2D images. Results from Berea sandstone and Grosmont carbonate indicated that using only 2D images our proposed method could effectively estimate 3D elastic moduli of rock samples.

Sign in / Sign up

Export Citation Format

Share Document