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Rock moduli boundary constraints
Rocks are usually not homogeneous, but are made up of multiple components such as mineral grains and pore space. On a larger scale, the bulk properties of rocks will be some weighted combination of the small-scale components. This averaging or upscaling step is needed if we wish to understand the behavior of our laboratory data or extract useful parameters from field data such as logs or seismic measurements. Understanding the boundary constraints is an important factor in this process.
Constant strain and constant stress limits
The simplest bounds are provided by the constant strain and constant stress limits. This method is equivalent to the series vs. parallel effective resistance of a resistor network. In the case that strains of the two materials making up our material are equal, as with the parallel plates in Fig. 1a, we get the upper (Voigt) limit. The response is controlled by the stiffer component.
where MV is the effective Voigt modulus, MA and MB are the component moduli, and A is the volume fraction of component A. In contrast, with the constant stress case (Fig. 1b), the soft component dominates the deformation and we get the lower (Reuss) limit.
where MR is the lower Reuss effective modulus. The average value between these two limits is often used in property estimation and is termed the Voigt-Reuss-Hill relation
Note that in the case for minerals plus pores, Mpore = 0 and MV decreases linearly with porosity. MR equals zero for all porosities.
An alternative approach, known as the Hasin-Shtrikman technique, is to fill space with concentric spheres. Material 1 is in the interior, and Material 2 forms a surrounding shell. Spheres such as these but of varying size are packed together to fill the entire medium (Fig. 2). The resulting upper and lower bounds ("+" vs. "–" respectively) for bulk and shear modulus are given by
where Ki, μi, and fi refer to the bulk and shear moduli and volume fraction of component i, respectively. The upper and lower bounds are derived by exchanging the stiff and soft components as "1" or "2."
The results of using Eqs. 1 through 5 to define boundaries are shown in Fig. 3. Using quartz as the first component and porosity as the second, the composite bulk modulus is plotted in Fig. 3a as a function of porosity. In one case, the pores are empty (black), in the other, water fills the pores and is the second component (blue). Because we used quartz as the solid component (see Isotropic elastic properties of minerals), these bounds should contain all possible values for sandstones (remember: for isotropic and homogeneous sandstones). If, on the other hand, our rock was made up of only quartz and calcite, we get bounds that appear in Fig. 3b. Note that the bounds have collapsed and produce only a narrow spread. This is a result of the two end components both being stiff and closer together. In cases such as these, a simple linear average can work well.
3 – (a) General bounds of a porous material made of quartz, both dry and saturated with water (HS = Hasin-Shtrikman). With extreme differences in material properties, the bounds can be very wide. (b) Example of bounds for a quartz-calcite mixture. Because the properties are comparable between the two minerals, the bounds act much more like simple linear averages.
|MA, MB||=||modulus of component a, b, etc., GPa or MPa|
|MO||=||reference oil molecular weight, g/mole|
|MR||=||Reuss bound modulus, GPa or MPa|
|MV||=||Voigt bound modulus, GPa or MPa|
|Kf 1, Kf 2||=||bulk modulus of fluid 1, 2, etc., GPa or MPa|
|KHS||=||Hasin-Shtrikman bound bulk modulus, GPa or MPa|
|f||=||frequency, s–1, Hz (cycles/s)|
|μ||=||shear modulus, GPa or MPa|
- Hashin, Z. and Shtrikman, S. 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11 (2): 127-140. http://dx.doi.org/10.1016/0022-5096(63)90060-7.
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