An introduction to Reuss and Voight averages for the moduli of mixtures of materials is presented. These bounds to the modulus of a mixture are explained in terms of series and parallel or iso-stress and iso-strain assumptions. The limitations of using these averages and the need for a more sophisticated model are then discussed.
The approach taken to handle these difficulties is to develop a method of interpolating between these two bounding limits. The SDEM modelling parameter L is introduced which describes a simple linear interpolation between the two limits. The parameter takes on values that vary between zero and one. Zero is the Reuss average (isostress) of the materials and one gives the Voight (iso-strain) average. This linear interpolation is used as the basis for a Staged Differential Effective Medium (SDEM) model. The inclusion of the effective medium modelling is key to the later results obtained.
The implications of the model are then discussed. SDEM models include critical concentration models as a special case. The first integration step is made with L=0, a Reuss average. At the critical concentration the modeling parameter is appropriately changed (nonzero) and a second integration step is performed. If even more integration steps are required they are then made. The derived critical porosity model for a dry rock reproduces the literature results. The brine-saturated models are however new results.
The critical porosity model for a dry rock is then used to derive a relation between the SDEM modelling parameter and the Biot coefficient. This gives a simple relation between the two parameters and the associated porosity changes. An alternative interpretation for the modeling parameter is therefore as the rate of change of the Biot coefficient with porosity.
The critical porosity solutions obtained for dry and brine saturated rocks also allow the effects of fluid saturation to be examined. The SDEM modelling parameter is eliminated between the two models and Gassman’s equation is obtained. This is a completely general result and will still be obtained at the end of any integration step. The derived SDEM model may therefore be viewed as consistent with Gassman’s equation which also allow the effects of multiple mineral moduli to be modelled.
For three-dimensional models, the Reuss and Voight averages represent bounds on the moduli of the material mixture (They will be close to the actual modulus of the mixture only for very special geometries of the materials). In real mixtures the modulus lies somewhere in between these bounds For example, the
oight-
euss-
ill (Hill 1952) average estimates the modulus by arithmetically averaging the Voight and Reuss bounds as shown in the following equations:
where the terms Mi and fi are the modulus and volume fraction of the ith component, respectively. Sometimes this is actually closer to the data. Performing an averaging of the Hashin-Shtrikman bounds is another obvious extension. The objective of this paper is to find refinements of these averaging techniques to determine more accurate values of the moduli of mixtures if additional textural information is included.
