## 6 Elastic Modeling Using Staged Differential Effective Medium (SDEM)  
Reuss lower bound and Voight upper bound have been utilized historically to associate the elastic moduli of rock components with the elastic moduli of the mixture. While the moduli of the mixtures are always located between these bounds [21], several researchers attempted to better locate moduli exact positions by adding more textural information on how grains and pores are structured within the rocks. One such attempt was the work by [PERSON] and [PERSON] [3], which extended the differential effective medium theory by introducing the concept of \"staging\" on how the process of sedimentation, lithification/compaction, and diagenesis combine to form sedimentary rocks. In the staged differential effective medium (SDEM) technique, inclusions of different length scale (aspect ratio) are gradually added to a host, neglecting interaction terms between the inclusions [20]. Similar to PCM to model resistivity, SDEM methodology models the elastic moduli of dual-porosity sandstones. In fact, PCM is a special case of SDEM, and these are general techniques for modeling the effect of mineralogy and texture on permeability, resistivity, and elastic measurements. The well-known critical porosity [22] model is inherently included in SDEM. To simulate the elastic moduli of a single-pore sandstone, we need a two-staged SDEM. The first step is to integrate the elastic moduli from host to the critical porosity with a length scale of L = 0, which mimics an isostress Reuss bound. The second step is to integrate from critical porosity to total porosity with a length scale of L#0 using:  
\[M=M_{c}\frac{1+\left(\frac{\partial_{c}-\varnothing}{\partial_{c}}\right). \left(\frac{M_{i}-M_{c}}{M_{c}}\right).L}{1-\left(\frac{\partial_{c}- \varnothing}{\partial_{c}}\right).\left(\frac{M_{i}-M_{c}}{M_{i}}\right).\left( 1-L\right)} \tag{7}\]  
where \(\varnothing_{c}\) is the critical porosity, \(M_{i}\) is the inclusion modulus, \(L\) is the length scale and related to this inclusion, and \(M_{c}\) is the isostress bound of the host and grain modulus at the critical porosity [20]. Dual-porosity sandstone can be simulated by adding an additional integration step to model the micropores [3]. Figure 2 illustrates the SDEM technique for a dual-porosity formation. The elastic moduli (bulk and shear) are plotted versus total porosity (\(\phi_{l}\)). The bound corresponding to length scale of \(L_{s}\) connects the elastic moduli at critical porosity (\(M_{c}\)) to elastic moduli at zero porosity (solid or matrix moduli). This bound is associated with intergranular pores. The bound corresponding to length scale of \(L_{sh}\), which does not pass through elastic moduli at critical porosity, is associated with the moduli change due to change in microporovosity. For a formation consisting of intergranular pores and micropores, the microporovosity bound intersects the intergranular bound at the intergranular porosity. Figure 1: A graphical representation of pore combination modeling (PCM). As explained in [PERSON] et al. [20], matrix elastic moduli, critical porosity, intergranular pore and micropore \(L\) parameters, and lithology exponents of each pore system can be obtained from the nonlinear optimization of velocity and resistivity core measurements. In the following, we present a new methodology to obtain these parameters from conventional well logs. The elastic moduli of the mixture (\(M\)) and total porosity, i.e., \(M\)-\(\phi_{t}\) are plotted in a linear plane; \(\phi_{i}\) is the intergranular porosity, \(\phi_{m}\) is the microporosity, and the sum \(\phi_{t}=\phi_{i+}\phi_{m}\), is the total porosity. Starting from point (\(\phi_{C},\ M_{c}\)) a bound with length scale of \(L_{i}\) illustrates the gradual decrease in total porosity due to sediment compaction. At this stage, intergranular pores are the only pore system and total porosity (\(\phi_{t}\)) equal to intergranular porosity of \(\phi_{i}\). Then, diagenesis comes into play and micropores start to evolve.