The theory and application of DEM-Gassmann rock physics model for complex carbonate reservoirs
Authors: Sun Sam Zandong , Wang Haiyang , Liu Zhishui , Li Yingwei , Zhou Xinyuan , Wang Zaoming 
Published: February 2012

The storage spaces of carbonate reservoirs in the Tarim Basin are dominated by secondary pore porosity such as dissolution caves, holes, and cracks (Song et al., 2001). Research has shown that pore shapes can significantly influence the velocities of elastic waves (Eshelby, 1957; Kuster and Toksöz, 1974; Xu and White, 1995; Yan et al., 2002; Sun et al., 2004). Empirical [Formula: see text] relationships such as Castagna's (1985) mudrock line, Han's relations (1986), and Greenberg-Castagna's (1992) relationship that ignore the effect of pore geometry are not applicable to carbonates (Xu and White, 1995; Wang et al., 2009). Cheng and Toksöz (1979) imaged various pore space structures using SEM, and proposed a pore aspect ratio spectrum that can be used to explain velocity prediction. Han (2004) measured velocities of 52 different carbonate rocks, indicating that caves influence seismic velocities due to high rigidity while micropores or cracks significantly decrease velocities of rocks due to the low rigidity. Regarding carbonate reservoirs that are dominated by secondary pore porosity, Eberli et al. (2003) discussed the velocity-porosity relationship for microporosity rocks, moldic-porosity rocks, interparticle porosity rocks, and densely cemented rocks. Wang et al. (2009) analyzed the practicality of three published models (Wyllie time-average equation, Gassmann's equation, and Kuster-Toksöz model) on velocity prediction for carbonate reservoirs in the Tarim Basin, proving that Kuster-Toksöz model is the best one. Meanwhile, they also pointed out that the Kuster-Toksöz model is a very high-frequency model, and is limited to dilute concentrations of the pores. Based on effective medium theories of Norris et al. (1985), Berryman (1992) proposed a new differential effective medium (DEM) model. The matrix in this model begins as phase 1 and is updated at each step when a new increment of phase 2 (inclusions) is added; the process continues until the expected proportion of the inclusions is reached. In this process, the former incremental inclusion (phase 2) and the matrix phase are taken as a new matrix, which makes the inclusion added at each step “dilute.” Unfortunately, DEM is also a high-frequency model. At high frequencies, there is not enough time for wave-induced pore pressure to equilibrate, so it is not applicable to consider the effect of pore fluids on elastic properties. Wu's (1992) arbitrary aspect ratio and Berryman's (1995) 3D special pores are introduced into Berryman's (1992) DEM model to calculate the elastic moduli of dry rock in this paper. Spheres, needles, and penny-shaped cracks are used to represent dissolution vugs (small-sized caves), holes (needle-shaped), and cracks that have developed in the Tarim Basin. These “special” pores are incrementally added into the matrix of carbonate rocks so that the inclusion added at each step is dilute. Then, Gassmann's equation is employed to calculate the elastic properties of the saturated rocks. This rock physics model in this paper is called the DEM-Gassmann's model.