# Modeling of Characteristics of Complex Microstructure and Heterogeneity at the Core Scale
## 1 Introduction  
Carbonate rocks have major economic significance; sixty percent of the world's oil reserves are in carbonate reservoirs, and the potential for additional gas reserves is considerable [1; 2; 3; 4]. As a result, studies focusing on the pore structure and petrophysical properties of carbonate are attracting widespread interest. Carbonate rocks are characterized by complex pore structures and strong heterogeneity [5]. For samples with the same porosity, their different pore structures greatly influence the ultrasonic wave velocity. Since the rock pore distribution is very complex and the pore structure is difficult to characterize quantitatively, many studies have used ideal geometries for approximation, with the most commonly used geometry being the ellipsoid, which employs the ratio of the short and long axes of the ellipsoid as its structural parameter [6; 7; 8]. Many studies have confirmed that the pore structure plays a key role in affecting the elastic wave velocity and elastic moduli [6; 9; 10; 11; 12; 13; 14; 15; 16], particularly in carbonate rocks with complex pore structures. Consequently, it is necessary to consider the effect of the pore structure in petrophysical modeling. Many models have considered the effect of local flow within pores arranged in different pore structures when elastic waves pass and have focused on changes in petrophysical properties. Among them, a traditional local flow mechanism model called the squirt model was developed to simulate the dispersion and attenuation of seismic wave velocities at the microscale (in nonhomogeneous spaces smaller than the wavelength) [17; 18; 19; 20; 21; 22; 23; 24]. The squirt flow mechanism assumes that when seismic waves pass through rock, the small \"compliant\" pores in the rock close, causing the fluid inside to be ejected into the large \"stiff\" pores, and the local flow phenomenon tendsto result in a large energy loss, which is the main cause of the induced large-scale seismic wave velocity [25]. This phenomenon of fluid flow between pores of different shapes and orientations occurs due to the pressure relaxation of seismic waves [26]. Squirt models are applicable to uniformly saturated rocks with low permeability (preventing the free flow of fluids within the pores) and low pressure (ensuring that the compliant pores are still open) [27]. With the development of computer hardware, simulations can provide more details of the pore structure [28]. In recent years, the development of multicore devices has significantly advanced current modelling capabilities, making numerical modelling based on the actual pore structure of rocks a popular research topic [21; 28; 29; 30; 31; 32]. Squirt flow models consider the squirt flow action of a compliant pore with a single aspect. For complex pore structures, numerous scholars have proposed improvement schemes, including the inclusion of medium stiffness pores [26], iterative inclusion of compliant pores with different aspect ratios [33], and squirt flow between stiff pores and compliant pores with different aspect ratios [34]. It is generally accepted that strong heterogeneity is a vital characteristic of carbonate rocks. The random medium model can be used to appropriately characterize this heterogeneity. This method allows us to determine the size and distribution of mineral particles, microstructures, and pores in samples, as well as to quantitatively evaluate the heterogeneity of the rock. At present, three kinds of random medium models are widely used, namely, Gaussian, exponential and von Karman autocorrelation function models [35; 36; 37], which are used to simulate the random distribution of rock particles with different geometric shapes. In addition, to meet various research needs, variable-scale reconstruction methods have been proposed. These include the Gaussian window function, simulated annealing, sequence classification, multipoint statistics, and their combinations [38; 39; 40; 41; 42; 43].