Abstract —This work is devoted to constructing parametric mathematical models of effective elastic properties of carbonate rocks based on the effective medium theory (petroelastic models). These models have recently been particularly in demand for exploration geophysics because they link the elastic properties of hydrocarbon reservoir rocks with the parameters of the void space—the shape and volume concentration of pores, cracks, and voids, and the degree of their connectivity. These parameters are determined from the measured velocities of elastic waves. However, the number of the unknown model parameters is, as a rule, larger than the number of the measured quantities so that such problems are underdetermined. In this case, the inverse problem of finding the model parameters may have an infinite number of solutions. Constraining the sought parameters with taking into account their physical meaning and existing experimental data can significantly increase the reliability of the obtained results and reduce the region of possible solutions of the inverse problem. In this work, we propose new approaches for constraining variations in the unknown model parameters which cannot be established from direct measurements, namely, for the connectivity of voids, crack porosity, and crack aspect ratio. Based on the correlation between the parameter of connectivity of voids and permeability, we developed an approach for constraining the range of variations in the parameter of connectivity of voids which relies on the Kozeny–Carman equation. The results of triaxial testing of rock samples on a servo hydraulic testing machine (press) are used for estimating the upper limit of crack porosity and assessing the crack shape. The characteristics of microstructure of the studied rocks obtained with the use of the established constraints increase the reliability of the constructed petroelastic models for carbonate rocks. These models can be subsequently used in various geophysical studies implying a relationship between the analyzed processes or properties and the microstructure of the rocks.

中文翻译：

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碳酸盐岩岩石弹性模型孔隙空间参数的界限

摘要：本文致力于基于有效介质理论（岩弹模型）构建碳酸盐岩有效弹性性质的参数化数学模型。这些模型最近特别需要勘探地球物理学，因为它们将油气藏岩石的弹性特性与空隙空间的参数联系起来——孔隙、裂缝和空隙的形状和体积浓度，以及它们的连通程度。这些参数是根据弹性波的测量速度确定的。然而，未知模型参数的数量通常大于测量量的数量，因此这些问题是不确定的。在这种情况下，寻找模型参数的逆问题可能有无数个解。考虑到它们的物理意义和现有的实验数据来约束所寻求的参数可以显着提高所获得结果的可靠性并减少逆问题的可能解的区域。在这项工作中，我们提出了新的方法来约束无法通过直接测量建立的未知模型参数的变化，即空隙的连通性、裂纹孔隙率和裂纹纵横比。基于空隙连通性参数与渗透率之间的相关性，我们开发了一种方法来约束空隙连通性参数的变化范围，该方法依赖于 Kozeny-Carman 方程。在伺服液压试验机（压力机）上对岩石样品进行三轴试验的结果用于估计裂缝孔隙率的上限和评估裂缝形状。使用已建立的约束条件获得的所研究岩石的微观结构特征增加了所构建的碳酸盐岩岩石弹性模型的可靠性。这些模型随后可用于各种地球物理研究，暗示所分析的过程或特性与岩石的微观结构之间存在关系。