PSI - Issue 66

Amani J. Majeed et al. / Procedia Structural Integrity 66 (2024) 212–220 Author name / Structural Integrity Procedia 00 (2025) 000–000

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The geometry of hydraulic fractures is influenced by various factors, including the initial stress conditions of the reservoir (both global and local), the mechanical properties of the reservoir rock (such as its heterogeneity and anisotropy), permeability, porosity, the presence of natural fractures, and the operational conditions during the injection process (such as the rate, volume, and pressure) (Liu et al., 2020; Tang et al., 2018). One of the most widely used hydraulic fracture models in the oil and gas industry is the Perkins-Kern-Nordgren (PKN) model, which was named after three contributors; Perkin, Kerns, and Nordgren (Nordgren, 1972). Although several more complex models have been introduced since that time, including pseudo-three-dimensional (P3D) and planar three-dimensional (PL3D) models. Nevertheless, researchers in hydraulic fracturing continue to widely use one-dimensional (1-D) models such as PKNs (Nordgren, 1972), Khristianovic-Geertsma-De Klerks (KGDs) (Geertsma and De Klerk, 1969; Zheltov, 1955). Due to the complexity of hydraulic fracturing processes, it is first necessary to analyze hydraulic fractures of simple geometries in order to develop efficient computational algorithms. Moreover, a three-dimensional (3D) geological models can also be utilized to acquire information about the distribution of rock mechanical properties and in situ stress. This includes parameters such as maximum horizontal stress, minimum horizontal stress, vertical stress, Young's modulus, Poisson's ratio, and tensile strength (Belyadi et al., 2019) (Nasiri, 2015). Nevertheless, Khristianovic-Geertsma-De Klerks (KGD) and Perkins-Kern-Nordgren (PKN) models are theoretical approaches used to describe fracture propagation in hydraulic fracturing and similar processes. Both models are essential for understanding hydraulic fracturing dynamics and are used extensively in the field of geomechanics and petroleum engineering to predict fracture behavior and optimize hydraulic fracturing operations, they assumed a 2D smooth elliptical growth of the fracture. The current study aims to accurately illustrate the progression of a fracture toward its final shape over time. Both models have been modified to provide an approximate propagation path for the fracture. As a result, a more realistic representation of the fracture shape and behavior is achieved. 2. Methodology Initially, it was thought that the hydraulic fracturing process would occur within a consistent and uniformly spread formation, leading to a symmetrical and two-winged fracture emanating from the injection point or line source of the fluid. In line with those assumptions, three fracture modeling techniques were devised; the Khristianovic-Geertsma De Klerks (KGD) model, the Perkins and Kern (PKN) model, and the radial fracture geometry or penny-shaped model. Where those models operate under the premise that the reservoir pressure, temperature, and horizontal stress remain constant (Belyadi et al., 2019), (Nasiri, 2015). The mentioned mathematical concepts will be discussed and explained in order to highlight the modifications made in this work. 2.1. Khristianovic-Geertsma-De Klerks (KGD) model In the KGD model, a 2D plane-strain approach is taken on a horizontal plane, with the assumption of a constant fracture height exceeding the fracture length. This model depicts the fracture with an elliptical horizontal cross-section and a rectangular vertical cross-section, where the fracture width remains consistent in the vertical direction independent of the fracture height (Nasiri, 2015); �� 0.48 � ��� � ������ � � � � � . � � � � ………………...……………………………………………………………… (1) ° � 1.32 � ������� � � � � � � � � . � � � � …………….………..……………………………………………………….... (2) ���� ��� � 0.96 � �� � ����� �� � � � �� �� � ����� �� � � � � � � � � ………………………………………………………………......... (3) Where, L is the fracture length, W is the fracture width, G is the Shear Modulus, is Poisson Ratio, Q is a pumping rate, μ is a fluid viscosity, σ is the stress, and Pw is the wellbore pressure