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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4

2.2. Perkins-Kerns-Nordgren (PKN) Model The PKN model postulates a constant fracture height irrespective of the fracture length. Within this model, a 2D plane-strain approach is utilized in the vertical plane, depicting the fracture with an elliptical cross-section in both horizontal and vertical directions. Unlike the KGD model, the PKN model presumes a notably smaller fracture height in comparison to the fracture length (Nasiri, 2015); ��� �� � � � � . � � � � ................................................................................................................................................ (4) ° � 4 � ������� � � � � �� � � � � � � � . � � � � ............................................................................................................................... (5) ��� 2.5 � � � � � ����� � � � � � � � � � . � � � � ............................................................................................................................. (6) 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 2.3. Development Models In the present work, tasks have been performed utilizing the Matlab program on KGD and PKN models to develop the models' shapes as realistically as feasible. Trials and errors were conducted to validate the modified models. Hence, the equations for two enhanced models KGD and PKN are presented below.

2.3.1 Develop KGD Model Model KGD, in the form of Eqs. 1 and 2, are modified as shown below: �� 0.48 � ��� � ������ � �

� � � . � � � � � 0.2 ���� ���� �� 0.2 ���� ��� � ....................................................................... (7) � � � . � � � � � 0.01 ���� ��� �� 0.01 ���� ��� � ................................................................ (8) � � � ........................................................................................................ (9)

° � 1.32 � ������� � � � � � ���� ��� � 0.96 � �� � ����� �� � � � �� �� � ����� �� � � � � �

2.3.2 Develop PKN Model Model PKN, in the form of Eqs. 4 and 5, are modified as shown below: �� 0.68 � �� � ������� � � � � . � � � � � 0.2 ���� ���� �� 0.2 ���� ��� � ................................................................................. (10) � � � . � � � � � 0.003 ���� ��� �� 0.003 ���� ��� � ....................................................................... (11) � � � . � � � � .................................................................................................................................... (12) Table 1 indicates fracture properties, operational conditions, and material parameters that can be utilized to depict fracture growth. ° � 2.5 � ������ � �� � � � ��� 2.5 � � � � � ����� � � � � � �