Moh Hasan

Department of Mathematics, University of Jember, Indonesia

Firdaus Ubaidillah

Department of Mathematics, University of Jember, Indonesia

Yunita Salsabila Imani

Department of Mathematics, University of Jember, Indonesia

Faizal Rifky Fahreza

Department of Mathematics, University of Jember, Indonesia

DOI: https://doi.org/10.19184/mims.v26i1.60043

ABSTRACT 

Rubber packers are important downhole tools used to isolate fluid zones and prevent leakage during oil and gas production. Understanding their deformation behavior is essential because the performance of the sealing system is strongly influenced by the expansion of the rubber element under pressure and mechanical loads. This study presents a numerical solution for the anchored rubber packer expansion model using the central finite difference method. The packer deformation is governed by a fourth-order ordinary differential equation with mixed Dirichlet-Neumann boundary conditions. A ghost point technique is applied to handle the zero-slope boundary conditions at the clamped ends. The resulting linear system is then solved using QR decomposition. We compared the numerical results with the exact analytical solution to observe the physical geometric transitions. The results show that the numerical method successfully captures the deformation shapes, including stress ripples in thinner rubber configurations. More importantly, the absolute error analysis reveals a fundamental difference in how the physical model is treated. The exact analytical formula strictly couples the mandrel force to the internal fluid pressure, locking it into a specific theoretical scenario. In contrast, the finite difference method allows us to decouple these variables. This gives us the flexibility to input the mandrel force as an independent mechanical load. When tested with different independent tightening forces, the numerical model successfully showed that the rubber expansion shrinks due to the tensile stress effect. This proves that the proposed numerical approach is a highly flexible and realistic tool for simulating complex, independent operational scenarios in petroleum engineering.

Keywords: Anchored packer, finite difference method, fourth-order differential equation, mixed boundary conditions, numerical analysis.

MSC2020: 65L10