Use of Eigenvalue Stability Analysis for Liner Inflow Test Interpretation

Summary The Well Integrity Management System is a central policy outlining well integrity requirements to ensure operational safety and safe production throughout well lifecycle. Among these requirements is a liner inflow test, particularly when set above the production packer to prevent any potential non-conformities such as Sustained Casing Pressure, tubular corrosion, an underground blowout and many other Well Integrity issues during production. Some Well Integrity Guidelines propose the use of Horner's technique to interpret these tests where predictions are made based on the Y-intercept of a semi-log scale after linearizing the original exponential decay function. This paper examines the use of Horner's technique for liner inflow tests by contrasting pressure build up equations and inherent assumptions made thereof against the fluid expansion phenomena due to the transient heat transfer. Alternatively, it examines the thermal expansion problem through the lens of stability analysis of Ordinary Differential Equations (ODEs) and provides an eigenvalue-based framework to assess these tests. While providing an overview on the limitations of Horner's method for thermal diffusion problem, this paper mainly focuses on the transient and steady-state temperature distribution equation and its solution considering both homogeneous and non-homogeneous boundary conditions. The stability analysis of ODE's and its relevance to the transient heat transfer are presented thereafter to arrive at the final test acceptance criteria. These building blocks act in an integrated way to derive the final acceptance criteria of the liner inflow test interpretation. This proposed technique introduces two acceptance criteria: positive decay rates (eigenvalues) and a consistently decreasing pattern of amplitudes ("wave heights") of spatial temperature variation described by Bessel's function. Positive decay rates ensure the transient term of the heat diffusion equation converges to zero as the test progresses. This occurs due to the negative exponent term, implying the stability conditions of ODE's being met. This convergence is further confirmed by the decreasing pattern of amplitudes of Bessel's oscillations, as the radial temperature variation follows a zeroth order Bessel's function of the first kind. To verify the validity of these criteria, field inflow test data is non-linearly matched to the exponential decay function using Python. This is the major advantage over Horner's technique as it avoids linearization. Decay rates of observed fluid volume remained consistently positive throughout the test, driving the transient temperature distribution term to zero. Amplitudes of spatial frequencies (n – modes) progressively decreased as well implying that higher-order temperature variations gradually faded as the test progressed. As a result, the final temperature distribution was converging to the steady-state term driven by the slower decay of low-order modes. This proposed method also employs Direction Fields (or Slope Fields) to visualize convergence and divergence depending on the sign of eigenvalues. Direction fields based on positive eigenvalues illustrate convergence, whereas for negative eigenvalues, direction fields accurately predict divergence, meaning flow does not cease but instead blows out from the origin, leading to instability. An equilibrium solution is reached when the exponential term in the temporal solution of heat conduction equation converges to zero with positive eigenvalues. This aligns with the stability analysis of ODE's and is the core premise of this proposed technique. Its application to field inflow test data and subsequent findings confirmed that the proposed method provides a more comprehensive and accurate framework to assess these tests. This is due to the consideration of the non-linear nature of the temperature distribution and its variation in both time and space domains.