Title : Use of stability analysis for liner inflow test interpretation
Abstract:
Objective/Scope: Liner Inflow testing ensures pressure containment throughout well life cycle as part of secondary barrier envelope. This paper aims to re-evaluate the current methodology used for Inflow Test interpretation by analyzing the underlying equations and introducing an eigenvalue-based framework. This approach assesses stability by determining if the function converges to zero, indicating cessation of fluid flow. In mathematical terms, inflow test interpretation aligns with examining system stability in ordinary differential equations (ODE) through eigenvalue analysis.
Methods, Procedures, Process: The transient heat equation, solved using separation of variables, introduces an exponential decay function. By calculating decay rates (eigenvalues) from observed fluid volume over elapsed time during inflow testing, system stability can be assessed. In ODEs, negative eigenvalues ensure stability by forcing the function to converge to zero. Similarly, for the reservoir-well system, positive eigenvalues in exponential decay function drive stability, as the exponent in the transient heat equation solution is inherently negative. This method treats the reservoir-well system holistically and evaluates its stability based on observed decay rates and eigenvalue trends.
Results, Observations, Conclusions: During inflow testing, fluid volume should exhibit an exponential decay pattern, with a consistently positive decay rate. To confirm thermal diffusion as the dominant mechanism, a non-linear exponential decay function was fitted to field inflow test data using Python. A good match on the model was evaluated based on initial fluid volume, evolution of eigenvalues, and residuals between predicted and observed fluid volume. While eigenvalues remained positive, their gradients were still evolving, and residuals stabilized only after 140-160 minutes. A strong match was achieved after 200- 220 minutes, accurately predicting fluid volume and ensuring consistent positive decay rates. Positive decay rate indicated flow convergence in the direction of fields, whereas negative rates led to divergence, blowing out from origin, leading to instability. Applying Excel trendline for exponential function yielded a good R2 score but failed to match initial fluid volumes, even after 300 minutes of test data. This highlights the limitations of traditional methods compared to the proposed eigenvalue-based framework.
Novel/Additive Information: This paper introduces a stability analysis framework distinct from the Horner technique. Stability analysis evaluates the direction of fields where solutions of original function converge to zero, whereas the Horner technique pseudo-mathematically makes prediction of original non-linear function from a Y-intercept after linearization of exponential decay function. Unlike pressure diffusion equations, thermal diffusivity equations do not equate the asymptote to the Y-intercept on a Horner plot, making stability analysis a more accurate alternative.
