Sensitivity to Initial Data Errors in Interpreting Temperature Logging of an Isolated Injection Well Segment

Abstract

This study considers the inverse problems inherent in interpreting temperature logging data from an isolated segment of the injection well in order to ascertain its operating period and the thermophysical properties of the oil reservoir.
The forward problem of thermal conductivity was reduced to a one-dimensional axisymmetric formulation within the oil reservoir layer, disregarding the vertical thermal exchange with neighboring layers.
The inverse problem of determining the well operating period was solved by reformulating the forward problem with regard to the temperature field derivative, which enabled the use of first-order optimization methods. Thus, Nesterov’s method was applied. An algorithm to automatically scale one of the method’s parameters (step length) was developed, and the optimal value of the second parameter (inertial step) was calculated. This increased the efficiency of the method by 10 – 15 % in solving the problem under consideration.
The algorithm’s stability against perturbations in the initial data on temperature and thermophysical properties was demonstrated. The sensitivity analysis revealed that a 1 % error in the temperature measurements results in a standard deviation of the solution, which is about 2 % from the true value of the well operating period. A similar level of error was seen when the thermal diffusivity was overor underestimated by approximately 15 %. The solution was little sensitive to variations in the heat transfer coefficient between the oil reservoir and the well at characteristic magnitudes; even with a twofold distortion, the error in the determination of the well operating period did not exceed 1.5 %. To mitigate the error in thermometry interpretation to 1 %, temperature measurements must have an error margin of no more than 0.25 %, alongside precisely specified thermophysical properties of the oil reservoir, or, alternatively, when temperature is measured accurately, the rock thermal diffusivity must be set within an error margin of less than 3 %, but it is nearly impossible under real conditions.
Increasing the number of temperature measurements diminishes the sensitivity to measurement errors, with the optimal efficacy achieved at 10 measurements, rendering further increments impractical.
Therefore, the algorithm’s stability and the solution’s sensitivity of the inverse problem of determining the reservoir thermal diffusivity for a given operating period of the well relative to temperature measurement errors were found. The results show that a 1 % error in temperature measurements leads to a standard deviation of about 6 % from the true value.