A Semi-Analytical Framework for Modeling Reservoir Temperature Under Multirate Flow

Abstract

Abstract Downhole temperature measurements provide valuable information that remains largely underexploited for reservoir characterization and surveillance, despite its potential to decisively impact incremental oil recovery early in field development. However, reservoir temperature is governed by complex interactions among pressure diffusion, thermodynamics, and coexisting heat-transport mechanisms. Modeling these interactions generally requires fully coupled simulators, whose computational demands make parameter inversion prohibitively expensive. This work presents an efficient semi-analytical solution that accommodates relevant transport mechanisms while accounting for multiple flow regimes and multirate schedules. The proposed framework followed a sequential formulation. First, the pressure problem was solved analytically in Laplace space. Numerical inversion algorithms were then employed to determine both the convective velocity field and the source-term of the underlying thermal problem in the time domain. Temperature evolution was subsequently computed along flow trajectories using the method of characteristics. Thermal diffusion into surrounding formations was incorporated analytically through an operatorsplitting technique, which enabled the accurate resolution of the steep thermal gradients associated with diffusive boundary layers at geological interfaces. Analysis of the temperature behavior revealed several unintuitive features arising from the coupled physics governing the system. For example, multirate flow schedules can produce temperature responses that differ substantially from constant-rate expectations, underscoring the importance of accounting for realistic rate histories when interpreting temperature data. Predictions were verified against both fully coupled numerical benchmark simulators and known analytical solutions. A sampling-based approach was then employed to perform parameter estimation for closed reservoir systems using the proposed semi-analytical framework as the forward model. Model bias and observation noise were explicitly considered, enabling the assessment of realistic scenarios that deviate from the assumed reservoir geometry and homogeneity. The results were demonstrated through both synthetic and field applications. Existing approaches to temperature modeling typically neglect either advective heat transport or thermal diffusion, and when both mechanisms are included, they often rely on steady-state assumptions. More flexible formulations accommodating multirate schedules rely on fully numerical solutions. This work introduced a semi-analytical framework that simultaneously accounts for advection and diffusion under multirate conditions. The formulation captures transient behavior, its transition toward boundary-dominated flow, and the resulting pseudosteady-state or steady-state regimes while enabling computationally efficient evaluations.