The aim of this study is to develop a numerical method providing the possibility of computing nonisothermal compositional flows faster than traditional finite-volume methods. A plane problem of oil, water, and gas flows is considered. The oil phase is represented by two components: the light fraction and the heavy fraction, which, similarly to water, can be gasified. The study takes into account both the nonlinearity of the oil flow law and the temperature dependence of the parameters of this law. This formulation of the problem is relevant for the simulation of the development of high-viscosity oilfields. To reduce the computational complexity of the problem, the streamline method with splitting with respect to physical processes is used. It separates the convective transport directed along the flow’s propagation from processes related to the heat conduction and gravity, which directions do not coincide with the convective flow. A distinctive feature of the proposed method is the joint solution of the pressure, energy-balance, and mass-component equations both on streamlines and on the initial grid. This feature allows us to correctly calculate oil flows with a complex temperature-dependent rheology. Numerical solutions of the system of flow equations on a two-dimensional grid and on streamlines are obtained by the IMPEC method. For the presented streamline method, we propose an algorithm accounting for the thermal conductivity and transition criteria between calculations on streamlines and on a two-dimensional grid. The developed software is verified by comparison with the analytical solutions and with the results of calculations by finite-volume methods on five-point and nine-point finite-difference stencils.