Linear non-adiabatic flow of an incompressible fluid in a porous layer – Review, adaptation and analysis of the available temperature models and solutions

This article will concentrate on the problem of 2D heat losses from a reservoir experiencing linear flow of an incompressible fluid with a uniform heat generation in the reservoir. This problem can be instantly attributed to some specific, industrially relevant cases, e.g. estimation of the transient temperature distribution in a hydraulic fracture. However, its main value is that it is an integral part of the full, transient temperature solution for the flow of low compressible fluids (i.e. liquids and, sometimes, gases) into horizontal wells. The full solution is essential for temperature transient analysis in smart wells – a monitoring approach of great potential, but in early development stage.

This article will first discuss the problem of the temperature transient analysis, the flow conditions and the assumptions required to sufficiently simplify the thermal model so that existing solutions can be applied, and their applicability confirmed. Secondly, we will discuss some closely related problems whose solutions are already available, along with the modifications required to apply them to our case. The article continues with a numerical study that was carried out to check the applicability of the existing (modified) solutions. We will either include, or discuss, all the relevant physical phenomena affecting the flowing fluid (Joule–Thomson effect, transient thermal expansion, friction losses, and multi-dimensional heat losses). Finally, a novel approach for the calculation of semi-steady state heat losses from a reservoir to the surrounding formation will be presented. It is based on the solution to the “hot fluid injection” problem with appropriate corrections.