Thermal conductivity model function of porosity: review and fitting using experimental data

Abstract

Thermal conductivity of porous rocks depends on a large variety of proper to rock parameters as well as external influences. Thus, it can generate difficulties in determining accurate thermal behavior of the rock. The rock parameters which influence the thermal conductivity are principally the porosity, the microstructure [1] and the mineral composition. However, these parameters, in turn, can be impacted by external influences such as temperature and pressure. An accurate determining of the thermal conductivity is crucial in oil and gas engineering or in geothermal application. For example, during thermal EOR or geothermal application, the porosity and/or the microstructure of the sedimentary rocks can vary due to the increase of temperature and pressure, and this modification must be quantified to be accounted for the thermal behavior of rocks. Many efforts have been done to estimate the thermal conductivity of sedimentary rocks in parallel to the experimental methods for its determination. These estimations have always been the subject of intensive studies, and a lot of data [2, 3] are obtained as well as models and methodologies to characterize the thermal conductivity of rocks [4–7]. Moreover, this type of estimation is well-known by other research communities. Indeed, we find the same formal analogy between Fourier, Ohm’s law, Darcy’s laws and thermal conductivity. For example, considering Darcy’s laws, the same problem is well-known and termed “upscaling” [8, 9] and consists in computation of the effective permeability considering a heterogeneous rock. Classically, the upscaling process can be related to percolation theory [10], which describes connectivity of objects within for example, a porous structure. We can also determine effects of this connectivity on macroscale properties such as thermal conductivity [11]. In particular, the fundamental contributions of Torquato who proposed strategies via rigorous microstructure-property relations [1, 12]. Finally, many technics are based on a porosity dependence and a link between the conceptual thermal conductivity of the non-porous rock, kR, and the thermal conductivity of the fluid saturated the porous rock kf. These technics are simple to implement especially when there is no precise information about the microstructure. In order to predict accurately the thermal efficiency of the geothermal installation or the oil recovery of a thermal EOR process, such as, for example, Steam Assisted Gravity Drainage (SAGD), very often the engineers invoke numerical simulations. Numerous reservoir simulators [13–16] allow to estimate the thermal conductivity as function of porosity, but these solutions are often based on a mixing laws which are quite simplistic models. The purpose of this paper is to propose a better methodology to predict a thermal conductivity of reservoir rocks depending on the porosity for a reservoir simulator which satisfy the following conditions