Heat Transfer in Porous Rocks Through Which Single-Phase Fluids Are Flowing
Authors: Adivarahan P. , Kunii D. , Smith J.M. 
Published: December 2007

Abstract
Effective thermal conductivities were measured for seven samples of porous rocks through which gases or aqueous salt solution were flowing, parallel and countercurrent to the flow of beat. The results are analyzed in terms of the contributions of the solid and fluid phases to the total energy transfer. The solid contribution for the sandstone samples studied was found to be a function of the porosity and an effective thermal conductivity for the solid phase. The fluid contribution was a function of the Peclet number. Based upon the equations presented, the effective conductivity of the system can be estimated from the properties of the fluid, the flow conditions, and the porosity and pore-size distribution of the rock.

Introduction
Heat-transfer characteristics of porous media containing fluids are needed for the design and evaluation of thermal methods of petroleum production. The study reported here considers the problem of countercurrent flow of fluid and heat in sandstone rocks, and is a continuation of previous studies concerned with parallel and radial flow of energy and fluid in both rocks and beds of fine particles. In addition to these earlier works, Asaad, Kimura, Preston, Zierfuss and Vliet and Waddams have reported effective conductivities for systems filled with stagnant fluids. There appears to be no published data on rocks through which fluids are flowing. The purpose of this study was to present such information and, particularly, to investigate the effects of properties of the rock and the flow rate on the results.

SCOPE
The rock samples were naturally occurring sandstones from different locations and were similar to petroleum-bearing formations. The properties of the seven samples showing the range in porosities and permeabilities are given in Table I. The mean pore sizes were calculated from mercury penetration data by averaging the radius with respect to pore volume. The details of this method and comparison with other procedures for estimating r are given elsewhere. Four fluids, nitrogen, carbon dioxide, helium and 10 weight per cent sodium-chloride solution were used with each sample. For Bartlesville sandstone, BA-5, runs were made at mean pressures from 15 to 250 psig using nitrogen. The other measurements were carried out at a mean pressure of 100 psig. The maximum temperature range encountered was 70 degrees to 300 degrees F. The flow rate of fluid ranged from about 1.0 to 130 lb/(hr)(sq ft), and this corresponded to a modified Reynolds-number range of 5 x 10(–4) to 1. .0. The variation in Prandtl number was 0.70 to 4.1.

EXPERIMENTAL WORK
The equipment consisted of apparatus to supply a constant flow of gas, or salt solution, at constant pressure to the bottom of a vertical test section, and metering and flow-regulation apparatus after the test section.
TABLE 1 - PROPERTIES OF ROCK SAMPLES
Mean Pore Sampel Porosity Radius Permeability K* Desingation ( x 10(5) ft) (md) [Btu/(hr)(ft)(degrees F)] Bartlesville Sandstone
BA-5 0.241 2.41 90 0.21 BA-9 0.219 2.40 27 0.32
Boise Sandstone
BO-Old 0.267 6.18 970 0.11 BO-New 0.298 5.62 1300 0.045
42 0.177 2.69 154 1.30 135 0.195 3.27 130 0.70
Alhambra Sandstone
AL 0.215 5.61 610 0.45
SPEJ
P. 290^