Relationship between thermal conductivity and water content of soils using numerical modelling P. C OSENZA , R. G UE ´ RIN & A. T ABBAGH UMR 7619 Sisyphe, Universite´ Pierre et Marie Curie and CNRS, case 105, 4 place Jussieu, 75252 Paris Cedex 05, France Summary There is no simple and general relationship between the thermal conductivity of a soil, , and its volumetric water content, , because the porosity, n, and the thermal conductivity of the solid fraction,  s , play a major part. Experimental data including measurements of all the variables are scarce. Using a numerical modelling approach, we have shown that the microscopic arrangement of water influences the relation between  and . Simulated values for n ranging from 0.4 to 0.6,  s ranging from 2 to 5 W m 1 K 1 and  from 0.1 to 0.4 can be fitted by a simple linear formula that takes into account n,  s and . The results given by this formula and by the quadratic parallel (QP) model widely used in physical property studies are in satisfactory agreement with published data both for saturated rocks and for unsaturated soils. Consequently, the linear formula and the QP model can be used as practical and efficient tools to investigate the effects of water content and porosity on the thermal conductivity of the soil and hence to optimize the design of thermal in situ techniques for monitoring water content. Introduction The determination and the monitoring of water content in soils and unsaturated superficial layers are among the funda- mental requirements in both agronomy and water manage- ment. One needs to know, on the one hand, the water reserve and, on the other, the rate of water flow through the unsat- urated layers. Many methods and techniques have been tested to determine the water content of a given soil volume, in particular those measuring electrical properties, conductivity or permittivity that are quick and cheap. Thermal properties also merit attention: they depend significantly on water con- tent, and measuring temperature is easy and much used in soil studies. Two independent thermal properties are involved in the dominant transfer process, namely conduction. The first, the volumetric heat capacity, C v , has been recognized as lin- early linked to the volumetric water content,  (de Vries, 1963), and is now in common use both in the laboratory and in the field to determine  with so-called ‘heat-pulse’ probes (Campbell et al., 1991). For the second, the thermal conduct- ivity , it has been recognized that the relationship, though monotonically increasing, can be more complicated. Despite continuous progress in the non-stationary step-pulse technique with needle probes (de Vries, 1952; Blackwell, 1954; Tabbagh & Jolivet, 1974; Tabbagh, 1985b; Larson, 1988), usable experi- mental data on the relation between  and  are scarce (Ochsner et al., 2001), perhaps because the mineralogy of the solid fraction and the air content have more pronounced influence than  (Smith & Byers, 1938; Smith, 1939; Tabbagh, 1976). This is in contrast to the case of dielectric permittivity (Topp et al., 1980). An increase in the conductivity with increasing water content can be the consequence of the decrease of the air content. An experimental study of the conductivity against water content would have thus to take into account air content and solid mineralogy, which would lead to much experimentation with at least three variables. This explains why recent studies with active thermal tech- niques use C v rather than . However, if one attempts to deduce  from natural variation in temperature in a soil then both properties have to be considered, and one needs a simple and reliable expression of the relationship between  and  or at least an unequivocal expression of @/@ to allow water content to be monitored. To reach this objective, we chose to use numerical simula- tions based on a physical analogy: if heat moves purely by conduction then our problem is mathematically the same as that in static electricity for electrical conductivity or dielectric permittivity, so the same modelling techniques can be applied. Consequently, we can ignore heat transfer by vapour flow (Hiraiwa & Kasubuchi, 2000) or water flow, i.e. by convection. As in the electrical analogue, we chose to apply the method of moments (MoM) (Tabbagh et al., 2000, 2002) without physical assumptions (except the unavoidable assumptions bearing on the discretization of the geometries of solid, water Correspondence: P. Cosenza. E-mail: cosenza@ccr.jussieu.fr Received 10 September 2002; revised version accepted 30 January 2003 European Journal of Soil Science, September 2003, 54, 581–587 # 2003 Blackwell Publishing Ltd 581