Correcting for finite probe diameter in the dual probe heat pulse method of measuring soil water content

Abstract

This paper provides a simple correction to the measurement of soil volumetric heat capacity (and soil water content) using a dual probe heat pulse device with instantaneous heating, to allow for rods of finite radius and heat capacity. The correction is based on the spatial weighting function of Knight et al. (2007) and checked using the identical cylinders perfect conductor (ICPC) theory of Knight et al. (2012). The dual probe heat pulse (DPHP) method uses two parallel rods to estimate soil volumetric heat capacity and hence volumetric water content. Most of the heat capacity of wet soil is due to its water content, and so the water content has a known relation to the heat capacity. A known pulse of heat applied to the emitter rod causes the temperature at the receiver rod to rise and then fall. The time and the size of the maximum temperature rise at the receiver are measured. Assuming that the thermal properties of the soil are spatially uniform, that the heat pulse is applied instantaneously, and that the rods have zero diameter and zero thermal capacity, the standard solution of the forward problem gives a formula for the maximum temperature rise in terms of the thermal capacity of the soil and the known parameters of the problem. Campbell et al. (1991) inverted this to give a solution of the inverse problem as a simple explicit formula for the soil heat capacity in terms of the maximum temperature rise and the known probe parameters. Kluitenberg et al. (1993) and Bristow et al. (1994) introduced a calculation method that accounts for the finite probe heating duration, but requires the evaluation of the exponential integral function. Knight and Kluitenberg (2004) derived a simple power series approximation which converges well as long as the heating duration is not too big compared to the time to maximum temperature. This method is simple enough to be implemented on a small computer chip in a device. However it has been observed that the measurements are less accurate when the soil is dry, and this is likely to be due to the finite radius and finite heat capacity of the rods being neglected in the existing analysis. The semi-analytical solution of Knight et al. (2012) assumes that each rod is a perfect conductor, and that the presence of the receiver does not affect the temperature field near the emitter. They derived a solution of the forward problem in the Laplace transform domain and inverted it numerically, finding that the probe radius and heat capacity have a significant effect on the time and magnitude of the maximum temperature at the receiver, and hence on the accuracy of the method. The spatial weighting function theory of Knight et al. (2007) for probes of zero radius provides an approximate method of calculating the apparent measured heat capacity of the soil-probe system as a weighted average of the heat capacity of the rods and the heat capacity of the surrounding soil, with the weights calculated by numerical integration and dependent on the probe radii. The weights need only be calculated once for a given probe geometry. This formula can easily be inverted to give a simple correction procedure which gives the soil thermal capacity from the measured apparent thermal capacity of the probe- soil mixture and the known probe heat capacity. As expected, a larger correction is necessary for dry soil than for wet soil, and the correction is zero when the soil and probe heat capacities happen to be the same. The procedure is simple enough to be implemented on a computer chip with limited memory and processing capacity. The accuracy of the procedure was checked by solving the forward problem of calculating the maximum temperature rise with known soil heat capacity and probe parameters using the semi-analytical solution of Knight et al. (2012). An iteration procedure was then used for the inverse problem to calculate the soil heat capacity given the measured temperature rise. The agreement with the approximate value given by the spatial weighting procedure is extremely good. It is relatively straightforward to extend the theory given here for instantaneous heating to the case of finite heating duration.