Identification of thermal conductivities by temperature gradient profiles: One‐dimensional steady flow

Abstract

The comparison model method (CMM) is applied to the identification of spatially varying thermal conductivity in a one‐dimensional domain. This method deals with the discretized steady‐state heat equation written at the nodes of a lattice, a lattice which models a stack of plane parallel layers. The required data are temperature gradient and heat source (or sink) values. The unknowns of this inverse problem are not nodal values but internode thermal conductivities, which appear in the node heat balance equation. The conductivities, e.g., the solutions to the inverse problem obtained by the CMM, are a one‐parameter family. In order to achieve uniqueness (which coincides with identifiability in this case), a suitable value of this parameter must be found. To this end we consider (1) parameterization, i.e., introducing equality constraints between the unknown coefficients, (2) the use of two data sets at least in a subdomain, and (3) self‐identifiability. Each of these items formally translates the available a priori information about the system, e.g., geophysical properties of the layers. Under the assumption that an admissible solution exists, we evaluate the effects on the solution of two types of data noise: additive noise affecting temperature and a kind of multiplicative noise affecting heat‐source terms. In the numerical examples we provide, parameterization is combined with the CMM in order to obtain the unique solution to a test inverse problem, the geophysical data of which come from a well drilled across Tertiary layers in central Italy. Finally, we consider data perturbed by pseudorandom noise. More precisely, we add noise to temperature gradients and obtain stability estimates which correctly predict the numerical results. In particular, if the layers where the parameterization constraint applies contain a strong source term (e.g., due to flowing water), the solution is relatively insensitive to noise. Also, for multiplicative noise affecting heat‐source terms, theoretical stability predictions are confirmed numerically. The main advantage of the CMM is its simple algebraic formulation. Its implementation in the field by means of a pocket calculator allows both a consistency check on the data being collected and estimates of the unknown values of conductivity.

Comments on “Steady- and transient-state inversion in hydrogeology by successive flux estimation” by P. Pasquier and D. Marcotte

“…(9) is a local equation applied separately at each point of the domain; therefore the CMM does not require the solution of a Cauchy problem through integration along characteristic lines, see Eq. (6). In other words it is not true that ''the CMM can be seen as an automated version of the tube method'' and ''consists of... a discrete solution of the Cauchy problem'' as stated respectively at pages 1935 and 1950 of [3].…”

“…A presentation of the CMM can be found in the original paper by Scarascia and Ponzini [8], and in subsequent papers by Ponzini and Lozej [5], Ponzini and Crosta [4] and Ponzini et al [6].…”

Thermal conductivity of sediments within the gas-hydrate-bearing interval at the JAPEX/JNOC/GSC et al. Mallik 5L-38 gas hydrate production research well

“…Index applied where suitable information about in situ ground temperatures and the local/regional geothermal heat flux is available (Blackwell, 1983;Ponzini et al, 1989). In theory, the average thermal conductivity of a specified sediment layer can be calculated directly, according to the formula…”

Modeling Groundwater Flow in Heterogeneous Porous Media with YAGMod

“…The CMM was proposed to identify T at every node of a discretization grid [18] and successively developed to directly compute internode transmissivities [19]. Further modifications were proposed by Ponzini and Crosta [20], Ponzini et al [26], Pasquier and Marcotte [27], Ponzini et al [28]. The CMM was applied to alluvial aquifers in Italy [14,16,21], Switzerland [22] and Canada [23,29,30] and to a carbonatic aquifer in southern Italy [17,31].…”