Strategy for thermal modelling

6.3.2 Modelling approach Introduction

The strategy for the thermal site descriptive modelling involves producing spatial statistical models of both lithology and thermal properties, and generating stochastic simulations to give spatial 3D realisations of thermal properties that are representative of the modelled rock domain. These realisa-tions are used to represent the rock domain statistically. The methodology is described in detail in /Back and Sundberg 2007, Back et al. 2007/.

There are different objectives for which the modelling approach can be used. In this report, the focus is on description. Of special interest for the description is to:

• determine the low percentiles of thermal conductivity,

• model how the thermal conductivity varies with scale, and

• produce realisations of the spatial distribution of thermal properties that can be used for subsequent modelling work, such as numerical temperature simulations for the thermal design of a repository (distances between canisters and tunnels).

For the description problem, no concern is given to specific locations in the rock mass; only the statistics of the rock domain of interest are addressed. The methodology for this type of problem is based on unconditional stochastic simulation (distributes simulated data spatially without honouring measurements at specific locations). Due to the large rock volumes and computer limitations, condi-tional stochastic simulation with high resolution can only be used for small parts of a rock domain.

The focus of the modelling approach is on thermal conductivity. In addition to thermal conductivity, the corresponding heat capacity distribution is determined based on the relationship between thermal conductivity and heat capacity for rock types in the Forsmark area (see section 6.2.5). The realisations of thermal conductivity are used as input to these calculations, and the result is a heat capacity distribution that represents the rock domain.

Outline of the methodology

The methodology, outlined in Figure 6-6, is applied separately for each rock domain. The simulation scale (1) is defined first. This scale determines how lithological data (2) should be prepared and if a change of support (5) is required for the thermal data (4) (change of support, or upscaling, refers to the change of the scale of data or simulated values). The lithological data acquired from boreholes and mapping of the rock surface need to be reclassified into thermal rock classes, TRCs (3). The main reason for this is that only a limited number of classes can be handled in the lithological simulations.

The lithological data are used to construct models of the transition between different TRCs, thus describing the spatial statistical structure of each TRC (7). The result is a set of transition probability models that are used in the simulation of TRCs (8). The intermediate result of this first stochastic simulation is a number of realisations of the spatial distribution of groups of rock types in each domain.

Based on the thermal data, a spatial statistical thermal conductivity model is constructed for each TRC (9). It consists of a statistical distribution and a variogram for each TRC. These are used in the stochastic simulation of thermal conductivity (10), which results in a number of equally probable realisations of thermal conductivity for the TRC.

In the next step, the realisations of TRCs (lithology) and thermal conductivity are merged (11), i.e.

each realisation of lithology is filled with simulated thermal conductivity values. The result is a set of realisations of thermal conductivity for each rock domain that considers both the difference in thermal properties between different TRCs, and the variability within each TRC. If the result is desired at a scale different from the simulation scale, upscaling of the realisations can be performed (12) to a scale not larger than the size of the simulation domain. In practice, upscaling should be made to a much smaller scale, preferably the canister scale. The results (13) can be presented in a number of ways, for example as 3D illustrations, histograms and statistical parameters for the rock mass, probabilities of encountering low thermal conductivity values, etc.

The methodology can also be used for other types of rock properties, once the appropriate upscaling procedure is determined.

6.3.3 Modelling assumptions

The modelling approach requires a number of assumptions in various steps of the modelling process.

The most important ones are listed below.

• It is assumed that thermal conductivity data (from TPS and SCA methods – section 6.2.1) represent the 0.1 m scale, which is the scale at which the initial simulations are performed (cells of cubic shape with 0.1 m sides).

• The 1 m scale is assumed to be sufficiently small to properly represent the subordinate rock types in the lithological simulations.

• The simulation volumes (50×50×50 m³ for scale 1 m) are assumed to be sufficiently large for the objectives of the simulations, see 6.3.2.

• Borehole information from Boremap and core samples used for measurements are assumed to be representative of the rock domain.

• Water movements are not considered in the modelling. It is assumed that modelling such effects could lead to non-conservative estimates of thermal properties.

• The modelling is performed using effective values of thermal conductivity (isotropic assumption) – this also applies to the upscaling methodology, where effective values are calculated using the SCA approach (described in section 2.3.4 in /Back and Sundberg 2007/).

• Geological interpretations, based on expert opinion, have been used in the simulations.

• For the purpose of lithological simulations, a rock domain is divided into thermal subdomains, each of which is assumed to be statistically homogeneous.

• In the lithological simulations, the transition between TRCs are assumed to follow a Markov Figure 6‑6. Schematic description of the approach for thermal conductivity modelling of a rock domain (λ represents thermal conductivity).

Lithological data

Stochastic simulation of TRCs

Stochastic simulation of λ

Upscaling n realisations

of geology

Defining Thermal Rock Classes (TRCs) within

the rock domain

Change of support

Transition probabilities of TRCs

Spatial statistical

structure of the TRCs Spatial statistical thermal model for each Expert TRC

knowledge

Merging of realisations

Thermal data Choice of

simulation scale

n·x realisations of λ (n for each TRC) x TRCs

Results n realisations of λ

x spatial statistical thermal models 1.

2. 3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

0.1 m scale

• Spatial correlation in thermal conductivity between different parts of a rock type body is assumed not to be “broken” by the presence of a different type of rock separating these parts.

• The presence of rock stress components is assumed to have no effect on the thermal properties.

Detailed information regarding the assumptions made for the lithological and the spatial statistical thermal models for each TRC is provided in the following sections.

6.3.4 Feedback from other disciplines

The rock domain model presented in section 5.4 provides the geological framework for the descrip-tion of thermal properties of the rock mass within the target volume at Forsmark. Two key rock domains have been identified inside the tectonic lens and target volume. These are the volumetrically more important domain RFM029, and the subordinate domain RFM045. The thermal properties of these two domains are evaluated here. A geological description of the two domains that addresses, for example, the dominant and subordinate rock types is provided in section 5.4.

Valuable cooperation with the geologists in the Forsmark modelling team has been established and maintained throughout the thermal modelling stages. Integration with geology was particularly close concerning the geological interpretations used as input in the stochastic simulations of litho logies.