## Introduction

Materials | References |
---|---|

Rock | |

Rock and clay | |

Quartz sand and silts | Midttømme and Roaldset 1998 |

Clay |

## Methodology

### Existing prediction method for k

_{r}is normalised thermal conductivity which is the function of saturation; k

_{sat}and k

_{dry}are the thermal conductivity of geomaterials at dry and saturation states, respectively; a is an empirical parameter; and S

_{r}is saturation degree. As a simple and easy-to-use model, Eq. (1) yields great performance for predicting k and has been referenced in many scientific studies (Balland and Arp 2005; Côté and Konrad 2005; Farouki 1981). Hence, the two extreme soil conditions above will be considered in further detail below for more general applications. Referring to Li et al. (2020), the thermal conductivity of solid (k

_{s}), air (k

_{a}) and water (k

_{w}) is set to 4, 0.02 and 0.5 W/(mK), respectively.

### Numerical method

#### Reconstruction of geomaterials

_{d}), which controls initial solid core content placement. Other solids are generated based on the growth probability of solid (P

_{i}) that contains four primary directions (i = 1, 2, 3, 4) and four secondary directions (i = 5, 6, 7, 8) (see Fig. 2a). The generation process of the solid phase will be terminated when the porosity (n) reaches the target content. Figure 2b demonstrates the flow chart of the whole modelling procedure. The anisotropic structure of geomaterials can be achieved by altering the value of P

_{i}. When P

_{i}in four primary directions is quadrupled that in four secondary directions, the rebuilt sample can be regarded as isotropic. As geomaterials tend to exhibit layering in the horizontal direction, the ratio of P

_{i}in the x-direction to that in the y-direction, labelled “Y”, often exceeds 1.

_{k}increases initially and then decreases with increasing porosity, reaching its maximum if n equals 0.5. Since different geomaterials represent various combinations of factors (e.g. porosity and different levels of layered structure), we selected a geomaterial specimen whose Y = 50 and n = 0.5 as a typical example to investigate the scale dependency of thermal properties. Findings from this particular case can be used to predict thermal conductivity and the corresponding anisotropy ratio for other heterogeneous geomaterials with different structures in practice.

#### Computation of thermal properties

_{m}) that represents the size of the sample in centimetre and millimetre (e.g. laboratory-scale) and the minimum scale (d) that is the grid cell size used in the simulations and is equivalent to the solid size.

_{0}is a prescribed temperature; n is the outward normal vector to the simulation domain. Each sample S is loaded with the defined two boundaries for homogenisation in the finite element analyses. When calculating the thermal conductivity along x-direction (k

_{x}), the Dirichlet boundary is applied on the boundaries perpendicular to the horizontal direction and the Neumann boundary is set at the surfaces parallel to the x-axis. In contrast, the two boundary conditions are swapped if the value of thermal conductivity along y-direction (k

_{y}) is needed to be computed.

_{k}) is defined as the ratio of the thermal conductivity in the x-direction (k

_{x}) to that in the y-direction (k

_{y}) since the geomaterials possess apparent layered structures in the horizontal direction (Jung et al. 2021).

## Results and discussion

### Anisotropic structure of geomaterials

Dimension of sample S (mm) | Number of sample S |
---|---|

2000 × 2000 | 1 |

1000 × 1000 | 4 |

500 × 500 | 16 |

400 × 400 | 25 |

250 × 250 | 64 |

100 × 100 | 400 |

50 × 50 | 1600 |

25 × 25 | 6400 |

### Scale dependency on thermal properties

_{k}of dry sample S under three typical dimensions (25, 100 and 400 mm). Results in Fig. 6a indicate that the range of the anisotropy ratio expands as sample T is gradually refined. As shown in Fig. 6b, the mean value of r

_{k}increases from 7.9 to 17.75 as decreasing sample dimension. In addition, the anisotropy ratio tends to be distributed in a normal distribution form as sample S refines. All samples with different dimensions extracted from sample T are prepared and tested to determine the specific relationship between thermal properties and observed scales. The FEM results of eight cases in Table 2 are discussed in detail.

_{x}, k

_{y}and r

_{k}) versus sample dimension by using mean value and median value exhibit similar trends. When the observed scale is relatively small, the properties in heat conduction have significant fluctuation. This can be attributed to the heterogeneity of the sample at a small scale inducing the heat transfer barriers or the flow channels with high connectivity. Both the average values of k

_{x}and r

_{k}have a similar trend as the increment of L

_{m}/d. In particular, the anisotropy ratio can reach up to 45 when the observed scale is 25 L

_{m}/d. On the contrary, the r

_{k}of sample T is 7.52, which is considerably lower than the datum at a small scale. The difference between data at small scales and results measured in situ tests demonstrates that direct upscaling of the results from small scales to large scales of interest in practical applications may lead to significant errors. Figure 7b also shows that the mean value of k

_{y}hardly changes with the sample dimension, which can be attributed to sample S with horizontal layered structures so that the mean thermal transfer along the vertical direction is not sensitive to the sample dimension. Comparing those data, one can conclude that the variation in thermal conductivity of SVE decreases with the increasing observed scale. It should be emphasised that when upscaling the small-scale (e.g. laboratory-scale) experimental or numerical tests on the geomaterial samples to field scale, the anisotropy in thermal conductivity may be altered by the scale dependency.

_{m}/d. Unlike dry geomaterials, the discrepancy of average thermal responses between different scales is not as apparent for saturated samples, implying a slight decrease when the sample dimension increases. This can be attributed to the different ratios of thermal conductivity of solids to that of pores (k

_{s}/k

_{p}). For dry geomaterials, the ratio of k

_{s}to k

_{a}is equal to 200 (= 4/0.02), while the ratio of k

_{s}to k

_{w}is merely equal to 8 (= 4/0.5) when the sample is saturated. A higher ratio of k

_{s}/k

_{p}can promote the effect of heterogonous structure on the anisotropy in thermal conduction. According to previous analyses, the thermal behaviours of geomaterials in practice probably lie between the results of two extreme conditions of geomaterials.

### Variation of thermal property

_{x}and k

_{y}) of saturated samples with the same sample dimension are higher than those of dry samples since k

_{w}is larger than k

_{a}. The mean values of thermal properties of saturated samples do not present remarkable changes with L

_{m}/d. However, the thermal conductivity of dry samples has more obvious scale-dependent characteristics (see Fig. 9a-1 to a-3). The greater ratio of thermal conductivity (k

_{s}/k

_{p}= 200) leads to more considerable changes in mean values of k

_{x}and r

_{k}but smaller than that of k

_{y}. The COV reduction is considerable when upscaling from a small scale to in situ scale, given in Fig. 9b.

_{x}and k

_{y}) decreases dramatically at relatively small scales where the sample dimension at dry conditions ranges from 25 to 250 L

_{m}/d. When the observed scale is larger than 250 L

_{m}/d, the COV has dropped to zero nearly. When the pores are occupied with water in geomaterials, the variation of COV in thermal conductivity is slight, ranging from 0.1 to 0 as L

_{m}/d increases. Notably, the COV of r

_{k}in saturated samples at larger scales is approximately close to 0, while that of dry samples gradually decreases with the increment of the observed scale (see Fig. 9b-1). Therefore, the variation of thermal properties is not negligible when the samples are tested at small scales. To ensure more accurate measurements, it is crucial to have an adequate number of laboratory samples, especially for anisotropic geomaterials with lower moisture content.

### Upscaling thermal properties

_{i}is the thermal conductivity of ith SVE, k

_{A}is the arithmetic average of SVEs, k

_{H}is the harmonic average of SVEs, k

_{G}is the geometric average of SVEs and k

_{Q}is the quadratic average of SVEs.

_{x}and k

_{y}calculated by four averaging models at dry conditions are larger than that at saturated conditions. The thermal conductivity of geomaterials is affected by the volumetric fraction and individual properties of each component with geomaterials. For dry samples, the pores are filled with air whose thermal conductivity equals 0.02 W/(mK) (Li et al. 2020). In contrast, the thermal conductivity of water occupying the entire pore space within the saturation sample is 0.5 W/(mK) (Li et al. 2020). Accordingly, for a particular type of geomaterial with the same porosity, the thermal conductivity of the geomaterial at the saturation state is higher than that at the dry condition. It can be observed that the values in Fig. 10c–d are higher than the corresponding values in Fig. 10a–b. Referring to Fig. 9, the COVs of r

_{k}of dry samples are higher than those of saturation samples, which indicates that the values of r

_{k}of SVEs at dry conditions exhibit significant variations. In other words, the deviations between k

_{x}and k

_{y}of SVE are more pronounced. After implementing four averaging schemes, the differences in ultimate average values between k

_{x}and k

_{y}of sample T at dry state are also more remarkable than those for sample T at saturation state, as shown in Fig. 10.

_{x}, whereas the arithmetic average model is more suitable for evaluating k

_{y}. As for saturated samples, the geometric average nearly coincides with the thermal conductivity from direct FEM over the entire sample T. Therefore, the minimum observed size should be limited to 1/8 (=250/2000) size of the macroscale sample of interest to derive relatively accurate results.

## Conclusions

_{x}and r

_{k}have similar decreasing trends with the increment of L

_{m}/d. The mean value of k

_{y}hardly changes with the sample dimension. Unlike dry geomaterials, the discrepancy of mean values of thermal properties between different scales is not as apparent for saturated samples, implying a slight decrease when the sample dimension increases. The thermal behaviours of geomaterials in practice probably lie between the results of two extreme conditions (dry and saturation) of geomaterials.

_{x}of dry samples, harmonic and arithmetic average models are more suitable for determining k

_{x}and k

_{y}, respectively. In addition, the geometric average model also performs better than the other three averaging methods for saturated geomaterials.

_{k}for various types of geomaterials, this study is likely not to perfectly replicate a geomaterial specimen with real-world properties but yield a series of statistical analyses with uncertainty qualification to provide a new perspective to explore the scale dependency of anisotropy in thermal conductivity. Besides, this work merely illustrated one typical in situ sample with a relatively high porosity (i.e. 0.5). It would be beneficial to acquire additional field-scale data to validate the effectiveness and applicability of upscaling schemes. Future work stemming from this preliminary study could involve the integration of more available field data representing geostatistical measurements to further narrow the gap between upscaling schemes and real-world energy and environment engineering applications. Moreover, comprehensive investigations of the influence of porosity on the scale dependency of anisotropy in thermal conductivity will be a focal point of future research.