Upscaling permeability for three-dimensional fractured porous rocks with the multiple boundary method

Upscaling permeability of grid blocks is crucial for groundwater models. A novel upscaling method for three-dimensional fractured porous rocks is presented. The objective of the study was to compare this method with the commonly used Oda upscaling method and the volume averaging method. First, the multiple boundary method and its computational framework were defined for three-dimensional stochastic fracture networks. Then, the different upscaling methods were compared for a set of rotated fractures, for tortuous fractures, and for two discrete fracture networks. The results computed by the multiple boundary method are comparable with those of the other two methods and fit best the analytical solution for a set of rotated fractures. The errors in flow rate of the equivalent fracture model decrease when using the multiple boundary method. Furthermore, the errors of the equivalent fracture models increase from well-connected fracture networks to poorly connected ones. Finally, the diagonal components of the equivalent permeability tensors tend to follow a normal or log-normal distribution for the well-connected fracture network model with infinite fracture size. By contrast, they exhibit a power-law distribution for the poorly connected fracture network with multiple scale fractures. The study demonstrates the accuracy and the flexibility of the multiple boundary upscaling concept. This makes it attractive for being incorporated into any existing flow-based upscaling procedures, which helps in reducing the uncertainty of groundwater models.     

A review of the thermal regime of the Eastern Alps with respect to the effects of paleoclimate and exhumation

Transient thermal signals such as Pleistocene surface temperature variations or exhumation of great rock volumes are important for the current thermal regime of the Eastern Alpine crust. In this study transient 1-D forward simulations and an analytical approach were used to estimate the order of magnitude of these effects. A comparison with numerical forward simulations and inverse analyses of steady-state heat conduction yields the following main conclusions with respect to the thermal regime of the Eastern Alps along the TRANSALP profile: (1) The change of surface temperatures in the past affects mainly the uppermost part of the Eastern Alpine crust. It results in a maximum thermal signature of more than − 6 K at a depth of 2 km. The deviations from a steady-state temperature gradient and heat flow in the region of the Tauern Window range from 0.3–4 K km^− 1 and 0–6 mW m^− 2, respectively, with maximum values at the surface. (2) Exhumation of the Eastern Alpine lithosphere may result in a thermal signature of up to 4 K at a depth of 1 km. The thermal signature increases further with depth to a maximum of approximately 80 K at a depth of 50 km. As the temperature gradient of the exhumation signal is almost zero at the base of the crust, Moho heat flow appears to be not critically perturbed. (3) The combined effect of exhumation and changing surface temperatures at the Tauern Window amounts to less than 15% of the steady-state temperatures at a depth of  8 km and to less than 10% at the base of Eastern Alpine root. The corresponding perturbation in heat flow is less than 20% at a depth of 4 km, approaching zero below 40 km.     

Heat Transport Processes in the Earth’s Crust

The heat of the Earth derives from internal and external sources. A heat balance shows that most of the heat provided by external sources is re-emitted by long-wavelength heat radiation and that the dominant internal sources are original heat and heat generated by decay of unstable radioactive isotopes. Understanding of the thermal regime of the Earth requires appreciation of properties and mechanisms for heat generation, storage, and transport. Both experimental and indirect methods are available for inferring the corresponding rock properties. Heat conduction is the dominant transport process in the Earth’s crust, except for settings where appreciable fluid flow provides a mechanism for heat advection. For most crustal and mantle rocks, heat radiation becomes significant only at temperatures above 1200°C.

The thermal regime of the Northeastern-German Basin from 2-D inversion

The thermal regime and the distribution of heat flow at the base of sedimentary basins is fundamental to the understanding of the process of basin evolution and the associated mobilization and migration of hydrocarbon and other fluids. For the Northeastern-German sedimentary basin, available information on structure, temperature, and thermal properties along a seismic DEKORP reflection profile allow high resolution 2-D forward and inverse simulations. This approach is attractive in situations where much information is available, if only with considerable uncertainty. In particular, this allows to introduce “soft” information into the analysis. In our case, forward simulations yield initial a priori estimates of the parameters while inversion calculations yield a posteriori estimates of the parameters and their uncertainty. The a priori parameters as well as their assumed uncertainty are input for a Bayesian parameter estimation scheme. In respect to the Northeastern-German sedimentary basin, the inverse analysis postulates a significant and characteristic a posteriori variation of thermal conductivity of the Zechstein unit along the entire profile as well as a generally large a posteriori thermal conductivity of the (pre-Permian) basement in the northern part of the basin. For inverse calculations, we used two alternative scenarios: One assumes the thermal conductivity of the Zechstein unit to be homogeneous along the profile while the other allows a lateral variation. A posteriori heat flow across the base of the model varies from 40 to 60 and 50 to 65 mW m⁻² for models in which values for thermal conductivity and radiogenic heat generation rate were either based on literature values or direct measurements, respectively.     

Coupled Process Models as a Tool for Analysing Hydrothermal Systems

Hydrothermal systems are characterised by complex interactions between heat transfer, fluid flow, deformation, species transport and chemical reactions. Numerical models can provide quantitatively constrained information in regions where acquisition of new data is difficult or expensive thus providing a means for reducing risks, costs, and effort during targeting, production, and management of resources linked to hydrothermal systems. Here we show how numerical simulations of hydrothermal processes can be used to better understand coupled reactive transport in modern geothermal systems and in ancient hydrothermal ore deposits. We give examples based on the Enhanced Geothermal System at Soultz-sous-Forêts in France, hydrothermal mineralisation at Mount Isa in Australia, and the geothermal resource at Hamburg-Allermöhe in Germany.     

3D finite volume groundwater and heat transport modeling with non-orthogonal grids,using a coordinate transformation method

Many popular groundwater modeling codes are based on the finite differences or finite volume method for orthogonal grids. In cases of complex subsurface geometries this type of grid either leads to coarse geometric representations or to extremely fine meshes. We use a coordinate transformation method (CTM) to circumvent this shortcoming. In computational fluid dynamics (CFD), this method has been applied successfully to the general Navier–Stokes equation. The method is based on tensor analysis and performs a transformation of a curvilinear into a rectangular unit grid, on which a modified formulation of the differential equations is applied. Therefore, it is not necessary to reformulate the code in total. We applied the CTM to an existing three-dimensional code (SHEMAT), a simulator for heat conduction and advection in porous media. The finite volume discretization scheme for the non-orthogonal, structured, hexahedral grid leads to a 19-point stencil and a correspondingly banded system matrix. The implementation is straightforward and it is possible to use some existing routines without modification. The accuracy of the modified code is demonstrated for single phase flow on a two-dimensional analytical solution for flow and heat transport. Additionally, a simple case of potential flow is shown for a two-dimensional grid which is increasingly deformed. The result reveals that the corresponding error increases only slightly. Finally, a thermal free-convection benchmark is discussed. The result shows, that the solution obtained with the new code is in good agreement with the ones obtained by other codes.     

The formation of iron hydroxide coatings in an Emscher Marl: inverse reactive transport modeling of reactive surface area

The possible effects of iron oxide coatings on the reactive surface area of calcite in column experiments have been studied and then modeled numerically. A column with six compartments separated by teflon filters was packed with Emscher Marl (essentially calcite). The marl was initially mixed with corundum as an internal standard. Hydrochloric acid with pH 3 was percolated through the column for a given period, after which the mineralogical changes were quantified by X-ray diffraction ex-situ. Then, the column compartments were re-filled and again percolated with HCl. This procedure was repeated five times. The losses and gains of calcite in the column compartments provided the data basis for modeling the entire experiment with the reactive transport code TOUGHREACT using a kinetic rate law. The experimental results showed that during the first period, loss of calcite in the first compartment is about 50 % less than that determined from the theoretical analysis. This showed the entrance of acid into the higher compartments through preferential flow paths (wormholes) which was observed at the boundary between sample and cell wall. This pattern could also be verified by the relatively high Peclet and low Damköhler numbers, showing a strongly advection-dominant system. Apart from calcite dissolution, structural Fe²⁺ released from calcite oxidized and formed iron hydroxide (FeOOH) coatings along preferential fluid pathways. Despite specific assumptions such as using pure calcite in the model, a comparison between modeling results and lab observations is instructive. The simulated calcite change patterns in the most compartments are consistent with the experiments. Some discrepancies are noted in the last compartment, which may bring the attention to a need for model improvements.     

New heat flow data from the immediate vicinity of the Kola super-deep borehole: Vertical variation in heat flow confirmed and attributed to advection

We present new heat flow values and other geothermal data in the upper crystalline crust in the immediate vicinity of the 12.4-km deep Kola super-deep borehole, NW Russia. Our results show a systematic vertical increase in geothermal gradient and heat flow density as deep as we could measure (1.6 km). Our results confirm earlier results on vertical heat flow trends of in the uppermost part of the Kola super-deep hole, and imply that the thermal regime is not in steady-state conductive conditions. In an area of 3-km × 5-km measurements were performed in 1–2-km deep boreholes surrounding the Kola super-deep hole and on core samples from these holes. Temperature logs are available from 36 holes. Core data exists from 23 boreholes with a total length of 11.5 km at a vertical resolution of 10 m. We carried out a very detailed study on thermal conductivity with regard to anisotropy, inhomogeneity and temperature dependence. Tensor components of thermal conductivity were determined on 1375 core samples from 21 boreholes in 3400 measurements. Additionally, we measured specific heat capacity, heat generation rate, density, porosity, and permeability on selected subsets of core samples. Heat flow from 19 boreholes varies between 31 and 45 mW m⁻² with an average value of 38 mW m⁻². In most boreholes the vertical heat flow profiles show a considerable variation with depth. This is consistent with observations in the upper part of the Kola super-deep borehole. We conclude that this variation is not caused by technical operations but reflects a natural process. It is considered to be due to a combination of advective, structural and paleoclimatic effects. Preliminary 3-D numerical modeling of heat and flow in the study area provides an indication of relative contributions of each of these factors: advective heat transfer turns out to have a major influence on the vertical variation of heat flow, although transient changes in surface temperature may also cause a significant variation. Heterogeneity of the rocks in the study area is less important.