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A Stochastic Study of Flow Anisotropy and Channelling in Open Rough Fractures

  • Sophie MarchandEmail author
  • Olivier Mersch
  • Michael Selzer
  • Fabian Nitschke
  • Martin Schoenball
  • Jean Schmittbuhl
  • Britta Nestler
  • Thomas Kohl
Original Paper

Abstract

The quantification of fluid flow in rough fractures is of high interest for reservoir engineering, especially for deep geothermal applications. Herein, rough self-affine fractures are stochastically generated with incremental shear displacement and geometrically described by two aperture definitions, the vertical aperture \( a_{\text{vert}} \) and the effective aperture \( a_{\text{eff}} \). In order to compare their effect on fracture flow, such as anisotropy and channelling, Local Cubic Law (LCL) model-based 2D fluid flow is simulated. The particularity of this approach is the combination of a stochastic generation of self-affine fractures with a statistical analysis (560 individual realizations) of the impact of the LCL’s aperture constraint on fracture flow. The results show that aperture definition affects the quantitative interpretation of flow anisotropy and channeling as well as the aperture distribution of the fractures with shearing. Higher values of mean aperture for a given fracture are found using \( a_{\text{vert}} \), whereas the aperture standard deviation is larger with \( a_{\text{eff}} \). In addition, flow anisotropy is significantly sensitive to aperture definition for small shear displacements and shows a relative higher dispersion with \( a_{\text{eff}} \). Thus, LCL prediction models based on \( a_{\text{vert}} \) are expected to lead to higher dispersion of anisotropy results with a higher uncertainty (factor ~ 2). Realizations based on \( a_{\text{vert}} \) lead to an enhanced clustering of high flow rates for higher shearing displacements. This channeling development results in higher total flow rates for these simulations. These findings support the direct calibration of pre-existing LCL anisotropy simulations based on \( a_{\text{vert}} \) towards more representative results using \( a_{\text{eff}} \).

Keywords

Fracture Roughness Aperture Stochastic Anisotropy Channelling 

List of Symbols

\( \alpha \)

Slope of the linear trend lines for \( I_{2} \) in terms of \( I_{1} \) (unitless)

\( \beta \)

Intercept of the linear trend lines for \( I_{2} \) in terms of \( I_{1} \) (unitless)

\( \Delta h \)

Height difference used to define \( p \) (in m)

\( \Delta \varvec{r} \)

Vector distance used to define \( p \) (in m)

\( \Delta x \)

Discretisation element length along the \( x \)-axis (in mm)

\( \Delta y \)

Discretisation element length along the \( y \)-axis (in mm)

\( \lambda \)

Arbitrary scaling factor used to define \( p \) (unitless)

\( \nu \)

Kinematic viscosity of the fluid (in m2 s−1)

\( \mu \)

Dynamic viscosity of the fluid (in kg m−1 s−1)

\( \rho \)

Fluid density (in kg m−3)

\( \sigma^{a} \)

Standard deviation of the aperture within a fracture (in mm)

\( \sigma_{\text{eff}}^{a} \)

Standard deviation of the aperture within a fracture based on the effective aperture definition (in mm)

\( \sigma_{\text{vert}}^{a} \)

Standard deviation of the aperture within a fracture based on the vertical aperture definition (in mm)

\( \sigma_{{N_{\text{offset}} }}^{I} \)

Standard deviation of the indicator \( I \) associated with the shearing \( N_{\text{offset}} \) (unitless)

\( \tilde{\sigma }_{{N_{\text{offset}} }}^{I} \)

Relative standard deviation for the indicator \( I \) at a given \( N_{\text{offset}} \) (unitless)

\( \tilde{\sigma }_{{N_{\text{offset}} }}^{{I_{1} }} \)

Relative standard deviation for the indicator \( I_{1} \) at a given \( N_{\text{offset}} \) (unitless)

\( \tilde{\sigma }_{{N_{\text{offset}} }}^{{I_{2} }} \)

Relative standard deviation for the indicator \( I_{2} \) at a given \( N_{\text{offset}} \) (unitless)

\( {\text{AF}} \)

Anisotropy factor (unitless)

\( {\text{AF}}_{\text{eff}} \)

Anisotropy factor based on the effective aperture definition (unitless)

\( {\text{AF}}_{\text{vert}} \)

Anisotropy factor based on the vertical aperture definition (unitless)

\( a \)

Arithmetic mean aperture (in m)

\( \bar{a} \)

Mean aperture within a fracture (in mm)

\( \bar{a}_{\text{eff}} \)

Mean aperture within a fracture based on the effective aperture definition (in mm)

\( \bar{a}_{\text{vert}} \)

Mean aperture within a fracture based on the vertical aperture definition (in mm)

\( a_{\text{e}} \)

Equivalent fracture aperture in the parallel-plate model (in mm)

\( a_{\text{eff}} \)

Effective aperture (in mm)

\( a_{ {\max} } \)

Maximum local aperture observed among the fractures (in mm)

\( a_{i} \)

Aperture value of the \( i^{th} \) element of a considered fracture (in mm)

\( a_{\text{vert}} \)

Vertical aperture (in mm)

\( a_{x,y} \)

Local aperture at \( \left( {x,y} \right) \) (in m)

\( {\mathcal{C}} \)

Set of elements belonging to one of the \( n/2 + 1 \) channels (unitless)

\( C_{{N_{\text{offset}} }}^{I} \)

Centroid value for all indicators \( I \) of one offset (unitless)

\( {\mathbf{f}} \)

Body force vector acting on the fluid (in kg m−2 s−2)

\( {\mathbf{g}} \)

Acceleration of gravity vector (in kg m−2 s−2)

\( {\mathbf{grad}}\left( \varvec{m} \right) \)

Gradient of the median \( m\left( {x,y,N_{\text{offset}} } \right) \) (unitless)

\( H \)

Hurst roughness exponent (unitless)

\( h_{\text{bot}} \)

Height of a fracture element on its bottom surface (in mm)

\( h_{\text{top}}^{0} \)

Height of a fracture element on its initial top surface before shearing (in mm)

\( h_{\text{top}}^{{N_{\text{offset}} }} \)

Height of a fracture element on its top surface after shearing of \( N_{\text{offset}} \) (in mm)

\( I_{1} \)

Relative proportion of channel area in a fracture (unitless)

\( I_{2} \)

Maximum channel length in a given fracture normalized by \( n \) (unitless)

\( L \)

Scaling for the total length of the fractures (in m)

\( L_{X} \)

Fracture width along the \( x \)-axis (in m)

\( L_{Y} \)

Fracture width along the \( y \)-axis (in m)

\( l \)

Scaling of the maximum surface amplitude of the fractures (in mm)

\( l_{w} \)

Channel width (in mm)

\( m\left( {x,y,N_{\text{offset}} } \right) \)

Height of the median between the top and bottom surface passing by the point \( \left( {x,y,N_{\text{offset}} } \right) \) (in mm)

\( \dot{m}_{x} \)

Finite derivative of \( m\left( {x,y,N_{\text{offset}} } \right) \) along the \( x \)-axis (unitless)

\( \dot{m}_{y} \)

Finite derivative of \( m\left( {x,y,N_{\text{offset}} } \right) \) along the \( y \)-axis (unitless)

\( N \)

Total number of elements constituting a fracture (unitless)

\( N_{\text{offset}} \)

Value of the shear displacement along the \( x \)-axis (in mm or in number of element)

\( n \)

Number of elements constituting the side length of a fracture (in number of element)

\( \varvec{n}\left( \varvec{m} \right) \)

Normal vector to the middle plane (unitless)

\( {\text{OF}} \)

Projection of the outgoing flow perpendicular to the pressure gradient (in m3 s−1)

\( {\text{OF}}_{\parallel } \)

Projection of the outgoing flow perpendicular to the pressure gradient parallel to the shearing direction (in m3 s−1)

\( {\text{OF}}_{ \bot } \)

Projection of the outgoing flow perpendicular to the pressure gradient perpendicular to the shearing direction (in m3 s−1)

\( p_{\text{d}} \)

Probability density function of the roughness distribution (in m−1)

\( p \)

Hydrodynamic pressure (in kg m−1 s−2)

\( Q \)

Volumetric flow rate (in m3 s−1)

\( Q1 \)

Lower quartile value (units of the variable considered)

\( Q3 \)

Upper quartile value (units of the variable considered)

\( \varvec{Q}_{\varvec{i}} \)

Volumetric flow rate vector at the ith element of the fracture (in m3 s−1)

\( Q_{X} \)

Projected volumetric flow rate on the \( x \)-axis (in m3 s−1)

\( Q_{j}^{x,y} \)

Total volumetric flow rate for each element of the fracture identified by the coordinates \( \left( {x,y} \right) \) in the direction \( j \) where \( j = \left\{ {X,Y} \right\} \) (in m3 s−1)

\( Re \)

Reynolds number (unitless)

\( Re_{ {\max} }^{\text{local}} \)

Maximum local Reynolds number observed (unitless) element surface of a considered fracture (in m2)

\( U \)

Flow velocity (in m s−1)

\( \varvec{u} \)

Flow velocity vector (in m s−1)

\( x \)

Fracture coordinate along the \( x \)-axis (in mm)

\( y \)

Fracture coordinate along the \( y \)-axis (in mm.)

Notes

Acknowledgements

The study is part of the Helmholtz portfolio project “Geoenergy”. The support from the program “Renewable Energies”, under the topic “Geothermal Energy Systems”, is gratefully acknowledged. We also thank the EnBW Energie Baden-Württemberg AG for supporting geothermal research at KIT.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology Institute of Applied GeosciencesKarlsruheGermany
  2. 2.Karlsruhe Institute of Technology Institute for Applied MaterialsKarlsruheGermany
  3. 3.Lawrence Berkeley National LaboratoryBerkeleyUSA
  4. 4.School and Observatory of Earth SciencesStrasbourgFrance

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