Stochastic modelling and upscaling of hydrodynamic transport in geological fractures

Characterizing hydrodynamic transport in fractured rocks is essential for carbon storage and geothermal energy production. Multiscale heterogeneities lead to anomalous solute transport, featuring breakthrough curve (BTC) tailing and nonlinear growth of plume spatial moments. We focus on purely advective transport within synthetic geological fractures with prescribed relative closure sigma Subscript a Baseline divided by left angle bracket a right angle bracket <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false" fence="false">⟨</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false" fence="false">⟩</mml:mo> </mml:math> $\sigma _a/\langle a \rangle$ and correlation length upper L Subscript italic c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">c</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $L_{\textit{c}}$ . We adopt a stochastic approach with multiple fracture realizations for each set of geometric parameters. Steady-state depth-averaged Stokes flow is solved under the lubrication approximation. Flow heterogeneity is organized over the correlation length upper L Subscript italic c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">c</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $L_{\textit{c}}$ . The ensemble-averaged velocity probability density functions (PDFs) are insensitive to upper L Subscript italic c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">c</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $L_{\textit{c}}$ but strongly influenced by sigma Subscript a Baseline divided by left angle bracket a right angle bracket <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false" fence="false">⟨</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false" fence="false">⟩</mml:mo> </mml:math> $\sigma _a/\langle a \rangle$ , particularly their low-velocity power-law scaling. A time-domain random walk (TDRW) simulation is used to compute plume spatial moments and outlet BTCs. The mean longitudinal plume position scales linearly with time at both early and late stages. The variance shows ballistic scaling at early times and a late-time behaviour controlled by the low-velocity power law of the velocity PDF, with exponent alpha <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>α</mml:mi> </mml:math> $\alpha$ strongly influenced by sigma Subscript a Baseline divided by left angle bracket a right angle bracket <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/cor