Fluid deformation in isotropic Darcy flow

The deformation of fluid elements governs many processes in porous media, including the transport, dispersion and mixing of solutes, chemical reactions and colloidal particles. Recently, it has been shown that even in strongly heterogeneous porous media, steady three-dimensional (3-D) Darcy flow with isotropic hydraulic conductivity admits only constrained kinematics. This is due to its inherently helicity-free nature, as the velocity is everywhere orthogonal to the vorticity. This property precludes braided streamlines and instead admits a pair of coherent 3-D streamfunctions, and the streamlines cannot wander freely throughout the flow domain. In this study, we consider the impact of these kinematic constraints upon fluid deformation at the Darcy scale. We show that the helicity-free condition corresponds to an orthogonal 3-D streamline coordinate system, which we use to derive an ab initio continuous time random walk framework for fluid deformation. We find that the helicity-free condition combined with the intermittent nature of shear events leads to fluid deformation that is limited to algebraic growth, with stretching ranging from sublinear to superlinear behaviour. Fluid deformation in 3-D isotropic Darcy flow is remarkably similar to that of two-dimensional (2-D) Darcy flow, and the structure of 3-D Darcy flow is fundamentally the same as two superposed 2-D Darcy flows. These results have implications for understanding flow and transport in heterogeneous porous media, and provide a basis for quantification of mixing and dispersion in isotropic 3-D Darcy flow.