Lagrangian coherent structures control solute dispersion in heterogeneous poroelastic media

Recent work has demonstrated that Lagrangian chaos can arise at the Darcy scale in even simple poroelastic media, relevant across geophysical, biophysical, and industrial domains. Key factors controlling the onset of chaotic advection in poroelastic media flow include medium heterogeneity, compressibility, and transient forcing. Here we focus on how the range of Lagrangian coherent structures (LCSs) present in poroelastic flows influences the transport of diffusive solutes. We use a minimal two-dimensional sinusoidally forced poroelastic computational model to show that LCS interactions impact the dispersion of solute plumes via the establishment of minimum flux manifolds, leading to strongly anomalous plume moments and solute residence time distributions. We interpret these results with fluid element stretching distributions and local Lyapunov exponents. Anomalous features persist even at low Péclet number. Our data show that solute dispersion in poroelastic systems subject to transient forcing can only be understood by resolving the LCSs of these complex flows.

Figure 1

Schematic of the PFHPM 2D model domain indicating the mean flow direction, pressure boundary conditions and heterogeneous conductivity field. The gray-scale shading indicates the value of the log-conductivity field $lnK\left(\mathbf{x}\right)$.

Figure 2

Pressure profiles and flow paths in the PFHPM and SHPM flows. Top: The dashed curve illustrates exponential decrease of the normalized amplitude of the pressure fluctuation ($|p_p/\mathcal{G}|$) for the equivalent homogeneous case. The upper axis is rescaled in terms of the dissipation length $x\sqrt{\mathcal{T}}$. Bottom: Flow paths of several tracer particles that comprise the four solute blobs originating at initial locations A, B, C, and D in the (black) steady flow and (colored) periodic flow. The tracer particle positions are plotted at intervals of $\delta t=P^{\prime }/32$. Note that the black circular symbols at the initial locations A, B, C, D are significantly larger than the initial solute blob size.

Figure 3

Relative stretching ${\xi }_i\left(t\right)$ in (a) steady and (b) periodic two-dimensional heterogeneous poroelastic Darcy flow with conductivity field correlation length $\ell =0.049$. Left inset: Probability density function of ${\xi }_i$ from an ensemble of 75 million variates ${\xi }_i\left(t_j\right)$ calculated from $2\times {10}^4$ tracer particles with time increments $\delta t=P^{\prime }/32$. Tracer particles are colored according to the local value of $\xi$ as per the legend (bottom right inset), where particles with local compression (blue) and stretching (red) greater than the threshold magnitude $\left|\xi \right|>{\xi }_{\text{crit}}$ are colored with larger stretching magnitudes depicted by more intense colors.

Figure 4

Annotated and colored Poincaré section of the PFHPM flow constructed by showing fluid tracer particle positions at integer multiples of the forcing period $P^{\prime }$. Various LCS types are shown corresponding to (i) regular region (light blue), (ii) KAM islands (light yellow), (iii) stochastic/chaotic regions (S, light pink, see text), hyperbolic points (H, orange dots), elliptic points (E), stable (red lines) and unstable (blue lines) manifolds, and the emptying region (light green). The unstable manifolds form part of the boundary of the emptying region, and the remainder of this boundary is comprised of “discharge regions” where tracer particles can flow through to the emptying region. Discharge points (purple) indicate the position of fluid tracer particles being advected through the emptying region at integer multiples of $P^{\prime }/10$. Regions of the forcing boundary with (without) purple discharge points indicate punctuated “outflow” (“inflow”) regions on this boundary where fluid flows out (in) during a forcing period.

Figure 5

Evolution of solute plumes comprised of $N_p={10}^5$ tracer particles released from initial locations C and D in (a) steady and (b) periodic flow. In each plate, the solute plumes are shown at two different times that vary with injection location: location C at $t/P^{\prime }=8,42$ and location D at $t/P^{\prime }=12,72$. Different tracer colors correspond to different Péclet numbers, ranging from $\infty$ (red), 8640 (blue), 864 (green), to 86.4 (purple).

Figure 6

Centroid distance $D_c$ between solute blobs in the steady and periodic flows as a function of the periodic blob centroid ${\overline{x}}_p$ for different initial locations [A (top left), B (bottom left), C (top right), and D (bottom right)] and Péclet numbers: $\infty$ (red), 8640 (blue), 864 (green), and 86.4 (purple). The background colors represent different LCSs in the periodic flow: regular region (light blue), KAM islands (light yellow), stochastic/chaotic regions (light pink), hyperbolic point (orange), and emptying region (light green).

Figure 7

Plume longitudinal spatial variance ${\sigma }_x^2$ for the steady (dashed line) and periodic (solid line) cases at initial locations A–D, and Péclet numbers ranging from 86.4 to $\infty$. Background shading is identical to that of Fig. 6.

Figure 8

Plume transverse spatial variance ${\sigma }_y^2$ for the steady (dashed line) and periodic (solid line) cases at initial locations A–D, and Péclet numbers ranging from 86.4 to $\infty$. Background shading is identical to Fig. 6.

Figure 9

Residence time distribution (RTD) of solute plumes for the steady (dashed line) and periodic (solid line) cases at four initial locations, with different Péclet numbers.

Figure 10

Spatial residence time distribution (RTD) of solute plumes as they exit the flow domain for the steady and periodic cases at initial locations A–D, with varying Péclet numbers from $\infty$ (red), 8640 (blue), 864 (green), to 86.4 (purple). The centers of the yellow circles indicate the exit locations of single nondiffusive particles released at the centers of mass of each of the four starting locations. Sample maximum values are indicated by horizontal lines for each Péclet number. The inset figures (i, ii, and iii) offer a magnified view to resolve details within the yellow circle.

Figure 11

Darcy flux goes from high pressure to low pressure, but due to porosity gradients a countering squeeze flux develops from lowered porosity in the low-pressure region. Circles are meant to suggest relative pore size.

Figure 12

(a) Lattice gyre FEM solute concentration field $C$ at $t=0.06$ (green distribution, log scale) for the $2\times 3$ lattice-gyre flow. Also shown are the initial $(t=0)$ flow field (blue vectors) and initial solute Gaussian blob with a 0.1% concentration diameter of 0.055, located at $(x,y)=(0.5,0.6)$. Asymmetry of the solute concentration field arises from the perturbation strength $\epsilon$. (b) Relative $L_2$ error between the FEM and DMM methods for the computed solute concentration field in the lattice gyre flow. The $L_2$ error is reported for different DMM operator splitting methods and temporal resolution.

Figure 13

(a) Schematic of a lamella-shaped scalar concentration field $c(z,t)$ subject to stretching in the $w$-direction and orthogonal compression in the $z$-direction. (adapted from Ref. [48]). (b) Evolution of the theoretical (red line) and numerical (blue dots) rescaled transverse blob variance ${\sigma }^2\left(t\right)/l_B^2$ with time toward unity (black dashed line).