Role of geomechanically grown fractures on dispersive transport in heterogeneous geological formations

A second order in space accurate implicit scheme for time-dependent advection-dispersion equations and a discrete fracture propagation model are employed to model solute transport in porous media. We study the impact of the fractures on mass transport and dispersion. To model flow and transport, pressure and transport equations are integrated using a finite-element, node-centered finite-volume approach. Fracture geometries are incrementally developed from a random distributions of material flaws using an adoptive geomechanical finite-element model that also produces fracture aperture distributions. This quasistatic propagation assumes a linear elastic rock matrix, and crack propagation is governed by a subcritical crack growth failure criterion. Fracture propagation, intersection, and closure are handled geometrically. The flow and transport simulations are separately conducted for a range of fracture densities that are generated by the geomechanical finite-element model. These computations show that the most influential parameters for solute transport in fractured porous media are as follows: fracture density and fracture-matrix flux ratio that is influenced by matrix permeability. Using an equivalent fracture aperture size, computed on the basis of equivalent permeability of the system, we also obtain an acceptable prediction of the macrodispersion of poorly interconnected fracture networks. The results hold for fractures at relatively low density.

Figure 1

Fracture centerline and aperture measurement. Black dots are nodes on the fracture wall (i.e., $f_n$, $f_{n+1}$, and $f_{n-1}$). White dots are the corresponding centerline nodes, $n_i$ and $n_{i+1}$. $f_n$ is a fracture tip. $a_i$ is the aperture of the fracture at $n_i$. The thick line represents a fracture line element on the centerline. The aperture assigned to the local line element, $a_{fi}$, is the average between $a_i$ and $a_{i+1}$.

Figure 2

An initial flaw distribution and a developed fracture set of a growth simulation. Flaws are grown within the propagation area; for flow simulations a model area is extracted from the original dataset.

Figure 3

Mechanically grown fracture networks at different growth stages (after 20, 50, and 80 iterations) with a maximum aperture size of 0.001 m. Colors are indicative of fracture aperture.

Figure 4

Average convergence error for different growth stages. $E^i$ is $\parallel {\langle c\rangle }^i-{\langle c\rangle }^{i-1}\parallel$ and ${\langle \rangle }^i$ denotes the breakthrough average of $i$ realizations.

Figure 5

Breakthrough curves for the model with different realizations at different fracture network growth stages. Bold lines indicate the average, and faint lines indicate the breakthrough curves for each realization.

Figure 6

Concentration profiles for a single fracture geometry realization and different matrix permeabilities at two different times. Bold lines are the profiles after 1 hour and the dashed lines are the concentration profiles after 5 hours.

Figure 7

Breakthrough curves for the model with different matrix permeabilities. Four different matrix permeabilities are compared: $k_m=10,8,6,4$ mD.

Figure 8

Breakthrough curves for seven realizations and a wider range of matrix permeability: $k_m=0.01,0.1,1,10$ mD. The bold lines are the average breakthrough curves, and the faint lines indicate the breakthrough curves for each realization.

Figure 9

Time derivative of breakthrough curves for the model with four different matrix permeabilities: $k_m=0.01,0.1,1,10$ mD. The bold lines are the ensemble averages of the seven realizations, and the faint lines indicate breakthrough curves of individual realizations.

Figure 10

Comparison of the breakthrough curves of three realizations between models with geomechanical aperture distribution (solid lines), and ones with equivalent aperture sizes (dotted lines). Flow simulations for each realization are conducted for two different rock matrix permeabilities: $k_m=4,10$ mD.

Figure 11

Aperture size histogram resulting from two different displacements of 0.001 m (white) and 0.0001 m (gray), at the model boundaries.

Figure 12

Breakthrough curves for different displacements of (a) $0.001$ m and (b) $0.0001$ m, and different matrix permeabilities: $k_m=4$ and 10 mD.

Figure 13

Breakthrough curves calculated for different aperture distributions: varying aperture (solid lines), equivalent aperture (dashed lines), and weighted average aperture (dotted lines), for a model with two matrix permeabilities, $k_m=4,10$ mD, and a displacement of 1 mm.