Fractures are the main flow path in rocks with very low permeability, and their hydrodynamic properties might change due to interaction with the pore fluid or injected fluid. Existence of minerals with different reactivities and along with their spatial distribution can affect the fracture geometry evolution and correspondingly its physical and hydrodynamic properties such as porosity and permeability. In this work, evolution of a fracture with two different initial spatial mineral heterogeneities is studied using a pore-scale reactive transport lattice Boltzmann method- (LBM-) based model. The previously developed LBM transport solver coupled with IPHREEQC in open-source Yantra has been extended for simulating advective-diffusive reactive transport. Results show that in case of initially mixed structures for mineral assemblage, a degraded zone will form after dissolution of fast-dissolving minerals which creates a resistance to flow in this region. This causes the permeability-porosity relationship to deviate from a power-law behavior. Results show that permeability will reach a steady-state condition which also depends on transport and reaction conditions. In case of initially banded structures, a comb-tooth zone will form and the same behavior as above is observed; however, in this case, permeability is usually less than that of mixed structures.
During reactive transport processes in tight rocks, fractures play an important role as it is the main flow path for species transport. Existence of fractured seals in geological CO2 sequestration and fractures in the host rock of nuclear waste disposal sites are some practical examples in which hydrodynamic properties of fractures can help to better understand the long-term evolution of the system [
The flow of incompressible fluids can be described by continuity and Navier-Stokes (NS) equations [
LBM has been used in this study to solve the Navier-Stokes equation. LBM describes the behavior of a collection of particles, and instead of macroscopic equation of fluid dynamics, it is based on the Boltzmann equation which describes dynamics of a gas on a mesoscopic scale [
In the above equations,
Equation (
Once these two steps are performed, distribution functions
As was mentioned, it can be shown that the LB equation can recover the NS equation using Chapman-Enskog analysis [
Reactive transport of chemical species can be described using the advection-diffusion-reaction (ADR) equation [
LBM can also be applied to solve the ADR equation. So, the LB equation for each species
In the above equations,
Figure
Flowchart of the LBM reactive transport model.
We do not include the reaction source-sink term in this step. In the next step, molar concentrations
All these steps are done using a PHREEQC wrapper file which is part of the LBM code used in this study and has been detailed in Patel’s work [
The geometry is updated using static update rules detailed in Patel et al. [
Once the streaming step is performed, species concentrations can be obtained using (
To investigate the effect of initial mineral distributions on fracture geometry evolution, the quartet structure generation set (QSGS) algorithm [
The geometry of a single fracture (in light gray) surrounded by reactive (in black) and nonreactive (in dark gray) minerals. In multispecies simulations, dark gray is kaolinite and black is calcite. The QSGS algorithm was used to grow nonreactive minerals inside a reactive mineral (in black) to form a mixed structure.
The geometry of a single fracture (in light gray) surrounded by reactive (in black) and nonreactive (in dark gray) minerals. In multispecies simulations, dark gray is kaolinite and black is calcite. The QSGS algorithm was used to grow nonreactive minerals inside a reactive mineral (in black) to form a banded structure.
In this section, it is assumed that the rock matrix around the fracture consists of two different minerals. For the single-species case, it is considered that one mineral is reactive (with molar volume of 2 × 10−3 m3/mol) and the other is inert. A single synthetic species
Figure
Normalized permeability versus normalized porosity at different Pe and Da numbers for initially mixed mineral assemblage.
Concentration (mol/L) of species
Figure
Concentration (mol/L) of species
We also ran a case in which distribution of inert minerals inside the reactive minerals form finer mixed structures. Figure
Normalized permeability versus normalized porosity at Pe = 26 and Da = 20 for initially mixed mineral assemblage where two different mineral sizes are used for nonreactive minerals.
Concentration (mol/L) of species
Velocity profile (m/s) for cases with (a) coarser and (b) finer nonreactive minerals when Pe = 26 and Da = 20. In both (a) and (b), porosity (after dissolution) is near 0.56 but permeability (after dissolution) is 1.56 × 10−9 m2 for (a) and 1.41 × 10−9 m2 for (b).
For the banded structures, the geometry illustrated in Figure
Normalized permeability versus normalized porosity at different Pe and Da numbers for initially banded mineral assemblage.
The contour maps in Figures
Concentration (mol/L) of species
Concentration (mol/L) of species
In the banded structure case as well as the mixed structures, it can be observed that the permeability increases until it reaches a constant value. Pe and Da numbers can affect the time it takes for the permeability to reach this steady-state condition, but it is the existence of nonreactive minerals that causes this plateau in permeability-porosity curve. This behavior in normalized permeability-porosity relationship indicates that the relation
Normalized permeability versus normalized porosity at different Pe and Da numbers when a single fracture is surrounded only by a single reactive mineral and no nonreactive mineral exists in the rock matrix.
In the previous section, effects of mineral spatial heterogeneity and different reactive transport conditions on fracture evolution were investigated when synthetic single-species reactive and nonreactive minerals were present in the model. In reality, however, the minerals are composed of different species and have more complicated kinetic rate equations which might affect the fracture textural evolution in ways that are not the same as evolutions related to the synthetic minerals. Therefore, in this section, simulations are performed to show the ability of the developed pore-scale reactive transport model to handle more complicated cases where different minerals with more sophisticated kinetic rate equations exist in the model. As discussed before, in this section, we simulate preferential dissolution of calcite minerals while kaolinite minerals are also present in a fractured caprock, under two different initial mineral distributions.
In this section, the geometry illustrated in Figure
Concentrations used for the initial and inlet solutions.
Parameter | Initial value in the fracture | Value at the inlet |
---|---|---|
pH | 7.4 | 3.9 |
Ca | 3.401 × 10−3 mol/kgw | 5.510 × 10−3 mol/kgw |
C | 6.975 × 10−3 mol/kgw | 1. 334 mol/kgw |
Na | 1 mol/kgw | 1 mol/kgw |
Cl | 1 mol/kgw | 1 mol/kgw |
Al | 1.777 × 10−6 mol/kgw | — |
Si | 1.777 × 10−6 mol/kgw | — |
In (
The kaolinite reaction is described as
We performed simulations at two different Pe numbers of 2.6 and 0.026. Ca and Al concentrations at three different times are shown in Figures
Ca concentrations (mol/L) at (a)
Ca concentrations (mol/L) at (a)
Al concentrations (mol/L) at (a)
Al concentrations (mol/L) at (a)
Normalized permeability versus normalized porosity at two different Pe numbers for initially mixed mineral assemblage (multispecies case).
Also, Ca and Al concentrations at the outlet are plotted in Figures
Effluent Ca concentrations versus time at two different Pe numbers.
Effluent Al concentrations versus time at two different Pe numbers.
As in Section
Ca concentrations (mol/L) at (a)
Ca concentrations (mol/L) at (a)
Al concentrations (mol/L) at (a)
Al concentrations (mol/L) at (a)
Normalized permeability versus normalized porosity at two different Pe numbers for an initially banded mineral assemblage (multispecies case).
It should be pointed out that in the banded structure, compared to the mixed-structure case, the permeability values are smaller indicating that when nonreactive minerals are distributed as banded structures, they act as a flow barrier and, as can be observed in Figures
Velocity profile (m/s) for cases with (a) mixed and (b) banded structures when Pe = 2.6 for a multispecies case. In both (a) and (b), porosity (after dissolution) is near 0.47 but permeability (after dissolution) is 1.58 × 10−9 m2 for (a) and 1.43 × 10−9 m2 for (b).
Also, similar to the mixed-structure case, while calcite dissolves and recedes from the main flow channel, the rate of calcite dissolution decreases due to buffering effect and lower velocity in the areas between two vertical kaolinite bands (comb-tooth zones). This will cause a decrease in Ca concentration after it reaches a maximum at the beginning (Figure
Effluent Ca concentrations versus time at two different Pe numbers.
Effluent Ca concentrations versus time for mixed and banded structures when Pe = 2.6.
Effluent Al concentrations versus time at two different Pe numbers.
In this study, a previously developed LBM transport solver coupled with IPHREEQC has been extended for simulating advective-diffusive reactive transport. The LBM was used to investigate the effects of initial mineral spatial heterogeneity on textural alteration of a fracture and consequently on its hydrodynamic properties. The LBM coupled with IPHREEQC geochemical solver enables us to model a broad range of different geochemical reactions. To simulate the effects of different mineral spatial distributions, synthetic models were constructed. Results showed that the generated degraded and comb-tooth zones [
Moreover, to show the ability of the LBM model for realistic cases, we simulated a case with calcite and kaolinite minerals which were assumed to represent two main minerals in a rock matrix around a single fracture inside a caprock. It was observed that the tortuosity in the degraded zone (in the case of initially mixed distributions) and distance from the main flow channel (in the case of initially banded structures) lowers the permeability of the fracture. Also, a decreasing trend in the effluent Ca concentration confirmed the effect of degraded and comb-tooth zones on decreasing the dissolution rate of calcite in those regions.
3D simulations need to be done in order to better understand the effects of initial mineral distributions on fracture hydrodynamic properties. The developed LBM can also be used to simulate the 3D geometries as well, and this will be published in a future work.
The applicability of the current LBM approach (without flow solver) used in this work has been previously demonstrated for ion exchange problems [
This benchmark shows how a single species A (with constant concentration) diffuses from left boundary into a square domain with size
Schematic of the domain used for single-species reaction-diffusion benchmark.
with the boundary conditions
The analytical solution of (
Steady-state concentrations, obtained by analytical (contour and solid lines) and LB (dashed lines) methods, for a single species diffusing through the left boundary of a square domain and reacting at the top boundary according to a first-order reaction while other boundaries are zero concentration gradients.
To further verify the implementation of a coupled LBM-IPHREEQC reactive transport model, a 1D simulation of advection-diffusion-reaction (ADR) was considered in which four different species A, B, C, and D are reacting according to the following decay reactions [
Sketch of domain used for 1D simulation of advection-diffusion-reaction of four species involved in a chain of decay reactions. The left boundary is maintained at constant concentration for different species, and the right boundary is a zero concentration gradient.
Breakthrough curves for four different species at the outlet of a domain where species A is injected into the domain and a chain of decay reactions occurs between four species. Different lines show the results obtained with the LBM approach, and symbols indicate the results simulated using PHREEQC.
The authors declare that there is no conflict of interest regarding the publication of this paper.
Funding for this study was provided by the PROTECT project (no. 233736) financed by the Research Council of Norway, Total E&P, and DEA which is gratefully acknowledged.