# Capability for Hydrogeochemical Modelling within Discrete Fracture Networks

^{*}

## Abstract

**:**

## 1. Introduction

^{3}.

## 2. Materials and Methods

#### 2.1. Equations for Solute Transport within DFNs

- ${P}_{R}$ [Pa] the residual pressure: ${P}_{R}={P}_{T}+{\rho}_{0}gz;$ (where g is the acceleration due to gravity, $z$ is the elevation above sea level, ${P}_{T}$ is the total pressure and ${\rho}_{0}$ is a reference density);
- ${c}_{i}$ [-] the mass fraction of solutes in the fracture;
- ${c}_{i}^{\prime}$ [-] the mass fraction of solutes in the matrix;
- ${e}_{h}$ [m] the effective hydraulic aperture in the fracture;
- ${e}_{t}$ [m] the transport aperture in the fracture;
- $\rho \left({c}_{i},{P}_{R}\right)$ [kg m
^{−3}] and $\mu \left({c}_{i},{P}_{R}\right)$ [kg m^{−1}s^{−1}] the fluid density and viscosity, respectively. These can be calculated using an empirical expression [16]; - $\overrightarrow{Q}$ [m
^{2}s^{−1}] the volume of water flowing per second, per unit width of the fracture; - $t$ [s] the time;
- $\overrightarrow{\nabla}$ the two-dimensional gradient operator within the fracture;
- $D$ [m
^{2}s^{−1}] the dispersion tensor within the mobile water in the fracture; this includes contributions from diffusion and from hydrodynamic dispersion; the latter has components parallel and perpendicular to the flow; - ${D}_{i}$ [m
^{2}s^{−1}] the intrinsic diffusion coefficient for diffusion into the rock matrix; - $w$ [m] the perpendicular distance from the fracture plane into the rock matrix;
- $\alpha $ [-] the capacity factor (when there is no sorption, this is the same as the porosity of the matrix).

#### 2.2. DFNs in ConnectFlow

- The groundwater pressures are continuous between two intersecting fractures.
- Groundwater fluxes are conserved at an intersection between fractures.

- A regular mesh of triangular elements is used to discretise individual fractures, as shown in Figure 2. The pressure and concentration are approximated on each element as a linear combination of basis functions. For each element, there is an associated node. For example, the pressure is given by$$P(\overrightarrow{x})={\displaystyle \sum}_{i}{\varphi}_{i}(\overrightarrow{x}){P}_{i}$$where ${P}_{i}$ is the pressure at node i, and the basis function ${\varphi}_{i},$ associated with node i, takes the value 1 at the node and 0 at all the other nodes. These nodes are referred to as “local nodes” in ConnectFlow.Each triangular finite element on the fracture can have a different hydraulic aperture, with values typically sampled from a probability distribution function.
- For the overall fracture network, the pressure and concentration are defined by their values at “global nodes” that exist on the intersections between fractures. The intersections are approximated using the lines formed by the boundaries of the triangular elements on the fracture (see Figure 2). The global nodes coincide with local nodes on the approximated intersection. The global problem does not have a separate mesh but rather relies on the underlying mesh on each fracture.Each global node I has a corresponding global basis function. This global basis function is approximated by the finite-element solution for the steady-state groundwater flow equations on the fracture in the case in which the residual pressure is specified to be $1$ at global node $I$ and $0$ at all the other global nodes on the fracture. The global flow and transport calculations can either be steady state or fully transient.

#### 2.3. Rock Matrix Diffusion

- The total diffusion length into the matrix ${w}_{max}={\displaystyle \sum}_{j=1,{n}_{fv}}{w}_{i}\left[m\right]$;
- The number of finite volume cells per global node, ${n}_{fv}$;
- The intrinsic diffusion coefficient, ${D}_{i}$ [m
^{2}/s]; - The porosity of the matrix or capacity factor,$\alpha $ [-].

#### 2.4. Representation of Solute Concentrations

- As a mass fraction, c, of a solute species, i, calculated from the mass of each species, M
_{i}, divided by the sum of the masses of water, M_{wat}, and the solute species, M_{j}.

- 2.
- Using reference waters, which are waters (solutions) with defined (and fixed) solute compositions. In this case, n reference waters are specified, each of these has a specified composition of m solutes (fixed for the duration of the calculation). Given a mixture of reference waters, the mass fraction of a solute species i, is calculated by multiplying the fraction of each reference water F
_{w}, by the mass fraction of species i contained within that reference water c_{i}_{,w}.$${c}_{i}={\displaystyle \sum}_{w=1,n}{F}_{w}{c}_{i,w}={\displaystyle \sum}_{w=1,n}\left(\frac{{F}_{w}{M}_{i,w}}{{M}_{wat}+{{\displaystyle \sum}}_{j=1,m}{M}_{j,w}}\right)$$ConnectFlow determines the proportion of each reference water with the assumption that the sum of all reference water fractions is one.$$\sum}_{w=1,n}{F}_{w}=1$$

#### 2.5. Reactive Transport

## 3. Results

#### 3.1. Analytical Solution for Transport within an Idealised Two Fracture System Including RMD

#### 3.2. Single Fracture Calculations

#### 3.3. Random Fracture Case

^{−9}m

^{2}/s) and standard deviation ln(T) = 2. This distribution is truncated below ln(T) = −25.3 (T ≈ 10

^{−11}m

^{2}/s) and above ln(T) = −16.1 (T ≈ 10

^{−7}m

^{2}/s). Transmissivity was not correlated with fracture size.

#### 3.4. Verification for Reactive Transport in DFN Models

#### 3.5. Meteoric Water Penetration at Olkiluoto Island

^{2}and is bounded by brackish seawater. The island is largely flat with elevations mostly below 10 m. Precipitation infiltrates down to the crystalline bedrock, driven by the surface topology. These groundwater flows occur within a network of connected fractures. The fractures were generated by various regional tectonic events occurring during the Precambrian period. Drill-hole fracture data was analysed in [23] to determine an empirical relation between the intensity and transmissivity of bedrock fractures, and depth.

^{−18}m

^{2}and a minimum porosity of 10

^{−4}is enforced in cells containing no fractures. The ECPM and DFN calculations have similar run-times for the resolutions used here.

#### 3.6. Laxemar Repository-Scale Model

## 4. Conclusions

- Transporting multiple solute species, coupled with the flow equation via the density and viscosity.
- An algorithm for calculating solute diffusion into the rock matrix (RMD) around each fracture.
- An interface with the iPhreeqc library to model chemical reactions involving solutes, rock minerals, and minerals on fracture/pore surfaces.
- The performance of ConnectFlow’s DFN module has also been significantly improved via parallelisation.

- Site-scale simulations of dilute water penetration at Olkiluoto including rock matrix diffusion.
- Repository scale calculations of dilute water penetration at Laxemar including calcite reactions and RMD. Over the 2000-year period simulated, freshwater ingress and calcite precipitation are both observed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**In ConnectFlow, DFN simulations are performed in two steps. Firstly, a series of calculations are done for each individual fracture (each calculation determines a global basis function). The global basis functions are then used to build the global equations which are then solved.

**Figure 2.**Example of ConnectFlow’s regular triangular finite element discretisation as used for fractures. Intersections are approximated using the lines formed by the boundaries of the triangular elements on the fracture. The regular finite element mesh can be solved very efficiently.

**Figure 3.**Schematic of transport within the fracture and the matrix. The blue arrows indicate advective-diffusive transport between sub-fractures; the black arrows indicate diffusive transport between the fracture and the matrix, and also within the matrix.

**Figure 4.**An idealised system of regularly spaced fractures embedded in a porous matrix (adapted from [15]).

**Figure 5.**(TOP) Stochastic fractures in a cuboid block, coloured by transmissivity. (BOTTOM) The equivalent CPM for the fractures, coloured by permeability. For clarity, cells whose permeability is less than 10

^{−18}m

^{2}have been removed. Reprinted with permission from [22], 2020, Posiva Oy.

**Figure 6.**Concentrations halfway along the block at x = 250 m. DFN and ECPM cases are both shown, both with and without RMD.

**Figure 7.**Comparison of carbon mass fraction for DFN and CPM cases (almost exact agreement) with and without chemical reactions. The mass fractions are measured halfway along the fracture. Reprinted with permission from [22], 2020, Posiva Oy.

**Figure 8.**The site-scale fracture network used for the DFN model for Olkiluoto. The outline of the island is included in black for context. Blues denote lower transmissivities and reds show higher transmissivities. Reprinted with permission from [22], 2020, Posiva Oy.

**Figure 9.**Horizontal slices of concentration at −410 m depth. These show the salinity evolution at the Olkiluoto site for a 2000-year period. Two models are shown, ECPM (

**bottom**) and DFN (

**top**). Both cases include rock matrix diffusion.

**Figure 10.**The fracture network used for the Laxemar calculations. Darker blues denote low transmissivites, and greens and yellows denote higher transmissivities.

**Figure 11.**Inorganic carbon mass fractions for a horizontal cross-section at −490 m elevation for the Laxemar model. The image on the left includes chemical reactions and the image on the right does not.

**Figure 12.**Average mass fraction of carbon (averaged over the central 1 km

^{2}of the model) versus depth for calculations with and without chemical reactions involving calcite.

Parameter | Symbol | Value |
---|---|---|

Dispersion length (longitudinal and transverse) | ${l}_{{D}_{long}},$${l}_{{D}_{trans}}$ | 100 m, 10 m |

Transmissivity | $T$ | 9.77 × 10^{−5} m^{2}/s |

Transport aperture | ${e}_{t}$ | 0.02 m |

**Table 2.**Retardation values from the single fracture example with different RMD parameters. Three retardations are reported: calculated (analytical) values from solving Equation (14), DFN with 5 RMD finite volumes, and DFN with 500 RMD finite volumes. The variant with the smallest diffusion coefficient has the shallowest penetration into the matrix and thus requires more finite volumes to resolve. Reprinted with permission from [22], 2020, Posiva Oy.

Diffusion Length (m) | Matrix Diffusion Coeff. (m ^{2} s^{−1}) | Matrix Porosity | Retardation (Calculated) | Retardation 5 Finite Vols. | Retardation 500 Finite Vols. |
---|---|---|---|---|---|

0.495 | 3.0 × 10^{−12} | 0.3 | 9.1 | 5.5 | 8.9 |

0.495 | 1.0 × 10^{−11} | 0.3 | 22.8 | 20.5 | 22.4 |

0.495 | 5.0 × 10^{−11} | 0.3 | 29 | 28.4 | 28.8 |

0.495 | 2.0 × 10^{−10} | 0.3 | 30 | 29.8 | 29.9 |

0.495 | 5.0 × 10^{−11} | 0.1 | 10.3 | 10.1 | 10.3 |

0.495 | 5.0 × 10^{−11} | 0.6 | 57 | 55.9 | 56.6 |

0.1 | 5.0 × 10^{−11} | 0.3 | 6.6 | 6.8 | 6.8 |

0.9 | 5.0 × 10^{−11} | 0.3 | 49.9 | 48.6 | 49.4 |

**Table 3.**Parameters used for the stochastic block case. The fracture area per unit volume is determined in the DFN model and the matrix diffusion length is assumed to be the inverse of this.

Parameter | Symbol | Value |
---|---|---|

Dispersion length (longitudinal and transverse) | ${l}_{{D}_{long}}$$,{l}_{{D}_{trans}}$ | 7 m, 1.4 m |

Matrix porosity | $\alpha $ | 10^{−4} |

Intrinsic diffusion coefficient | ${D}_{i}$ | 5 × 10^{−12} m^{2}/s |

Fracture area per unit volume (whole model) | $\sigma $ | 0.481 m^{2}/m^{3} |

Matrix diffusion length | ${w}_{max}$ | 2.08 m |

**Table 4.**Results for the random fracture case (reprinted with permission from [22], 2020, Posiva Oy). The travel time is the time taken for the midpoint of the block to reach 50% saline composition. The analytical retardation is calculated using the methods derived in [15] for an idealised fracture model (Figure 4) with the same rock matrix parameters and mean transport aperture as the random fracture model.

Result | Value |
---|---|

DFN travel time without RMD | 1.02 × 10^{8} s |

ECPM travel time without RMD | 8.55 × 10^{7} s |

DFN travel time with RMD | 4.08 × 10^{9} s |

ECPM travel time with RMD | 2.78 × 10^{9} s |

Retardation for ECPM | 32.6 |

Retardation for DFN | 40.1 |

Analytical retardation | 39.6 |

Parameter | Symbol | Value |
---|---|---|

Dispersion length (longitudinal and transverse) | ${l}_{{D}_{l}}$$,{l}_{{D}_{t}}$ | 100 m, 10 m |

Kinematic porosity (CPM) | ${\varphi}_{f}$ | 0.01 |

Transport aperture (DFN) | ${e}_{t}$ | 0.01 m |

Density of fresh and saline water | ${\rho}_{wat}$$,{\rho}_{sol}$ | 998.2 kg/m^{3}, 1026 kg/m^{3} |

Matrix porosity | $\alpha $ | 0.3 |

Matrix diffusion coefficient | ${D}_{i}$ | 5 × 10^{−11} m^{2}/s |

Matrix diffusion length | ${w}_{max}$ | 0.495 m |

**Table 6.**Compositions of the dilute water and sea water used in the single-fracture calcite transport calculation (after charge balancing).

Component | Seawater | Dilute Water |
---|---|---|

C [kg/kg] | 2.01 × 10^{−5} | 1.47 × 10^{−6} |

Ca [kg/kg] | 6.69 × 10^{−5} | 4.92 × 10^{−6} |

Cl [kg/kg] | 4.06 × 10^{−3} | 3.54 × 10^{−8} |

Na [kg/kg] | 2.59 × 10^{−3} | 2.30 × 10^{−8} |

H [kg/kg] | 1.62 × 10^{−6} | 1.68 × 10^{−7} |

O [kg/kg] | 7.97 × 10^{−5} | 7.22 × 10^{−6} |

E [-] | −1.52 × 10^{−16} | −2.03 × 10^{−19} |

pH [-] | 7.98 | 9.91 |

pe [-] | −4.45 | −6.58 |

Property | Value | Units |
---|---|---|

Number of fractures | 29,012 | |

Number of sub-fractures | 461,687 | |

Total fracture area per unit volume (P_{32}) | 2.02 × 10^{−2} | m^{−1} |

P_{32} for Hydrozone (averaged over model) | 2.56 × 10^{−3} | m^{−1} |

Tessellation length | 20 | m |

**Table 8.**Properties used in the Olkiluoto model for the two reference waters. Properties with an asterisk are specified at standard temperature and pressure. The density and viscosity are calculated from the salinity using an empirical expression from [16].

Description | Saline Concentration (kg/m^{3}) * | Density (kg/m^{3}) * | Viscosity (Pa.s) * | Salinity (g/kg) | |
---|---|---|---|---|---|

Ref. water A | Brine | 77.50 | 1051.4 | 1.128 × 10^{−3} | 73.71 |

Ref. water B | Meteoric | 2.42 | 999.9 | 1.005 × 10^{−3} | 2.420 |

**Table 9.**Rock matrix diffusion related parameters. The flow wetted surface is calculated during the upscaling process from the fracture area within each cell.

Property | Symbol | Value |
---|---|---|

Porosity of matrix | $\alpha $ | 0.005 |

Intrinsic diffusion coefficient | ${D}_{i}$ | 6 × 10^{−14} m^{2}/s |

Flow wetted surface (ECPM) | $\sigma $ | 3.238 m^{2}/m^{3} (0 to −50 m depth)0.692 m ^{2}/m^{3} (−50 m to −150 m)0.698 m ^{2}/m^{3} (−150 m to −400 m)0.372 m ^{2}/m^{3} (below −400 m) |

Matrix diffusion length | ${w}_{max}$ | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sigma $}\right.$ (Random fractures) |

Number of RMD finite volumes | ${n}_{fv}$ | 5 |

Finite volume lengths | ${w}_{i}$ | $\raisebox{1ex}{${w}_{max}$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$ |

**Table 10.**Boundary conditions used in the Olkiluoto models. Note that the physical system would have flow through the sides of the model, but no-flow boundary conditions have been used as a simplification.

Surface | Concentration | Pressure |
---|---|---|

Top | Initial value | Initial value |

Sides | Initial value | Zero flow |

Bottom | Initial value | Zero flow |

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**MDPI and ACS Style**

Applegate, D.; Appleyard, P. Capability for Hydrogeochemical Modelling within Discrete Fracture Networks. *Energies* **2022**, *15*, 6199.
https://doi.org/10.3390/en15176199

**AMA Style**

Applegate D, Appleyard P. Capability for Hydrogeochemical Modelling within Discrete Fracture Networks. *Energies*. 2022; 15(17):6199.
https://doi.org/10.3390/en15176199

**Chicago/Turabian Style**

Applegate, David, and Pete Appleyard. 2022. "Capability for Hydrogeochemical Modelling within Discrete Fracture Networks" *Energies* 15, no. 17: 6199.
https://doi.org/10.3390/en15176199