# Barite Scaling Potential Modelled for Fractured-Porous Geothermal Reservoirs

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Reservoir Flow

#### 2.2. Reactive Transport

`pitzer.dat`database was used to account for the high ionic strengths of the aqueous solutions.

#### 2.3. Scenarios

## 3. Results

#### 3.1. Reservoir Simulation Scenarios

#### 3.2. Scenario Analysis

#### 3.3. Scaling Score

## 4. Discussion

#### 4.1. Simulation Results

#### 4.2. Scenario Analysis

#### 4.3. Scaling Score and Implications for Geothermal Systems

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Abbreviation | Description | Unit |

EGS | Enhanced Geothermal System | − |

LND | Landau | − |

NGB | North German Basin | − |

URG | Upper Rhine Graben | − |

A | Aquifer cross-sectional area | ${\mathrm{m}}^{2}$ |

$\mathrm{Da}$ | Damköhler number | − |

H | Aquifer thickness | $\mathrm{m}$ |

I | Ionic strength | $\mathrm{M}$ |

$\mathrm{IAP}$ | Ionic activity product | − |

J | Injectivity | ${\mathrm{m}}^{3}/\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

$\mathcal{K}$ | Permeability | ${\mathrm{m}}^{2}$ |

${K}_{\mathrm{sp}}$ | Solubility constant | − |

$\mathcal{L}$ | Permeability/injectivity loss | $1/\mathrm{s}$ |

N | Precipitation potential | $\mathrm{mol}/{\mathrm{m}}_{\mathrm{sol}}^{3}$ |

P | Pressure | $\mathrm{Pa}$ |

Q | Flow rate | ${\mathrm{m}}^{3}/\mathrm{s}$ |

R | Reaction rate | $\mathrm{mol}/{\mathrm{m}}_{\mathrm{rock}}^{3}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

S | Specific reactive surface area | ${\mathrm{m}}^{2}/{\mathrm{m}}_{\mathrm{rock}}^{3}$ |

$\mathrm{SF}$ | Scaling factor for reactive surface area | − |

$\mathrm{SR}$ | Saturation ratio ($=\mathrm{IAP}/{K}_{\mathrm{sp}}$) | − |

${S}_{\mathrm{S}}$ | Specific inner rock surface area | ${\mathrm{m}}^{2}/{\mathrm{m}}_{\mathrm{rock}}^{3}$ |

$\mathcal{T}$ | Transmissivity | ${\mathrm{m}}^{3}$ |

T | Temperature | $\mathrm{K}$ |

V | Flow proxy ($=Q/2\pi H$) | ${\mathrm{m}}^{2}/\mathrm{s}$ |

$\mathcal{X}$ | Scaling score | − |

$\delta $ | Fracture aperture half-width | $\mathrm{m}$ |

c | Solute concentration | $\mathrm{mol}/{\mathrm{m}}_{\mathrm{rock}}^{3}$ |

$\mathrm{g}$ | Gravitational acceleration | $\mathrm{m}/{\mathrm{s}}^{2}$ |

k | Rate constant | $\mathrm{mol}/{\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

${k}^{\prime}$ | Volumetric rate constant ($=k\xb7S$) | $\mathrm{mol}/{\mathrm{m}}_{\mathrm{rock}}^{3}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

$\mu $ | Dynamic viscosity | $\mathrm{Pa}/\mathrm{s}$ |

$\phi $ | Porosity or volume fraction | − |

q | Darcy flow velocity | $\mathrm{m}/\mathrm{s}$ |

${q}^{\prime}$ | Normalised Darcy flow velocity | $\mathrm{m}/\mathrm{s}$ |

r | Radial distance from well-centre | $\mathrm{m}$ |

${r}_{\mathrm{e}}$ | Range of influence | $\mathrm{m}$ |

${r}_{\mathrm{w}}$ | Well radius | $\mathrm{m}$ |

$\rho $ | Density | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

s | Water column | $\mathrm{m}$ |

t | Simulation time | $\mathrm{s}$ |

${\tau}_{\mathrm{A}}$ | Advective time | $\mathrm{s}$ |

${\tau}_{\mathrm{R}}$ | Reactive time | $\mathrm{s}$ |

v | Pore velocity ($=q/\phi $) | $\mathrm{m}/\mathrm{s}$ |

x | Distance | $\mathrm{m}$ |

0 | Subscript: initial value at $t=0$ | − |

1 | Subscript: value at $t>0$ | − |

$\mathrm{c}$ | Subscript: characteristic | − |

$\mathrm{eq}$ | Subscript: equilibrium | − |

$\mathrm{f}$ | Subscript: fluid | − |

$\mathrm{frac}$ | Subscript: fracture layer | − |

i | Subscript: variable entity (solute, layer, etc.) | − |

j | Subscript: model grid node | − |

m | Subscript: mineral phase | − |

$\mathrm{por}$ | Subscript: porous layer | − |

$\mathrm{rad}$ | Subscript: radial | − |

$\mathrm{res}$ | Subscript: reservoir | − |

## Appendix A. Numerical Implementation of the Radial Diverging Flow Field

## Appendix B. Derivation of the Radial Equilibrium Length

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**Figure 1.**Conceptual model of radially diverging groundwater flow near an injection well of a geothermal system. (

**a**) Injection flow components are split into two parts for the sedimentary–porous aquifer (upper layer) and the granitic–fractured aquifer with variable sizes and different amounts of horizontal fractures (lower layer). The boundary is the range of influence, i.e., where the induced pressure difference moves towards zero. (

**b**) Produced fluids, originating from the deep reservoir, are cooled and depressurised along the flow path through the heat exchanger. Thermodynamically disequilibrated fluids are then re-injected via the injection well, where scaling and formation damage are anticipated.

**Figure 2.**Reservoir simulation results of scenario 1. (

**a**) Conceptual view of the reservoir model with the injection well at the origin. (

**b**) Porosity change along the horizontal flow axis at the end of the simulation, as well as the previously calculated and simulated equilibrium lengths. (

**c**) Effective permeability change in the layers and the reservoir over the duration of the simulation.

**Figure 3.**Reservoir simulation results of scenario 2. (

**a**) Conceptual view of the reservoir model with the injection well at the origin. (

**b**) Porosity change along the horizontal flow axis at the end of the simulation as well as the previously calculated and simulated equilibrium lengths. (

**c**) Effective permeability change in the layers and the reservoir over the duration of the simulation.

**Figure 4.**Previously calculated (${r}_{\mathrm{eq},\mathrm{calc}}$, Equation (11)) and simulated equilibrium lengths (${r}_{\mathrm{eq},\mathrm{sim}}$) are correlated. ${r}_{\mathrm{eq},\mathrm{calc}}$ can thus be used as a predictive parameter. ${Q}_{i}$ refers to the flow rate of the respective porous or fractured layer.

**Figure 5.**Permeability loss of the fracture layer (${\mathcal{L}}_{\mathrm{frac}}$) is dependent on the fracture half-aperture ($\delta $), amount of fractures (${n}_{\mathrm{frac}}$), flow rate (${Q}_{\mathrm{frac}}$), and fluid sample, i.e., fluid chemistry. The relationships are shown for all scenarios after the final simulation time of ten years.

**Figure 6.**Effective reservoir permeability loss (${\mathcal{L}}_{\mathrm{res}}$) decreases as the flow rate ratio of the fracture layer (${Q}_{\mathrm{frac}}/{Q}_{\mathrm{tot}}$) increases. Only the scenario with the base injection flow rate of $100\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{3}/\mathrm{h}$ are shown after the final simulation time of ten years. The average losses for each input fluid sample are indicated as dashed lines.

**Figure 7.**(

**a**) The porous layers show clear signs of being transport-limited with respect to loss due to scaling, whereas the fracture layers are rather reaction-limited. (

**b**) Initial ion ratios for the considered samples versus the precipitation potential (${N}_{\mathrm{eq}}$), i.e., the maximum amount of barite that can precipitate, provided thermodynamic equilibrium is reached, show a positive correlation. (

**c**) The final layer permeability loss can be fitted with an analytical scaling score (Equation (19)). The upper dashed line for the fracture layers is a linear regression using an exponent $a=1.9$. The lower dashed line is for the porous layers and uses an exponent of $a=1.5$.

**Table 1.**Chemical compositions of the geothermal fluids samples given as total elemental concentrations. All concentrations are given in (mM), except for $\mathrm{pH}$. LND is a sample from the Landau geothermal reservoir [31]. URG $2000\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$ and URG $3000\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$ are averaged samples from the Upper Rhine Graben at respective depths [30]. The samples have been further modified to achieve equilibrium with respect to the common reservoir minerals barite, quartz, anhydrite, celestite, and calcite at reservoir conditions; chloride was fitted to reach charge balance [1].

Sample | LND | URG 2000 m | URG 3000 m |
---|---|---|---|

$\mathrm{K}$ | 106 | $78.8$ | 133 |

$\mathrm{Na}$ | 1270 | 894 | 1590 |

$\mathrm{Ca}$ | 196 | 104 | 177 |

$\mathrm{Mg}$ | $3.25$ | $4.23$ | $4.29$ |

$\mathrm{Sr}$ | $10.6$ | $2.7$ | $8.83$ |

$\mathrm{Ba}$ | $0.11$ | $0.0189$ | $0.0953$ |

$\mathrm{Fe}$ | $0.403$ | $1.84$ | $3.73$ |

$\mathrm{Cl}$ | 1790 | 1180 | 2100 |

$\mathrm{Br}$ | $2.84$ | $1.29$ | $2.61$ |

$\mathrm{S}\left(6\right)$ | $3.02$ | $5.93$ | $3.61$ |

$\mathrm{C}\left(4\right)$ | $4.14$ | $4.04$ | $7.42$ |

$\mathrm{Si}$ | $2.12$ | $1.09$ | $1.89$ |

$\mathrm{pH}$ | $5.41$ | $5.66$ | $5.28$ |

I | 2010 | 1320 | 2310 |

**Table 2.**The reservoir simulation variables and ranges that were considered for the exhaustive scenario analysis. Each unique parameter combination represents a scenario (${3}^{4}=81$ scenarios).

Variable | Range |
---|---|

Sample | [LND, URG 2000 m, URG 3000 m] |

${Q}_{\mathrm{tot}}$ | [50, 100, 200] ${\mathrm{m}}^{3}/\mathrm{h}$ |

${Q}_{\mathrm{frac}}/{Q}_{\mathrm{tot}}$ | [$0.1$, $0.5$, $0.9$] |

${n}_{\mathrm{frac}}$ | [1, 10, 100] |

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Tranter, M.; De Lucia, M.; Kühn, M.
Barite Scaling Potential Modelled for Fractured-Porous Geothermal Reservoirs. *Minerals* **2021**, *11*, 1198.
https://doi.org/10.3390/min11111198

**AMA Style**

Tranter M, De Lucia M, Kühn M.
Barite Scaling Potential Modelled for Fractured-Porous Geothermal Reservoirs. *Minerals*. 2021; 11(11):1198.
https://doi.org/10.3390/min11111198

**Chicago/Turabian Style**

Tranter, Morgan, Marco De Lucia, and Michael Kühn.
2021. "Barite Scaling Potential Modelled for Fractured-Porous Geothermal Reservoirs" *Minerals* 11, no. 11: 1198.
https://doi.org/10.3390/min11111198