# Multiphase Multicomponent Numerical Modeling for Hydraulic Fracturing with N-Heptane for Efficient Stimulation in a Tight Gas Reservoir of Germany

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}H

_{2n+2}:C

_{5}-C

_{10}) [26,27]. Through this scheme, the use of water, as well as the environmental issues associated with it, can be largely minimized. The proposed fluid can be handled as a liquid at surface conditions and is compatible with the formation fluid as it is a petroleum product and hydrocarbon in nature. The density (ranging between 626–730 kg/m

^{3}) and viscosity (0.2–0.702 mPa·s) is also considerably less compared with water, therefore, the fracture closure and flowback can be significantly faster than water-based fluid. A single component, such as n-heptane, or a combination of different components according to different reservoir conditions can be used. As it is considerably less dense than water, the hybrid fluid concept, whereby initiating fracturing with 30–40 m

^{3}water is followed by light alkanes injection, can help reduce large surface injection pressure requirements. The phase behavior of the individual components of the proposed fluid are shown in Figure 2.

## 2. Materials and Methods

#### 2.1. N-Heptane as Frac-Fluid

^{3}which is much lower than water. Therefore, to avoid excessive surface injection pressure requirements, a hybrid fluid concept will be utilized in this work, in which fracturing is initiated by injecting a small volume of water followed by an n-heptane injection.

#### 2.2. Numerical Modeling

^{plus}, was then coupled with TOUGH2MP to model hydraulic fracturing in different reservoirs such as oil, gas, and geothermal [37,38,39,40].

^{plus}(full 3D rock mechanical simulator) and TMVOC (reservoir simulator). In this approach, a fracture model, MM fluid flow model, and proppant transport model are implemented in the THM coupled FLAC3D

^{plus}-TMVOC framework [38,39]. In addition, fracture elements residing in the host matrix elements in a pre-defined path perpendicular to least principal stress are considered.

#### 2.2.1. Mechanical Deformation

^{plus}formulation, which relies upon the elasto-plasticity theory, is used. In this regard, the displacement increment in a time interval is determined by the solution of the equation of motion (Equation (1)). The strain and stress increments in a time step can be determined using the continuum and constitutive equations (Equations (2) and (3)) [30].

^{3}); ${g}_{i}$: gravitational acceleration (m/s

^{2}); ${v}_{i}$: velocity (m/s); $t$: time (s); $\Delta \epsilon $: strain increment (-); $u$: displacement (m) $\mathsf{\Delta}\mathsf{\sigma}\prime $: effective stress increment; $D$: physical matrix; $\nu $: Poisson coefficient (-);$E$: Elastic modulus (Pa); $v$: Poisson’s ratio (-); and $i,j\in \left(1,2,3\right)$.

^{3}/m

^{3}); $\Delta {\epsilon}_{o}$: over reduced strain (-); and ${w}_{resd}$: residual width (m).

#### 2.2.2. Multi-phase Multi-Component (MM) Fluid Flow and Proppant Transport

^{3}); ${F}_{\beta}$: mass flux (mol/m

^{2}/s); $\beta $: liquid/gas/NAPL (non-aqueous phase liquids) phase; ${x}_{\beta}^{\kappa}$: component k mole fraction in phase β (-); and ${q}^{\kappa}$: source or sink (mol/m

^{3}/s).

^{3}).

^{2}); ${k}_{r\beta}$: relative permeability of phase β; ${\mu}_{\beta}$: viscosity of phase β (Pa·s); $\nabla {P}_{\beta}$: pressure gradient (Pa/m); and $g$: gravity (m/s

^{2}).

^{3}); ${\rho}_{f}$: fluid density (kg/m

^{3}); ${\mu}_{a}$: apparent viscosity (Pa·s); and $R{N}_{p}$: particle Reynold’s number: $\frac{{d}_{p}{v}_{p}{\rho}_{f}}{{\mu}_{ap}}$.

^{plus}-TMVOC coupling. The data sharing between software and specific calculations performed at a time step can be observed in detail in Figure 7. The computing time of the simulation depends upon the size of 3D model, number of grid blocks, isothermal or non-isothermal process, number of components for injection and production, etc., and therefore may range from a few hours to a few days.

## 3. Verification of Developed Numerical Model

- Fracturing ability;
- Non-isothermal flow;
- Proppant transport.

#### 3.1. Fracturing Ability

^{3}of fresh water with five injection-shutin cycles, was carried out to create a fracture area of more than 0.5 km

^{2}[49].

#### 3.2. Non-Isothermal Flow

#### 3.3. Proppant Transport

^{3}, with guar gel in the model at a height of about 0.914 m from the base (Figure 12). The simulation was carried out for a period of 150 s and injection rate and proppant concentration were maintained at 0.252 × 10

^{−3}m

^{3}/sec and 479.36 kg/m

^{3}, respectively [52].

## 4. Case Study

^{3}. A bottom-hole temperature and pressure of 423.15 K and 67 MPa, respectively, were recorded. Hydraulic fracturing was performed in the year 2000 targeting the Wustrow, Dethlingen and Havel formations. However, there was no significant productivity increase, and the production continued only for a few months with intermittent shutin periods recovering only around 2 million m

^{3}gas. Considering homogenous conditions, a quarter (1/4) 3D geometric model of the reservoir was generated. Figure 15a,b present the 3D model and pressure and stress profile of the reservoir.

^{3}fluid and 120-ton 3465 kg/m

^{3}density proppants were injected. The injection continued for a period of 150 min. The injection rate remained low until 63 min and then gradually increased to 4.89 m

^{3}/min. A proppant slug at 5 ppg (pounds/gallon) (=599 kg/m

^{3}) was initially placed to remove the existing tortuosity. Afterwards, the proppant injection began at 96 min and was increased to a maximum proppant concentration of 7 ppg (=839 kg/m

^{3}) at the end. As per the main frac data, the simulation of the fracking operation was performed with the developed model. Figure 16 presents the bottom-hole pressure (BHP) match of the simulated frac job with measured data.

^{3}of injection fluid, the radiator shaft on the blender sheared off. Thus, the operation had to be stopped and commenced the following day. Due to the presence of still x-linked gel in the tubing, high tubing pressures were recorded and the first 2 m

^{3}of fluid could only be injected at 0.2 m

^{3}/min. The crosslinker was switched off and displacement was carried out with linear gel, however, the injection rates remained below 0.6 m

^{3}/min until the first 60 min. Consequently, a pressure match was not feasible due the abnormal pressure recorded, as depicted in section A of Figure 16. Afterwards, the pressure finally dropped, and the injection rate was increased. It can be observed that a reasonable pressure match was obtained during the main course of injection (Figure 16, section B).

^{3}to 56 m

^{3}(Figure 18).

#### 4.1. Sensitivity Analysis with N-Heptane as Frac-Fluid

#### 4.1.1. Injection Rate

^{3}/min were considered to analyze their effect on fracture propagation. The injection program is discussed in Table 6.

^{3}/min almost tripled the fracture volume with more fracture half-length, height and width. As it was not an ultra-tight gas reservoir, the leakoff of injected fluid was higher. At lower injection rates, fluid had more time to percolate into the surrounding matrix, resulting in lower fracture volume. Therefore, increasing the injection rate increased the fracturing rate. In addition, injecting gelled fluid from the beginning will also reduce excessive fluid leakoff.

#### 4.1.2. Fluid Viscosity

^{3}/min and 8 m

^{3}/min (Table 7). As was expected, increasing viscosity increased the simulated reservoir volume, and fractures with maximum widths of 0.9–1.4 cm can be generated with higher injection rates and fluid viscosities (Figure 20).

#### 4.1.3. Injection Period

^{3}/min and 8 m

^{3}/min are considered in this section. An increase of about 15 m in half-length, with increasing injection time and viscosity, was observed (Figure 21). Similarly, an increase in fracture height of 16 m for 6 m

^{3}/min and 30 m for 8 m

^{3}/min was observed when the injection period was increased from 1.25 to 2 h. For the case of 8 m

^{3}/min and 0.15 Pa·s fluid, a fracture half-length of 105 m and a height of 138 m was achieved at 1.75 h. Further injection had no effect on the fracture geometry apart from increasing width. For 0.25 Pa·s and 0.35 Pa·s fluids, similar fracture heights of 154 m and 162 m at 1.75 and 2 h for an 8 m

^{3}/min injection rate, respectively, were obtained. For an injection rate of 6 m

^{3}/min and 0.15 Pa·s viscosity, 81 m half-length and 126 m fracture height fracture were created.

#### 4.1.4. Reservoir Permeability

^{3}was generated with lowest leakoff in case 3 due to the lower leakoff. Therefore, pure fluid at lower injection rates can be utilized in such reservoirs for fracturing.

#### 4.1.5. Fluid Flowback

^{3}/min was maintained for 77 min, during which 375 m

^{3}of injection fluid and 67 tons of proppant were injected. The details of fluid injection schedule are provided in Table 9.

^{3}) at the end of injection, after which the pumping was stopped.

#### 4.2. Design Proposals

^{2}); $d$: distance between fracture element and perforation (m); $A$: fracture element area (m

^{2}); ${c}_{p}$: proppant concentration (-); ${c}_{max}$: maximum proppant concentration; $k$: reservoir permeability (m

^{2}); ${x}_{f}$: fracture half-length (m) and $n$: total fracture elements (-).

^{3}/min for a duration of 105 min were proposed. Two fracture designs are presented in this paper, in which fractures with weighted dimensionless conductivity of 30 and 44 are created with n-heptane. Due to quick fracture closure and leakoff, better proppant placement and improved borehole-fracture hydraulic connection was established in comparison with previous frac job. The fracture geometry and design parameters are presented in Figure 27.

## 5. Discussion

^{3}/min injection rate. The leakoff is low in ultra-tight reservoirs, so pure n-heptane can be utilized for fracture propagation and the gelled fluid may be required for the proppant transport period.

^{3}/min, can generate a fracture with weighted dimensionless conductivity of 44 (Fcd: 180) and more than 100 m half-length fracture. While proposal 2, with a lower injection rate (i.e., 6 m

^{3}/min), can create weighted dimensionless fracture conductivity of 30 (Fcd: 123). In both designs, excellent hydraulic connection between borehole and fracture is predicted.

## 6. Conclusions

^{3}/min, and injecting gelled fluid will result in frac jobs with better fracture geometries in wellbore x. The alternative frac-fluid can be chosen according to different reservoir conditions and desired fracture geometry and conductivity. The developed numerical model with the capability to simulate hydraulic fracturing process in full 3D approach with MM fluid flow can be utilized to perform hydraulic stimulation with a variety of injection fluids. By promoting the use of alternative fluids, environmental hazards associated with water-based fluids can be minimized, in addition to efficient hydraulic fracturing with better fracture conductivities and borehole-fracture connection. Fluid recovery and reuse for subsequent fracturing operations can make its application economically attractive and even comparable with conventional water-based fluid fracking.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Phase behavior of individual components of n-alkanes. Last point (in black circle) on each curve represents respective critical point [28].

**Figure 3.**Phase behavior of n-heptane [28].

**Figure 7.**Software coupling and data sharing. (${\sigma}_{s}$: stress state, $u$: fluid velocity,$\rho $: density, $\mu $: viscosity, $w$: width, and ${S}_{f}$: fluid saturation).

**Figure 8.**The (

**a**) 3D quarter geological model, and (

**b**) stress state and pressure profile in the model, s_min: minimum horizontal stress, s_ver: vertical stress, and s_max: maximum horizontal stress.

**Figure 9.**Fracture pressure comparison for simulation results and published data [39], sim: simulated, meas: measured, and inj_rate: injection rate.

**Figure 13.**Proppant distribution contours at 10 and 100 s, (

**a**) numerical [51] and (

**b**) simulated (developed numerical model).

**Figure 15.**The (

**a**) 3D geometric (1/4) model, and (

**b**) pressure and stress in the model, S_yy: minimum horizontal stress, S_zz: vertical stress.

**Figure 16.**History match (BHP_sim) with measured BHP, injection rate and proppant schedule plotted for the main fracturing treatment.

**Figure 17.**Fracture half-width, proppant concentration, viscosity, and temperature profiles at shutin (150 mins) and closure (600 mins).

**Figure 19.**The (

**a**) fracture half-width (m) profiles for 4–6–8 m

^{3}/min injection rates at the end of injection, and (

**b**) fracture height, half-length and volume for injection rates 4–6–8 m

^{3}/min.

**Figure 20.**Fracture half-width (m) for different injection strategies for different injection rates and fluid viscosities.

**Figure 21.**Fracture heights, half-lengths and volumes for different injection times for 0.15 Pa·s, 0.25 Pa·s and 0.35 Pa·s fluids at 6 and 8 m

^{3}/min injection rates.

**Figure 22.**The (

**a**) fracture half-width (m) for the three cases, and (

**b**) fracture height, half-length and volume.

**Figure 25.**Fracture volume for the two cases at different times; w: water-based, and hf: n-heptane hybrid fluid.

**Figure 26.**Saturation contours of reservoir gas and injection fluids after one and seven days of flowback.

**Table 1.**Important properties of n-heptane [29].

Name | C_{n}H_{2n+2} | Critical Temperature, K | Critical Pressure, MPa | Boiling Point, K | Molecular Weight, g/mol | Density @ 289 K, kg/m^{3} |
---|---|---|---|---|---|---|

n-heptane | C_{7}H_{16} | 540.2 | 2.74 | 371.6 | 100.2 | 689.5 |

Model Properties | |
---|---|

Porosity | 38.5% |

Permeability | 1.6 × 10^{−11} m^{2} |

Initial pressure | 101.3 × 10^{3} Pa |

Temperature | 295.15 K |

Parameter | Unit | Value |
---|---|---|

Injection rate | m^{3}/sec | 0.252 × 10^{−3} |

Proppant density | kg/m^{3} | 2650 |

Liquid density | kg/m^{3} | 1035 |

Proppant concentration in injection fluid | kg/m^{3} | 479.36 |

Fluid viscosity | Pa·s | 0.35 |

Proppant diameter | mm | 0.6 |

Power law coefficient | - | 0.65 |

Parameter for viscosity correction | - | 1.8 |

Maximal proppant concentration | - | 0.65 |

Formation | Top Depth (m) | Young’s Modulus (Pa) | Poisson’s Ratio (-) | Porosity (-) | Permeability (m^{2}) | Density (kg/m^{3}) |
---|---|---|---|---|---|---|

Z1–An | 4490 | 2.90 × 10^{10} | 0.275 | 0.041 | 9.720 × 10^{−21} | 2940 |

Z1–Rk | 4535 | 2.85 × 10^{10} | 0.253 | 0.044 | 4.84 × 10^{−17} | 2840 |

Kupfe | 4538 | 2.89 × 10^{10} | 0.260 | 0.059 | 3.15 × 10^{−19} | 2715 |

He–Me | 4540 | 2.94 × 10^{10} | 0.255 | 0.040 | 1.05 × 10^{−16} | 2705 |

Mu–Me | 4554 | 3.06 × 10^{10} | 0.206 | 0.068 | 2.78 × 10^{−16} | 2655 |

Ni–Me | 4574 | 3.72 × 10^{10} | 0.220 | 0.001 | 1.00 × 10^{−21} | 2750 |

Da–Me | 4598 | 3.31 × 10^{10} | 0.230 | 0.081 | 1.154 × 10^{−16} | 2574 |

Ba–Me | 4609 | 3.01 × 10^{10} | 0.197 | 0.110 | 1.165 × 10^{−16} | 2500 |

Wu–Me | 4627 | 2.93 × 10^{10} | 0.2025 | 0.110 | 4.887 × 10^{−16} | 2493 |

Eb–Me | 4654 | 2.76 × 10^{10} | 0.200 | 0.110 | 8.430 × 10^{−16} | 2707 |

De–fo | 4670 | 2.65 × 10^{10} | 0.1975 | 0.113 | 5.79 × 10^{−16} | 2668 |

Ha–Sa | 4723 | 2.61 × 10^{10} | 0.2267 | 0.113 | 9.504 × 10^{−16} | 2668 |

Al–Su | 4816 | 2.92 × 10^{10} | 0.25 | 0.064 | 1.067 × 10^{−20} | 2750 |

**Table 5.**Relative permeability and capillary pressure parameters for three-phase multi-component flow of fluid.

Relative Permeability [41,47] | Capillary Pressure [41,48] | ||
---|---|---|---|

S_{rg} | 0.04 | ${\alpha}_{g\mathrm{a}}$ | 11 |

S_{ra} | 0.05 | ${\alpha}_{aw}$ | 12 |

S_{rw} | 0.32 | n | 1.84 |

n | 3 | m | 0.45 |

Injection Schedule (75 min) | |
---|---|

0–8 min | Water |

8–30 min | Light n-heptane |

30–75 min | Gelled n-heptane (0.15 Pa·s) |

Injection Schedule (75 min) | |
---|---|

0–8 min | Gelled-water |

8–75 min | Gelled-n-heptane |

75 min (300 m^{3} Injected at 4 m^{3}/min) | |
---|---|

Original-1 | Water + n-heptane + gelled fluid n-heptane (0.15 Pa·s) |

Original-2 | Gelled water + gelled n-heptane (0.15 Pa·s) |

Permeability × 0.01 [m^{2}] | Water + n-heptane + gelled n-heptane (0.15 Pa·s) |

1. Conventional (375 m^{3}) | 2. Hybrid-N-Heptane (375 m^{3}) | ||||||
---|---|---|---|---|---|---|---|

Injection Fluid | Total Injected (m^{3}) | Injection Time (min) | Injection Fluid | Total Injected (m^{3}) | Injection Time (min) | ||

Stage-1 | Water | 195 | 0–40 | Stage-1 | Water | 30 | 0–6 |

Stage-2 | Gelled-water | 180 | 40–77 | Stage-2 | n-heptane | 166 | 6–40 |

Stage-3 | Gelled n-heptane | 179 | 40–77 |

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## Share and Cite

**MDPI and ACS Style**

Mehmood, F.; Hou, M.Z.; Liao, J.; Haris, M.; Cao, C.; Luo, J.
Multiphase Multicomponent Numerical Modeling for Hydraulic Fracturing with N-Heptane for Efficient Stimulation in a Tight Gas Reservoir of Germany. *Energies* **2021**, *14*, 3111.
https://doi.org/10.3390/en14113111

**AMA Style**

Mehmood F, Hou MZ, Liao J, Haris M, Cao C, Luo J.
Multiphase Multicomponent Numerical Modeling for Hydraulic Fracturing with N-Heptane for Efficient Stimulation in a Tight Gas Reservoir of Germany. *Energies*. 2021; 14(11):3111.
https://doi.org/10.3390/en14113111

**Chicago/Turabian Style**

Mehmood, Faisal, Michael Z. Hou, Jianxing Liao, Muhammad Haris, Cheng Cao, and Jiashun Luo.
2021. "Multiphase Multicomponent Numerical Modeling for Hydraulic Fracturing with N-Heptane for Efficient Stimulation in a Tight Gas Reservoir of Germany" *Energies* 14, no. 11: 3111.
https://doi.org/10.3390/en14113111