# Pumping Schedule Optimization in Acid Fracturing Treatment by Unified Fracture Design

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Workflow

#### 2.2. Optimization of the Fracture Geometry Parameters

- For a specific acid volume and an initial guess for final ${x}_{Opt}$ (${x}_{Old}$), the average ideal fracture width is as [4]:$$\overline{{w}_{i}}=\frac{\mathcal{X}{V}_{acid}}{2\left(1-\varnothing \right){x}_{Old}{h}_{f}}$$
- One of the important parameters to estimate the fracture conductivity is the average ideal fracture width. A geostatistical method was employed to calculate the fracture conductivity [28] (Equation (3)). In this method, the effect of the spatial behavior of the formation’s permeability and elastic properties are considered in the calculations. Since the fracture conductivity is approximately proportional to ${\overline{{w}_{i}}}^{2.5}$, it is important, therefore, that the acid-etched width be accurately determined [29].$$\begin{array}{c}{k}_{f}={C}_{1}\mathrm{exp}\left(-{C}_{2}{\sigma}_{c}\right)\\ {C}_{1}=4.48\times {10}^{9}\left[0.1756{\left(erf\left(0.8{\sigma}_{D}\right)\right)}^{3}{\overline{{w}_{i}}}^{2.49}\right]\times [1+(1.82erf(3.25({\lambda}_{D,x}-\\ 0.12\left)\right)-1.31erf\left(6.71\left({\lambda}_{D,z}-0.03\right)\right)\left)\sqrt{exp\left({\sigma}_{D}\right)-1}\right]\times [0.22{\left({\lambda}_{D,x}{\sigma}_{D}\right)}^{2.8}+\\ {0.01{\left(\left(1-{\lambda}_{D,z}\right){\sigma}_{D}\right)}^{0.4}]}^{0.52}\\ {C}_{2}=[14.9-3.78ln\left({\sigma}_{D}\right)-6.81ln\left(E\right)]\times {10}^{-4}\end{array}$$
- After that, the proppant number for acid fracturing is determined, as reported in [30]. An equivalent proppant number was used for calculating the optimum dimensionless fracture conductivity.$${N}_{A}=\frac{2{k}_{f}{V}_{f}}{k{V}_{r}}$$$${C}_{FD\text{\_}Opt}=\{\begin{array}{cc}1.6& {N}_{A}<0.1\\ 1.6+exp\left[\frac{-0.583+1.48Ln\left({N}_{A}\right)}{1+0.142Ln\left({N}_{A}\right)}\right]& 0.1\le {N}_{A}\le 10\\ 1.6& {N}_{A}>10\end{array}$$
- In this step, the optimal fracture half-length and optimal fracture width are calculated as in [30]:$${x}_{Opt}={\left(\frac{{k}_{f}{V}_{f}}{2{C}_{FD\text{\_}Opt}k{h}_{f}}\right)}^{0.5}$$$${w}_{Opt}={\left(\frac{{C}_{FD\text{\_}Opt}k{V}_{f}}{2{k}_{f}{h}_{f}}\right)}^{0.5}$$
- Finally, the calculated fracture half-length and fracture width from Equations (6) and (7) are compared with initial ${x}_{Old}$ and $\overline{{w}_{i}}$. This process continues until the estimated values reach a stable condition.

#### 2.3. Fracture Propagation Model

#### 2.4. Acid Model

#### 2.4.1. Fluid Velocity Components and Pressure

#### 2.4.2. Calculate the Acid Concentration

#### 2.4.3. Rocks Displacement on the Fracture Surfaces

#### 2.4.4. Boundary Condition

^{−}$\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ and $\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$) is zero. The fluid velocity component along the $y$ axis located in the fracture surfaces is based on fluid leak-off velocity changes [34]. The leak-off rate depends on a leak-off coefficient and is proportional to the treatment time’s root inverse [6].

#### 2.5. Leak-Off Model

## 3. Results and Discussion

#### 3.1. Acid Model Validation

#### 3.2. Model Establishment

#### 3.3. Optimization of Injection Parameters

^{3}(16% acid HCl) was selected. It should be mentioned that in all graphs of this section, the red point on each shape indicates the convergence condition.

#### 3.3.1. Effect of Flow Rate on Optimization Results

#### 3.3.2. Effect of Acid Volume on Optimization Results

#### 3.4. Parametric Study

#### 3.4.1. Effect of Formation Permeability

#### 3.4.2. Effect of Injection Fluid Rheology

#### 3.4.3. Effect of the Acid Type

#### 3.4.4. Effect of the Acid Percentage

#### 3.4.5. Effect of the Young’s Modulus

#### 3.4.6. Effect of the Formation Closure Stress

## 4. Conclusions

- Due to the large number of calculations, the simulations were performed with the proposed method for a specific case. The results have shown that the concentration of acid decreases along the fracture length and fracture walls. Acid-etched width and consequently, conductivity, decrease along the fracture length.
- The flow rate optimization for 16% gelled acid shows that the optimization results are influenced by the acid transport behavior within the fracture.
- The behavior of acid volume against minimum error (for a given volume) was investigated, and it was observed that the optimal flow rate increases with increasing acid volume.
- The parametric study shows that when formation permeability is decreased, the optimal fracture half-length and average fracture width increase and decrease, respectively. In this case, the optimal flow rate increases.
- Fluid viscosity is a controllable parameter during acid fracturing operations. As the fluid viscosity increases, the optimal flow rate and volume of the injected acid decrease. Therefore, increasing the fluid viscosity, to a certain extent, can improve the results of the acid fracturing treatment optimization. On the other hand, its excessive increase has no economic justification.
- Sensitivity analyses on three types of acid systems show that the optimal flow rate for straight acid is higher than for the other two types of acid. It was also observed that for retarded acids, the optimal conditions are reached only at a high acid volume.
- The acid percentage is an influential parameter on the results. For a 15% acid concentration, the required flow rate is higher than 5% and 10%. The acid volume must be increased to achieve optimal conditions for a sample with a high acid concentration.
- The parametric study shows that the optimal flow rate and acid volume increase and decrease, respectively, for high Young’s modulus. In addition, the effect of closure stress was also investigated and it was observed that for a sample with high closure stress, low flow rate and high acid volume are required.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${C}_{D}$ | Dimensionless acid concentration, dimensionless |

${C}_{FD\text{\_}Opt}$ | Optimum fracture conductivity, dimensionless |

${C}_{L}$ | Leak-off coefficient, Ft/$\sqrt{\mathrm{Min}}$ [m/$\sqrt{\mathrm{S}}$] |

${C}_{c}$ | Compressibility fluid-loss coefficient, Ft/$\sqrt{\mathrm{Min}}$ [m/$\sqrt{\mathrm{S}}$] |

${C}_{eqm}$ | Acid equilibrium concentration, $\mathrm{moles}/{\mathrm{m}}^{3}$ |

${C}_{i}$ | Injected-acid concentration, $\mathrm{moles}/{\mathrm{m}}^{3}$ |

${c}_{t}$ | Total compressibility, 1/psi [m.s^{2}/kg] |

${C}_{v}$ | Viscous fluid-loss coefficient, Ft/$\sqrt{\mathrm{Min}}$ [m/$\sqrt{\mathrm{S}}$] |

${C}_{v,wh}$ | Viscous fluid-loss coefficient with wormhole, Ft/$\sqrt{\mathrm{Min}}$ [m/$\sqrt{\mathrm{S}}$] |

${C}_{w}$ | Wall-building fluid-loss coefficient, Ft/$\sqrt{\mathrm{Min}}$ [m/$\sqrt{\mathrm{S}}$] |

$\overline{c}$ | Mean acid concentration |

${D}_{eff}$ | Effective acid diffusion coefficient, Ft^{2}/min [m^{2}/s] |

$E$ | Young’s modulus, psi [kg/m.s^{2}] |

${E}_{f}$ | Reaction rate coefficient, $\frac{\mathrm{kg}\mathrm{moles}\mathrm{HCl}}{{\mathrm{m}}^{2}\mathrm{s}{\left(\frac{\mathrm{kg}\mathrm{moles}\mathrm{HCl}}{{\mathrm{m}}^{3}\mathrm{acid}\mathrm{solution}}\right)}^{{\mathrm{n}}^{\prime}}}$ |

${f}_{r}$ | Fraction of acid to react before leaking off, dimensionless |

${G}_{m}$ | Constant for the mean acid concentration profile |

${h}_{f}$ | Fracture height, Ft [m] |

$k$ | Formation permeability, md [m^{2}] |

$K$ | Consistency index, lb.s^{n}/Ft^{2} [Pa.s^{n}] |

${k}_{f}$ | Fracture permeability, md [m^{2}] |

$M{W}_{acid}$ | Molecular weight of the acid, $\mathrm{gr}/\mathrm{moles}$ |

$n$ | Power in the power-law, dimensionless |

${n}^{\prime}$ | Reaction order, dimensionless |

${N}_{A}$ | Acid number, dimensionless |

${N}_{Pe}$ | Peclet number |

${N}_{R{e}^{\ast}}$ | Reynolds number |

$p$ | Fluid net pressure, psi [kg/m.s^{2}] |

$\mathsf{\Delta}p$ | Pressure difference between fracture and formation, psi [kg/m.s^{2}] |

$q$ | Injected flow rate, bbl/min [m^{3}/s] |

${Q}_{ibt}$ | Number of PV’s injected at wormhole breakthrough, dimensionless |

${S}_{p}$ | Spurt loss coefficient, Ft [m] |

$t$ | Injection time, s |

${t}_{0}$ | Time for acid to reach a particular point in the fracture, min [s] |

$u$ | Velocity in the $x$ direction, Ft/min [m/s] |

$v$ | Velocity in the $y$ direction, Ft/min [m/s] |

${V}_{acid}$ | Acid volume, bbl [m^{3}] |

${V}_{f}$ | Induced fracture volume, Ft^{3} [m^{3}] |

${v}_{i}$ | Velocity vector, Ft/min [m/s] |

${v}_{L}$ | Leak-off velocity, Ft/min [m/s] |

${V}_{r}$ | Reservoir drainage volume, Ft^{3} [m^{3}] |

$w$ | Velocity in the $z$ direction, Ft/min [m/s] |

$\overline{w}$ | Averaged fracture width in pad stage, in [m] |

${\overline{w}}_{Acid}$ | Estimated average acid-etched width, in [m] |

$\overline{{w}_{i}}$ | Average ideal fracture width, in [m] |

${w}_{Opt}$ | Optimal fracture width, in [m] |

$w{k}_{f}$ | Fracture conductivity, md-ft [m^{3}] |

${x}_{f}$ | Estimated fracture half-length, Ft [m] |

${x}_{Old}$ | Initial guess for final ${x}_{Opt}$, Ft [m] |

${x}_{Opt}$ | Optimal fracture half-length, Ft [m] |

## Greek

$\beta $ | Gravitational dissolving power, dimensionless |

$\overline{\beta}$ | Constant in Equation (8), dimensionless |

${\lambda}_{D,x}$ | Dimensionless horizontal correlation length, dimensionless |

${\lambda}_{D,z}$ | Dimensionless vertical correlation length, dimensionless |

${\lambda}_{m}$ | Eigenvalues for the mean acid concentration profile |

${\sigma}_{c}$ | Closure stress, MMpsi [kg/m.s^{2}] |

${\sigma}_{D}$ | Dimensionless standard deviation of permeability, dimensionless |

$\varnothing $ | Formation porosity, dimensionless |

$\rho $ | Fluid density, lbm/Ft^{3} [kg/m^{3}] |

${\rho}_{rock}$ | Formation rock density, lbm/Ft^{3} [kg/m^{3}] |

$\upsilon $ | Poisson ratio, dimensionless |

$\mu $ | Fracture-fluid viscosity of Newtonian fluid, cp [kg/m.s] |

${\mu}_{a}$ | Acid viscosity, cp [kg/m.s] |

${\mu}_{Oil}$ | Oil viscosity, cp [kg/m.s] |

$\mathcal{X}$ | Volumetric dissolving power, dimensionless |

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**Figure 3.**Comparison between the calculated acid concentration along the fracture at different Peclet numbers using the proposed model and the Terrill, 1965 method [36].

**Figure 4.**Simulated acid concentration within the fracture domain for a flow rate of 10 bbl/min and 400 bbl acid volume. (

**a**) fracture half-length vs. height (

**b**) fracture half-length vs. width.

**Figure 5.**Acid-etched width (

**a**) and fracture conductivity (

**b**) distribution in the fracture domain for a flow rate of 10 bbl/min and 400 bbl acid volume.

**Figure 6.**The average acid-etched width (

**a**) and fracture half-length (

**b**) versus injection flow rate.

**Figure 8.**Minimum computational error for a given volume (

**a**). Error derivative changes versus acid volume (

**b**).

**Figure 10.**The influence of formation permeability on the optimal injection flow rate for a 500 bbl Acid Volume. (

**a**) K = 0.001 md and (

**b**) K = 0.1 md.

**Figure 11.**The influence of formation permeability on the optimal injection flow rate for a 1000 bbl acid volume. (

**a**) K = 0.001 md and (

**b**) K = 0.1 md.

**Figure 12.**The impact of fluid viscosity on the of flow rate (

**a**) and acid volume (

**b**) for a 540 bbl acid volume.

**Figure 13.**Results of flow rate optimization for three acid systems types investigated for a 540 bbl acid. (

**a**) straight acid, (

**b**) gelled acid, and (

**c**) emulsified acid.

**Figure 14.**Variations of fracture half-length (

**a**) and average etched width (

**b**) with flow rate for three acid system types.

**Figure 15.**Results of acid volume optimization for three acid systems types. (

**a**) straight acid, (

**b**) gelled acid, and (

**c**) emulsified acid.

**Figure 16.**The effect of acid concentration on the flow rate optimization results for a 540 bbl acid volume. (

**a**) 5 wt % acid, (

**b**) 10 wt % acid, and (

**c**) 15 wt % acid.

**Figure 17.**The effect of acid concentration on acid volume optimization results. (

**a**) 5 wt % acid, (

**b**) 10 wt % acid, and (

**c**) 15 wt % acid.

**Figure 19.**The impact of Young’s modulus on the flow rate (

**a**) and acid volume (

**b**) for a 540 bbl acid volume.

**Figure 21.**The influence of formation closure stress on the flow rate (

**a**) and acid volume (

**b**) for a 540 bbl acid volume.

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

Formation properties | ||

Young’s Modulus (E) | 6 | MMPsi |

Poisson Ratio ($\upsilon $) | 0.25 | - |

Porosity ($\varnothing $) | 0.071 | - |

Permeability (k) | 0.4 | md |

Wormhole breakthrough pore volume (${Q}_{ibt}$) | 1.5 | - |

Layer Thickness (H) | 50 | m |

Closure Stress (${\sigma}_{c}$) | 4200 | Psi |

Total compressibility (${C}_{t}$) | 1.983 × 10^{−5} | ${\mathrm{Psi}}^{-1}$ |

Reservoir Oil Viscosity (${\mu}_{Oil}$) | 1.66 | cp |

Formation Rock Density (${\rho}_{Rock}$) | 2600 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Reservoir Temperature (T) | 246 | ^{°}F |

Reservoir Pressure (${p}_{r}$) | 3000 | Psi |

Fracturing Pressure (${p}_{f}$) | 4300 | Psi |

Acid Properties | ||

Density (${\rho}_{Acid}$) | 1000 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Acid initial concentration (${c}_{i}$) | 4.4 (16%) | $\mathrm{moles}/{\mathrm{dm}}^{3}$ |

Spurt loss (${S}_{p}$) | 0 | m |

Fraction of acid to react before leaking off (${f}_{r}$) | 0.3 | - |

Reaction order (${n}^{\prime}$) | 0.63 | - |

Reaction rate coefficient (${E}_{f}$) | 0.3263 | $\frac{\mathrm{kg}\mathrm{moles}\mathrm{HCl}}{{\mathrm{m}}^{2}\mathrm{s}{\left(\frac{\mathrm{kg}\mathrm{moles}\mathrm{HCl}}{{\mathrm{m}}^{3}\mathrm{acid}\mathrm{solution}}\right)}^{\mathrm{n}\prime}}$ |

Parameters of UFD method | ||

Volumetric dissolving power ($\mathcal{X}$) | 0.082 | - |

Drainage radius (${r}_{e}$) | 1900 | Ft |

Dimensionless horizontal correlation length (${\lambda}_{D,x}$) | 1 | - |

Dimensionless vertical correlation length (${\lambda}_{D,Z}$) | 0.05 | - |

Dimensionless standard deviation of permeability (${\sigma}_{D}$) | 0.4 | - |

**Table 2.**Constant coefficients for Equations (29) and (30) to calculate ${\lambda}_{m}$ and ${G}_{m}$.

$\mathit{m}$ | ${\mathit{g}}_{\mathbf{0}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{0}}$ | ${\mathit{g}}_{\mathbf{1}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | ${\mathit{g}}_{\mathbf{2}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | ${\mathit{g}}_{\mathbf{3}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | ${\mathit{h}}_{\mathbf{1}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | ${\mathit{h}}_{\mathbf{2}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

0 | 1.68231 | −2.26693 | 6.7544 | −1.8408 | 6.7593 | −4.6274 |

1 | 5.67053 | −0.696 | 17.2931 | −2.9304 | 1.0032 | −3.4376 |

2 | 9.66842 | −0.39587 | 10.7745 | −0.5564 | −5.7028 | −0.4705 |

3 | 13.66772 | −0.27662 | 7.9375 | −0.1358 | −9.15 | −0.5668 |

4 | 17.6674 | −0.21305 | 6.34331 | −0.0373 | −12.4496 | −0.71196 |

$\mathit{m}$ | ${\overline{\mathit{g}}}_{\mathbf{0}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | ${\overline{\mathit{g}}}_{\mathbf{1}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | ${\overline{\mathit{g}}}_{\mathbf{2}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | ${\overline{\mathit{g}}}_{\mathbf{3}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | ${\overline{\mathit{h}}}_{\mathbf{1}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | ${\overline{\mathit{h}}}_{\mathbf{2}\mathbf{,}\mathit{m}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ |

0 | 9.10378 | −2.38279 | 14.9298 | −8.97017 | −7.08188 | −1.18392 |

1 | 0.53126 | 1.88909 | −12.5375 | 8.13482 | 4.01538 | 0.35148 |

2 | 0.15272 | 0.39035 | −1.6607 | 0.68079 | 1.0394 | 0.5154 |

3 | 0.06807 | 0.0733 | −0.4172 | 0.11131 | 0.58639 | 0.14123 |

4 | 0.03737 | 0.01901 | −0.1503 | 0.02756 | 0.35277 | 0.05623 |

**Table 3.**Property of three acid systems types studied [37].

Acid Types | ${\mathit{\mu}}_{\mathit{a}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\left(\mathbf{k}\mathbf{g}\mathbf{/}\mathbf{m}\mathbf{.}\mathbf{s}\right)$ | ${\mathit{D}}_{\mathit{e}\mathit{f}\mathit{f}}\mathbf{\left(}\mathbf{c}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{s}\mathbf{\right)}$ | n | $\mathit{K}\left(\mathbf{P}\mathbf{a}\mathbf{.}{\mathbf{s}}^{\mathbf{n}}\right)$ |
---|---|---|---|---|

Straight | 1 | 0.0000213 | 1.0 | 0.00109 |

Gelled | 15 | 0.000008 | 0.65 | 0.05 |

Emulsified | 30 | 2.64 × 10^{−}^{8} | 0.675 | 0.315 |

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## Share and Cite

**MDPI and ACS Style**

Lotfi, R.; Hosseini, M.; Aftabi, D.; Baghbanan, A.; Xu, G.
Pumping Schedule Optimization in Acid Fracturing Treatment by Unified Fracture Design. *Energies* **2021**, *14*, 8185.
https://doi.org/10.3390/en14238185

**AMA Style**

Lotfi R, Hosseini M, Aftabi D, Baghbanan A, Xu G.
Pumping Schedule Optimization in Acid Fracturing Treatment by Unified Fracture Design. *Energies*. 2021; 14(23):8185.
https://doi.org/10.3390/en14238185

**Chicago/Turabian Style**

Lotfi, Rahman, Mostafa Hosseini, Davood Aftabi, Alireza Baghbanan, and Guanshui Xu.
2021. "Pumping Schedule Optimization in Acid Fracturing Treatment by Unified Fracture Design" *Energies* 14, no. 23: 8185.
https://doi.org/10.3390/en14238185