# Study on Critical Drawdown Pressure of Sanding for Wellbore of Underground Gas Storage in a Depleted Gas Reservoir

^{1}

^{2}

^{*}

_{2}Transformation and Storage in Deep Formations)

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Test on Mechanical Properties of Reservoir Rock

^{3}to 2.4 g/cm

^{3}, and the porosity range was 7.8% to 27.9%; the permeability range of from 0.65 to 973.37 mD. The mechanical properties tests were performed using the MTS 815 rock mechanics test system, as shown in Figure 1a. The acquired stress-strain curve and the core images before and after the test in well #1 are presented in Figure 1b. The results indicate that the triaxial compressive strength ranged from 55.9 MPa to 143.9 MPa, and the Poisson’s ratio ranged from 0.07 to 0.19, while the Young’s modulus ranged from 4.5 GPa to 9.4 GPa. As shown in Figure 1c, by fitting the maximum principal stress and confining pressure obtained by the experiment, the Mohr circle of the corresponding rock sample can be obtained, and then the friction angle and cohesion strength can be obtained.

_{p}and Δt

_{s}are the compressional sonic travel time and shear sonic travel time, respectively.

_{max}), the minimum horizontal principal stress (SH

_{min}), and the vertical principal stress (SV), were estimated using the empirical equation proposed by [43]. The estimated crustal stress values of the three wells were adopted as the initial conditions for the inversed analysis of the in-situ crustal stress in the target formation in the next section, in order to improve the convergence efficiency.

#### 2.2. Mathematical Model of the Yield Criterion for Simulation

_{1}is the first invariant of the Cauchy stress and J

_{2}is the second invariant of the deviatoric part of the Cauchy stress. The constants A, B are determined from experiments. If assuming that the Drucker–Prager yield surface circumscribes the Mohr–Coulomb yield surface, then the expressions for A and B are as follows [44]:

_{p}) is adopted as the failure criterion of the rock near the wellbore [44],

_{px}, ε

_{py}, ε

_{pz}and are the plastic strains in the three principal stress directions, dimensionless. According to the previous studies, the well is believed to be sanding when the equivalent plastic strain exceeds the critical strain limit (CSL), which was usually defined as from 3‰ to 8‰ [30,45,46]. In this study, the CSL of 5‰ is used.

## 3. Results and Discussion

#### 3.1. Inversion of the In-Situ Stress Distribution

#### 3.2. Simulation on Sanding Prediction of Well Failure in UGS

^{3}at vertical height) was applied to the inner wall to ensure the stability of the wellbore and simulate the process of balanced drilling. Then the static fluid column pressure in the wellbore was decreased to simulate the gas production process. The amplitude function of the equivalent plastic strain around the wellbore was investigated. When the equivalent plastic strain exceeded the CSL, the wellbore was considered to be at high risk of sanding. The maximum principal stress and the equivalent plastic strain distribution of the wells when exceeding the CSL are presented in Figure 10. The maximum equivalent plastic strain distributed in the direction of maximum principal stress. As usual, compressive stress is negative. The corresponding differential pressures of the gas production are the critical pressure of the wellbore sanding, which are 5.59 MPa, 3.98 MPa and 4.01 MPa for well #1, well #2 and well #3 when the pressure of the gas storage is 30 MPa, respectively. During the gas injection and withdrawal of UGS, sand onset occurred at well #2 and #3 in the gas production when the differential pressure of the increased to 3.8 MPa and 4.1 MPa. The simulation results showed good agreements with the field-measured benchmark data.

#### 3.3. Influencing Factors of the Critical Pressure Difference for Sanding in UGS

## 4. Conclusions

- (1)
- Based on the experimental results, the functional relationship between key rock mechanics and rock density, such as compressive strength, elastic modulus, and Poisson’s ratio, was established. Coupling with the well-logging curves, the mechanical properties of the rock in the coring well could be calculated accurately.
- (2)
- The 3D geological model of the reservoir for UGS was transformed to a finite element geomechanics model, and was used as inputs for the inversed analysis of in-situ crustal stress. Adopting the measured in-situ stress in the samples of Well #1 as the target parameter, the crustal stress of the reservoir was determined coupling with the genetic algorithm and geomechanical simulation on Abaqus software.
- (3)
- Based on the stress distribution obtained by inversed analysis, the sub model of the zone near the wellbore was established. Numerical simulations on the well drilling and the cycling of high-speed injection-withdrawal were conducted. Taking the CSL of 5‰ as the sanding criterion of the wellbore, the CDPs of the gas production in the UGS were predicted, which are 5.59 MPa, 3.98 MPa, and 4.01 MPa for well #1, well #2 and well #3 when the pressure of the gas storage is 30 MPa, respectively. The simulation results showed good agreement with the field-measured benchmark data of well #2 and well #3.
- (4)
- The effects of moisture contents (ranging from 10% to 40%), and cycling times of gas injection and withdrawal (ranging from 40 to 200 cycling times) on the critical differential pressure were simulated and analyzed. The results indicated that the CDP decreased with increase of the moisture content and the cycling times.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D | Three dimensional |

ATWC | Advanced thick-walled cylinder () |

BEM | Boundary Element Method |

CSL | Critical strain limit |

CDP | Critical differential pressure |

DEM | Discrete Element Method |

FDM | Finite Difference Method |

FEM | Finite element method |

GA | Genetic algorithm |

NG | Natural gas |

PR | Poisson’s Ratio |

TWC | Thick-wall cylinder strength |

UCS | Unconfined compressive strength |

UGS | Underground gas storage |

YM | Young’s Modulus |

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**Figure 1.**Triaxial compression test on the drilled core from the target formation. (

**a**) MTS 815 rock mechanics test system. (

**b**) Stress-strain curve and images before and after the test of one core sample in well #1. (

**c**) The fitted Mohr circle and the calculation of the friction angle and cohesion strength.

**Figure 2.**Well logging and geomechanical properties profile of the target wells. (

**a**) Well #1. (

**b**) Well #2. (

**c**) Well #3.

**Figure 3.**FEM geomechanical modeling of the target reservoir. (

**a**) Geological model and well location. (

**b**) Geometry of FEM geomechanical model. (

**c**) FEM geomechanical model mesh.

**Figure 4.**Mapped rock properties based on the FEM model of the target reservoir. (

**a**) Density (unit: g/cm

^{3}) distribution of the target formation. (

**b**) Young’s Modulus (unit: GPa) distribution of the target formation. (

**c**) Poisson’s Ratio distribution of the target formation.

**Figure 7.**Simulation results of distribution of three principal stresses. (

**a**) Vertical principal stress. (

**b**) Maximum horizontal principal stress. (

**c**) Minimum horizontal principal stress.

**Figure 8.**Extraction of the sub-model with stress field surrounding the target well. (

**a**) Geological model of target formation. (

**b**) Model of the zone near well. (

**c**) Schematic of wellbore and boundary conditions. (

**d**) Schematic of crustal stress for the sub.

**Figure 10.**Maximum principal stress and the equivalent plastic strain distribution of the wells at the CSL.

**Figure 11.**Maximum principal stress and the equivalent plastic strain distribution of the wells at CSL for different moisture contents.

**Figure 12.**Maximum principal stress and the equivalent plastic strain distribution of the wells at CSL for different cycling times of gas injection and withdrawal.

**Figure 14.**Critical differential pressure of gas production for different cycling times of gas injection and production.

Sanding Index | Equation | Threshold |
---|---|---|

Porosity [23] | ψ = Volume_{pore}/Volume_{total} | Varying with the reservoir types; where Volume_{pore} and Volume_{total} are the volume of the pore and the rock, respectively. |

Acoustic wave travel time [24] | $\Delta {t}_{c}=\frac{1}{{V}_{P}}$ | where V_{p} is the velocity of the P-wave;95 μs/ft <Δt _{c} < 105 μs/ft, Slight sanding;Δt _{c} ≥ 105 μs/ft, Severe sanding |

Combination modulus E_{c} [25] | ${E}_{c}=\frac{9.94\times {10}^{8}{\rho}_{r}}{\Delta {t}_{c}{}^{2}}$ | where ρ_{r} is the rock density;E _{c} ≥ 2.0 × 10^{4} MPa, No sanding;2.0 × 10 ^{4} MPa ≥ E_{c} ≥ 1.5 × 10^{4} MPa, Slight sanding;E _{c} < 1.5 × 10^{4} MPa, Severe sanding |

Index B_{i} [26] | ${B}_{i}=K+\frac{4}{3}G$ | where K and G are the volumetric modulus and shear modulus; B _{i} > 20 GPa, No sanding;20 GPa > B _{i} ≥ 1.4 × 10^{4} MPa, Slight sanding (but will sanding seriously after water breakthrough);B _{i} < 14 GPa, Severe sanding |

Schlumberger’s ratio [27] | $R=\frac{\left(1-2\mu \right)+\left(1+\mu \right){\rho}^{2}}{6\left(1-\mu \right){\left(\Delta {t}_{p}\right)}^{4}}$ | where μ is the Poisson’s ratio; R ≤ 5.9 × 10 ^{7} MPa^{2}, Sanding |

CDP Model | Equation | Nomenclature | Failure Mode |
---|---|---|---|

Unconfined compressive strength(UCS)/2 [30] | $\Delta {P}_{w}^{cr}=L\times {\sigma}_{UCS}$ | L is the empirical constant of 0.3~0.5, σ_{UCS} is the uniaxial compressive strength | Compressional failure |

Nordgren’s model [31] | $\begin{array}{l}\Delta {P}_{w}^{cr}={J}_{2}-c{\overline{J}}_{1}\\ {\overline{J}}_{1}=\left({\sigma}_{1}+{\sigma}_{2}+{\sigma}_{3}\right)/3-p\\ {J}_{2}=\left({\left[{\sigma}_{1}-{\sigma}_{2}\right]}^{2}+{\left[{\sigma}_{2}-{\sigma}_{3}\right]}^{2}+{\left[{\sigma}_{3}-{\sigma}_{1}\right]}^{2}\right)/6\end{array}$ | c is the material constant for non-linearity | |

Almisned’s model [32] | $\Delta {P}_{w}^{cr}=\mathrm{max}\left\{\begin{array}{c}\frac{\frac{3\mu}{1-\mu}{P}_{ob}-{\sigma}_{UCS}{S}^{a}}{2B}\\ \frac{\left(3-\frac{3\mu}{1-\mu}\right){P}_{ob}-{\sigma}_{UCS}{S}^{a}}{2B}\\ \frac{\frac{\mu}{1-\mu}{P}_{ob}-{\sigma}_{UCS}{S}^{a}}{B}\end{array}\right\}$ | B is the Biot’s constant, P_{ob} is the overburden pressure, s and a are Hoek Brown material constants | Shear failure |

Morita et al.’s model [33] | $\begin{array}{l}\Delta {p}_{w}^{cr}=\frac{1}{3-2\left(\frac{1-2\mu}{1-\mu}\right)}\\ \left(\begin{array}{l}-3\overline{{\sigma}_{H}}-2{T}_{0}+\frac{2{T}_{0}\left(3+\alpha \right)}{\alpha}\times \\ \left\{{\left[1+\frac{{B}_{0}+2{B}_{1}{T}_{0}}{2{T}_{0}\frac{1-\mu}{E}\frac{3+\alpha}{3+2\alpha}\sqrt{\frac{1}{6}\left({\alpha}^{2}+4\alpha +6\right)}}\right]}^{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\left(2\alpha +3\right)$}\right.}-1\right\}\end{array}\right)\end{array}$ | ${C}_{0}=\frac{3C}{\sqrt{9+12{\mathrm{tan}}^{2}\varphi}}$$,{C}_{1}=\frac{\mathrm{tan}\varphi}{\sqrt{9+12{\mathrm{tan}}^{2}\varphi}}$$,\alpha =\frac{6{C}_{1}}{\frac{1}{\sqrt{3}}-2{C}_{1}}$$,{T}_{0}=\frac{{C}_{0}}{\frac{1}{\sqrt{3}}-2{C}_{1}}$ | |

Vaziri et al.’s model [34] | $\begin{array}{l}\Delta {P}_{w}^{cr}\le {P}_{0}-\frac{2{\sigma}_{v}-\lambda \times TWC}{2-A}-{P}_{0}\frac{A}{2-A}\\ A=\frac{\left(1-2\mu \right)B}{1-\mu}\end{array}$ | σ_{v} is the vertical stress, λ is a factor depending on thick-wall cylinder strength (TWC) test, | |

$\Delta {P}_{w}^{cr}=(Ck+{P}_{0})-\sqrt{{(Ck+{P}_{0})}^{2}-2Ck{P}_{0}}$; $k=\frac{4\mathrm{cos}\varphi}{1-\mathrm{sin}\varphi}$ | C is the cohesive force, φ is the frictional angle, P_{0} is the pore pressure | Tensile Failure |

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**MDPI and ACS Style**

Song, R.; Zhang, P.; Tian, X.; Huang, F.; Li, Z.; Liu, J.
Study on Critical Drawdown Pressure of Sanding for Wellbore of Underground Gas Storage in a Depleted Gas Reservoir. *Energies* **2022**, *15*, 5913.
https://doi.org/10.3390/en15165913

**AMA Style**

Song R, Zhang P, Tian X, Huang F, Li Z, Liu J.
Study on Critical Drawdown Pressure of Sanding for Wellbore of Underground Gas Storage in a Depleted Gas Reservoir. *Energies*. 2022; 15(16):5913.
https://doi.org/10.3390/en15165913

**Chicago/Turabian Style**

Song, Rui, Ping Zhang, Xiaomin Tian, Famu Huang, Zhiwen Li, and Jianjun Liu.
2022. "Study on Critical Drawdown Pressure of Sanding for Wellbore of Underground Gas Storage in a Depleted Gas Reservoir" *Energies* 15, no. 16: 5913.
https://doi.org/10.3390/en15165913