# A New Analytical Method for Calculating Subsidence Resulting by Fluid Withdrawal from Disk-Shaped Confined Aquifers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}injection using Hankel-transformed thin plate theory. These methods yield simplified expressions as compared to previous results derived using the superposition principle on surface uplift from a uniform pressure field. Hence, closed-form formulas for the subsidence at the well location are re-derived, while the formulas for the subsidence field are deducted by both methods and the mathematical relation between the two methodologies is discussed. Additionally, innovative closed-form asymptotic solutions for radial subsidence distribution are deduced for scenarios involving deep aquifers. These solutions demonstrate exceptional accuracy when aquifer depth exceeds aquifer diameter, exhibiting independence from formation permeability and fluid viscosity. The study explores the influence of physical parameters on the subsidence field.

## 1. Introduction

## 2. Mathematical Formulation

_{e}of the production well within the aquifer formation given by:

_{f}being the fluid bulk modulus, K the rock bulk modulus, α is the Biot coefficient, and C

_{m}is the coefficient of uniaxial compaction:

**Case**r

_{e}< R

**Case**r

_{e}> R

_{i}is the initial value of the pressure in the aquifer before production starts.

#### 2.1. Subsidence Derived from the Theory of Selvadurai, 2009 [28] and Li et al., 2015 [29]

_{i}caused by pumping:

_{0}is the zeroth order Bessel function of the first kind, and ξ is a radial wavenumber.

**Case**r

_{e}< R.

**Case**r

_{e}> R

_{e}, is of the general form:

_{e}/R and D/R, that is, in terms of the radius of influence and aquifer depth scaled by the aquifer radial extent.

**Case**r

_{e}< R

**Case**r

_{e}> R

**Case**r

_{e}< R

**Case**r

_{e}> R

^{γ}= 1.78, and both Equations (15) and (16) are linear in time owing to Equation (1). That is, they are linear in the produced fluid volume Qt. We may also note that the scaled subsidence w/w

_{0}is given by a single formula in both cases:

#### 2.2. Subsidence from the Theory of Geertsma, 1973 [14]

_{i}is not zero. In particular, for the profiles of interest we have:

**Case**r

_{e}< R

**Case**r

_{e}> R

_{i}) should also be added. Upon integrating Equations (20) and (21) one arrives at Equations (13) and (14).

_{3}is defined by

_{0}takes the form

**Case**r

_{e}< R

**Case**r

_{e}> R

## 3. Results

_{e}> R. When the rock is strong (case 1), the radius of influence reaches the boundary earlier. For Figure 2b,d the radius of influence affects the solution in the same way but the larger depth smooths out the influence of the aquifer size on the subsidence profiles.

_{e}/R. The ratio D/R, depth over aquifer size, is formed purely by geometry. The second ratio r

_{e}/R involves the dynamics of fluid withdrawal, as the radius of influence depends on time, and also depends on the elastic moduli of the aquifer rock.

_{e}/R = {1/7, 3/7, 5/7, 1, 9/7, 11/7} in the indicated colors. Given that r

_{e}encodes also the influence of elastic moduli, porosity, permeability of the aquifer rock, and the viscosity of the fluid, the ratio r

_{e}/R quantifies the ‘stage’ of evolution with respect to all these parameters. E.g., for a given production time, a more permeable or strong formation, will be in a more advanced stage, i.e., it will have a larger ratio r

_{e}/R, than a less permeable or weak formation.

_{0}|, where w

_{0}is the subsidence at r = 0 (well location) for each case, is given as function of the scaled radial distance r/R. We observe that when the aquifer is shallow, i.e., D/R << 1, the subsidence distribution is essentially negligible outside the radial extent of the aquifer (Figure 3a,b). One the other hand, for the case of deep aquifer, i.e., D/R near or larger than 1, the effect of the aquifer size on the subsidence distribution is virtually insignificant (Figure 3c,d). Equations (15)–(17) state the asymptotic solutions that illustrate mathematically such observations. In Figure 3d, we also include the solution (17) with crossed circle marks, which shows that the asymptotic solutions are in fact an excellent approximation already for D/R ~ 2. An implication of this fact is that the subsidence radial distribution is insensitive to variations in the permeability of the rock and the viscosity of the fluid.

## 4. Conclusions

- We derive the analytical asymptotic solutions for the radial distribution of subsidence, w(r), for the case of deep aquifer, showing that these solutions amount to an excellent approximation even when the aquifer depth is larger than the aquifer diameter, and that w(r) is independent of the formation permeability and the fluid viscosity;
- Finally, we express and analyze the influence of the physical parameters on the subsidence field through two similarity ratios, namely, the well radius of influence over aquifer radius and aquifer depth over aquifer radius.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Surface subsidence radial distribution: (

**a**) shallow aquifer–strong rock, (

**b**) deep–strong, (

**c**) shallow–weak, (

**d**) deep–weak. The letter “d” is short for days.

**Figure 3.**Scaled subsidence radial distribution as a function of the scaled distance: influence of aquifer depth (D/R) and stage of evolution (r

_{e}/R).

**Table 1.**Input data used in the analyses [9].

Variable | Values | |||

Geometric Properties | ||||

Case 1 | Case 2 | Case 3 | Case 4 | |

Radius of aquifer, R (m) | 3000 | 3000 | 3000 | 3000 |

Aquifer thickness, H (m) | 100 | 100 | 100 | 100 |

Aquifer depth, D (m) | 200 | 1000 | 200 | 1000 |

Aquifer Formation Properties | ||||

Porosity, φ (-) | 0.19 | 0.19 | 0.19 | 0.19 |

Permeability, k (m^{2}) | 1.9 × 10^{−13} | 1.9 × 10^{−13} | 1.9 × 10^{−13} | 1.9 × 10^{−13} |

Poisson ratio, ν (-) | 0.2 | 0.2 | 0.2 | 0.2 |

Biot coefficient, α (-) | 0.8 | 0.8 | 0.8 | 0.8 |

Rock bulk modulus, K (GPa) | 8 | 8 | 0.8 | 0.8 |

Fluid Properties and Pumping Parameters | ||||

Dynamic viscosity, μ (Pa.s) | 0.001 | 0.001 | 0.001 | 0.001 |

Fluid bulk modulus, K (GPa)_{f} | 2.1 | 2.1 | 2.1 | 2.1 |

Withdrawal rate, Q (m^{3}/day) | 100 | 100 | 100 | 100 |

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**MDPI and ACS Style**

Gravanis, E.; Sarris, E.N.
A New Analytical Method for Calculating Subsidence Resulting by Fluid Withdrawal from Disk-Shaped Confined Aquifers. *Water* **2023**, *15*, 3175.
https://doi.org/10.3390/w15183175

**AMA Style**

Gravanis E, Sarris EN.
A New Analytical Method for Calculating Subsidence Resulting by Fluid Withdrawal from Disk-Shaped Confined Aquifers. *Water*. 2023; 15(18):3175.
https://doi.org/10.3390/w15183175

**Chicago/Turabian Style**

Gravanis, Elias, and Ernestos N. Sarris.
2023. "A New Analytical Method for Calculating Subsidence Resulting by Fluid Withdrawal from Disk-Shaped Confined Aquifers" *Water* 15, no. 18: 3175.
https://doi.org/10.3390/w15183175