# Software for Estimation of Stochastic Model Parameters for a Compacting Reservoir

^{*}

## Abstract

**:**

_{m}of reservoir rocks, and the parameter tgβ, which indirectly describes the mechanical properties of the overburden. The calculations were performed on leveling measurements of land subsidence, as well as on the geometry of the compaction layer and pressure changes in aquifers. The estimation of model parameters allows the prediction of surface deformations due to planned fluid extraction. An algorithm with a graphical user interface was implemented in the Scilab environment. The use of SubCom v1.0 is presented using the case of an underground hard coal mine. Water drainage from rock mass accompanying coal extraction resulted in compaction of the aquifer, which in turn led to additional surface subsidence. As a result, a subsidence trough occurred with a maximum subsidence of 0.56 m.

## 1. Introduction

- verify the efficacy of a parameter estimation method based on a stochastic model;
- implement an algorithm in open-source software;
- model subsidence due to dewatering in an underground coal mine.

## 2. Compaction and Subsidence Model

#### 2.1. Principles of a Model Based on the Influence Function

_{0}− p(t)—pore pressure decrease, M

_{0}—primary thickness of layer, A—geometry of reservoir element, R

^{2}= (x − s)

^{2}+ (y − t)

^{2}—distance in the horizontal plane between a reservoir element (x,y) and a calculation point on the surface (s,t) and r—radius of influence, which takes the form of:

_{m}—compaction coefficient, defined by the expression:

#### 2.2. Program Framework

_{i}), thickness (M

_{i}), depth (H

_{i}) and coordinates describing the arbitrary geometry of a single element (${X}_{i}^{j},{Y}_{i}^{j}$).

#### 2.3. Calculations for Testing Data

## 3. Study Area

#### 3.1. Geological Background

^{3}with an average water hardness in the range 0.32–0.78 °n. These are very soft waters of the Cl-HCO

_{3}-Na type, whose characteristics should be associated with the fractures and 11–23% porosity. The filtration coefficient of this layer is 10

^{−8}m/s to 10

^{−5}m/s, with an average of 10

^{−6}m/s in the mine area. The primary pressure acting on the top of the Carboniferous layer amounts to approx. 7.5 MPa [43]. The depression cone was monitored with 44 piezometric boreholes located in the vicinity of the mine (Figure 5). Surface deformations could be observed along with the development of the depression cone (Figure 5).

#### 3.2. Subsidence Due To Fluid Withdrawal

## 4. Results

#### 4.1. Dimension Testing Of Aquifer Elements

#### 4.2. Estimation Results of Compaction Coefficient C_{m} and tgβ

_{m}and tgβ to be determined. The estimation results in profile A-A and the deviation are shown in Figure 7. The compaction coefficient determined by the software for the Jurassic aquifer was C

_{m}= 8.0 × 10

^{−4}± 0.1 MPa

^{−1}at the range expressed by tgβ = 0.50 ± 0.05 at correlation coefficient R

^{2}of 0.77. According to Doornhof [46], the compaction coefficient for sandstone may oscillate between 5.0 × 10

^{−4}MPa

^{−1}and 15.0 × 10

^{−4}MPa

^{−1}, which indicates high reliability of the obtained results. The tgβ value is responsible for the range of surface compaction. For instance, a value of tgβ of 0.10–0.20 was assumed in modeling for the gas field in Groningen, which is located at a depth of over 2000 m [47]. The determined value of tgβ = 0.50 for the Jurassic strata in the coal mine at 500–600 m also seems credible.

## 5. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

^{2}) may be defined.

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**Figure 1.**The idea of a computational model based on the influence function and graphical interpretation of the tgβ parameter. The dark orange arrow visualizations the influence function, which represents the deformation process in rock mass.

**Figure 4.**Determination of optimal parameter values related to different initial conditions. The testing was carried out for modeling data; optimal parameters are represented by a star on the graph. The iterations are represented by colored dots on each track.

**Figure 5.**Location of the hard coal mine and the regional tectonics of the area. Piezometric boreholes and a large dewatering trough in the deep extraction area are shown. Condition as of 2011.

**Figure 6.**Division of aquifer into cuboid reservoir elements whose bases measured 1000 m, 200 m, and 50 m. Owing to the large number of base elements measuring 50 m, a buffer was used for profile A-A. Changes in reservoir element pressure are shown in the color scale.

Type of Data | Symbol | Description | Data visualization |
---|---|---|---|

Profile | Id, X, Y, S | Information about surface subsidence on the profile | |

Aquifer | ${X}_{i}^{j},{Y}_{i}^{j}$ | The geometry of a single aquifer element in the same coordinate system as the profile | |

Hi | Depth | ||

Mi | The thickness of a single aquifer element | ||

Δpi | Pressure decrease in the aquifer, determined for the bottom of the aquifer element |

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**MDPI and ACS Style**

Witkowski, W.T.; Hejmanowski, R. Software for Estimation of Stochastic Model Parameters for a Compacting Reservoir. *Appl. Sci.* **2020**, *10*, 3287.
https://doi.org/10.3390/app10093287

**AMA Style**

Witkowski WT, Hejmanowski R. Software for Estimation of Stochastic Model Parameters for a Compacting Reservoir. *Applied Sciences*. 2020; 10(9):3287.
https://doi.org/10.3390/app10093287

**Chicago/Turabian Style**

Witkowski, Wojciech T., and Ryszard Hejmanowski. 2020. "Software for Estimation of Stochastic Model Parameters for a Compacting Reservoir" *Applied Sciences* 10, no. 9: 3287.
https://doi.org/10.3390/app10093287