# Modeling of Gas Flows in Underground Gas Storage Facilities

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## Abstract

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## 1. Introduction

- Balancing gas supply and demand during a given period to compensate for fluctuations associated with the change of seasons (summer/winter periods);
- Gas balancing in the gas pipeline system;
- Optimization of the gas transmission system (GTS);
- Providing consumers with gas under conditions of optimal operation of underground gas storage facilities (UGS); and
- Provision of gas supply in case of failures or malfunctions at UGS production sites or in the gas transmission system.

- Gas self-flowing injection;
- Gas self-flowing withdrawal;
- Injection of gas by compression;
- Gas withdrawal by compression; and
- Gas reduction.

- Geometric, geological, geophysical and accumulating parameters of reservoirs;
- Parameters of ground equipment and piping, as well as instrumentation diagrams of gas collection and preparation;
- Parameters of wells and types of opening of their bottomhole zones; and
- Technological projects of cyclic operation (technological indicators of operation).

^{3}per day [2]. Among the possible options for the operation of underground storage facilities [3], the most efficient rely on integrated indicators, which take into account fuel and energy costs, as well as the parameters of reliability of GTS operation. Optimization of GTS operation encourages the development of optimal strategies for their interaction with UGS, as well as the interaction of UGS groups with each other, in order to maximize their joint energy-saving potential. To achieve this, it is necessary to make full use of the potential of periods of compressorless withdrawal (injection) of gas:

- Subject to the existing or projected mode of operation of the GTS; and
- If necessary, by regulating the process of gas withdrawal from storage facilities by reducing the pressure in the main gas pipeline.

- Companies that deal directly with geophysical services (Schlumberger, Weatherford, Reimer and Integra); and
- Oil and gas companies, including their structure geophysical service units (ExxonMobil, Baker Hughes, Gazpromgeofizika, Surgutneftegeofizika, etc.).

- Companies that are directly involved in the development and sale of software for geophysical services (Paradigm and Rock Flow Dynamics); and
- Large service companies that are involved in software development and creation, among other activities (Schlumberger and CGGVeritas).

- Schlumberger: 30%;
- CGGVeritas, which includes Fugro-Jason and Hampson-Russell Software & Services: 15%;
- Paradigm: 10%; and
- Roxar: 7–10% [36].

- There are no software modules for calculating the modes of multishop compressor stations with different types of GPA, variable flow parts and multistage gas compression; and
- There is no optimal planning of UGS operating modes integrated with the GTS as a single hydraulic complex.

- Models of operation of underground storage facilities in the modes of gas injection and withdrawal; and
- System models of UGS operation in gas injection and withdrawal modes as a part of GTS as a single hydraulic complex.

- Models that take into account the main internal and external factors influencing the parameters of gas flows in the facilities to achieve a given degree of adequacy in terms of gas-dynamic and non-stationary filtration processes;
- A sufficient level of differentiation of the UGS model to avoid solving incorrect mathematical problems at the stage of their integration into a single thermohydraulic system; and
- A sufficient level of complexity of the models to ensure the solution of operational control tasks with maximum speed and sufficient accuracy.

## 2. Modeling and Operation of Underground Storage Facilities

#### 2.1. Object of Study

- A gas-saturated volume that is sufficient to store the required volume of active gas;
- Satisfactory filtration-capacity properties; and
- A gas operation mode.

- The rational ratio of buffer and active gas;
- The number of wells;
- The maximum and minimum pressures in the reservoir; and
- The compressor station capacity.

- Gas mode (without intrusion of reservoir waters into the gas-saturated volume or their minor influence); and
- Elastic water-pressure mode (when reservoir waters enter the gas-saturated volume).

#### 2.2. Operational Problems

#### 2.3. Modeling Problems

- Non-stationarity of the system, i.e., constant change in the parameters of filtration and gas-dynamic processes;
- Significant inertia, i.e., filtration processes continue for several months after the cessation of gas injection and extraction;
- A significant time delay in identifying the situation; for identification, it is necessary to have operational data at a significant time interval, part of which is measured late and not at the same time and with insufficient frequency;
- Multicriteria functioning; the quality of functioning of each object of the technological chain “reservoir—gas pipeline—outlet” is characterized by its individual parameters;
- Uniqueness: the individuality of the structure and operating conditions of the technological equipment (as there are no two identical gas storage facilities);
- Evolvability: object parameters, topology and composition of the object continuously change over time;
- Functional situationality: the purpose of functioning and methods of UGS management depend on the specific situation; and
- Incomplete information and significant uncertainty: a lack of measurements of many variables.

- Reservoir: bottomhole zones of wells; and
- Gas-gathering station: booster compressor station (BCS).

- A stable process of realization of each separate model, as well as a steady process of coordination of mode parameters on model borders;
- Formulation and algorithmic implementation of the minimum complexity of the main direct and inverse optimization problems of planning, identification of model parameters and study of the capacity of facilities and the system as a whole; and
- Analysis of current regimes and evaluation of the effectiveness of pre-design solutions, etc.

## 3. Mathematical Models of Underground Storage Facilities

- Correct formulation of tasks;
- An automated process of UGS model construction with changes of facility models and topology of graph diagrams;
- Development of universal procedures for the implementation of models;
- Analysis and interpretation of modeling results and comparative analysis of UGS operation options; and
- Development of UGS functional support without adjustment of existing (basic) mathematical methods and software, etc.

#### 3.1. Mathematical Model of UGS Structure

- Formation of a unified system of classification and coding of facilities of gas transmission and storage;
- System connection of concepts, classes of objects and relations between objects of the subject area of gas storage;
- End-to-end addressing of all facilities of the system, i.e., the code of the same object coincides across all diagrams;
- Object orientation of all developed piping and instrumentation diagrams; and
- Creation of a graphical database (DB) of integrated object-oriented information structures, which are then used in computer and analytical complexes in order to support the adoption of control decisions at all levels of UGS management within the GTS of Ukraine.

#### 3.2. Mathematical Model of Gas Filtration in Reservoirs

- Geophysical information obtained during the operation of the UGS reservoir at the stage of the gas field;
- Structural maps (Figure 6);
- Logging maps of wells;
- Available UGS operational information;
- Project information for the creation of UGS;
- Technical and technological information of UGS equipment manufacturers; and
- Detailed piping and instrumentation diagrams, etc.

_{i}are placed.

_{i}) of region Ω satisfies the boundary conditions:

- The Dirichlet condition on Γ
_{i}:

- The Neumann condition on Γ:

- ${\nu}_{x}=\mathrm{cos}\left(\nu ,x\right),{\nu}_{y}=\mathrm{cos}\left(\nu ,y\right)$—vector components external normal ν to the area $\Omega \subset {R}^{2}$;
- $k\left(x,y,p\right)$—permeability coefficient, m
^{2}; - $m\left(x,y\right)$—porosity coefficient;
- $h\left(x,y\right)$—the thickness of the gas-saturated layer, m;
- $q\left(t\right)$—source function, m
^{3}/s; - z—gas compressibility coefficient;
- μ—coefficient of dynamic viscosity of the gas, Pa·s; and
- p
_{0}—gas pressure under atmospheric conditions, Pa.

- ${q}_{i}$—gas withdrawal from the i-th well, m
^{3}/s; - $\delta \left(x\right)$—Dirac delta function;
- $\eta \left(t-{t}_{ji}\right)$—single Heaviside function; and
- V—the volume of the gas-saturated reservoir, m
^{3}.

#### 3.3. Gas Gathering System

- υ—gas flow rate, m/s;
- D—inner diameter of the pipeline, m;
- h—elevation of pipeline, m;
- λ—coefficient of hydraulic resistance;
- $g$—freefall acceleration, m/s
^{2}; - $x$—current coordinate $x\in \left[0,l\right]$, m,
- l—the length of the pipeline, m; and
- ρ—gas density under operating conditions, kg/m
^{3}.

#### 3.3.1. Modeling the Process of Gas Movement in Well Gas Pipelines and Working Columns

_{mp}) and the pressure at the bottomhole zone (p

_{pw}), the formula obtained by integrating Equation (6) is used:

- ${q}_{at}$—the flow rate of the well under standard conditions, m
^{3}; - ${\rho}_{at}$—gas density under standard conditions, kg/m
^{3}; - $L$—the height of the working column, m;
- $R$—gas constant, Dj/kgK;
- $T$—gas temperature in degrees Kelvin; and
- ${F}_{csc}$—the cross-sectional area of the working column, m
^{2}.

#### 3.3.2. Model of Gas Inflow to the Bottomhole Zone of the Well

#### Stationary Mode of Gas Inflow to the Well

- ${p}_{0},{q}_{0},{\rho}_{0}$—values of pressure, well flow rate and gas density, respectively, under normal conditions; and
- $F$—filtration surface area.

_{r}) and bottomhole (P

_{b}) pressures are related by the following relation [57,58]:

- ${p}_{0}$ i ${T}_{0}$—pressure and temperature under normal conditions; ${p}_{0}$ = 1013⋅10
^{5}Πa; ${T}_{0}$ = 273 K; - ${T}_{r}$—the average temperature of the gas in the reservoir, K;
- ${r}_{w}$—the radius of the well along the bit, m; and
- ${R}_{k}$—the radius of the drainage zone, m.

- ${R}_{wz}$—radius of the bottomhole zone, m;
- ${h}_{x}$—the height of the part of the casing that is perforated;
- ${r}_{k1}{l}_{k1}{n}_{01}$ and ${r}_{k2}{l}_{k2}{n}_{02}$—perforation channels, where ${r}_{ki}\mathrm{and}{l}_{ki}$ are the radii and lengths of perforation channels, respectively; and
- ${n}_{0i}$—perforation density ($i=1,2$).

#### Non-Stationary Mode of Gas Inflow to the Well

- 1.
- At the initial moment of time, $t=0$, $p={p}_{0}=const$, and at the boundary of the region:

- ${p}_{0}$—initial pressure;
- ${p}_{a}$—atmospheric pressure;
- ${R}_{aw}$—the area of gas inflow to the well; and
- $\frac{\partial p}{\partial r}=0$—the condition of impermeability of the outer region of the gas flow to the well.

- 2.
- The radius of the outer circle (${S}_{0}$) is $a$, and the radius of the concentric inner circle ($\tilde{s}$) is $b$. Boundary conditions are naturally set as follows:
- The initial pressure distribution (${P}_{0}$) is constant;
- At the external border (${S}_{0}$) condition, $\partial P/\partial r=0$; and
- At the internal border, $P={P}_{2}\equiv const$.

#### Determination of Gas Temperature at the Wellhead

- $\Pi $—the average geothermal gradient in the area from the bottomhole to the wellhead, °C/m;

- ${A}_{p}$—thermal efficiency (${A}_{p}=1/427$);
- ${D}_{dr}$—throttle effect;
- ${\lambda}_{p}$—thermal conductivity of the rock;
- ${T}_{fp}$—reservoir temperature;
- ${T}_{s}$—soil temperature; and
- ${h}_{n}$—the depth of the constant temperature zone, m.

- $p-\left[\mathrm{MPa}\right]$, $G+53.9{\rho}_{s}q$—weight consumption of gas, kgF/h;
- $\tau $—well operation time, hours;
- $h$—effective reservoir capacity, m;
- ${c}_{p}$—specific heat capacity of gas;
- ${c}_{n}$—volume thermal conductivity of rocks; and
- ${\rho}_{s}$—average gas density.

#### Balance Models of Reservoirs

#### 3.4. Model of the Gas Pumping Unit

- $n$—revolutions of the centrifugal supercharger (CSC), 1/s;
- $q$—gas consumption through CSC, m
^{3}/s; - ${\eta}_{pol}$—polytropic efficiency of CSC;
- ${q}_{p}^{n}$—nominal fuel gas consumption, m
^{3}/s; - $\epsilon $—degree of compression;
- ${N}_{e}^{n}$—rated power of the gas turbine unit (GTU), W;
- ${K}_{Ne}$—coefficient of technical condition of gas turbine units (GTUs);
- ${K}_{t}$—coefficient that takes into account the influence of atmospheric air temperature;
- ${t}_{0}$—air temperature at the entrance to the gas turbine, °C;
- ${t}_{0}^{n}$—nominal air temperature at the entrance to the gas turbine, °C;
- ${p}_{a}$—absolute atmospheric pressure depending on altitude H, ata;
- ${N}_{i}$—internal power of CSC, W;
- ${Q}_{n}$—nominal lower specific volume heat of combustion of fuel;
- ${\eta}_{e}^{n}$—nominal efficiency of GTU;
- ${\eta}_{m}$—mechanical efficiency;
- ${K}_{N}$—technical condition according to capacity;
- ${z}_{pr}$, ${R}_{pr}$, ${T}_{pr}$—gas parameters at which the characteristics of the supercharger are experimentally determined;
- ${\gamma}_{c}$—specific weight of gas under standard conditions ($P$ = 0.1033 MPa; $T$ = 293 K), kg/m
^{3}; - ${n}_{n}$—rated speed of the supercharger, 1/s; and
- ${\phi}_{k},k=1\xf73$—empirically established functions.

- Deviation of the gas parameters at the inlet to the supercharger $\left({z}_{in},R,{T}_{in}\right)$ from their consolidated values;
- Deviation of the actual speed of the supercharger ($n$) from its nominal value (${n}_{n}$).

- The position of the operating points on the characteristics of CSC to ensure the pump-free operation of the GPU;
- Minimum and maximum volumetric capacity of CSC;
- Speed of rotation of the shaft CSC (${n}_{min}\le n\le {n}_{max}$);
- Maximum capacity of the GTU of GPU;
- The maximum outlet pressure of CSC, which is determined by the strength of the pipelines at the outlet of the CSC;
- The maximum temperature at the outlet of CSC, which is determined by the insulating coating of the pipelines;
- The minimum value of the pressure at the outlet of each CSC;
- Conditions associated with the specified level of stability of the GPU (distance from the surge zone); and
- Conditions of coordination of the scheme of connection of CSC with the inlet and outlet pipelines and main gas pipelines.

- Area of regulation by CSC revolutions;
- The maximum permissible limits for the performance of aggregates; and
- Limits on the volume performance of centrifugal superchargers.

- ${q}_{{k}_{ij}}$—the flow rate in ${k}_{ij}$ supercharger of the $j$-th shop of the $i$-th group;
- ${Q}_{i}$—the productivity of the $i$- th group;
- $\mathrm{\Delta}{P}_{j{s}_{j}}$—pressure increase in the ${s}_{j}$-th stage of the $j$-th workshop;
- $\mathrm{\Delta}{P}_{i}$—pressure increase in the $i$-th group of ${m}_{i}$ consecutively working shops; and
- ${k}_{ij}$—the number of degrees of gas compression in the $j$-th workshop of the $i$-th group.

#### 3.5. Integrated UGS Model

- Piping and instrumentation diagram of GTS + UGS;
- Piping and instrumentation diagram of the system “bottomhole—wells—gas collection system”;
- Piping and instrumentation diagram of the compressor station;
- Structural diagram of the reservoir;
- Geological structure of the reservoir;
- Technological indicators of underground storage operation;
- Technical characteristics of UGS facilities;
- Reports of geological and technological operation of underground storage facilities (operational indicators and gas-dynamic studies of wells);
- Base of models of gas flows in facilities; and
- Database of measured data.

- (1)
- Reservoir model;
- (2)
- Model of the technological chain, which consists of a model of gas flows of bottomhole zones of wells, a model of a well and a model of a gas gathering system; and
- (3)
- Technological chain “gas gathering station—main gas pipeline”, with a compressor station as the main object of the chain.

- Mass flow balance:

- Heat balance:

- Bottomhole zone $\left({i}_{\Gamma i},{i}_{zi}\right)$
- Well $\left({i}_{zi},{i}_{si}\right)$
- Well casing $\left({i}_{si},{i}_{0i}\right)$; and
- Well pipeline $\left({i}_{0i},{i}_{shi}\right)$.

## 4. Discussion

- The need to provide access to the adaptation parameters of each facility separately;
- The requirement to identify and analyze the work of individual technologically related facilities;
- The need to solve all the necessary direct and reverse tasks in the operational dispatching management of processes at the UGS within a satisfactory time frame; and
- The requirement for the development of mathematical methods with minimal intervention in the existing mathematical methods.

## 5. Summary and Conclusions

- Hydraulic models of underground storage facilities are often built on the basis of analysis of the results of petrophysical and geological modeling. This approach does not guarantee the required accuracy of the properties of the hydrodynamic model, as the geological information is fragmentary and heterogeneous in detail, reliability and completeness, with an unknown degree of adequacy. The correctness of the geological model is often clarified in the process of hydrodynamic modeling.
- The main source of information support for the process of building models of UGS facilities is the available industrial operational information. In many cases, it cannot be unambiguously interpreted due to the impossibility of a separate study of many factors influencing their behavior. Thus, the volume of the pore space of the reservoir depends on its geometric dimensions, porosity distribution and, in part, on the distribution of gas saturation and permeability. The inaccuracy of estimating one parameter is transferred to the inaccuracy of estimating other parameters.
- To construct hydraulic equivalents, multiple physical explanations are often available for the same array of available observed data.
- The input data for hydrodynamic modeling contain random and systematic errors that do not allow for a correct physical explanation; therefore, it is often impossible to identify them.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

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**Figure 1.**Schematic representation of the piping and instrumentation diagram of underground gas storage.

**Figure 5.**Underground gas storage facilities on a fragment of the piping and instrumentation diagram of the GTS of Ukraine.

Topics | Source Position in References |
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The results of a comparative analysis of theoretical studies and experimental data for varying permeabiliies of porous media are presented | [6] |

Studies of fluid filtration processes in reservoir systems with uniform permeability are presented | [7] |

The Cauchy problem with limited and continuous initial data is considered for the analysis of the behavior in a long time interval of processes in a heterogeneous porous medium with a singular critical density | [8] |

Models and methods of analysis of fluid filtration processes in porous media are proposed | [9] |

Real examples of hydrodynamic systems and methods of analysis of ideal and real fluid flows are considered | [10] |

Well stimulation using reservoir engineering concepts, as well as topics such as reservoir characterization, hydraulic fracturing, matrix acidification and chemical treatment; reservoir damage, which refers to the loss of reservoir productivity, was also comprehensively investigated | [11] |

Porous media, considering Darcy’s law and hydrodynamic equations of fluid flow, as well as methods of potential theory and two- and three-dimensional problems of filtration in media with uneven permeability | [12] |

The basics of flow-through porous materials, including the structure and properties of porous materials; statics of liquids in porous media; physical and mechanical theory of flow; stable laminar flow of homogeneous liquids; transitional laminar flow of homogeneous liquids; simultaneous flow of immiscible liquids; problems of moving boundaries, movement and deposition of solid bodies; simultaneous laminar flow of mixing liquids; and theories of models with phase changes | [13] |

Phenomena generated by filtration flows of various natures in porous media, as well as methods for their analysis | [14] |

Basic equations of the mechanics and thermophysics of multiphase media of various structures, as well as methods of describing interphase interaction in dispersed media | [15] |

Models and methods of thermohydraulic analysis of gas flows in pipeline systems and porous media | [16] |

The problem of modeling fluid flow and heat transfer in geological fractured reservoirs, as well as the formulation of the physics of their hydraulic behavior is given; the reliability of the study is verified by simulators that combine cracks as one-dimensional elements embedded in the rock matrix | [17] |

Petrophysics of productive zones is used for the study of wells using a set of well-logging data, together with the analysis of their cores; such results are critical for effective field development and effective management of oil and gas fields | [18] |

The problems asscociated with interpreting the results of seismic exploration for the development of an underground gas storage facility | [19] |

The methods of researching wells in heterogeneous porous media | [20] |

Features of testing wells in low-permeability reservoirs; the hypothesis of nonlinear filtering, which is present in low permeability reservoirs, is accepted; in such cases, the behavior of the debit in the case of filtering does not obey Darcy’s law; the result of this work is a tool for reliable prediction of the productivity of wells and a technique for interpreting the results of well tests under nonlinear filtering conditions | [21] |

Assessment of the effectiveness of various methods of increasing the productivity of underground gas storage wells, as well as comparison of the productivity of wells of increased diameter in the interval of the productive formation | [22] |

Methods for researching the filtration parameters of the near-outbreak zone of reservoirs to obtain data of gas-dynamic studies in stationary and non-stationary modes of use of wells | [23] |

Well models for many numerical methods, in particular, the standard finite element method, the control volume finite element method and the mixed finite element method | [24] |

The theoretical foundations of mathematical modeling of reservoir systems are outlined, and analytical and numerical methods of solving filtration equations using computers are described; recommendations for the construction of mathematical and computer models, as well as their analysis and examples of software, are given | [25] |

An iterative method is proposed for evaluating the operational reliability of the UGS in a depleted reservoir under various scenarios of hydrocarbon injection and extraction | [26] |

A multiobjective optimization model was built to identify an optimal operation scheme for a gas transmission network; this model balances two conflicting optimization goals: maximizing the given rate of gas supply to nodes and minimizing the cost of electricity consumption by the compressor station | [27] |

Stellated pipeline networks, cascade dendritic pipeline networks and insertion dendritic pipeline networks, which are three common connection structures of connecting pipelines; a versatile mixed-integer linear programming model is establishes that considers terrain and obstacle conditions, with the aim of minimizing the total investment | [28] |

A model for the analysis of the risks of breaching the integrity of wells is proposed to ensure the safety of UGS operation | [29] |

Hydrogeochemical modeling to identify the potential risks of underground hydrogen storage in depleted gas fields | [30] |

Intelligent wells equipped with an interval control valve (ICV) are considered, which are used to significantly improve the production of hydrocarbons in oil and gas fields | [31] |

Smart well technology, which allows for determination and control the flow of oil or gas that a well can produce, taking into account the geometry and potential of the formation, as well as criteria related to the performance curves of the oil and gas well; a significant improvement in oil or gas production and produced water control is achieved by applying the developed integrated optimization approach, whereby all parameters of the interval control valves are optimized together during the operating process | [32] |

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## Share and Cite

**MDPI and ACS Style**

Iwaszczuk, N.; Prytula, M.; Prytula, N.; Pyanylo, Y.; Iwaszczuk, A.
Modeling of Gas Flows in Underground Gas Storage Facilities. *Energies* **2022**, *15*, 7216.
https://doi.org/10.3390/en15197216

**AMA Style**

Iwaszczuk N, Prytula M, Prytula N, Pyanylo Y, Iwaszczuk A.
Modeling of Gas Flows in Underground Gas Storage Facilities. *Energies*. 2022; 15(19):7216.
https://doi.org/10.3390/en15197216

**Chicago/Turabian Style**

Iwaszczuk, Natalia, Myroslav Prytula, Nazar Prytula, Yaroslav Pyanylo, and Aleksander Iwaszczuk.
2022. "Modeling of Gas Flows in Underground Gas Storage Facilities" *Energies* 15, no. 19: 7216.
https://doi.org/10.3390/en15197216