Splitting methods play a significant role in the coproduction of tight reservoirs which are characterized by vertical multilayer superimposition. It directly affects the accuracy of reservoir performance analysis and detailed descriptions. However, conventional splitting methods are limited to a few factors and static factors without considering the effect of layer parameter change. In this study, sensitivity analysis was carried out on five factors that affect the production splitting in coproduction wells. The research shows that in the production process, multiple parameters have a direct impact on the production of layers. Different parameters, which have to be included to split production, have different scale effects on layer production. Comparing the results of the KH method with the numerical simulation results, the limitation of the KH method for yield splitting is illustrated. A novel dynamic splitting method for production (DPSM) was proposed. This method is based on two primary methods, which are the multifactor static method for production splitting of gas (GPSM) and water (WPSM) and use the catastrophe theory and material balance equation (MBE) and obtain the final results by iterative method. The advantage of this method is that more accurate results in the production process are obtained by selecting eight factors, which contain 6 static factors and 2 dynamic factors, for research. It is more in line with the production practice that the ultimate results of production splitting vary with the production process. The accuracy and practicality of the results had been verified by numerical simulation. This method has practical significance for production splitting in tight gas reservoirs.

Tight gas reservoirs are the main targets of unconventional development [

The conventional method of production splitting is to obtain geological parameters such as permeability and effective thickness for qualitative research without suspending production. At present, conventional production splitting techniques focus on four types of methods: parameter methods, numerical simulation methods, production profile testing methods, and other improved methods derived from these methods.

The traditional parameter method named KH is mainly to split production by permeability (K) and effective thickness (H). Only two static factors are considered in this method. There is a big deviation in the calculation results. Many scholars have modified this method: the KNK method considers well-controlled geological reserves (N) in layers [_{r} method considering effective permeability (K_{r}) [

Numerical simulation is an important means to understand the reservoir. Through history simulation, the properties and states can be obtained, and then the production of the reservoir can be split in each stage of development. With the development of computer technology, numerical simulation has developed greatly. Prabowo et al. were the first to study the dynamic response of well production and pressure by establishing a numerical simulation model without layer interflow [

The production profile test method uses combining logging tools to record the practical production rates of each section. Then, we can obtain the splitting coefficient of reservoir production by interpreting the recorded data. Generally, this method is the most accurate [

Researchers have proposed many new production splitting methods on the basis of various mathematical theories and basic seepage laws [

The above method that we discussed cannot well combine dynamic and static data in the calculation process. In addition, most of the methods also do not consider the effect of water production. The calculation results are either complicated or cannot be consistent with the practical results of dynamic changes. Therefore, a simple and accurate method is urgently needed in the practical production splitting process.

In this paper, the results of sensitivity analysis show that the production splitting is controlled by multiple factors and it changes dynamically over time. A novel and more accurate dynamic splitting method considering dynamic and static factors was proposed based on catastrophe theory, MBE, and iterative calculation. It is proved that DPSM is more accurate and more practical for tight gas reservoir by comparing with the KH method and original splitting method based on the catastrophe theory.

In the process of production, the production of layers is jointly affected by many factors. The variation of subsurface parameters will also lead to the error of splitting [

In this method, the catastrophe theory, material balance equation, and iterative calculation method are used in the calculation process. Calculation takes into account 2 dynamic factors: gas saturation and effective permeability and 6 static factors: effective thickness, porosity, perforating height, sandstone content, reservoir density, and gas reservoir depth. The rationality and accuracy of this method are guaranteed by the multifactor analysis method and the comprehensive application of various theories. It needs to be highlighted that the iterative calculation produces splitting results at each instantaneous moment, and the results are more in line with the reality of dynamic changes of parameters in practical production.

There are some assumptions that need to be specified in the calculation. It is considered that the layer production is controlled by the same factors during the calculation. The effect of different fluid properties on production is ignored. Most importantly, we do not consider communication between layers, so the influence factors act independently in the layers.

The catastrophe theory assumes that the states of catastrophe events are determined by a number of state variables [

The most commonly used models of the catastrophe theory include fold catastrophe, cusp catastrophe, swallowtail catastrophe, oval umbilicus catastrophe, hyperbolic catastrophe, butterfly catastrophe, and parabolic catastrophe. In this paper, we just use cusp catastrophe (equations (

In this study, we decompose the overall target to construct a multilevel architecture diagram by studying the state variables and control variables of the event. Normalization formula of different models are used to quantify the operation. The final evaluation result of the multilevel architecture diagram is obtained.

The layer production of multilayer commingled producing wells is regarded as controlled by three control variables, which are reserve characteristics, development characteristics, and geological characteristics. These control variables are controlled by several factors as state variables at the same time, which constitute a subsystem. The reserve characteristics are controlled by effective thickness, ,porosity and gas saturation, which constitute the swallowtail catastrophe model. The development characteristics are controlled by two control variables, effective permeability, and perforating height, which constitute the cusp catastrophe model. The geological characteristics are controlled by the sandstone content, reservoir density, and gas reservoir depth, which constitute the swallowtail catastrophe model. Reservoir characteristics, development characteristics, and geological characteristics constitute the swallowtail catastrophe model.

In this study, it is assumed that the influencing factors of gas production and water production of layers are the same. Moreover, we can use the same factors to build similar systems of catastrophe analysis for gas and water. Figure

The schematic of the catastrophe system for gas and water. Final target values are yellow, development feature target values are pink, static variables are blue, and dynamic variables are green.

The catastrophe system of GPSM

The catastrophe system of WPSM

The equilibrium surface is constituted by choosing the most disadvantage of factors. Maximum values of some affecting factors which have a positive effect on layer production are selected, e.g., the thickness, porosity, saturation, and effective permeability in the reservoir. The other factors are negative to the production and are selected as the minimum, for example, perforating height, sandstone content, reservoir density, and gas reservoir depth. So, we can get the equilibrium surface successfully.

The effect factors of the system need to be normalized so as to simplify the complexity of subsequent calculation (equation (

The weight of each factor in the subsystem of reserve characteristics, development characteristics, and geological characteristics can be calculated by the normalization formula. The weighted average value for factor weight in the subsystem is calculated, so that we can get the object value to the whole system. Similarly, the final object value of the system is obtained by the normalization formula and weighted average method, which is the weight of the comprehensive factor in each layer. The calculation formula is shown below (equations (

The production weight of layers by comparing the final object value of layers and the equilibrium surface can be obtained. The production weight of layers is the splitting coefficient which was calculated by the catastrophe theory. The splitting coefficient calculation is shown in equation (

The WPSM and GPSM have the same calculation process due to the similar catastrophe system. We can use the same formula when using WPSM.

The material balance theory is one of the basic theories to study reservoir flow and many scholars have optimized the MBE in many ways [

The relative permeability curves of each core are obtained from experimental data and field well test interpretation results [

Changes in parameters occur throughout the production process. The traditional splitting method of the catastrophe theory only works at the time nodes which are defined by the data point. It will show strong errors at other points in time. In this study, we mainly consider the variation of water saturation and effective permeability with the production process in tight gas production. Figure

Calculation process of DPSM.

In Figure

In order to obtain an accurate response of DPSM through simple calculation and simplify the complexity of numerical simulation, a homogeneous conceptual model is established which is characterized by a double layer. Figure

Concise diagram of the mechanism model.

The effect of reservoir fluid on the production splitting is not discussed in this study. We assume that the fluids of layers have the same properties in this conceptual model. Fluid properties are set out in Table

Key parameters of the fluid properties.

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

Water compression coefficient (MPa^{−1}) | |

Water viscosity (cP) | 0.476 |

Water density (kg/m^{3}) | 1000 |

Gas density (kg/m^{3}) | 0.761 |

The initial parameters of the two reservoirs.

Layer | Layer 1 | Layer 2 |
---|---|---|

The thickness of the gas layer (m) | 10 | 10 |

Gas saturation (%) | 60 | 55 |

Porosity (%) | 12 | 8 |

Perforating height (m) | 10 | 10 |

Effective permeability (mD) | Dynamic calculation result | |

Sandstone content (%) | 86 | 81 |

Reservoir density (g/cm^{3}) | 2.36 | 2.36 |

Middle depth of layer (m) | 2305 | 2319 |

Absolute permeability (mD) | 0.08 | 0.10 |

Gas-water relative permeability curve.

Figure ^{3}/day.

The effect of the wellhead production rate on splitting.

Factors used in DPSM are analyzed by numerical simulation to illustrate the necessity of considering the process of production splitting. The layer production ratio is affected by the fluid flow process, which proves the necessity of studying the dynamic parameters. This study also proves the limitations of the KH method and indicates that a dynamic production splitting method with multifactor analysis is needed.

The effect of permeability of layers on the production ratio of the coproduction well is analyzed by the single-factor control method. We set the upper permeability as a constant value of 0.12 mD and set the lower layers as 0.30 mD, 0.40 mD, and 0.60 mD to achieve the purpose of setting the permeability ratio of two layers as 2, 3, and 4, respectively. Figure

Sensitivity analysis. (a) The effects of permeability, (b) the effect of effective thickness, (c) the effect of porosity, (d) the effect of saturation, and (e) the effect of perforation thickness.

In Figure

Permeability determines the flow capacity of fluid. Through the analysis of permeability, it is found that the permeability of layers is an important factor affecting the ratio of layer production. With the increase of permeability difference, the ratio of production is larger. We also found that when the permeability ratio of the upper and lower layers is constant, the production ratio of the layer still changes. So, we think that there are other factors that affect the production ratio of layers. It is necessary to study other factors which affect the production ratio. It has great significance for the production splitting.

We set the same parameters in the layers of the conceptual model and only change the value of effective thickness to achieve the purpose of studying the sensitivity of effective thickness. We set the upper effective thickness as a fixed value of 2 m and set the lower layer as 0.66 m, 0.50 m, and 0.40 m to achieve the purpose of setting the effective thickness ratios of two layers as 2, 3, and 4. As shown in Figure

The effective thickness represents the size of the production section and the size of the reserves of layers. This directly controls the production ratio near to the ratio of effective thickness. In the production process, the gas near the well is produced before the pressure reaches the boundary. The dominant layer has a large rate of production and needs more replenishment. However, the gas flow in layers is slow and cannot be replenished quickly and the production ratio drops at the beginning. After a period of time, the production ratio increases due to the increased flow provided by the layers. This sensitivity analysis indicates that the effective thickness is also an important factor controlling the layer production ratio. The fluid flow stage also affects the production ratio of the layers. We must consider the effect of the fluid flow and effective thickness when splitting the production.

Through the method of controlling variates, only the porosity of layers in the conceptual model is set to change and the effect of the porosity change on the production splitting is observed. We set the upper porosity as a fixed value of 0.12 and set the lower layers as 0.60, 0.40, and 0.30 to achieve the purpose of setting the porosity ratios of two layers as 2, 3, and 4, respectively. As shown in Figure

These changes are due to the reserves controlled by porosity. After reserves near the wellbore are exhausted, reserves far away from the wellbore have started to be tapped. High-porosity reservoirs provide faster. When the pressure wave reaches the boundary, the production of layers is directly controlled by reserves and the production ratio tends to be the ratio of reserves. This proves that the porosity of the reservoir can have a great effect on the production of layers. The effect of porosity must be considered when splitting production.

Gas saturation in tight gas reservoirs is generally less than 60%. We set the upper gas saturation as a fixed value of 0.60 and set the lower layers as 0.30, 0.20, and 0.15 to achieve the purpose of setting the gas saturation ratios of two layers as 2, 3, and 4, respectively. The effect of saturation on production splitting is analyzed by observing the influence of the initial gas saturation on the production ratio. As shown in Figure

In the early stage of production, due to the high water saturation, the effective permeability of the gas phase is too low to not flow out. After a period of time, layer 2 begins to produce gas due to the water saturation decreases. Because of the continuous decreases of water saturation, the relative permeability of gas in layer 2 gradually increases. The gas production increases, and the production ratio decreases. The large difference between the production ratio indicates that saturation has a great impact on layer production. We can indicate that the saturation is dynamic varying with the development, so it is necessary to consider its effects.

In the early stage of production, due to the high water saturation, the relative permeability of the gas phase is too low to not flow out. After a period of time, layer 2 begins to produce gas due to the water saturation decreases. Because of the continuous decreases of water saturation, the relative permeability of the gas phase in layer 2 gradually increases. The gas production increases, and the production ratio decreases. The large difference between the production ratio indicates that saturation has a great impact on layer production. We can indicate that the saturation is dynamic varying with the development, so it is necessary to consider its effects when splitting production.

The effect of perforation thickness on practical production splitting was studied by setting the difference of perforation thickness. Moreover, in order to ignore the impact of different perforation positions on production, we chose the middle position as far as possible. The upper perforation thickness is set as a fixed value of 10 m, and the lower layers were set as 8 m, 6 m, 4 m, and 2 m to achieve the purpose of setting the gas saturation ratios of two layers as 5/4, 5/3, 5/2, and 5/1, respectively. Figure

In Figure

The above research shows that in the production process, multiple parameters have a direct impact on the production of layers. Different parameters have different scale effects on layer production. In the process of production splitting, multiple factors are needed to be analyzed synthetically. The production splitting method constructed by considering many factors is more reasonable and accurate.

The KH method (equation (

Numerical simulation results show that the ratio of production is not always the same as the ratio of KH in numerical simulation (Figures

Figure

The numerical simulation results of controlling the same KH ratio,

In order to prove the accuracy of DPSM, the results of DPSM, KH method, and numerical simulation are compared. The detailed calculation process and the data are given above, and the calculation is ran by a computer. Figure

Comparison of results between the DPSM, KH method, and numerical simulation.

In Figure

The KH method only calculates the splitting coefficient based on layer permeability and effective thickness, which generates a rough result. Moreover, because the selected parameters are static parameters, they cannot be adapted to the change of the practical yield splitting coefficient in the production process. The error increases as the production process progresses. The original method based on catastrophe theory has a higher accuracy because of having more parameters for comprehensiveness. Similarly, because the selected parameters are static, the accuracy in the initial stage of production is higher and it is opposite in the later stage of production.

The new method proposed in this paper draws on the characteristics of the original splitting method based on the catastrophe theory and uses the MBE to calculate the dynamic parameters of the production process. This method achieves the purpose of calculating the splitting coefficient dynamically. Compared with the KH method and the old splitting method based on the catastrophe theory, the method in this paper inherits the advantages of the other two methods. It increases the accuracy of instantaneous time by taking into account many factors. Through the application of dynamic parameters, the precision in the production process is increased. It is simple and intuitional for the evaluation of production splitting. This method provides a new idea for production splitting, and it will directly affect the accuracy of reservoir performance analysis and fine reservoir description.

For a multilayer coproduction well in a tight gas reservoir, the production ratio of layers is controlled by many factors. The effect of different factors on production is shown in the direction and degree of action. The splitting method that only considers less factors, such as the KH method, is not accurate and cannot meet the practical production needs. In the process of production splitting, the effect of various factors on the layer production ratio should be considered as much as possible

During the production of the reservoir, reservoir parameters will change as it shows that the production of a small layer changes with time. The production splitting results based only on static data, such as the result of the KH method, cannot accurately represent the practical production ratio during the production process. It is necessary to combine dynamic factors with static factors to obtain accurate production splitting results

A novel DPSM method to split the production of multilayer coproduction wells is proposed. Compared with the traditional KH method and static method which is based on the catastrophe theory, the proposed method takes into account both dynamic and static factors. And the calculation result changes dynamically with the production process, which is more in line with the numerical simulation results, and keeps the error within 10%. This method has a certain practical value and can be used for the coproduction of a tight gas reservoir and can improve the accuracy of reservoir performance analysis and detailed descriptions

In the process of production splitting, the combination of the catastrophe theory, material balance equation (MBE), and iterative calculation method can achieve dynamic results while synthesizing multiple factors. It is proved that the catastrophe theory has a great prospect of production splitting. This prepares the ground for production splitting of more factors through catastrophe theory. It shows the feasibility of production splitting by integrating multiple theories and provides a new idea by using multiple theories

Cumulative water production, m^{3}

Gas-bearing area, km^{2}

Average thickness, m

Average porosity, %

Average gas saturation, %

Initial gas saturation, %

Underwater volume ratio, 1.005

Effective permeability, mD

Relative permeability, mD

Absolute permeability, mD.

The data used to support the findings of this study are intersected within the article.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (51774256).