Study on Double Pressure Funnels and Gas-Liquid Two-Way Mass Transfer after Fracturing in Shale Gas Reservoirs

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Introduction
With the rapid development of horizontal well drilling and multistage fracturing technology, unconventional natural gas has been effectively developed on a large scale [1,2]. Sichuan Basin is the main shale gas producing area in China and one of the fastest growing shale gas production areas in the world [3,4]. Since 2010, high-production shale gas resources have been found in Changning-Weiyuan, Zhaotong, Fuling, and other blocks, and three marine shale gas industrial-demonstration areas have been established [5]. In the long-term practice of shale gas development, unusual phenomena such as "the lower the flowback of the gas well, the higher the production after fracturing," and "when the gas well is opened after shut-in for a period, the gas production increases but the liquid production decreases" [6] have been found. Also, the development system such as well shut-in after fracturing and small nozzle gas testing are preliminarily formed. Researchers up to now have carried out many experiments and numerical simulation studies in shale gas development [7][8][9][10][11], but they have not obtained a systematic understanding to explain the unusual phenomena found in engineering practices, nor have they quantitatively given the well shut-in time and drainage system. Shale gas development generally goes through fracturing, well shut-in, and flowback stages before production stage. The essence of internal changes in shale reservoirs during these different stages is the redistribution of pressure and the transfer of liquid and gas. At present, researches on shale gas fracturing, well shut-in, and drainage mainly focus on three aspects: flow mechanism, complex fracture network formed by multistage fracturing of horizontal wells, and simulation and optimization during drainage stages.
Shale gas seepage mechanism covers the range of molecular scale, micro and nanopores, and macroscale [12], and there are migration modes such as slip, adsorption, diffusion, and seepage [13,14]. The purpose of the study on shale gas seepage mechanism is to provide mathematical models for the development of shale gas at various stages of gas fracturing, well shut-in, and drainage. The most important one is modified Darcy's law. For example, the modified apparent permeability can describe the slippage, pressure-sensitive, and diffusion effects of shale gas that deviate from the traditional cognition [15][16][17].
The network structure and area generated after fracturing determine the flow boundary range of gas-liquid twophase flow in well shut-in and drainage. Since it is impossible to directly measure the fracture-network type and SRV area after fracturing, it is important to use fracture propagation model for simulation and calculation. Common fracture propagation simulation includes two-dimensional, quasi three-dimensional, plane three-dimensional, and full threedimensional models [18,19]. The most practical and reliable model for shale gas multistages fracturing is the plane threedimensional model [20,21]. Another method to obtain the fracture morphology and SRV area after fracturing is the fracturing construction-data inversion [22]. By establishing the wellhead-pressure-drop curve chart after the hydraulicfracturing stop-pump, the parameters such as fracture network morphology, fracture half-length, and SRV area permeability can be obtained through curve fitting.
The researches of shale gas well shut-in and drainage optimization are mainly experiments and numerical simulations. Yu et al. [23] applied nonintrusive EDFM (embeddeddiscrete fracture model) to couple fracture and reservoir models, and numerical simulation with and without considering nature fractures is conducted in shale gas development. As shown in complex fracture (considering nature fractures) system, it has large drainage area and highdrainage efficiency, therefore the cumulative gas and water production will be larger than simple fractures (without considering nature fractures) system. Chen et al. [24] used the structured grid near the fracture to simulate the unified flowback of multilayer fracturing; Eltahan et al. [25] conducted 3D-reservoir simulation of complex fractured network that intermeshes with formation matrix and studied well shut-in impact on the productivity and recovery of shale oil reservoirs; Wijaya and Sheng [26] conducted a flow-geomechanical model to study the relationship between imbibition and well shut-in; it shows only if imbibition dominated during recovery in shale reservoir, well shut-in tends to improve both the ultimate oil recovery and net present value (NPV); Tao et al. [27] designed a water spontaneous imbibition apparatus, though clay mineral content measurement and salt ion concentration diffusion experiment, the optimal shut-in time for type I and type II shale reservoir are 20 days and 15 days, respectively. Wu et al. [28] proposed a two-phase flow model considering formation damage caused by clay expansion and gave a semianalytical-solution model based on Laplace transform, but this method is mainly used for production data analysis and prediction of tight gas wells.
The three aspects involved in the study of shale gas fracturing, well shut-in and drainage in the existing literature are independent of each other. The numerical simulation of well shut-in and drainage bases on the assumption of constant initial pressure and use structured grid finite difference. It is obvious that the structured grids are difficult to characterize the complex fracture network of horizontal well multistage fracturing, especially during fracturing, a large amount of fracturing fluid enters the formation, and the pressure near the wellbore is several times of the original formation pressure, so the assumption that the initial pressure keeps constant is unreasonable. At the same time, the existing simulation results can only analyze the impact of some parameters on well shutin and drainage and have not summarized reasonable rules and effective optimization methods from the simulation results.
In this paper, the unstructured PEBI grid is used to replace the structured grid, and the finite volume method is used to discrete shale gas-water two-phase equation to realize the accurate characterization of the fracture. The pressure distribution after fracturing is obtained by using the multidimensional instantaneous source function as the initial pressure condition for well shut-in and drainage simulation, so the gas-liquid two-phase simulation of shale is carried out. According to the simulation results, the understanding of "double pressure drop funnels" and "gas-liquid two-way mass transfer" is proposed, and some phenomena in the process of shale gas development are well explained. Meantime, it also provides a theoretical basis for the optimization of drainage system in shale gas production.

Theory and Optimization Method
Large scale fracturing of shale gas results in a large amount of fracturing fluid entering the formation. During fracturing stage, high pressure near the fracture gradually spreads to the far distance from formation with low formation pressure, and a large reversed pressure difference is formed (defined as reversed pressure funnel). Also during the drainage stage after fracturing, the high-pressure area near the fracture and the low-pressure area near the wellbore also form a pressure funnel due to pressure difference, resulting in the phenomenon of double-pressure funnels, and the liquid follows the seepage law from high to low pressure and migrates to the far distance from formation. On the other hand, the area near the fracture is occupied by large amount of fracturing fluid, so the gas concentration is close to 0, while the far distance from formation is not affected by the fracturing fluid so the gas concentration is close to 100%. Due to the concentration difference, the gas diffuses from the far distance from formation to the vicinity of the 2 Geofluids fracture, causing the phenomenon of gas-liquid two-way mass transfer. The mechanism of double pressure funnels and gas-liquid two-way mass transfer are special phenomena in shale gas development (Figures 1 and 2).

Pressure Distribution Caused by Fracturing Fluid
Injection. There is a vertical-fractured well in the closedrectangular formation, and the well location and boundary are shown in Figure 3. We assume that there is a fracture in the rectangular area, the fluid in the formation is slightly compressible, and the half-length of the fracture is x f . Compared with the pressure buildup test after shale gas production for a long time, the injection time of fracturing fluid during fracturing construction is shorter. Therefore, the fracture liquid is assumed to be injected instantaneously (τ = 0). The pressure distribution of the vertical-fracture well can be derived by using the instantaneous source function [29] and Newman product method [30]. That is, where χ = k/ðϕμC t Þ is pressure coefficient, (m 2 /s); k is formation permeability (μm 2 ); ϕ is the formation porosity; h is the effective thickness of the formation, (m); μ is the fluid viscosity in the formation, (MPa·s); C t is the comprehensive compression coefficient, (MPa -1 ); p i is the original formation pressure, (MPa); x f is the half-length of the fracture, (m); G is the total amount of injected fracturing fluid, (m 3 ); ðx w , y w Þ is the well location, (m); ðx e , y e Þ is the side length of the rectangular boundary, (m).

Mathematical
Model of Gas-Liquid Two-Phase Flow with PEBI Grid. After the multistage fracturing of horizontal wells forms a complex fracture network, the shale is divided into two systems: bedrock and fracture. The shale desorption only occurs in the bedrock when formation pressure is low. The desorbed gas enters the fracture network from the bedrock through diffusion, and the adsorption effect can be ignored. Considering that shale gas is mainly natural gas dominated by methane, the black oil model can be used for gas-liquid two-phase seepage. Meanwhile, the PEBI grid can accurately characterize the internal boundaries of wells and fractures in the multistage fracturing of horizontal wells. The PEBI grid is an unstructured grid [31], each grid is adjacent to multiple grids, and the number of adjacent grids is not fixed. As shown in Figure 4, there are 6 grids numbered 0-5 around the grid i. Due to the orthogonality of PEBI grid, finite volume method is used to discretize the shale gasliquid two-phase process.
Consider any grid i in the PEBI grid shown in Figure 5. Assuming the grid that adjacent to the grid i is j, and ∑j sign represents the summation of all adjacent grids of grid i, the implicit discrete equations of water and gas phase equations are where T ij,l = λ ij,l G ij is the conductivity coefficient (m 3 / Pa.s), and the geometric factor G ij is the ratio of the area ω ij (the adjacent surface of two adjacent grids i and j) multiply permeability K ij to the distance between the center points of these two grids d ij [32]. That is wherein λ ij,l is the ratio of relative permeability to the viscosity and volume coefficient Other variables in Equations (2) and (3) are defined as follows: where Z is the height calculated from a certain datum (m); k rl , μ l , γ l , and B l ðl = w, gÞ are the relative permeability, viscosity (Pa.s), underground fluid gravity, and volume coefficient of the l phase, respectively; f is the solution variables of p, S w , and S g ; ϕ is porosity; C g is the gas compression coefficient, (Pa -1 ); V i is the volume of the i th grid unit, (m 3 ); Δt = t n+1 − t n is the time step length difference

Calculation and Analysis
To demonstrate the utility of the proposed theory in this paper, a shale gas well is taken as an example. The parameters used in the calculation are given in Table 1 for 20 stages fractured-horizontal well in the center of the rectangular reservoir.

Double Pressure Funnels Phenomenon of Flowback after
Fracturing. Large amount of fracturing fluid enters the formation during fracturing, and a high-pressure area is formed near the wellbore. When flowback after fracturing, on the one hand, a pressure funnel is formed between the highpressure area of fracturing fluid near the fracture and the original formation pressure, on the other hand, a funnel is formed near the fracture where the fracturing fluid flows back, which results in double pressure funnels phenomenon. In order to calculate the double pressure funnels, the analytical solution is combined with the PEBI grid numerical simulation. The initial pressure distribution calculated by Equation (1) is used in the PEBI grid numerical simulation. Assume the flowback rate is 400 m 3 /d (the flowback rate of single fracture is 20 m 3 /d), Figure 6 shows the 5-day single fracture double pressure funnels diagram. Figure 7 is the pressure change curve of single fracture along the fracture direction in the formation. There are two kinds of flow phenomena when fracturing fluid flows back. One is flowing from fracture to the wellbore (corresponding to zone 1 in Figure 7), and the other is flowing from fracture to the far distance from formation (corresponding to zone 2 in Figure 7). Zone 1 in Figure 7 is a pressure funnel area near the fracture when the fracturing fluid flows back to the wellbore, and zone 2 is another pressure funnel area formed by the high-pressure area caused by fracturing fluid near the fracture and the original formation pressure. Figure 8 displays the formation pressure distribution of 20 fractures in 5 days after flowback. The red area (63.84 MPa) in Figure 8 is the high-pressure area caused by the injection of fracturing fluid between the two fractures, and the blue area (34.61 MPa, which is lower than the original formation pressure) is the pressure distribution near the fractures. Figure 9 illustrates the 3D pressuredistribution diagram of Figure 8. The double pressure funnels phenomenon can be found near each fracture during fracturing fluid flowback.

Gas-Liquid Two-Way Transfer Phenomenon after
Fracturing. In multiphase system, the motivation of gas satisfies Fick's diffusion law. The velocity v a caused by diffusion [28,33] can be expressed as where D m is gas diffusion coefficient (m 2 /s), and c = m/v is the gas concentration (kg/m 3 ). From the definition of concentration, for the gas diffusion in the formation, the concentration means density of the gas in the formation. According to the equation of state of the real gas, the density of the gas can be expressed as [30,34] According to the definition of gas compression coefficient [30], According to Equations (14) and (15), the velocity of gas migration V a caused by diffusion can be rewritten as where M is the unit molar mass of the gas mixture (kg/ kmol), R = 8:314 is the universal gas constant kg/(kmol.k), z is the deviation factor of real gas, and T, ρ, and c g is the gas temperature (k), density (kg/m 3 ), and compression coefficient (MPa-1), respectively. By comparing the gas seepage equation, the gas migration velocity is the sum of the seepage velocity and the diffusion velocity. In this way, the gas phase conductivity coefficient λ ij,g from Equation (5) under the PEBI grid can be defined

Geofluids
Using the data in Table 1 and unstructured PEBI grid numerical simulation program, the diffusion of gas in fracturing fluid is calculated. Free gas content V g in the for-mation (m 3 /T). Figure 10 shows the change of water-gas ratio near the fracture with well shut-in time under different free gas content of shale.

Geofluids
It can be seen from Figure 10 that during the stop-pump and well shut-in period, the gas content in the water near the fracture is zero when the pump is just stopped. With the increase of the shut-in time, the gas content in the water gradually increases, but the change rate of the gas content in the water decreases with the increase of time. When the shut-in time is long, the gas content in the water becomes a constant. It can also be seen in Figure 10 that the free gas content in the formation also affects the water-gas ratio. When V g is small, the water-gas ratio curve near the fracture becomes constant in a short time. For example, if V g = 1, the water-gas ratio curve approaches constant after 3 days shutin. If V g = 5, the water-gas ratio curve still rising after 15 days shut-in. At the same time, if V g is large, the water-gas ratio near the fracture is large. Figure 11 is the gas and water saturation distribution for 1 day well shut-in. Figure 12 is the gas and water saturation distribution for 5 days well shut-in. The red area represents S g = 1, and the blue area represents S g = 0. It can be seen with time increasing, gas accumulates near the wellbore. It shows that gas can move from high-pressure area to low-pressure area by diffusing.

Theoretical Explanation for Low Flowback of Fracturing
Fluid but High-Gas Production. If large amount of fracturing fluid flows back, it means the formation permeability is good. Because the viscosity of gas is far less than that of liquid, according to the conventional seepage theory, the gas production should be high. However, an unusual phenomenon which is "if less fracturing fluid flows back, the gas production will be high" exists in shale gas development. There have been no reasonable theoretical explanations. The proposed double pressure funnels and gas-liquid two-way transfer theory can well explain this phenomenon.
After fracturing of shale reservoir, a large reconstruction area is formed. After fracturing fluid enters the formation, a pressure imbalance area with high pressure near the wellbore and low pressure at far distance from formation is formed. The fracturing treatment effectively improves the permeability of the formation. The fracturing fluid flows to the far distance from formation through seepage, which is equivalent to increasing the average pressure of the formation. On the other hand, the initial concentration of gas in the fracturing fluid near the wellbore is almost zero, and the high-concentration gas in the fracturing reconstruction affected area enters the vicinity of the fracture through diffusion, resulting in the phenomenon of gas-liquid two-way mass transfer. When flowing back after fracturing, with the production of liquid and gas, the pressure near the wellbore decreases, and the pressure funnel in zone 1 in Figure 13 gradually advances to the far distance from formation. At the same time, the fracturing fluid in zone 2 also continues to advance to the far distance from formation, further improving the average formation pressure. If there is less flowback liquid, the double funnel phenomenon exists for a long time, the formation maintains high pressure for a long time, and the gas production is high. On the contrary, if there is more flowback fluid, the double funnel phenomenon will soon disappear; the formation is difficult to maintain high pressure, so the gas production will be less.

Determination of the Reasonable Shut-In Time.
There are many qualitative descriptions of shale gas shut-in time but few quantitative calculation methods. The theory of double pressure funnels and gas-liquid two-way mass transfer in this paper can quantitatively give the reasonable well shut-in time. During the shut-in period of shale gas well after fracturing, due to the phenomenon of pressure funnel between the wellbore and the formation, the formation pressure redistributes with time. Figure 14 shows the pressure distribution under different well shut-in times calculated by the parameters in Table 1. The pressure near the wellbore

Geofluids
flattens with the increase of time during the shut-in period, and the pressure also diffuses outward. The relative difference between the value of the pressure isoline and the original formation pressure equals to 0.1% is defined as the pressure boundary line. The pressure requirements of the well shut-in are determined by the moving speed of the pressure boundary line.
Free gas will diffuse due to concentration difference. When large amount of fracturing fluid enters near the wellbore, the concentration of gas in the fracturing fluid is zero. If the content of free gas in the formation is high, the gas will diffuse from the low-pressure area of the formation rich in free gas to the fracturing fluid in the high-pressure area due to concentration difference, and at the same time, a part of the gas displaced by water will also diffuse to the fracturing fluid near the wellbore. When the shut-in time is long enough, the fracturing fluid and gas reach a new equilibrium. The time for gas diffusion reaches equilibrium varies with the content of free gas.
The moving speed of the pressure boundary line and the diffusion equilibrium time of free gas are the two conditions for determining the reasonable well shut-in time. Through calculation and combined with the field experience of shale gas development in Zhejiang oilfield of PetroChina and some other fields of Sinopec, the reasonable well shut-in time should meet the following requirements: the moving speed of the pressure boundary line is less than (0.1 m/d). The water-gas ratio change rate with time near the fracture  In order to ensure the reliability of the calculation, a high-precision pressure gauge is installed at the wellhead during fracturing, with the sampling frequency of 1 Hz. The wellhead pressure is converted to the bottom-hole pressure using the wellbore two-phase flow program. The measured bottom-hole pressure must be consistent with the calculated bottom-hole pressure, ensuring the effectiveness of the algorithm.

Relationship between Shale Gas Flowback Rate and Gas
Productivity. Shale gas drainage is water-gas two-phase seepage of complex fracture network [35] (including diffusion). Figure 15 is the gas-water two-phase seepage curve used in the calculation, which is calculated by using the PEBI grid program. The initial pressure distribution in the numerical simulation is calculated by using Formula (1) from the data in Table 1. For 20 days well shut-in time and the free gas content V g = 5 (m 3 /T), according to the gas diffusion equation, the change of gas concentration during the well shut-in after pressure can be calculated. The calculation method provided here is also applicable to dual media. The main flow channel [36] can be used to equivalent the dual media to gas-water two-phase seepage in the fracture network.
Based on the above calculation, as shown in Figure 16, the production curve with time in one year is simulated when the flowback rate is 5%, 10%, and 20%, respectively. The results shown that: (1) at the initial stage of flowback, the maximum daily output of gas is 214000 m 3 , and the maximum daily output is less affected by the flowback rate (2) after that, the daily gas production decreased rapidly, and the influence of different liquid flowback rates on the daily gas production increased significantly; (3) after about 100 days, the daily gas production tends to be stable. When the production reaches one year, the daily gas production corresponding to the fracturing fluid flowback rate of 20%, 10%, and 5% is 47700 m 3 , 5800m 3 , and 72700 m 3 , respectively. The calculation results prove the conclusion that less fracturing fluid flowback will produce more gas.

Summary and Conclusions
In this paper, the instantaneous source function and Newman product method are employed to obtain the formation pressure distribution during multistage fracturing of horizontal wells. It is taken as the initial pressure-distribution condition of gas-liquid two-phase numerical simulation, and the double pressure funnels and gas-liquid two-way mass transfer are investigated in detail. The main conclusions in this work are summarized as follows: (1) Double pressure funnels phenomenon exists during fracturing fluid flowback. After fracturing of shale gas well, a large amount of fracturing fluid accumulates near the fracture, forming a high-pressure area far higher than the original formation pressure, and a funnel is formed between the high-pressure area and the original formation pressure. During production stage, another pressure funnel is formed near the fracture. Eventually, there appear double pressure funnels around each fracture (2) Gas-liquid two-way mass transfer phenomenon exists between free gas and fracturing fluid. The funnel formed by the high-pressure area and the original formation pressure makes the liquid flow into the formation through seepage. At the same time, the gas concentration of fracturing fluid is far lower than that in the formation. Therefore, the gas diffuses from the formation to the vicinity of the fracture through diffusion, forming gas-liquid two-way mass transfer (3) Double pressure funnels and gas-liquid two-way mass transfer explain the production phenomenon of "less fracturing fluid flowback but high-gas production." Two conditions for the reasonable well shut-in time are also given, and they are the moving speed of the pressure boundary line should be less than 0.1 m/d, and the water-gas ratio near the fracture should be less than (1/d) with time (4) It is found from PEBI grid numerical simulation that, when shale gas well is shut-in for 20 days and produce for 1 year, the daily gas production corresponding to fracturing fluid flowback rates of 20%, 10%, and 5% is 47700 m 3 , 5800 m 3 , and 72700 m 3 , respectively. The simulation results prove the production phenomenon of less fracturing fluid flowback but high-gas production (5) The fracturing fluid injected during fracturing leads to the formation of a high-pressure zone near the well, which is equivalent to supplementing the formation with a certain amount of energy, thus increasing the formation pressure during drainage. If the initial production of flowback is too large, the bottom-hole pressure will drop quickly and the production pressure difference will be large, which may lead to risks such as sand production and fracture closure. According to the double pressure funnels study, reasonable production can be determined to optimize shale gas production The advantage of the study is as follows: using PEBI grid to realize exact characterize of fracture, and the analytical solution of pressure distribution after fracturing is used for numerical simulation of well shut-in and drainage process. The double pressure funnels and gas-liquid two-way mass transfer theory is put forward, some phenomena in shale gas development 20% fowback 10% fowback 5% fowback Figure 16: Change of gas production with time under different flowback rates after fracturing. 10 Geofluids are well explained, and the theory basis of reasonable well shutin time and drainage optimization is provided. For further research of the theory and application, the disadvantage and limitation of recent study are as follows: Disadvantage. If free gas content in formation is too low, the accuracy of the method is relatively low.
Limit. For the calculation, rectangle reservoir model is applied, but for other shape models, the method needs to improve.