# Research on Productivity Evaluation Method Based on Shale Gas Well Early Flow Stage Data

Authors: Yize Huang; Xizhe Li; Wei Guo; Xiaohua Liu; Wei Lin; Chao Qian; Mengfei Zhou; Yue Cui; Xiaomin Shi
Journal: Geofluids
 (2023)

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## Abstract

Since the accurate early gas well production regime is related to the production period and final productivity, it is crucial to make better use of early flow stage production data for gas well productivity evaluation. Absolute open flow potential (AOF) and estimated ultimate recovery (EUR) are essential parameters for evaluating the productivity of shale gas wells. This study establishes new AOF calculation methods for the early production stage. The analytical model can calculate the AOF only by using stable pressure and production data in the flowback stage, which greatly improves the efficiency of productivity evaluation. Three methods have, respectively, calculated the productivity of the shale gas wells above 3500 m in the Luzhou block. The results show that well L2-3 has the highest AOF, averaging278.1×104m3/d, whereas well Y2-8 has the lowest AOF, averaging 100.2×104m3/d. Different AOF calculation methods are identified for gas wells in different stages of production. For gas wells in the initial unstable flow stage, a pseudogas production index method is recommended. A water production index analysis method, with lower evaluation results, is proposed for gas wells in the flowback stage. A modern production decline analysis method is found to be preferred for calculating the EUR of deep shale gas wells. Well L2-3 has the highest average EUR of 1.26×108m3, whereas well Y2-8 has the lowest average EUR of 0.42×108m3. The Blasingame method is recommended for medium-to-high-production gas wells, whereas a normalized pressure integral method is suggested for low-production wells. A strong exponential quantitative link between the AOF and the EUR shows that a fracture system’s initial productivity has a significant impact on a well’s EUR. The findings of this study enrich the productivity evaluation system, increase the accuracy of productivity evaluation results, and provide theoretical support for deep shale gas wells.


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## Body

## 1. Introduction

Shale gas is an unconventional energy source with high development potential [1–3]. However, the reservoir of natural fractures and the cross-scale flow mechanism created by artificial fracturing have led to the production features of high peak production and decline rate. Shale has the characteristics of ultralow porosity and permeability [4]. Geological and engineering factors have caused shale gas wells to be in an unstable flow stage for a long time, making productivity evaluation particularly difficult. Productivity evaluation is the most important approach for monitoring well dynamics, predicting well productivity, calculating reservoir parameters, and evaluating fracturing effects in the development process of gas wells [5]. Establishing a suitable controlled pressure and production system at an early stage of development becomes especially valuable in order to obtain high gas production with lengthy production periods under such complex geological conditions. The AOF and EUR in the early stage play an essential role in the production regime [6].The productivity of a gas well in conventional gas reservoirs is typically evaluated using the absolute open flow potential (AOF) as an evaluation index, which is the gas production when the flow pressure at the bottom of the well is zero. It can be obtained by analyzing test data using the productivity well test method [7]. The adsorbed and free gas within the shale matrix are not taken into account by the AOF in the test gas stage, which only accounts for productivity within the reservoir-reformed zone. Therefore, the estimated ultimate recovery (EUR) has become another mainstream evaluation index.Currently, the methods for evaluating shale gas well productivity are classified as follows: production decline analysis method, analytical method, numerical simulation method, and machine learning method. Many scholars proposed various empirical formulas and derivative methods [8–11] for analyzing the production decline of a single well under constant pressure production conditions, such as Arps, PLE, SEPD, and YM-SEPD [12–18]. The production decline analysis method has been expanded by introducing the material balance pseudotime and normalized pseudopressure functions [19–25], which have been widely used in the productivity evaluation of tight gas reservoirs such as shale gas, as the analysis object has been extended from a single gas well production analysis to a two-parameter analysis of production and pressure [26–31]. However, the limitations of applying the production decline method for shale gas flow stages have also caused some uncertainties in the productivity evaluation results [32–34].As research scholars continue to study shale gas transport laws in the subsurface, the analytical method of productivity evaluation can better describe the flow mechanisms in different stages of shale gas development [35–39]. By using the mass transfer control equation, which takes into account the existence of distinct flow mechanisms like the Knudsen diffusion [40–45], molecular surface diffusion, adsorption-desorption, the Klinkenberg effect, the high-speed non-Darcy effect, Darcy flow, and the effective stress sensitivity of shale gas in the reservoirs [46–55], an analytical model describing the cross-scale transport of molecules is created. However, due to the many idealized assumptions in the analytical model, the analytical method cannot accurately reflect the complex flow mechanisms and development laws of shale gas in the actual formation [56–58]. With the continuous integration of the artificial intelligence with oil and gas field production sites, machine learning methods are gradually emphasized in the productivity evaluation of shale gas wells [59–62].However, shale gas has distinct geological properties, complex flow mechanisms, advanced reservoir reforming technologies, and preferred gas well production methods; the productivity index is affected by both engineering and geological considerations [63–65]. Thus, accurate results are not easily obtained [66–68]. Therefore, establishing a new and accurate productivity evaluation method for deep shale gas wells is the focus of the current research in shale gas reservoir engineering. Conventional productivity test methods mainly include back pressure tests (BPTs), isochronal tests, and modified isochronal tests. A productivity equation is fitted to stable pressure and production data obtained from multiple production systems, and the AOF is calculated by combining the productivity curve. On the one hand, verifying the results is difficult because the AOF is the ultimate production of gas wells [69, 70]. Although the production data used in these methods is readily available, a period of decline in gas well production is required before the EUR evaluation, and there is a time lag. The complex flow mechanism of shale gas leads to underfitting, and multiple fitting parameters yield multiple solutions [71, 72]. An analytical method requires detailed geological parameters of the gas reservoir, a transfer equation can describe the cross-scale flow mechanism of shale gas, and analytical solutions of production can be obtained using the Laplace transform and Green’s function [73–75]. However, a model for an analytical method is established with many idealized assumptions, which cannot accurately reflect the actual flow of shale gas in a subsurface reservoir. These errors can significantly affect the accuracy of productivity evaluation [76, 77]. Briefly, there are still many challenges in the current research on shale gas well productivity evaluation.In summary, owing to geological and engineering factors, shale gas wells have been in the early unstable flow and flowback stages for a long time [78]. In this study, a pseudogas production index method and a water production index analysis method are constructed in order to produce more accurate results for gas well productivity evaluation utilizing the production data in these stages and to decrease their delay. The core principles of production decline analysis (PDA) are defined concurrently in order to enhance the shale gas well productivity evaluation system and get reliable productivity forecast indices. The calculation results and application prospects of each method are examined using three shale gas wells from the Luzhou block of the Sichuan Basin of China as examples. The research results are expected to provide theoretical support for methods for evaluating shale gas well productivity as well as scientific guidance for the formulation of development technology policies.


## 2. Theory of Productivity Evaluation Method of Deep Shale Gas Well

### 2.1. Empirical Method

The traditional AOF calculation method makes reference to test well analysis of production data utilizing a BPT method. Exponential, binomial, and one-point procedures make up the majority of BPT techniques [7].The exponential equation is an empirical relationship between the gas well production and the constant flow pressure proposed by Rawlins in 1936 [6], which is expressed in
(1)qg=CpR2−pwf2n.Logarithms of the two sides of Equation (1) are taken to yield
(2)lgqg=lgC+nlgpR2−pwf2,(3)lgpR2−pwf2=1nlgqg‐1nlgC.From Equation (3), it can be interpreted that pR2−pwf2n and qg are linearly related on a bilogarithmic plot with a slope of 1/n and an intercept of −1/nlgC. The line represents the exponential productivity curve.Using a BPT to obtain the constant production and pressure, an exponential capacity curve can be easily drawn, the slope of which provides the value ofn; C can be obtained from
(4)C=qgpR2−pwf2n.Atpwf=0, substituting the initial reservoir pressure, pR, into Equation (5) yields qAOF of the gas well.
(5)qAOF=CpR2n,where qg is the gas production (104 m3/d), pR is the initial reservoir pressure (MPa), pwf is the flow pressure at the bottom of the well (MPa), C is the productivity equation coefficient (104 m3/d/MPa2n), and n is the seepage index.


### 2.2. Pseudogas Production Index Method

Production data from deep shale gas wells are characterized by early and rapidly declining and long-term unstable flow stages, making it difficult to reach the quasistable flow stage [77, 78]. When a well is in the unstable flow stage, its pseudogas production index versus material balance time bilogarithmic plot is a straight line with a slope of 0 to −1/2 [69, 70]. The production data of wells in the unstable flow stage are fit to a straight-line section to obtain the early pseudogas production index. The main benefit of the pseudogas production index approach over the traditional method is that it broadens the application conditions by calculating AOF without requiring production to reach the quasistable flow stage and providing future prediction findings. The pseudogas production index is defined in
(6)qgΔmp=qgmpi−mpwf,(7)mpk=2∫0ppkμugkZkdp.According to Sun et al., the early-stage pressure meets the conditions for using thep2 method, assuming that μZ is a constant. Therefore, p2 is used to substitute mp [79].The material balance time is defined in(8)tm=Npqg.The pseudogas production index for the first day is obtained by fitting a straight-line section to the early unstable flow stage data, and the results are substituted into Equation (9) to calculate the AOF.
(9)qAOF=qgΔmptmbearlympi,where k=i or wf, mpk is the pseudopressure when the pressure is pk (MPa2/(mPa s)), μgk is the gas viscosity when the pressure is pk (mPa s), Zk is the gas deviation factor when the reservoir pressure is pk, Np is the cumulative production (m3), and tm is the material balance time (t).The lack of test production from shale gas wells in a block makes early productivity evaluation difficult [3]. Therefore, this study establishes a pseudogas production index method to calculate the AOF based on the analysis of production data in the early unstable flow stage. However, because this method requires gas well production data, it is applicable to deep shale gas wells in the early unstable flow stage.


### 2.3. Flowback Water Production Index of Analytical Method

Fracturing fluid flowback is performed after the completion of gas well fracturing. A shale gas well’s hydraulic fracture flowback model is shown in Figure1. After hydraulic fracturing, the well is saturated, and the fracturing fluid then flows from the microfracture into the main fracture and finally into the wellbore to the surface. h is the fracture height, w is the fracture width, and n is the number of fractures. The flowback rates of shale gas wells worldwide are low, only 10%–40%, among which those of deeper shale gas wells are relatively higher [79, 80]. For a nonflowback fracturing fluid, the following are generally accepted:
(1)
A fracturing fluid enters the micropores in the matrix because of imbibition(2)
A fracturing fluid is retained in fractures that rapidly close within a short periodFigure 1
Hydraulic fracture flowback model of shale gas well.In this study, water is considered to constitute a large proportion of a fracturing fluid and used to represent it. The variations in the pressure, water production, and gas production with time are shown in Figure2. In the early stage (Figure 2(a)), the mobile fluid in the fracture system is single-phase water, gas production is not seen, the pressure progressively changes, and the cumulative water production rises, signifying an unstable flow stage. In the middle stage (Figure 2(b)), the gas production in the fracture system gradually increases, and both the pressure and water production reach a peak. The water in one part of the fracture is still in the unstable flow stage, whereas the water in the other part of the fracture enters the pseudostable flow stage. The gas starts to break through into the fracture, and this stage belongs to the transition flow stage.Figure 2
Flowback and production characteristic curves of shale gas well.When pressure reaches the fracture barrier in the latter stage (Figure2(c)), the entire fracture system appears to be in a two-phase flow. The water in the matrix is difficult to flowback because of the imbibition effect, and at this point, if there is no energy supplement from the reservoir, the flow is anticipated to enter the suggested stable flow stage [81, 82].Based on the flowback characteristics of the Luzhou deep shale gas wells, the flowback water production index of analytical method is based on the data of stage a in Figure2. Stage a is in the early stage of gas well flowback. After a period of well soak, a large amount of fracturing fluid and movable water is discharged from the fracture. At this stage, it is assumed that no gas is produced. The water in the matrix-fracture system is assumed to behave as a bilinear flow. The analytical model for the flowback stage is established as expressed in Equation (10), and the boundary conditions within the model correspond to constant production. The model assumptions and the detailed derivation process are presented in Appendix A.
(10)∂2p∂x2=φμct0.0853k∂p∂t,px,0=pi,∂p∂xx⟶0=qwBμ0.0853khwn,p∞,t=pi.Solving Equation (10) yields the following relationship between water production and pressure:
(11)qw=nhwpi−pwf2Bw0.0853πkφcttμw.Using the same approach and assuming that the fluid in the matrix-fracture system during the production stage is a single-phase gas, an analytical bilinear flow model for gas production during the production stage is expressed in(12)∂2mp∂x2=φμctk∂mp∂t,mx,0=pi,∂m∂xx⟶0=qgμpscT0.0853khwnTsc,m∞,t=pi.Solving Equation (12) yields the relationship between gas production and pseudopressure as follows:
(13)qg=nhwmpi‐mpwfTsc2pscTBg0.0853πkφcttug.The gas well AOF is provided by(14)qAOF=nhwTscmpi2pscTBg0.0853πkφcttμg.By combining Equations (11) and (14), the formula for calculating the AOF following simplification is expressed in
(15)qAOFqw=TscBwmpiμwpscTBgpi‐pwfμg.The AOF is calculated using Equation (16) by obtaining the relevant parameters and the values of flowback rate and pressure based on the gas well data.
(16)qgAOF=qwpi−pwfTscBwmpiμwpscTBgμg.In practical applications, the data related to the above equations are difficult to obtain accurately and complex to use. Therefore, this study establishes a productivity evaluation method using a water production index. For the curve smoothing part, the relationship between cumulative water production and pressure changes over a specific period of time is determined using the characteristic curve of the shale gas flowback stage. The water production index,Jw, is calculated using the cumulative water volume per unit time and unit pressure.
(17)Jw=Qwpi‐pwft.Equation (17) is substituted into Equation (16), which is simplified as
(18)qAOF=JwTscBwmpiμwpscTBgμg,where qw is the water production (m3/d); h is the fracture height (m); w is the fracture length (m); pi is the initial reservoir pressure (MPa); pwf is the flow pressure at the well bottom (MPa); Bw is the water volume factor, with a value of 1.02; k is the permeability (mD); φ is the porosity (%); ct is the reservoir compressibility coefficient; μw is the water viscosity, with a value of 1.3 mPa·s; qAOF is the AOF (104 m3/d); Tsc is the standard state temperature, with a value of 15°C; psc is the standard state pressure, with a value of 0.101325 MPa; Bg is the gas volume factor, with a value of 3.0×10−3m3/m3; μg is the gas viscosity, with a value of 0.0135 mPa·s; and Jw is the water production index.


## 2.1. Empirical Method

The traditional AOF calculation method makes reference to test well analysis of production data utilizing a BPT method. Exponential, binomial, and one-point procedures make up the majority of BPT techniques [7].The exponential equation is an empirical relationship between the gas well production and the constant flow pressure proposed by Rawlins in 1936 [6], which is expressed in
(1)qg=CpR2−pwf2n.Logarithms of the two sides of Equation (1) are taken to yield
(2)lgqg=lgC+nlgpR2−pwf2,(3)lgpR2−pwf2=1nlgqg‐1nlgC.From Equation (3), it can be interpreted that pR2−pwf2n and qg are linearly related on a bilogarithmic plot with a slope of 1/n and an intercept of −1/nlgC. The line represents the exponential productivity curve.Using a BPT to obtain the constant production and pressure, an exponential capacity curve can be easily drawn, the slope of which provides the value ofn; C can be obtained from
(4)C=qgpR2−pwf2n.Atpwf=0, substituting the initial reservoir pressure, pR, into Equation (5) yields qAOF of the gas well.
(5)qAOF=CpR2n,where qg is the gas production (104 m3/d), pR is the initial reservoir pressure (MPa), pwf is the flow pressure at the bottom of the well (MPa), C is the productivity equation coefficient (104 m3/d/MPa2n), and n is the seepage index.


## 2.2. Pseudogas Production Index Method

Production data from deep shale gas wells are characterized by early and rapidly declining and long-term unstable flow stages, making it difficult to reach the quasistable flow stage [77, 78]. When a well is in the unstable flow stage, its pseudogas production index versus material balance time bilogarithmic plot is a straight line with a slope of 0 to −1/2 [69, 70]. The production data of wells in the unstable flow stage are fit to a straight-line section to obtain the early pseudogas production index. The main benefit of the pseudogas production index approach over the traditional method is that it broadens the application conditions by calculating AOF without requiring production to reach the quasistable flow stage and providing future prediction findings. The pseudogas production index is defined in
(6)qgΔmp=qgmpi−mpwf,(7)mpk=2∫0ppkμugkZkdp.According to Sun et al., the early-stage pressure meets the conditions for using thep2 method, assuming that μZ is a constant. Therefore, p2 is used to substitute mp [79].The material balance time is defined in(8)tm=Npqg.The pseudogas production index for the first day is obtained by fitting a straight-line section to the early unstable flow stage data, and the results are substituted into Equation (9) to calculate the AOF.
(9)qAOF=qgΔmptmbearlympi,where k=i or wf, mpk is the pseudopressure when the pressure is pk (MPa2/(mPa s)), μgk is the gas viscosity when the pressure is pk (mPa s), Zk is the gas deviation factor when the reservoir pressure is pk, Np is the cumulative production (m3), and tm is the material balance time (t).The lack of test production from shale gas wells in a block makes early productivity evaluation difficult [3]. Therefore, this study establishes a pseudogas production index method to calculate the AOF based on the analysis of production data in the early unstable flow stage. However, because this method requires gas well production data, it is applicable to deep shale gas wells in the early unstable flow stage.


## 2.3. Flowback Water Production Index of Analytical Method

Fracturing fluid flowback is performed after the completion of gas well fracturing. A shale gas well’s hydraulic fracture flowback model is shown in Figure1. After hydraulic fracturing, the well is saturated, and the fracturing fluid then flows from the microfracture into the main fracture and finally into the wellbore to the surface. h is the fracture height, w is the fracture width, and n is the number of fractures. The flowback rates of shale gas wells worldwide are low, only 10%–40%, among which those of deeper shale gas wells are relatively higher [79, 80]. For a nonflowback fracturing fluid, the following are generally accepted:
(1)
A fracturing fluid enters the micropores in the matrix because of imbibition(2)
A fracturing fluid is retained in fractures that rapidly close within a short periodFigure 1
Hydraulic fracture flowback model of shale gas well.In this study, water is considered to constitute a large proportion of a fracturing fluid and used to represent it. The variations in the pressure, water production, and gas production with time are shown in Figure2. In the early stage (Figure 2(a)), the mobile fluid in the fracture system is single-phase water, gas production is not seen, the pressure progressively changes, and the cumulative water production rises, signifying an unstable flow stage. In the middle stage (Figure 2(b)), the gas production in the fracture system gradually increases, and both the pressure and water production reach a peak. The water in one part of the fracture is still in the unstable flow stage, whereas the water in the other part of the fracture enters the pseudostable flow stage. The gas starts to break through into the fracture, and this stage belongs to the transition flow stage.Figure 2
Flowback and production characteristic curves of shale gas well.When pressure reaches the fracture barrier in the latter stage (Figure2(c)), the entire fracture system appears to be in a two-phase flow. The water in the matrix is difficult to flowback because of the imbibition effect, and at this point, if there is no energy supplement from the reservoir, the flow is anticipated to enter the suggested stable flow stage [81, 82].Based on the flowback characteristics of the Luzhou deep shale gas wells, the flowback water production index of analytical method is based on the data of stage a in Figure2. Stage a is in the early stage of gas well flowback. After a period of well soak, a large amount of fracturing fluid and movable water is discharged from the fracture. At this stage, it is assumed that no gas is produced. The water in the matrix-fracture system is assumed to behave as a bilinear flow. The analytical model for the flowback stage is established as expressed in Equation (10), and the boundary conditions within the model correspond to constant production. The model assumptions and the detailed derivation process are presented in Appendix A.
(10)∂2p∂x2=φμct0.0853k∂p∂t,px,0=pi,∂p∂xx⟶0=qwBμ0.0853khwn,p∞,t=pi.Solving Equation (10) yields the following relationship between water production and pressure:
(11)qw=nhwpi−pwf2Bw0.0853πkφcttμw.Using the same approach and assuming that the fluid in the matrix-fracture system during the production stage is a single-phase gas, an analytical bilinear flow model for gas production during the production stage is expressed in(12)∂2mp∂x2=φμctk∂mp∂t,mx,0=pi,∂m∂xx⟶0=qgμpscT0.0853khwnTsc,m∞,t=pi.Solving Equation (12) yields the relationship between gas production and pseudopressure as follows:
(13)qg=nhwmpi‐mpwfTsc2pscTBg0.0853πkφcttug.The gas well AOF is provided by(14)qAOF=nhwTscmpi2pscTBg0.0853πkφcttμg.By combining Equations (11) and (14), the formula for calculating the AOF following simplification is expressed in
(15)qAOFqw=TscBwmpiμwpscTBgpi‐pwfμg.The AOF is calculated using Equation (16) by obtaining the relevant parameters and the values of flowback rate and pressure based on the gas well data.
(16)qgAOF=qwpi−pwfTscBwmpiμwpscTBgμg.In practical applications, the data related to the above equations are difficult to obtain accurately and complex to use. Therefore, this study establishes a productivity evaluation method using a water production index. For the curve smoothing part, the relationship between cumulative water production and pressure changes over a specific period of time is determined using the characteristic curve of the shale gas flowback stage. The water production index,Jw, is calculated using the cumulative water volume per unit time and unit pressure.
(17)Jw=Qwpi‐pwft.Equation (17) is substituted into Equation (16), which is simplified as
(18)qAOF=JwTscBwmpiμwpscTBgμg,where qw is the water production (m3/d); h is the fracture height (m); w is the fracture length (m); pi is the initial reservoir pressure (MPa); pwf is the flow pressure at the well bottom (MPa); Bw is the water volume factor, with a value of 1.02; k is the permeability (mD); φ is the porosity (%); ct is the reservoir compressibility coefficient; μw is the water viscosity, with a value of 1.3 mPa·s; qAOF is the AOF (104 m3/d); Tsc is the standard state temperature, with a value of 15°C; psc is the standard state pressure, with a value of 0.101325 MPa; Bg is the gas volume factor, with a value of 3.0×10−3m3/m3; μg is the gas viscosity, with a value of 0.0135 mPa·s; and Jw is the water production index.


## 3. Theory of EUR Evaluation Method of Deep Shale Gas Well

Shale gas EUR evaluation methods include the material balance, PDA empirical, and modern PDA methods. The PDA method does not need to consider a reservoir geological model and a production system and uses only production data; the method has the characteristics of simplicity and practicality. These methods have been developed into mature commercial software (e.g., Harmony) and are the mainstream methods for EUR evaluation. This study describes the basic principles and application conditions of productivity evaluation methods, establishes an EUR evaluation process for deep shale gas wells, and evaluates the EURs of the three shale gas wells in the Luzhou block.Gas wells must continuously produce in order for an empirical method to work. An empirical method cannot be used to produce a system with changeable bottom-hole flow pressure [71]. However, most gas wells adopt such a system for actual production. In addition, the actual production process for a single-phase gas well has both variable pressure and production, owing to the compressibility of the gas. For the PDA of gas wells under variable production and pressure conditions, Blasingame et al. and Jieming et al. presented a material balance pseudotime function and a normalized pseudopressure [30, 31]. This is the greatest advantage of modern PDA methods.The fundamental idea behind this approach is to use a material balance pseudotime parameter to examine the transition of an evaluation object from liquid to gas phases. Using normalized pressure parameters, the pseudopressure flow equation of a gas well is changed into a normalized pseudopressure flow equation. Finally, the application condition of the PDA method is altered from being constant pressure to variable pressure. By simplifying the flow equation and fitting it with actual production data, physical parameters of the actual formation can be obtained, and the gas well productivity can be effectively predicted [32].The Blasingame method uses the material balance time and a pseudotime function to obtain the material balance pseudotime function and combines the normalized pseudopressure to extend the Fetkovich method.The pseudotime function of the material balance is defined in(19)tca=μgiCtiqg∫0tqgμgCtdt.The normalized pseudopressure is defined in(20)pp=μgiZpi∫0ppμgZdp.The pseudo-pressure-normalized production is defined in(21)qgΔpp=qgppi‐ppwf,where ugi and ug are the gas viscosities at pressures pi and p, respectively (mPa·s); Cti and Ct are the total reservoir compressibility coefficients at pressures pi and p, respectively (MPa−1); t is the production time (d); tca is the material balance pseudotime (d); qg is the gas production (m3/d); and pp is the normalized pseudopressure under reservoir conditions p (MPa).The Blasingame approach uses three characteristic curves to collectively fit the data to compute the EUR in order to improve calculation accuracy and combat result distortion caused by the low precision of the production data. These are the pseudo-pressure-normalized production curveqg/Δpp, pseudo-pressure-normalized production integral curve qg/Δppi, and pseudo-pressure-normalized production derivative curve.The Agarwal-Gardner (A-G) method and the normalized pressure integral (NPI) method are also the main methods for PDA [33]. Based on the Blasingame decline analysis method, the A-G method adds a dimensionless parameter relationship to unstable well test analysis to establish a PDA-type curve. This type of curve fitting analysis process is the same as the Blasingame method. This method mainly analyzes the production and time relationship data. The distinction between the Blasingame and A-G approaches, which decreases the many solutions of the fitting, is in the definitions of the dimensionless parameters. The dimensionless production of the A-G technique is converted to a dimensionless pressure using the type curve of the NPI method. The horizontal axis of a type curve is still a dimensionless time. The process of the type-curve fitting analysis of the NPI method is similar to those of the other two methods.In addition to calculating the EUR, modern PDA methods can quantify and analyze other reservoir parameters, such as permeability and skin coefficient [34]. The results of the type-curve fitting are subsequently used to qualitatively identify the gas flow stage and production status of a gas well.The EUR evaluation methods for shale gas wells are summarized in Table1.Table 1
Summary of production decline analysis methods.
Method categoryMethod subcategoryApplicable conditionsAdvantages/disadvantagesArpsExponential decline curveHarmonic decline curveHyperbolic decline curveQuasistable flow/single well/constant pressure production systemUnderfitting with data in early decline stage/high productivity prediction resultFetkovichTheoretical plate method of normalized decline curve fittingTotal production data fitting/constant pressure onlyNew empirical modelPLEDSEPDDuongImproved hyperbolic declineUnstable flow/quasistable flow/single-well/constant-pressure production systemEntire production data fitting/uncertainty of prediction results is high when production data are fewModern PDA modelBlasingameA-G and NPIEntire production flow/single-well/variable pressure production systemEntire production data fitting/many fitting parameters lead to existence of multiple solutionsFlow material balanceFitting method of material balance equation for material balance pseudotime functionQuasistable flow/variable pressure production systemObtain linear relationship to calculate geological reserves in unstable stage


## 4. Results and Discussions

### 4.1. Results of Pseudogas Production Index Method

We evaluated three deep shale gas wells in the Luzhou block—L2-3, Y4-5, and Y2-8—using the established pseudogas production index method.First, bilogarithmic curves of the pseudogas production index and the material balance time were plotted using the production data of the three gas wells in the unstable flow stage. It is noticeable from Figure3 that the data points vary regularly over a straight line with a slope of 0 to −1/2. The early pseudogas production index values of the three gas wells were obtained by fitting the intersection of the straight-line section exhibited in the early unsteady flow stage with the vertical coordinate axis.Figure 3
Production curves of pseudogas production index method: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)From Figure3, the pseudogas production indexes are 0.053×104 m3/d/MPa2 for L2-3, 0.016×104m3/d/MPa2 for Y2-8, and 0.026×104m3/d/MPa2 for Y4-5. The measured parameters and pseudogas production indexes of the three gas wells were substituted into Equation (9), and the AOF results were calculated, which are listed in Table 2.Table 2
Calculation parameters and results of pseudogas production index method.
Well namepi (MPa)mpi (MPa2/(mPa s))qg/Δmp (104 m3/d/MPa2)qAOF (104 m3/d)L2-372.0051840.053274.75Y2-879.0062410.01699.85Y4-575.0056250.026146.25The results in Table2 suggest that a large productivity gap occurs among the wells, where well L2-3 has the largest AOF of 274.75×104m3/d, whereas well Y2-8 has the lowest AOF of 99.85×104m3/d. This suggests that well L2-3 has better reservoir conditions and a well-developed fracture system than the other two wells, and the initial productivity of this gas well is high.


### 4.2. Results of Flowback Water Production Index of Analytical Method

The three deep shale gas wells in the Luzhou block were analyzed using the water production index analytical method. Shale gas was produced using a pressure-controlled method during the flowback stage. Therefore, a smooth section of the pressure curve variation was selected, and the water production time, pressure value, and accumulated water production were substituted into Equation (17) to calculate the water production index, Jw. As shown in Figure 4, the water production index, Jw, for well L2-3 is 132.45 m3/d/MPa, and those calculated for the remaining two wells, Y2-8 and Y4-5, are 39.72 m3/d/MPa and 62.23 m3/d/MPa, respectively.Figure 4
Flowback curve of L2-3 well.The AOFs of the gas wells were calculated using the water production index analytical method, and the results are listed in Table3. The AOF of well L2-3 is the highest at 262.54×104m3/d, whereas that of well Y2-8 is the lowest at 90.89×104m3/d. The water production index analytical method only requires stable production data in the flowback stage, to deduce the AOF, and the required data are accurate and easily available, which significantly reduces the calculation cost and avoids the calculation errors caused by irregular data.Table 3
Calculation parameters and results of water production index analytical method.
Well nameReservoir pressurePi (MPa)Temperature (°C)Jw (m3/d/MPa)qAOF (104 m3/d)L2-372138132.45262.54Y2-87914539.7290.89Y4-57514062.23132.93


### 4.3. Results of Empirical Method

Conventional BPTs are conducted for shale gas wells, which require that both the wellhead pressure and gas production should be stable for at least 5 days and their fluctuations should not exceed 5%. However, deep shale gas well reservoirs have developed micro-nanopores with extremely low porosity and permeability and no natural productivity. Their complex fracture system must be obtained by a large-scale reservoir fracturing and reforming technology, owing to the industrial gas flow and economic benefits [4]. Currently, many shale gas wells are produced using pressure-controlled methods, which generally fail to obtain the required test pressure and test production. Among the three shale gas wells in the block considered in this study, only well L2-3 met the requirements for a BPT well analysis. An exponential empirical method was used to calculate the AOF. The initial reservoir pressure, pr, of well L2-3 was 72 MPa, and the remaining parameters were chosen as listed in Table 4.Table 4
Calculation parameters and results of empirical method.
Back pressure testNozzle diameter (mm)pwf (MPa)qg (104 m3/d)pr2−pwf2 (MPa2)Productivity point 1553.6018.891721.444Productivity point 2752.3024.291859.216Productivity point 31049.8431.282110.18Figure5 illustrates that the slope of the productivity curve of well L2-3 is 0.4 and the intercept is 2.72. According to Equation (5) for the exponential empirical method, qAOF=289.84×104m3/d for well L2-3.Figure 5
Productivity curve of L2-3 using empirical method.


### 4.4. Results of Numerical Simulation Method

In this study, dynamic simulations of ultimate gas production were performed for the three gas wells based on the properties of the deep shale gas reservoirs in the Luzhou block as well as the wellbore parameters and production systems of the gas wells. The AOFs of the gas wells were obtained from the simulated inflow performance relationship (IPR) curves when the bottom-flow pressure was zero. The CH4 content in the gas recovered from the three gas wells exceeded 98%, with extremely low amounts of CO2 and N2. The detailed input parameters of the model are listed in Table 5.Table 5
Calculation parameters of numerical simulation method.
ParameterWell nameL2-3Y2-8Y4-5Drainage area (ha)152.1155.30136.50Original gas in place (106 m3)288.13402.64297.97Horizontal length (m)156018531751Facture half length (m)251530Temperature (K)411418413Reservoir pressure (MPa)727975Net pay (m)222520Total porosity (%)4.94.64.5Initial gas saturation (%)546061Permeability (mD)0.270.210.19As shown in Figure6, the intersection of an IPR curve with the x-axis is the AOF of the gas well. The results of the numerical simulation are very similar to those calculated by the pseudogas production index method and the water production index analytical method. Among them, well L2-3 has the highest AOF of 297.00×104m3/d, whereas well Y2-8 has the lowest AOF of 110.00×104m3/d.Figure 6
Numerical simulation results of IPRs of shale gas wells.


### 4.5. Contrast and Verification

We compared the AOF results obtained using the various calculation methods. As shown in Figure7, well L2-3 has the highest AOF, with an average of 278.1×104m3/d. Well Y2-8 has the lowest AOF, with an average of 100.2×104m3/d, whereas well Y4-5 has an average AOF of 142.4×104m3/d.Figure 7
Comparison results of different calculation methods.By comparing the AOFs calculated by the different methods, Figure7 shows that the results calculated using the water production index analytical method are small, which is caused by using the pressure square, instead of the pseudopressure. This is because the initial reservoir pressure in deep shale is high, typically greater than 40 MPa. However, as a gas well continues to produce, the pressure rapidly decreases to 20–30 MPa and subsequently remains stable. This suggests that a small portion of the pressure in the initial stages of gas production is overestimated by the analytical method. The accuracy of the analytical pseudogas production index method and water production index method for computing the AOFs is confirmed by the findings of the various approaches, which generally show minimal difference. In the absence of productivity test data, this study suggests that different methods should be used to calculate the AOFs for gas wells in different production stages. For gas wells in the flowback stage, the water production index analysis method is recommended, and for early gas-producing wells in the unstable flow stage, the pseudogas production index method is suggested to evaluate well productivity.


### 4.6. Results of EUR Evaluation Method

There are multiple ways to determine the EUR of gas wells, as explained in Section3, and the results must be based on a thorough investigation of the fitting effects of various approaches. To better fit the dynamic production curves of shale gas wells, this study uses modern PDA methods to calculate the EUR, which mainly include the Blasingame, A-G, and NPI methods. The production curves of the three gas wells in the block exhibit good characteristics and are in the continuous or fluctuating decline stage, thus meeting the conditions for the modern PDA methods. This paper briefly describes the calculation process and presents the analysis and discussion of the calculation results.


#### 4.6.1. Production Data Preprocessing

Shale gas well production data can show various incorrect data points, which are errors caused by on-site testing. In particular, data points near the times when a well is switched on and off typically have large errors, and researchers must manually filter and remove such erroneous information.


#### 4.6.2. Calculation Parameters

Modern PDA methods mainly analyze gas production and wellhead tubing pressure, and the latter parameter needs to be converted using the wellbore tubular flow into the wellbore flow pressure for modification. In addition, various parameters such as the gas well temperature, initial reservoir pressure, net pay, porosity, and gas saturation are required, and the values of all parameters are listed in Table6.Table 6
Calculation parameters of modern PDA method.
ParameterWell nameL2-3Y2-8Y4-5Wellhead temperature (K)657065Sandface temperature (K)411418413Reservoir pressure (MPa)727975Net pay (m)222520Total porosity (%)4.94.64.5Measured depth (m)560061006260True vertical depth (m)389140574129Maximum deviation angle (°)92.696.593.9Initial gas saturation (%)546061Gas impurityCO2 (%)1.331.460.98H2S (%)000.03N2 (%)0.560.560.61Deviation factor (Z)1.5Volume factor (Bg)3Gas viscosity (mPa·s)0.0135


#### 4.6.3. Fitting Type Curve and Calculating EUR

The pseudo-pressure-normalized production curveqg/△pp, pseudo-pressure-normalized production integral curve qg/Δppi, and pseudo-pressure-normalized production derivative curve were obtained from the production data of the three gas wells. The three curves were fitted using the type curves of the Blasingame, A-G, and NPI methods. The results in Figures 8–10 show that the Blasingame method fits the data of L2-3 and Y4-5 the best, whereas the NPI method fits the data of Y2-8 the best.Figure 8
Fitting of Blasingame method type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)Figure 9
Fitting of A-G method type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)Figure 10
Fitting of NPI method-type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)As shown in Figures8–10, the three shale gas wells reach the pseudostable flow stage, and the three methods are fitted well. Based on Figures 8(b) and 8(c) and 9(b) and 9(c), we conclude that, in the early unstable flow stage, the actual production curves of Y2-8 and Y4-5 deviate slightly from the type curve, and the actual curves gradually approach the type curve downward. This suggests that the skin coefficients in the early stages of Y2-8 and Y4-5 gradually increase, and the fracturing fluid contaminates the surrounding gas wells, which negatively affects the gas production and leads to a relatively lower AOF and EUR. From Figures 8(a) and 9(a), the gas production of well L2-3 is reliable and without contamination.The EURs of the gas wells were calculated using a commercial software, and the results are provided in Table7.Table 7
Calculation results of modern production decline analysis methods.
Well nameEUR (108 m3)Average EUR (108 m3)BlasingameA-GNPIFMBL2-31.331.261.251.211.26Y2-80.440.430.410.410.42Y4-50.810.970.940.840.89According to Table7, the average EURs of L2-3 and Y2-8 are the highest and lowest at 1.26×108m3 and 0.42×108m3, respectively, while the average EUR of Y4-5 is 0.89×108m3. The results demonstrate that the modern PDA methods are suitable for the gas wells in the block. A set of EUR evaluation processes for deep shale gas wells was proposed for this method, and the EURs of the three gas wells in the block were fitted and calculated. The calculation results presented some differences between the various methods, which indicates the uncertainty of the EUR evaluation. The Blasingame method is recommended for medium-to-high-production gas wells, and the NPI method is suggested for low-production wells. The A-G method does not fit well for the Luzhou deep shale gas well production data and is not recommended.Based on the productivity evaluation results of the three shale gas wells in the block, the relationship curve between the AOF and EUR was drawn and obtained quantitatively, which is shown in Figure11.Figure 11
Relationship curve between AOF and EUR of three gas wells.Based on the fitting results, this study concludes that the AOFs and EURs of the deep shale gas wells in the Luzhou block have a good exponential relationship andR2=0.99. The error analysis of the calculated value of the model and the actual value is shown in Table 8.Table 8
Error analysis results.
Well nameFitting modelEUR (108 m3)qAOF (104 m3/d) (model value)qAOF (104 m3/d) (actual value)Error analysis (%)L2-3y2=65.44+15.51e1.97x21.26270.3278.12.81Y2-80.42100.32100.20.12Y4-50.89139.4142.42.11This is because water production in the flowback stage effectively characterizes the initial productivity of the fracture system, and high water production implies good fracture development. The productivity of a shale gas well is determined by the original geological reserves and the fracture system’s channel conductivity. There must be a quantitative relationship between initial productivity and the EUR. Theoretical analysis suggests that a high initial productivity of the fracture system implies a high EUR of a gas well. As a result, achieving optimal gas well development early on and achieving significant initial production productivity are critical issues for shale gas to effectively increase production. Geological and engineering factors influence both of these key parameters of production productivity evaluation. Furthermore, research on the dominant productivity control factors is the foundation for developing a model for high-production shale gas wells, which requires additional in-depth research.


## 4.1. Results of Pseudogas Production Index Method

We evaluated three deep shale gas wells in the Luzhou block—L2-3, Y4-5, and Y2-8—using the established pseudogas production index method.First, bilogarithmic curves of the pseudogas production index and the material balance time were plotted using the production data of the three gas wells in the unstable flow stage. It is noticeable from Figure3 that the data points vary regularly over a straight line with a slope of 0 to −1/2. The early pseudogas production index values of the three gas wells were obtained by fitting the intersection of the straight-line section exhibited in the early unsteady flow stage with the vertical coordinate axis.Figure 3
Production curves of pseudogas production index method: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)From Figure3, the pseudogas production indexes are 0.053×104 m3/d/MPa2 for L2-3, 0.016×104m3/d/MPa2 for Y2-8, and 0.026×104m3/d/MPa2 for Y4-5. The measured parameters and pseudogas production indexes of the three gas wells were substituted into Equation (9), and the AOF results were calculated, which are listed in Table 2.Table 2
Calculation parameters and results of pseudogas production index method.
Well namepi (MPa)mpi (MPa2/(mPa s))qg/Δmp (104 m3/d/MPa2)qAOF (104 m3/d)L2-372.0051840.053274.75Y2-879.0062410.01699.85Y4-575.0056250.026146.25The results in Table2 suggest that a large productivity gap occurs among the wells, where well L2-3 has the largest AOF of 274.75×104m3/d, whereas well Y2-8 has the lowest AOF of 99.85×104m3/d. This suggests that well L2-3 has better reservoir conditions and a well-developed fracture system than the other two wells, and the initial productivity of this gas well is high.


## 4.2. Results of Flowback Water Production Index of Analytical Method

The three deep shale gas wells in the Luzhou block were analyzed using the water production index analytical method. Shale gas was produced using a pressure-controlled method during the flowback stage. Therefore, a smooth section of the pressure curve variation was selected, and the water production time, pressure value, and accumulated water production were substituted into Equation (17) to calculate the water production index, Jw. As shown in Figure 4, the water production index, Jw, for well L2-3 is 132.45 m3/d/MPa, and those calculated for the remaining two wells, Y2-8 and Y4-5, are 39.72 m3/d/MPa and 62.23 m3/d/MPa, respectively.Figure 4
Flowback curve of L2-3 well.The AOFs of the gas wells were calculated using the water production index analytical method, and the results are listed in Table3. The AOF of well L2-3 is the highest at 262.54×104m3/d, whereas that of well Y2-8 is the lowest at 90.89×104m3/d. The water production index analytical method only requires stable production data in the flowback stage, to deduce the AOF, and the required data are accurate and easily available, which significantly reduces the calculation cost and avoids the calculation errors caused by irregular data.Table 3
Calculation parameters and results of water production index analytical method.
Well nameReservoir pressurePi (MPa)Temperature (°C)Jw (m3/d/MPa)qAOF (104 m3/d)L2-372138132.45262.54Y2-87914539.7290.89Y4-57514062.23132.93


## 4.3. Results of Empirical Method

Conventional BPTs are conducted for shale gas wells, which require that both the wellhead pressure and gas production should be stable for at least 5 days and their fluctuations should not exceed 5%. However, deep shale gas well reservoirs have developed micro-nanopores with extremely low porosity and permeability and no natural productivity. Their complex fracture system must be obtained by a large-scale reservoir fracturing and reforming technology, owing to the industrial gas flow and economic benefits [4]. Currently, many shale gas wells are produced using pressure-controlled methods, which generally fail to obtain the required test pressure and test production. Among the three shale gas wells in the block considered in this study, only well L2-3 met the requirements for a BPT well analysis. An exponential empirical method was used to calculate the AOF. The initial reservoir pressure, pr, of well L2-3 was 72 MPa, and the remaining parameters were chosen as listed in Table 4.Table 4
Calculation parameters and results of empirical method.
Back pressure testNozzle diameter (mm)pwf (MPa)qg (104 m3/d)pr2−pwf2 (MPa2)Productivity point 1553.6018.891721.444Productivity point 2752.3024.291859.216Productivity point 31049.8431.282110.18Figure5 illustrates that the slope of the productivity curve of well L2-3 is 0.4 and the intercept is 2.72. According to Equation (5) for the exponential empirical method, qAOF=289.84×104m3/d for well L2-3.Figure 5
Productivity curve of L2-3 using empirical method.


## 4.4. Results of Numerical Simulation Method

In this study, dynamic simulations of ultimate gas production were performed for the three gas wells based on the properties of the deep shale gas reservoirs in the Luzhou block as well as the wellbore parameters and production systems of the gas wells. The AOFs of the gas wells were obtained from the simulated inflow performance relationship (IPR) curves when the bottom-flow pressure was zero. The CH4 content in the gas recovered from the three gas wells exceeded 98%, with extremely low amounts of CO2 and N2. The detailed input parameters of the model are listed in Table 5.Table 5
Calculation parameters of numerical simulation method.
ParameterWell nameL2-3Y2-8Y4-5Drainage area (ha)152.1155.30136.50Original gas in place (106 m3)288.13402.64297.97Horizontal length (m)156018531751Facture half length (m)251530Temperature (K)411418413Reservoir pressure (MPa)727975Net pay (m)222520Total porosity (%)4.94.64.5Initial gas saturation (%)546061Permeability (mD)0.270.210.19As shown in Figure6, the intersection of an IPR curve with the x-axis is the AOF of the gas well. The results of the numerical simulation are very similar to those calculated by the pseudogas production index method and the water production index analytical method. Among them, well L2-3 has the highest AOF of 297.00×104m3/d, whereas well Y2-8 has the lowest AOF of 110.00×104m3/d.Figure 6
Numerical simulation results of IPRs of shale gas wells.


## 4.5. Contrast and Verification

We compared the AOF results obtained using the various calculation methods. As shown in Figure7, well L2-3 has the highest AOF, with an average of 278.1×104m3/d. Well Y2-8 has the lowest AOF, with an average of 100.2×104m3/d, whereas well Y4-5 has an average AOF of 142.4×104m3/d.Figure 7
Comparison results of different calculation methods.By comparing the AOFs calculated by the different methods, Figure7 shows that the results calculated using the water production index analytical method are small, which is caused by using the pressure square, instead of the pseudopressure. This is because the initial reservoir pressure in deep shale is high, typically greater than 40 MPa. However, as a gas well continues to produce, the pressure rapidly decreases to 20–30 MPa and subsequently remains stable. This suggests that a small portion of the pressure in the initial stages of gas production is overestimated by the analytical method. The accuracy of the analytical pseudogas production index method and water production index method for computing the AOFs is confirmed by the findings of the various approaches, which generally show minimal difference. In the absence of productivity test data, this study suggests that different methods should be used to calculate the AOFs for gas wells in different production stages. For gas wells in the flowback stage, the water production index analysis method is recommended, and for early gas-producing wells in the unstable flow stage, the pseudogas production index method is suggested to evaluate well productivity.


## 4.6. Results of EUR Evaluation Method

There are multiple ways to determine the EUR of gas wells, as explained in Section3, and the results must be based on a thorough investigation of the fitting effects of various approaches. To better fit the dynamic production curves of shale gas wells, this study uses modern PDA methods to calculate the EUR, which mainly include the Blasingame, A-G, and NPI methods. The production curves of the three gas wells in the block exhibit good characteristics and are in the continuous or fluctuating decline stage, thus meeting the conditions for the modern PDA methods. This paper briefly describes the calculation process and presents the analysis and discussion of the calculation results.


### 4.6.1. Production Data Preprocessing

Shale gas well production data can show various incorrect data points, which are errors caused by on-site testing. In particular, data points near the times when a well is switched on and off typically have large errors, and researchers must manually filter and remove such erroneous information.


### 4.6.2. Calculation Parameters

Modern PDA methods mainly analyze gas production and wellhead tubing pressure, and the latter parameter needs to be converted using the wellbore tubular flow into the wellbore flow pressure for modification. In addition, various parameters such as the gas well temperature, initial reservoir pressure, net pay, porosity, and gas saturation are required, and the values of all parameters are listed in Table6.Table 6
Calculation parameters of modern PDA method.
ParameterWell nameL2-3Y2-8Y4-5Wellhead temperature (K)657065Sandface temperature (K)411418413Reservoir pressure (MPa)727975Net pay (m)222520Total porosity (%)4.94.64.5Measured depth (m)560061006260True vertical depth (m)389140574129Maximum deviation angle (°)92.696.593.9Initial gas saturation (%)546061Gas impurityCO2 (%)1.331.460.98H2S (%)000.03N2 (%)0.560.560.61Deviation factor (Z)1.5Volume factor (Bg)3Gas viscosity (mPa·s)0.0135


### 4.6.3. Fitting Type Curve and Calculating EUR

The pseudo-pressure-normalized production curveqg/△pp, pseudo-pressure-normalized production integral curve qg/Δppi, and pseudo-pressure-normalized production derivative curve were obtained from the production data of the three gas wells. The three curves were fitted using the type curves of the Blasingame, A-G, and NPI methods. The results in Figures 8–10 show that the Blasingame method fits the data of L2-3 and Y4-5 the best, whereas the NPI method fits the data of Y2-8 the best.Figure 8
Fitting of Blasingame method type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)Figure 9
Fitting of A-G method type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)Figure 10
Fitting of NPI method-type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)As shown in Figures8–10, the three shale gas wells reach the pseudostable flow stage, and the three methods are fitted well. Based on Figures 8(b) and 8(c) and 9(b) and 9(c), we conclude that, in the early unstable flow stage, the actual production curves of Y2-8 and Y4-5 deviate slightly from the type curve, and the actual curves gradually approach the type curve downward. This suggests that the skin coefficients in the early stages of Y2-8 and Y4-5 gradually increase, and the fracturing fluid contaminates the surrounding gas wells, which negatively affects the gas production and leads to a relatively lower AOF and EUR. From Figures 8(a) and 9(a), the gas production of well L2-3 is reliable and without contamination.The EURs of the gas wells were calculated using a commercial software, and the results are provided in Table7.Table 7
Calculation results of modern production decline analysis methods.
Well nameEUR (108 m3)Average EUR (108 m3)BlasingameA-GNPIFMBL2-31.331.261.251.211.26Y2-80.440.430.410.410.42Y4-50.810.970.940.840.89According to Table7, the average EURs of L2-3 and Y2-8 are the highest and lowest at 1.26×108m3 and 0.42×108m3, respectively, while the average EUR of Y4-5 is 0.89×108m3. The results demonstrate that the modern PDA methods are suitable for the gas wells in the block. A set of EUR evaluation processes for deep shale gas wells was proposed for this method, and the EURs of the three gas wells in the block were fitted and calculated. The calculation results presented some differences between the various methods, which indicates the uncertainty of the EUR evaluation. The Blasingame method is recommended for medium-to-high-production gas wells, and the NPI method is suggested for low-production wells. The A-G method does not fit well for the Luzhou deep shale gas well production data and is not recommended.Based on the productivity evaluation results of the three shale gas wells in the block, the relationship curve between the AOF and EUR was drawn and obtained quantitatively, which is shown in Figure11.Figure 11
Relationship curve between AOF and EUR of three gas wells.Based on the fitting results, this study concludes that the AOFs and EURs of the deep shale gas wells in the Luzhou block have a good exponential relationship andR2=0.99. The error analysis of the calculated value of the model and the actual value is shown in Table 8.Table 8
Error analysis results.
Well nameFitting modelEUR (108 m3)qAOF (104 m3/d) (model value)qAOF (104 m3/d) (actual value)Error analysis (%)L2-3y2=65.44+15.51e1.97x21.26270.3278.12.81Y2-80.42100.32100.20.12Y4-50.89139.4142.42.11This is because water production in the flowback stage effectively characterizes the initial productivity of the fracture system, and high water production implies good fracture development. The productivity of a shale gas well is determined by the original geological reserves and the fracture system’s channel conductivity. There must be a quantitative relationship between initial productivity and the EUR. Theoretical analysis suggests that a high initial productivity of the fracture system implies a high EUR of a gas well. As a result, achieving optimal gas well development early on and achieving significant initial production productivity are critical issues for shale gas to effectively increase production. Geological and engineering factors influence both of these key parameters of production productivity evaluation. Furthermore, research on the dominant productivity control factors is the foundation for developing a model for high-production shale gas wells, which requires additional in-depth research.


## 4.6.1. Production Data Preprocessing

Shale gas well production data can show various incorrect data points, which are errors caused by on-site testing. In particular, data points near the times when a well is switched on and off typically have large errors, and researchers must manually filter and remove such erroneous information.


## 4.6.2. Calculation Parameters

Modern PDA methods mainly analyze gas production and wellhead tubing pressure, and the latter parameter needs to be converted using the wellbore tubular flow into the wellbore flow pressure for modification. In addition, various parameters such as the gas well temperature, initial reservoir pressure, net pay, porosity, and gas saturation are required, and the values of all parameters are listed in Table6.Table 6
Calculation parameters of modern PDA method.
ParameterWell nameL2-3Y2-8Y4-5Wellhead temperature (K)657065Sandface temperature (K)411418413Reservoir pressure (MPa)727975Net pay (m)222520Total porosity (%)4.94.64.5Measured depth (m)560061006260True vertical depth (m)389140574129Maximum deviation angle (°)92.696.593.9Initial gas saturation (%)546061Gas impurityCO2 (%)1.331.460.98H2S (%)000.03N2 (%)0.560.560.61Deviation factor (Z)1.5Volume factor (Bg)3Gas viscosity (mPa·s)0.0135


## 4.6.3. Fitting Type Curve and Calculating EUR

The pseudo-pressure-normalized production curveqg/△pp, pseudo-pressure-normalized production integral curve qg/Δppi, and pseudo-pressure-normalized production derivative curve were obtained from the production data of the three gas wells. The three curves were fitted using the type curves of the Blasingame, A-G, and NPI methods. The results in Figures 8–10 show that the Blasingame method fits the data of L2-3 and Y4-5 the best, whereas the NPI method fits the data of Y2-8 the best.Figure 8
Fitting of Blasingame method type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)Figure 9
Fitting of A-G method type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)Figure 10
Fitting of NPI method-type curve and production data: (a) L2-3, (b) Y2-8, and (c) Y4-5.
(a)(b)(c)As shown in Figures8–10, the three shale gas wells reach the pseudostable flow stage, and the three methods are fitted well. Based on Figures 8(b) and 8(c) and 9(b) and 9(c), we conclude that, in the early unstable flow stage, the actual production curves of Y2-8 and Y4-5 deviate slightly from the type curve, and the actual curves gradually approach the type curve downward. This suggests that the skin coefficients in the early stages of Y2-8 and Y4-5 gradually increase, and the fracturing fluid contaminates the surrounding gas wells, which negatively affects the gas production and leads to a relatively lower AOF and EUR. From Figures 8(a) and 9(a), the gas production of well L2-3 is reliable and without contamination.The EURs of the gas wells were calculated using a commercial software, and the results are provided in Table7.Table 7
Calculation results of modern production decline analysis methods.
Well nameEUR (108 m3)Average EUR (108 m3)BlasingameA-GNPIFMBL2-31.331.261.251.211.26Y2-80.440.430.410.410.42Y4-50.810.970.940.840.89According to Table7, the average EURs of L2-3 and Y2-8 are the highest and lowest at 1.26×108m3 and 0.42×108m3, respectively, while the average EUR of Y4-5 is 0.89×108m3. The results demonstrate that the modern PDA methods are suitable for the gas wells in the block. A set of EUR evaluation processes for deep shale gas wells was proposed for this method, and the EURs of the three gas wells in the block were fitted and calculated. The calculation results presented some differences between the various methods, which indicates the uncertainty of the EUR evaluation. The Blasingame method is recommended for medium-to-high-production gas wells, and the NPI method is suggested for low-production wells. The A-G method does not fit well for the Luzhou deep shale gas well production data and is not recommended.Based on the productivity evaluation results of the three shale gas wells in the block, the relationship curve between the AOF and EUR was drawn and obtained quantitatively, which is shown in Figure11.Figure 11
Relationship curve between AOF and EUR of three gas wells.Based on the fitting results, this study concludes that the AOFs and EURs of the deep shale gas wells in the Luzhou block have a good exponential relationship andR2=0.99. The error analysis of the calculated value of the model and the actual value is shown in Table 8.Table 8
Error analysis results.
Well nameFitting modelEUR (108 m3)qAOF (104 m3/d) (model value)qAOF (104 m3/d) (actual value)Error analysis (%)L2-3y2=65.44+15.51e1.97x21.26270.3278.12.81Y2-80.42100.32100.20.12Y4-50.89139.4142.42.11This is because water production in the flowback stage effectively characterizes the initial productivity of the fracture system, and high water production implies good fracture development. The productivity of a shale gas well is determined by the original geological reserves and the fracture system’s channel conductivity. There must be a quantitative relationship between initial productivity and the EUR. Theoretical analysis suggests that a high initial productivity of the fracture system implies a high EUR of a gas well. As a result, achieving optimal gas well development early on and achieving significant initial production productivity are critical issues for shale gas to effectively increase production. Geological and engineering factors influence both of these key parameters of production productivity evaluation. Furthermore, research on the dominant productivity control factors is the foundation for developing a model for high-production shale gas wells, which requires additional in-depth research.


## 5. Conclusion

AOF and EUR are critical parameters for assessing productivity. In this study, the productivity of three deep shale gas wells from the Luzhou block in the Sichuan Basin of China is calculated by using the established method. The utility and potential of the various methods are discussed and analyzed. The main conclusions of this study are as follows:(1)
A pseudogas production index method and a water production index analytical method are introduced in this study as new methods for calculating the AOF. The water production index analytical method only requires stable production data in the flowback stage to derive the AOF, which are simple to obtain, reducing the calculation costs and avoiding the calculation errors caused by multiple parameters. The results show that wells L2-3 and Y2-8 have the highest and lowest AOFs, averaging278.1×104m3/d and 100.2×104m3/d, respectively, and the average AOF of well Y4-5 is 142.4×104m3/d. The AOF calculation methods for gas wells in different stages of production are different. The pseudogas production index method is recommended for gas wells in the early unstable flow stage. The water production index analytical method is suggested for gas wells in the flowback stage, These methods focus on the accuracy of the selected data, mainly including identification of the flow phase, pressure, gas production, water production, and time; otherwise, it will cause errors in the results(2)
The AOFs of the three shale gas wells are calculated using an exponential method and a numerical simulation method, and the results are comparable to those of the pseudogas production index method and the water production index analytical method. Consequently, the accuracies of the new methods and the enhancement of their reliability are confirmed(3)
The areas near the wellbores of Y2-8 and Y4-5 are contaminated with the fracturing fluid, which is detrimental to obtaining a high AOF and EUR. The EUR calculation results show that the average EUR of L2-3 is the highest at1.26×108m3 and the lowest at 0.42×108m3. Based on the fitting results, the Blasingame method is recommended for medium-to-high-production gas wells, whereas the NPI method is suggested for low-production wells(4)
The AOFs and EURs of the gas wells have a strong exponential relationship andR2=0.99. The AOF of a gas well characterizes the maximum productivity of the shale fracture system. The results suggest that the magnitude of gas production is proportional to the initial productivity of the fracture system. The original geological reserves and the fracture system channel conductivity are critical factors for determining whether a gas well is highly productive. A high initial productivity of the fracture system implies a high EUR of a single well



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*Source: 2529837-2023-03-01.xml*