#### Abstract

The estimation of reserves and performance prediction are two vital tasks for the development of gas reservoirs where the evaluation of gas in place or well-controlled reserves, as the foundation of the performance analysis of gas wells, turns to be exceedingly significant. Advanced production data analysis or modern rate transient analysis (RTA) methods mainly depend on the iterative calculations of material balance quasitime () and type curve fitting, the essence of which is to update the average reservoir pressure data time and again. The traditional Arps’ decline models are of empirical nature despite the convenience and applicability to the constant bottomhole pressure (BHP) condition. In order to avoid the implicit iteration, this paper develops an explicit method for estimating the average reservoir pressure on the basis of dynamic material balance equation (DMBE), termed “flow integral method,” which can be applied to various gas production systems under boundary-dominated flow (BDF). Based on the flow integral method and the decline parameter evaluation, we employ the hyperbolic decline model to model the gas well performance at a constant BHP. The analytical formulations of decline rate and decline exponent are deduced from the DMBE and the static material balance equation (SMBE) considering the elastic compressibilities of rock pore and bound water. The resulting decline parameter method for explicit estimation of gas reserves boasts a solid and rigorous theory foundation that production rate, decline rate, and average reservoir pressure profiles have reference to each other, and its implementation steps are explained in the paper. The SMBE can, combined with the estimated pressure profile by the flow integral method, also be used to determine gas reserves which is not limited to the constant-BHP condition and can calibrate the estimates of the decline parameter method. The proposed methods are proven effective and reliable with several numerical cases at different BHPs and a field example.

#### 1. Introduction

The production performance analysis of gas wells runs through the gas reservoir development. The variations of production rate, bottomhole pressure (BHP), cumulative gas production, and average reservoir pressure with time are the basic data for reservoir engineers to implement the analysis and prediction of gas well performance, among which measurements of average reservoir pressure, nonetheless, are far from easy. Therefore, the static material balance relationship ( vs. ) is generally used to correlate the average reservoir pressure with the cumulative gas production by the use of reserves . The determination of average reservoir pressure, from a certain perspective, is equivalent to the determination of gas reserves. The ways of estimating gas in place or gas reserves [1–3] mainly include analogy, volumetric method, material balance analysis, pressure transient analysis (PTA), rate transient analysis (RTA), history matching, and numerical simulation. The volumetric method [4–6] is often used in geological modeling of gas reservoirs or gas fields, and the estimated reserves by static data have relatively low accuracy. The traditional material balance method [7–9] is widely used, but it requires a certain number of measurement points of formation pressure. Nowadays, PTA and RTA have found their extensive applications in determining the well-controlled reserves. Well testing must be carried out to obtain the accurate pressure data requisite for PTA [10–12] at the expense of production. The daily production data necessary for RTA [13–15], however, are convenient to glean and are analyzed by rigorous theoretical models for which these cost-effective methods have developed rapidly in recent years.

The research on RTA can be traced back to the statistical law of rate data of oil and gas wells presented by Arps [16] who empirically summarized the rate-time relationship into three decline types, namely, exponential, hyperbolic, and harmonic. The conventional hyperbolic decline model (HDM) as a classical tool to analyze the production data of constant-BHP cases is easy to apply, but its theoretical foundation is not fully reckoned with.

Rate decline analysis came into widespread use during the early 1980s when Fetkovich [17, 18] proved that dimensionless rate against time for decline curve analysis during late BDF is subject to the exponential decline law under the constant-BHP condition. Fetkovich’s type curves, composed of the dimensionless curves generated by the analytical solution in the transient flow period and Arps’ empirical curves in the BDF stage, have created a new direction of modern decline curve analysis where the analytical solution to unsteady flow model and the log-log plot as used in well test analysis are presented. Fetkovich’s approach removes the empiricism of Arps’ models, but its application is still confined to the slightly compressible fluid because it reckons without the change in gas properties.

Following Fetkovich’s investigation, Carter [19, 20] explained the variations of gas viscosity and compressibility with pressure by introducing the product ratio and proposed a new set of type curves for gas reservoir analysis. The approximate drawdown parameter takes into account the effect of initial pressure drawdown on the product of gas viscosity and compressibility which enables Carter’s approach to be applied to constant-BHP gas flow systems.

In order to address the influence of the changes in production scenarios on data analysis, Blasingame et al. [21] introduced an idea to analyze the production data for variable rate/variable pressure drop systems by using the time function of constant pressure analog which can be obtained by the constant rate analog (or material balance pseudotime ) for variable-rate flow under BDF conditions. This approach, for gas well analysis, entails the iterative calculations of to determine the parameters and as shown in Equation (4). After that, Palacio and Blasingame [22] demonstrated that the relation of dimensionless rate vs. dimensionless time for the decline curve analysis during BDF coincides with Arps’ harmonic decline model based on the material balance time (or pseudotime) function, and then, a new kind of type curves was developed. The innovation of Blasingame’s approach is that it not only considers the changes in fluid properties but also fully explains the variations of production schedules as long as the boundary effect can dominate the fluid flow. In addition, other advanced production decline analysis methods such as the normalized pressure integral method (NPI) [23], Agarwal and Gardner’s type curves [24, 25] are also applied to the production data analysis of gas wells. The differences between these approaches and Blasingame’s type curves (the most representative RTA method) mainly lie in the plot functions; hence, they, in essence, are all consistent with the BDF solutions [26, 27] whether to place emphasis on pressure analysis or rate analysis.

The type curve analysis approaches need an iterative procedure for the determination of gas reserves or material balance pseudotime before curve fitting and parameter interpretation. In addition to type curve analysis, some researchers also proposed other RTA methods to conduct reserve estimation and performance prediction without curve fitting. Blasingame and Lee [27], for instance, presented an iterative method for the determination of gas in place, termed variable-rate reservoir limit testing (VRRLT) of gas wells, where SMBE and iterations of average reservoir pressure are combined to estimate reserves of the volumetric gas reservoir. Zhang et al. [28] developed the VRRLT for abnormally pressured gas reservoirs by considering the compressibilities of rock pore and bound water in the mathematical model of gas flow and SMBE and proposed the selection strategy of initial iteration value.

Mattar and McNeil [29, 30] put forward the “flowing” material balance procedure only applicable to constant-rate cases. This approach, however, may not apply to gas reservoirs because the variation of gas property parameters (such as viscosity, deviation factor, and compressibility) with pressure is not considered and its theoretical basis is not rigorously demonstrated, too.

Mattar et al. [31, 32] developed the dynamic material balance (DMB) procedure, an extension of the flowing material balance, to cope with the rate fluctuations of the gas well. This approach is derived from the “stabilized flow” solution of the gas flow model and the material balance equation of the volumetric gas reservoir. Mattar’s dynamic material balance procedure boasts a much stricter fundamental basis than the previous flowing material balance by the use of pseudovariables to tackle the changes in production rate and gas properties. The dynamic material balance equation (DMBE), afterwards, was strictly deduced by Zhang et al. [33] who extended the DMB to abnormally pressured gas reservoirs by incorporating the compressibility effects of irreducible water and rock into SMBE and the gas flow model. Similarly, SMBE, DMBE, and iterations of material balance pseudotime or average reservoir pressure can be combined to estimate gas reserves where the selection strategy of the initial value can refer to Zhang et al. [28].

The VRRLT and DMB function well without type curve fitting, but they still entail the iterations of material balance pseudotime or average reservoir pressure for determination of reserves in an implicit fashion. In order to avoid the iterative procedure, some researchers began to attempt more direct and explicit approaches for reserve estimation with the unremitting application. Ye and Ayala [34] proposed a method to linearize the gas diffusion equation by using density function () and pseudotime factor (i.e., depletion-driven time rescaling factor ). The density function, in reality, ignores the viscosity term in pseudopressure, while the treatment of pseudotime factor is consistent with the traditional definition of pseudotime. They evade the iteration of gas reserves for the first time and use the coincidence of liquid solution and numerical simulation results (or gas well data) in the early stage when changes in gas properties are not obvious to select the dimensionless reservoir radius in liquid-type curves, and then, the differences during BDF are employed to determine the and profiles. An approximate relation of to is developed to construct the material balance plot, namely, vs. , for estimation of reserves where the slope of log-log plot of vs. is approximately reckoned a constant . This approach is targeted at the rate-time analysis of the gas well in the volumetric gas reservoir under a constant-BHP specification.

Stumpf and Ayala [35] also presented an explicit method for determining gas in place under constant-BHP conditions by the use of vs. straight line based on the hyperbolic window (hyperbolic decline period) identification. They stated that the hyperbolic window only exists in the early period of BDF when the decline exponent can be considered as a constant determined by the initial pseudopressure drawdown ratio and the integral mean value of with respect to pseudopressure so the rate data conform to the hyperbolic decline law. The hyperbolic window can be determined by adjusting the rate and time multipliers via the hyperbolic type curve fitting. This approach, however, seems little easy in terms of the whole analysis process and inapplicable to geopressured gas reservoirs where the elastic influences of irreducible water and rock are not negligible.

Alom et al. [36] proposed a type curve fitting method to analyze production data during BDF by using the modified pseudotime function defined by the cumulative gas production . This approach, similar to Blasingame’s type curves in principle, uses polynomials to match the relation of to , but in fact, the relationship between and time is hard to determine because the profile of average reservoir pressure and the relation of vs. are unknown.

Zhang et al. [37] developed the material balance-quasipressure approximation condition method for determining gas reserves which combines the material balance condition derived from the material-balance principle with the quasipressure approximation condition obtained by solving the mathematical model of gas flow through porous media. This approach also circumvents the determination of material balance pseudotime and works well for both gas reservoirs with abnormal pressure and those with normal pressure under BDF conditions.

Recently, Wang and Ayala [38] extended the hyperbolic decline model under constant-BHP conditions presented by Stumpf and Ayala [35] to the special variable-BHP condition and deduced the decline exponent for the volumetric gas reservoir. Following the analysis steps of Stumpf and Ayala, they also employed the type-curve fitting and ~ straight-line relation to implement estimation of reserves. It is worth noting that the ratio of pseudopressure at BHP to that at average reservoir pressure is postulated constant. Under this condition, the hyperbolic decline model can be applied to all data during BDF rather than only the early BDF. These approaches, however, reckoned without the elastic effects of irreducible water and rock. Furthermore, the decline exponent in the hyperbolic decline model (Stumpf and Ayala; Wang and Ayala) is evaluated in terms of variable on average, that is, is deemed the integral mean of with respect to pseudopressure from BHP to initial pressure, but in reality, is a function of average reservoir pressure (larger than BHP).

Jongkittinarukorn et al. [39] presented a correlation between production rate, decline rate, and decline exponent that relates vs. curve to reserve estimation, based on the viscosity-compressibility ratio defined by Carter [20] and the stabilized gas flow equation proposed by Ansah et al. [40, 41], where , a dimensionless pressure, is defined as ; denotes decline rate and is decline exponent. They considered the changes in decline exponent for the volumetric gas reservoir at constant-BHP conditions. The synthetic data for validation, nevertheless, is generated by Ansah et al.’s model which is the rationale of Jongkittinarukorn’s approach in the first place.

To sum up, the current rate transient analysis methods can be roughly divided into five categories: (1) traditional Arps’ decline curve analysis; (2) Fetkovich’s and Carter’s type curves for constant-BHP production data analysis; (3) modern decline curve analysis represented by Blasingame’s type curves capable of analyzing the production system under variable-rate/variable-BHP conditions; (4) the variable-rate reservoir limits testing [27, 28] and dynamic material balance method [31–33] based on the iterations of material balance pseudotime without the curve fitting; (5) some explicit methods [34, 35] to avoid pseudotime calculations. The first method is simple to use, but the theoretical basis is not rigorous. The second is limited to constant-BHP conditions, and curve fitting is complicated. The third and fourth methods are effective in dealing with the BDF data under various production conditions such as constant rate, constant pressure, and even variable rate/variable pressure, but the repeated iterations on average reservoir pressure or pseudotime are indispensable. The fifth category of methods can accomplish reserve estimation in an iteration-free way which has gradually become a research hotspot in the field of gas reservoir performance analysis; however, their applicability to complicated production schedules and abnormally pressured reservoirs remains to be investigated.

Therefore, taking into consideration the compressibility effects of rock and irreducible water, this article is aimed at exploring the explicit methodologies for determining the average reservoir pressure and reserves of a gas reservoir, presenting the theoretical proof of the applicability of the traditional hyperbolic decline model to the gas well with a constant-BHP specification, and eliminating the empiricism of HDM and the defect that it fails to estimate gas in place. These methodologies can evade the iteration calculations of or and lay the foundation for the performance analysis of gas wells.

The overall layout of this paper is as follows. Firstly, the variable-flowrate solutions (or correlations between rate and pressure during BDF) are introduced. Secondly, the static material balance equation is introduced to deduce the analytic expressions of decline rate and decline exponent; then, we explain the rationales behind the traditional hyperbolic model to characterize the production decline data of constant-BHP gas well and the fundamentals of the decline parameter method available for reserve estimation. Thirdly, the flow integral method is presented for estimating the average reservoir pressure by a reference point during BDF based on the flow integral equation derived from the DMBE, and the implementation steps of the decline parameter method are described. Then, several case investigations from both simulated and realistic data are presented to demonstrate and validate the proposed methodologies. Finally, the innovations of the article are outlined and the conclusions are drawn.

#### 2. Model Development

##### 2.1. Correlations of Rate vs. Pressure during BDF

To linearize the diffusivity equation of gas flow, it is usually necessary to introduce pseudovariables such as

where (Pa) denotes pseudopressure, (Pa) is initial reservoir pressure, (Pa·s) represents natural gas viscosity, (Pa·s) denotes gas viscosity at , is gas deviation factor or compressibility factor, denotes gas compressibility factor at , (s) is time, (s) denotes pseudotime, (Pa^{-1}) represents system compressibility, (Pa^{-1}) represents system compressibility at , represents irreducible water saturation, (Pa^{-1}) denotes gas compressibility, (Pa^{-1}) is rock compressibility, and (Pa^{-1}) is water compressibility. All the symbols in this paper are explained in international standard units (SI) for convenience.

By solving the mathematical model of gas flow with variable rates for a gas well in the center of a closed reservoir, one obtains the solution of boundary-dominated flow (BDF) represented by
where or (Pa) denotes the pseudopressure at , (Pa) denotes the pseudopressure at bottomhole pressure (BHP) , and (m^{3}/s) is the surface production rate of the gas well; (s) represents material balance pseudotime, (Pa/m^{3}) denotes the slope of the vs. straight line, and (Pa·s/m^{3}) is its intercept which is generally reckoned a constant during BDF; (m^{3}) is gas in place or gas reserves; and when the international system of units is used.

The relation of pseudopressure at the average reservoir pressure to pseudopressure at BHP under boundary-dominated flow is subject to the dynamic material balance equation [33] given by where (Pa) is the average reservoir pressure, and (Pa) denotes the pseudopressure at .

According to the definition of pseudopressure, Equation (1), we can rewrite Equation (9) as

Equation (10) has the same variable “” as Equation (4).

##### 2.2. Conventional Decline Parameters

###### 2.2.1. Decline Rate

Taking the derivative of Equation (10) with respect to time, we have

When a gas well operates at a constant BHP, is equal to 0; then, Equation (11) is transformed into

Taking into account the compressibilities of rock pore and bound water, the static material balance equation (SMBE) of gas reservoirs captures the intrinsic relationship between average formation pressure and cumulative gas production, which is given by
where denotes the unified pressure function in the SMBE, Pa; is the cumulative gas production, m^{3}; and represents the irreducible water saturation at .

Differentiating Equation (13) with respect to gives

Differentiating Equation (14) with respect to yields where denotes the compressibility function dependent on , , and . Though defined in Equation (17) is different from defined in Equation (3) in form, they are identical in essence and both can be transformed into when and approximate to 0.

According to the chain rule, the derivative of the average reservoir pressure with respect to time can be expressed as

By substituting Equations (15) and (16) into Equation (18), one obtains

The decline rate is defined as

Substituting Equations (12) and (19) into Equation (20) gives

Equation (21) shows that the decline rate dependent upon is a function of and decreases with time because continues to fall with the falling formation pressure.

###### 2.2.2. Decline Exponent

The decline exponent is defined as

Substituting Equation (21) into Equation (22) yields

By putting Equation (19) into Equation (23), we obtain

From Equation (1), it follows

Substituting Equations (9) and (25) into Equation (24) gives

The decline exponent in Equation (26) is similar to the formulations given by Chen and Teufel [42, 43] and the difference lies in the new compressibility function in the paper. Equation (26) can be further reduced to where denotes the derivative of the logarithm of with respect to the logarithm of , and represents the instantaneous pseudopressure level.

It is found that the decline exponent is the product of and , all of which are dependent on which decreases with time. With falling , falls; however, tends to rise as a whole. Therefore, changes slowly with time during the depletion of formation energy and ultimately approaches 0 when gas production is subject to the exponential decline. An equivalent constant is employed to replace the dynamic decline exponent given the slow change in , and the hyperbolic decline model is used to match the production data of the gas well at a constant BHP in the paper.

Note that in Equation (9) is considered a constant in the above derivation process and it indeed holds true for a quite long time during BDF. In fact, the value of , nevertheless, also varies slightly with time.

###### 2.2.3. Estimation of Gas Reserves

The conventional hyperbolic decline model developed by Arps is delineated by three parameters (, , and ), and the production rate and decline rate are represented by
where denotes the initial gas flowrate during BDF, m^{3}/s; is the initial decline rate, s^{-1}; and represents the equivalent decline exponent.

From Equation (21), it follows

According to Equation (10), it follows
where is the average formation pressure at a certain time , Pa; denotes the production rate at , m^{3}/s; and is the decline rate at , 1/s.

Based on Equations (32) and (33), can be written as

From Equation (34), it follows
where denotes the calculated reserves from the production data (rate, decline rate, and average reservoir pressure) at , m^{3}.

Equation (35) reveals that gas reserves can be estimated according to the intrinsic relationship between production rate, decline rate, and average reservoir pressure, but it needs to know the relation of formation pressure to time. Measurements of average reservoir pressure are expensive and time-consuming so it is unrealistic to glean or record the whole profile of . The lack of data prompts us to infer the unknown data points from a limited number of measurement points of average reservoir pressure. What follows in the passage will introduce a method to determine the profile merely based on a certain measurement point during BDF.

##### 2.3. The Flow Integral Method for Determination of Average Reservoir Pressure

From Equation (10), it follows

If the formation pressure at is known, then the production rate at that time can be expressed as

Combining Equation (36) with Equation (37), we obtain
where is the average reservoir pressure at , Pa; *p*_{wf,k} denotes the bottomhole pressure at , Pa.

Equation (38), termed “flow integral equation,” indicates that the average reservoir pressure at (i.e., ) can be solved by the integral operation of with respect to . To speed up the solution process, we utilize the 6th-degree polynomial to approximate the nonlinear function , that is,

From Equation (39), it follows where is the antiderivative of with respect to , is an arbitrary constant (for convenience, let in the paper), and () denotes the coefficient of the polynomial function used to approximate .

Substituting Equation (40) into Equation (38), the rearranged equation is represented by

Equation (41) is applicable to all cases under BDF, not limited to the constant-BHP condition. If the BHP remains unchanged, then this equation collapses to

The right-side terms of Equation (41) are known and its left side is a 7th-degree polynomial; thus, can be easily solved in the range of . The explicit method for estimating the average reservoir pressure data, based on the flow integral equation and one measure point during BDF, is called the “flow integral method” in the paper.

##### 2.4. The Decline Parameter Method for Determination of Gas Reserves

The profile of average reservoir pressure can be determined by the flow integral method so that the terms related to in Equation (35) can also be determined. The profiles of , , and can, therefore, be used to conduct the explicit estimation of gas reserves only based on production data. The specific steps are as follows: (1)Sort out the gas reservoir and fluid property parameters, and determine the multinomial coefficients in (2)Use the HDM to fit the rate-time data of gas well at a constant BHP and determine three model parameters (, , and )(3)Use Equation (31) to determine the decline rate (4)Glean the available data and select a certain data point at during BDF to calculate the values of average reservoir pressure with time by the flow integral method(5)Use , , and to determine based on Equation (35). The average value of is the estimated reserves given bywhere denotes the number of data points.

The above iteration-free method, termed the “decline parameter method,” can explicitly calculate gas reserves of a constant-BHP system based on the hyperbolic decline parameters and the flow integral equation. The static material balance relationship can be also employed to estimate gas in place after the profile of is obtained by the flow integral method. The intercept of vs. straight line is , and its slope is related to gas reserves defined as

The value of determined by the SMBE is comparable to estimated by the decline parameter method.

#### 3. Model Validation

In this section, several synthetic cases with different BHPs and a field case are used to illustrate the effectiveness and applicability of the proposed methodologies above. Furthermore, the modified Stumpf method is also introduced for comparison.

##### 3.1. Synthetic Cases: Gas Reservoir with Normal Pressure

A commercial numerical simulator generates the production data of a gas well at constant BHPs centered in a circular reservoir. The simulated gas reservoir has an original pressure () of 25 MPa, a temperature () of 70°C (or 343.15 K), and a reference depth of 2500 m. The molar mass of natural gas () is 18 g/mol, and the pseudocritical temperature and pressure determined by Sutton’s correlations [44, 45] are 201.34 K and 4.60 MPa, respectively. Hall and Yarborough’s approach [46] is used for calculations of -factor and gas compressibility, and Londono et al.’s correlations [47, 48] for gas viscosity. Other property parameters are shown in Table 1.

Figure 1 illustrates the relationships of vs. and vs. for the simulated gas reservoir with normal pressure, and their nonlinearities may result in large error when real pressure or pressure-squared approach is adopted. Equation (45), a 6th-degree polynomial as represented by the dotted line in Figure 1, is used to approximate the relation of to , and the polynomial boasts high precision within the range of 0.2 to 25 MPa.

The curve of vs. shown in Figure 2 reveals the static material balance relationship, and vs. in Figure 3 reflects the change in the decline rate to some extent. The maximum value of the instantaneous decline exponent dependent on is displayed in Figure 4, and the relationship between the pseudopressure and is shown in Figure 5.

There are 2064 time steps set in the numerical simulation, and the gas well is produced under various constant-BHP conditions for 7000 days (about 19.16 years). Four simulated production schedules correspond to four synthetic cases as shown in Table 2.

###### 3.1.1. Case 1-1

The decline parameter method is employed to estimate the gas reserves for Case 1-1 with a BHP of 20 MPa. The solution steps are as follows:

Step 1. Determination of property parameters.

Table 1, Figures 1–5, and Equation (45) display the corresponding results.

Step 2. Determination of hyperbolic decline model.

Equation (30) is used to match the rate-time data of the first 6000 days, and three model parameters are determined as follows: , , and . The production decline law characterized by the hyperbolic decline model almost coincides with the actual production rate profile as shown in Figure 6 where an equivalent decline exponent constant could scarcely give rise to large error in the production rate. The instantaneous decline exponent would approximate to 0 only when the rate is very low and close to 0 though it varies slowly with time.

Step 3. Calculation of decline rate.

Equation (31) is employed to estimate the corresponding decline rate at each time point , as shown in Figure 7.

Step 4. Estimation of average reservoir pressure.

The flow integral method is utilized to determine the average reservoir pressure profile. A reference point is necessary for that method. Take as an example here. Equation (42) is adopted to determine the average reservoir pressure at each time point; then, the predicted profile is shown in Figure 8.

Step 5. Estimation of gas reserves.

The value of in Equation (35) can be calculated by the decline rate in Step 3 and the average reservoir pressure in Step 4. Figure 9 shows the profile of estimated reserves, and the resulting is with an error of 1.041%.

Step 6. Verification by the static material balance equation.

The profile in Step 4 can also be used to draw vs. curve as shown in Figure 10 where and MPa/(10^{4} m^{3}). The value of calculated by Equation (44) is with an error of -0.313%. The small difference between and validates the decline parameter method.

###### 3.1.2. Case 1-2

As previously mentioned, Equation (30) is used to match the 6000 d rate profile for Case 2-1 with a constant BHP of 15 MPa, and three model parameters are determined as follows: , , and . The production rate predicted by the hyperbolic decline model is consistent with the real production profile as shown in Figure 11.

Similarly, take the and at 1000 d as the reference data. Figure 12 displays the estimated reserve profile, and the resulting value of is with an error of 0.952%. In addition, the straight line of vs. has a negative slope of MPa/(10^{4} m^{3}) and determined by the SMBE is with an error of -0.946%.

###### 3.1.3. Case 1-3

Figure 13 shows the real and predicted profiles of production rate with a BHP of 10 MPa where the dashed line is obtained by the hyperbolic decline model with , , and . The ultimate of determined by the decline parameter method, as shown in Figure 14, compares favourably with of determined by the SMBE.

###### 3.1.4. Case 1-4

The real and predicted profiles of production rate for Case 1-4 with a BHP of 5 MPa are exhibited in Figure 15 where the dotted line derives from the hyperbolic decline model with , , and . Figure 16 shows the estimated profile of . Similar to the previous cases, of coincides with of 1.867302158973 × 10^{8} m^{3}.

Table 3(a) presents the calculation results of the decline parameter method for the synthetic gas reservoir with normal pressure where Error1 denotes the calculation error of obtained by the decline parameter method and Error2 represents the corresponding error of determined by the static material balance equation.

For the sake of contrast, the modified Stumpf method is developed in the appendix following the research of Stumpf and Ayala [35]. It is necessary for this method to evaluate the average decline exponent () on the basis of Equations (28) and (29); then, the straight line of vs. can be drawn. Its slope and intercept can be used to estimate the decline parameters such as , , and (see Appendix). The corresponding analysis results for numerical cases with normal pressure are illustrated in Table 3(b) and Figures 33 and 34 (see Appendix), where denotes the value of gas reserves calculated by the modified Stumpf method and Error3 represents its error of estimation.

It can be seen from Tables 3(a) and 3(b) that the decline parameter method provides desirable estimations of gas reserves and decline parameters though the results of the modified Stumpf method are also acceptable except Case 1-4. The differences between two methods lie in the calculation approach of decline exponent and the processing of production data. The former seems more effective in the performance analysis of the relatively low-pressure gas reservoir than the latter which is mainly applicable to early BDF period.

##### 3.2. Synthetic Cases: Gas Reservoir with High Pressure

The gas reservoir with a high initial pressure of 80 MPa is simulated at the temperature of 120°C. Table 4 presents the reservoir and gas properties for the high-pressure cases. Four constant-BHP schedules are generated as shown in Table 5.

Figures 17–21 exhibit the relation curves of and vs. , vs. , vs. , vs. , and vs. for the synthetic cases with high pressure, respectively. The nonlinear relationship of vs. within the range of MPa in Figure 17 can approximately be represented by the following polynomial:

###### 3.2.1. Case 2-1

Figure 22 displays the simulated production profile and the predicted profile by the HDM with three parameters: , , and for Case 2-1 with a BHP of 64 MPa. The estimated profile of is shown in Figure 23, and is . The value of estimated by the SMBE is which compares well with the result of the decline parameter method.

###### 3.2.2. Case 2-2

The real and predicted profiles of production rate for Case 2-2 with a BHP of 52 MPa are illustrated in Figure 24 where the dashed line is obtained by the hyperbolic decline model with , , and . Figure 25 shows the estimated profile of determined by Equation (35). Similarly, of coincides with of .

###### 3.2.3. Case 2-3

Figure 26 shows the real and predicted profiles of production rate for Case 2-3 with a BHP of 40 MPa in which the dashed line represents the matched hyperbolic decline model with , , and . Then, the ultimate of determined by the decline parameter method, as shown in Figure 27, is comparable with of determined by the vs. curve.

###### 3.2.4. Case 2-4

As mentioned earlier, Equation (30) is used to match the rate profile of the first 6000 days for Case 2-4 with a constant BHP of 24 MPa, and three model parameters are determined as follows: , , and . The production rate predicted by the hyperbolic decline model is in line with the real production profile as shown in Figure 28. The decline parameter method is applied, and Figure 29 displays the estimated reserve profile with the resulting of in line with of determined by the SMBE.

The computation performance of the decline parameter method for the synthetic gas reservoir with high pressure is summarized in Table 6(a). The corresponding analysis results of the modified Stumpf method are displayed in Table 6(b) and Figures 35 and 36 (see Appendix). Two methods both give similar estimations of gas reserves, but the latter shows an increasing error trend with a decrease in .

It is found, from the previous synthetic cases with whether normal pressure or high pressure, that an equivalent decline exponent is capable of matching the production decline data of the gas well with a constant BHP though the instantaneous decline exponent as a function of changes slowly with time. The numerical cases fully demonstrate that (1)The decline parameters of HDM are affected by the BHP of the gas well, and the initial production rate during BDF and the equivalent decline exponent are more sensitive to BHP values than the initial decline rate . The lower the bottomhole flowing pressure the gas well is produced at, the greater and are, while the change in is little obvious(2)The flow integral method, an explicit approach for calculations of average reservoir pressure based on the flow integral equation, can estimate the profile effectively and conveniently because only one data point during BDF is necessary, and in reality, these estimations are not confined to the constant-BHP situations(3)The decline parameter method, an explicit technique for calculations of gas in place (or gas reserves) based on the conventional decline parameters in HDM can reliably evaluate reserves of the production system with a constant BHP. Furthermore, the value of gas reserves calculated by the static material balance equation is available for the mutual verification with the estimate of the decline parameter method(4)The production data matching in the decline parameter method could be superior to the analytic calculations of , , and in the modified Stumpf method based on the straight line of vs. which is only suitable for early BDF data. The former extends the range of application of the latter. These approaches, however, can be verified against each other(5)The decline parameter method is valid for both high-pressure and normal pressure gas reservoirs. The calculation errors for the high-pressure system, however, are slightly larger than those of the low-pressure system which may be attributed to the more decrease in formation pressure and a bit severer fluctuation in during BDF for high-pressure gas reservoirs

##### 3.3. Field Case: Gas Well A in West Virginia

Case 3, a field example provided by Fetkovich et al. [18], reflects the production scenario of *Gas Well A* in West Virginia. Fraim and Wattenbarger [49] also investigated this case and digitized the raw rate-time data as shown in Figure 30. The hyperbolic decline model is used to fit the rate profile, and the resulting three parameters are as follows: , , and .

Since the specific gravity of the natural gas is 0.57, we obtain the pseudocritical temperature and pressure are 191.22 K and 4.61 MPa, respectively, according to Sutton [44, 45]. Other property parameters are shown in Table 7.

As revealed by the dotted line in Figure 31, the relation of to for Case 3 can be represented by the following polynomial in the pressure range of 0.2 to 30:

According to Fetkovich et al., the average reservoir pressure at 200 days is 3268 psi (or 22.532067 MPa), and the corresponding production rate is estimated to be by the matched hyperbolic model. Then, the formation pressure profile is determined by the flow integral method. Figure 32 displays the estimated reserve profile by implementing the above steps of the decline parameter method.

The average value of for Case 3 leads to a of ; however, is not estimated because of unavailable cumulative production. The negligible difference between in the paper and the estimate for gas in place of Mscf (or ) predicted by Fraim and Wattenbarger [49] using normalized time matching demonstrates the applicability and feasibility of the decline parameter method.

#### 4. Conclusions

The flow integral equation during BDF is derived in the paper from the dynamic material balance equation of the gas reservoir; then, the flow integral method based on it is proposed for explicit determination of the average reservoir pressure profile. The decline parameter method, a novel approach for explicit estimation of gas reserves, is developed to evaluate the reserves of the gas well at a constant BHP on the basis of the conventional hyperbolic decline model and the predicted formation pressure data by the flow integral method; and the static material balance equation, on the other hand, can be used to verify the results. These methods are suitable for both abnormally pressured gas reservoirs and normal pressure systems because the compressibilities of rock and bound water are considered in the rationales. The following conclusions are drawn: (i)The hyperbolic decline model with a constant decline exponent can accurately delineate the production rate data of the gas well at a constant BHP, although the actual value varies slowly with time(ii)The values of BHP influence the decline parameters (, , and ) in HDM, and the initial decline rate is relatively insensitive to BHPs compared to the other two parameters. The smaller the BHP is, the larger and are(iii)The flow integral method, not limited to the constant-BHP conditions, is effective in estimating the profile of average reservoir pressure for various gas production systems during BDF in an explicit fashion(iv)The decline parameter method, capturing the intrinsic correlation between production rate, decline rate, formation pressure, and gas reserves, is mainly applicable to the production well with a constant BHP and can also provide results of reference for those gas wells with slight fluctuations in BHP(v)The production data matching in the decline parameter method could be superior to the analytic calculations of , , and in the modified Stumpf method based on the straight line of vs. which is limited to relatively early BDF period. These approaches, nonetheless, can be verified against each other(vi)The static material balance equation, combined with the estimated formation pressure data by the flow integral method, is also capable of determining the reserves accurately, which can also be employed to verify the decline parameter method

#### Appendix

#### Modified Stumpf Method

Both Stumpf and Ayala [35] and Wang and Ayala [38] used the linear relationship between and to determine gas reserves. Considering the different pseudopressure definition in this paper from Stumpf and Ayala and Wang and Ayala, we need to rederive the expressions of initial production rate (), initial decline rate (), pressure-averaged decline exponent (), and gas in place ().

Similar to the average value of presented by Stumpf and Ayala, we introduce a pressure-averaged decline exponent () given by

The relation of cumulative gas production () to time () in the hyperbolic decline model can be written as

Due to

Equation (A.2) is transformed into

From Equation (A.4), it follows

According to Equations (9) and (21), the initial production rate () and the initial decline rate () can be expressed as where denotes the reservoir pressure at the start of BDF.

Then, is given by

Substituting Equation (A.8) into Equation (A.5) gives

We denote the slope and the intercept of vs. straight line by and , respectively, that is,

So divided by is equal to

From Equation (A.12), it follows

From Equation (A.11), it follows

Substituting Equations (A.13) and (A.14) into Equation (A.8) yields

As Stumpf and Ayala [35] mentioned, approximate to during early BDF, therefore, Equation (A.13) can be rewritten as

The above discussion on decline parameters follows the idea of Stumpf and Ayala; however, the vs. straight line is employed instead of vs. . Equations (A.14) through (A.16) underlie the “modified Stumpf and Ayala method.”

The analysis results of the modified Stumpf method for numerical examples are illustrated in Table 3(b), Table 6(b), and Figures 33–36.

**(a) Case 1-1**

**(b) Case 1-2**

**(c) Case 1-3**

**(d) Case 1-4**

**(a) Case 1-1**

**(b) Case 1-2**

**(c) Case 1-3**

**(d) Case 1-4**

**(a) Case 2-1**

**(b) Case 2-2**

**(c) Case 2-3**

**(d) Case 2-4**

**(a) Case 2-1**

**(b) Case 2-2**

**(c) Case 2-3**

**(d) Case 2-4**

#### Data Availability

The gas property data used for numerical simulation and filed application are generated by Londono and Hall-Yarborough correlations.

#### Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

#### Acknowledgments

This study has been supported by the Department of Middle East E&P and Department of Asia-Pacific E&P, Research Institute of Petroleum Exploration and Development, PetroChina. This work is supported by the National Science and Technology Major Project of China (Grant No. 2017ZX05030-003 and 2016ZX05015002).