A COUPLED PSEUDO-PRESSURE/DENSITY APPROACH TO DECLINE CURVE ANALYSIS OF NATURAL GAS RESERVOIRS

Transcription

1 The Pennsylvania State University The Graduate School John and Willie Leone Family Department of Energy and Mineral Engineering A COUPLED PSEUDO-PRESSURE/DENSITY APPROACH TO DECLINE CURVE ANALYSIS OF NATURAL GAS RESERVOIRS A Thesis in Energy and Mineral Engineering by Jaidev M. Gokhale Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December 2014

2 ii The thesis of Jaidev M. Gokhale was reviewed and approved* by the following: Luis F. Ayala H. Associate Professor of Petroleum and Natural Gas Engineering Associate Department Head for Graduate Education Thesis Advisor John Yilin Wang Assistant Professor of Petroleum and Natural Gas Engineering Shimin Liu Assistant Professor of Energy and Mineral Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT Natural gas is quickly gaining popularity as a clean and obtainable energy resource. With increasing exploitation of this resource the calculation of original gas in place (OGIP) becomes a crucial first step to developing natural gas reservoirs in a profitable manner. Once an estimate for reserves has been established key economic decisions can be made to maximize profitability from a given field. Furthermore, the calculation of OGIP serves to appraise reservoir value and influences key decisions on asset takeovers. Due to the importance of OGIP to influence key decisions of economics and asset development, careful consideration must be given into the development of techniques that are used to calculate OGIP. Decline curve analysis has been a preferred tool used by the industry to assist in the calculation of OGIP. This study reviews existing methods of decline curve analysis and provides a new, explicit method to calculate OGIP. The proposed approach demonstrates the ability to map the Arps decline parameters to a rigorous boundary dominated flow equation. Previously, the determination of Arps decline parameters relied predominantly on empirical methods. This study presents a physically justifiable approach to calculate the Arps decline exponent prior to the analysis of production data. Furthermore, the utilization of the mapped Arps decline parameters to predict OGIP using a straight-line analysis technique for cases of constant reservoir drawdown is also investigated through numerical and field case studies.

4 iv TABLE OF CONTENTS List of Figures... v List of Tables... vii Nomenclature... viii Acknowledgements... xi Introduction... 1 Chapter Literature Review... 3 Chapter Model Development The α-based Deliverability Equation for BDF Flow Mapping the Arps Hyperbolic Decline parameters Explicit Reserves Analysis Using the α-based Deliverability Equation Chapter Results and Discussions Case Study Case Study Case Study Case Study Field Case: West Virginia Gas Well A Conclusions References Appendix A Linearity of Pseudo-Pressure Versus Density Data Appendix B Derivation of Straight Line Analysis Method Using the α-based Deliverability Equation Appendix C Derivation of α=2 for Ideal Gasses and α=1 for Liquids Appendix D Selection of the Hyperbolic Window Using the Harmonic Stem... 68

5 v LIST OF FIGURES Figure 1-1.: Analysis of boundary dominated flow production data using the straight-line analysis technique Figure 3-2.: Selection of boundary dominated flow data through Universal type curve matching Figure 4-1.: Al-Hussainy and Ramey (1966) deliverability dquation and alpha based deliverability equation versus numerical simulation for Case1a Figure 4-2.: Al-Hussainy and Ramey (1966) deliverability equation and alpha based Deliverability Equation versus numerical simulation for Case1b Figure 4-3.: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based deliverability equation versus numerical simulation for Case1c Figure 4-4.: Calculation of alpha and Arps decline exponent for Case Study Figure 4-5.: Al-Hussainy and Ramey (1966) deliverability equation and alpha Based deliverability equation versus numerical simulation for Case2A Figure 4-6.: Al-Hussainy and Ramey (1966) deliverability equation and alpha based deliverability equation versus numerical simulation for Case2B Figure 4-7.: Al-Hussainy and Ramey (1966) deliverability equation and alpha based deliverability equation versus numerical simulation for Case2C Figure 4-8.: Case 3a-re=175ft, straight line analysis using the alpha based straight-line analysis approach Figure 4-9.: Case 3b-re=350ft, straight line analysis using the alpha based straight-line analysis approach Figure 4-10.: Case 3c-re=700ft, straight line analysis using the alpha based straight-line analysis approach Figure 4-11.: Case 3d-re=200ft, straight line analysis using the alpha based straight-line analysis approach Figure 4-12.: Case 3e-re=500ft, straight line analysis using the alpha based straight-line analysis approach Figure : Case 3f-re=800ft, straight line analysis using the alpha based straight-line analysis approach Figure 4-14.: Case 4a-Pwf=100psia, straight line analysis using the alpha based straightline analysis approach

6 vi Figure 4-15.: Case 3d-re=200ft, straight line analysis using the alpha based straight-line analysis approach Case 4b-Pwf=1600psia, straight line analysis using the alpha based straight-line analysis approach Figure 4-16.: Case 4c-Pwf=2500psia, straight line analysis using the alpha based straightline analysis approach Figure 4-17.: Case 4d-Pwf=3000psia, straight line analysis using the alpha based straightline analysis approach Figure 4-18.: Case 4e-Pwf=3600psia, straight line analysis using the alpha based straightline analysis approach Figure 4-19.: Case 4f-Pwf=4000psia, straight line analysis using the alpha based straightline analysis approach Figure 4-20.: Field Case, West Virginia Gas Well A...47

7 vii LIST OF TABLES Table 4-1.: Reservoir and fluid properties for Case Study 1, constant drawdown condition and varying reservoir extents Table 4-2.: Reservoir and fluid properties for Case Study 2, constant reservoir extent and varying draw down conditions Table 4-3.: Reservoir and fluid properties for Case Study 3, constant drawdown condition and varying reservoir extents Table 4-4.: Results of Case Study Table 4-5.: Reservoir and fluid properties for Case Study 4, constant reservoir extent and varying draw down conditions Table 4-6.: Results of Case Study 4. (Predicted OGIP for all cases is 6.05 BCF)

8 viii Nomenclature A Reservoir Area, ft 2 B B gi b b pss C A c g c ti D D i G G p (t) h k m(p ) m(p wf ) m(p i ) MW OGIP P P Fluid intrinsic parameter Initial gas formation volume factor, cf/scf Arps Decline Exponent Pseudo-steady state coefficient Dietz Shape Factor Gas isothermal compressibility, (psi) 1 Initial isothermal gas compressibility,psi 1 Arps decline rate, 1/day Arps initial decline rate, 1/day Original Gas In Place, scf Cumulative gas production, scf Formation thickness, ft Formation permeability, md Reservoir pseudo-pressure, psia2 cp Well flowing pseudo-pressure, psia2 cp Initial pseudo-pressure, psia2 cp Molecular weight, lb/lbmol Original Gas In Place, scf Pressure, Psia Average reservoir pressure, psia

9 ix P sc P pi P pwf Pressure at standard conditions, 14.7 psia Initial Blasingame pseudo pressure, psia Well flowing Blasingame pseudo pressure, psia b q Dd Dimensional flow parameter q g max Maximum flow rate obtainable at the initial time under full draw-down condition, scf d q gi q gsc R r m r m r ρ r w T T sc t DAd t t a Z g α γ Arps initial boundary dominated flow production, scf/day Gas flow rate at standard conditions, T = 60F and P = 14.7 Psia, scf d Universal gas constant Reservoir drawdown equivalency Reservoir drawdown Initial density to bottom-hole density ratio Well Radius, ft Reservoir temperature Temperature at standard conditions, 60F, 520R Dimensional time parameter Time, days Palacio material balance pseudo-time, days Gas compressibility factor Fluid parameter that captures linearity of pseudo-pressure and density, ( psia2 ) cp (ft3) lb Euler constant

10 x λ Average viscosity-compressibility ratio between average reservoir pressure and bottom-hole pressure. ρ ρ i ρ wf μ gi Reservoir density, lb ft 3 Initial reservoir density, lb ft 3 Well flowing density, lb ft 3 Initial gas viscosity, cp

11 xi ACKNOWLEDGEMENTS I would like to express my appreciation to the friends and family that have accompanied me on this journey. The completion of this work would not have been possible without their continuous support. I would like thank Dr. Ayala for his support as a friend, a mentor and a professor. Dr. Ayala has taught me many lessons that have guided me in not only my academic life but also my professional and personal life. I am very grateful for all of his support and advice. I would like to thank all my parents for their encouragement and love. Our weekly conversations always served to brighten my week and provide me with energy to move forward. I would like to especially thank my mom for her encouragement in pursuing this field. I would also like to thank my wife and best friend, Mansi, for always being there for me. Mansi was always there to support me and make laugh. I am thankful for her companionship, love and support through our journey. I would like thank Anand, my little brother, for being a great friend and good young man. I hope you work hard in your academic pursuits and I look forward to supporting you in all of your endeavors. Finally, I would to thank my grandfather, Abu, for being an inspiration to me. I will always remember our stories, our good times, and all of golf that we played together.

12 1 Introduction Natural gas is a vital global resource that is rapidly gaining popularity for energy generation and industrial applications. In recent years, the consumption of natural gas has seen a sharp increase due primarily to the realization and exploitation of natural gas reserves. The EIA predicts an average daily gas consumption of 72.6 BCF/day for 2014 (EIA, 2014). Furthermore, the average daily consumption of natural gas for 2014 has increased from the 2013 average and is expected to increase further in As natural gas seeks dominance in energy markets as a clean and obtainable energy source, it becomes important to consider the profitable development of natural gas reservoirs. To produce natural gas reservoirs in a profitable manner, it is important to determine the amount of gas in place. The determination of reserves is an essential calculation that is important in making economic decisions and planning asset production. One of the preferred methods for the determination of gas reserves is the use of decline curve analysis. Decline curve analysis is centered on the study of the decline in rate in a producing gas reservoir. Early methods of decline curve analysis included the use of Arps empirical relations. Through the evolution of decline curve analysis models, more sophisticated and rigorous methods have been proposed to determine original gas in place (OGIP). The prominent methodologies of decline curve analysis are dominated with implicit techniques for the estimation of OGIP. These techniques are versatile and robust in the sense that they can handle variable pressure production constraints in addition to a constant pressure

13 2 production constraint. However the requirement for iterations creates a process that at times can be difficult to implement. Further investigation into explicit methods are beneficial for the application of analyzing production data produced under a constant pressure production constraint. The explicit method reduces computational time while also providing a straight-forward analysis procedure.

14 3 Chapter 1 Literature Review The study of decline curves was largely initiated through the initial contributions of Arps. Arps (1945) published an empirical study that categorized the decline of rate in a reservoir as either exponential, hyperbolic or harmonic. Additionally, Arps was also able to identify mathematical relationships for each mode of decline. The work of Arps was a pivotal contribution to understanding the decline of reservoir rates and was widely used throughout the industry, despite its empirical origins. Fetkovich (1980) provided physical evidence that supported the empirical findings of Arps. In addition, Fetkovich was able to analytically justify the calculation of the hyperbolic exponent for gas reservoirs under full drawdown should fall in the range: 0 < b < 0.5, (Fetkovich,1980; Fetkovich et al. 1985). The work of Fetkovich represented an essential evolutionary step for decline curve analysis and further extended the analysis of hyperbolic decline in natural gas wells. Classical decline curve analysis defines the decline in natural gas reservoirs as the rate at which the production declines over time. Mathematically decline, D, is represented as: D = 1 dq q dt Equation 1

15 4 Arps (1945) identified that initial decline in a producing reservoir could either be a constant or decreasing as a function of time. To capture the behavior of initial decline, Arps introduced the Arps decline exponent,b. The Arps decline exponent is: b = d (1 D ) dt Equation 2 Hyperbolic decline is said to occur when the decline in the reservoir changes as a function of time. Through the integration of equations 1 and 2, the Arps hyperbolic decline equation is: q gi q gsc = (1 + bd i t) 1 b Equation 3 The expression for Arps hyperbolic decline is given by equation 3, where the initial production is given by, q gi, the Arps hyperbolic exponent is b, the intial decline is D i and time is t. This expression served as an early method to capture the behavior of natural gas reservoirs. The early beginnings of decline curve analysis are largely represented through the work of Arps. Modern methods of decline curve analysis have since seen considerable developments in comparison to the early methods of Arps. These methods have adopted definitions of pseudopressure and pseudo-time to effectively acknowledge the strong dependency of gas fluid properties on pressure. In addition, the methods have also invoked an iterative calculation that results in accurate determination of OGIP through the analysis of production data. This section will focus on some of the prominent methods that are typically used throughout the industry for the calculation of OGIP.

16 5 The Palacio-Blasingame (1993) introduced the implicit straight-line method and presented an iterative approach to calculating OGIP. Palacio-Blasingame acknowledged that prior methods were able to at best approximate the variation of gas fluid properties with changes in pressure (Palacio-Blasingame, 1993). As a result, the method defines and adopts two new pseudofunctions, namely rate normalized pseudo-pressure and material balance pseudo-time. The method identifies the following equation for the analysis of production data: q g P pi P pwf b a,pss = ( m a b a,pss ) t a Equation 4 which can be re-arranged to yield a straight-line analysis equation as follows P pi P pwf = (m a )t a + 1/b a,pss q g Equation 5 where the slope of straight-line analysis yields the product of initial compressibility and OGIP. m a = 1 Gc ti Equation 6 The two new pseudo-functions include, rate normalized pseudo-pressure: P p = μ gic ti P dp P i μ g Z g in addition to material balance pseudo-time t a = μ gic ti P i 0 P q g μ g (P )c t (P ) dt 0 t Equation 7 Equation 8 The Palacio-Blasingame method involves the above straight-line analysis equation and implements an iterative method to converge upon an OGIP estimate through the analysis of production data. The incorporation of rate normalized pseudo-pressure and material balance pseudo-time allow the model to honor the dependency of gas fluid properties on pressure. The model has been shown to be reliable and is typically employed throughout the industry for gas reserve estimation.

17 6 Other notable methods in decline curve analysis include the Flowing Material Balance (FMB), (Mattar and Anderson, 2003). The FMB method is similar to that of Palacio-Blasingame (1993) and also incorporates the pseudo-function, material balance pseudo-time. One favored advantage of this method is the ability to perform production analysis in the traditionally familiar graphical format of an extrapolation to the x-axis. This is reminiscent of the traditional reserves estimation technique of plotting P/Z vs. Gp to obtain a reserve estimate. However, the FMB is a rigorous procedure that incorporates the dependency of gas fluid properties on changes in pressure. The dependency of gas fluid properties on pressure variation is accounted for by the use of pseudo-pressure (Al-Hussainy et al, 1966) and material balance pseudo time. The model proposes the following straight-line equation for the implicit analysis of production data: q m(p i ) m(p wf ) = ( 1 ) Q G b n + 1 Equation 9 pss b pss Where the normalized cumulative production function, Q n is 2q t P i t a Q n = (c t μ i Z i ) (m(p i ) m(p wf )) Equation 10 And material balance pseudo-time is t a = μ gic ti P i t q g μ g (P )c t (P ) dt 0 Equation 11 Also pseudo-pressure as defined by Al-Hussainy is P m(p) = 2 P dp μ g Z g 0 Equation 12

18 7 The Flowing Material Balance is comparable to the implicit procedure presented by Palacio-Blasingame and is also widely used throughout the industry for reserve calculations. Both methods incorporate the use of pseudo-functions and invoke an iterative procedure targeted to yield OGIP. Another model that is also widely recognized is that of Ismadi et al., (2012). Ismadi presented a methodology where static and dynamic material balances are plotted simultaneously to achieve an estimate on OGIP with better confidence. The static OGIP is the traditional dry gas material balance statement that produces the OGIP estimate upon extrapolation to the x-axis. The dynamic material balance, is a concept that has been extended from the Mattar and Anderson methodology. The model proposes the following equation for the dynamic material balance: q m(p i ) m(p wf ) = ( 1 ) Q G b n + 1 Equation 13 pss b pss Their normalized cumulative production function, Q n, is defined as: Q n = ( m(p) m(p wf) m(p i ) m(p wf ) ) OGIP Equation 14 Additionally, pseudo-pressure in its current application is defined as: The static material balance is: P m(p) = 2 P dp μ g Z g 0 G p Equation 15 P av = P i (1 Z av Z i OGIP ) Equation 16

19 8 Ismadi demonstrated that the two methods were able to converge upon an OGIP and therefore provide an increased confidence on OGIP estimates. The method is closely related to the Flowing Material Balance method of Mattar and Anderson (2003). Therefore, it compares closely with the Flowing Material Balance as well as the implicit procedure of Palacio- Blasingame. More recently, Zhang and Ayala (2013) presented the rescaled exponential and density based decline model. One key advantage of the density based decline model is the ability to produce OGIP estimates without the use of pseudo-time functions. Similar to other methods discussed, the density based decline model utilizes an iterative approach to converge upon the OGIP estimate. The straight-line analysis equation proposed by Zhang and Ayala is: λ r p OGIP = 1 OGIP λ G p + 1 Equation 17 e q gsc q gi The initial density to bottom-hole density ratio is defined as r ρ = ρ i ρ wf ρ i Equation 18 In addition, λ is calculated as λ = μ gic gi μ gc g μ gi c gi = ρ ρ wf 2θ m(ρ ) m(ρ wf ) Equation 19 where θ = RT MW g Equation 20

20 The method lends itself well to straight-line analysis where the OGIP is directly found as the inverse of the slope of the best fit line through the production data. The model compares well to the other methods discussed. The modern methods of decline curve analysis are able to accurately describe the decline behavior of natural gas reservoirs, however rely heavily on the need for an iterative procedure to calculate OGIP. Recently, several methods have been proposed that accurately describe the decline of rate in a natural gas reservoir without the need of iteration. Ye and Ayala (2012) demonstrated an explicit density based approach that was capable of calculating OGIP. The explicit approach does not require any iterations and therefore provides a convenient and straight-forward analysis approach. The model uses the following equation for analysis of constant well flowing pressure production data: q gsc (t) 1 b = D i(b 1) 1 b Equation 21 b G p (t) + q q gi gi The initial density to bottom-hole density ratio is defined as r ρ = 1 r p r p Equation 22 The fluid intrinsic parameter describes the strength of the viscosity-compressibility coupling for a fluid, and is determined by: B = dlnμ gc g dlnρ Equation 23 From where the Arps hyperbolic exponent can be calculated, independent of rate-time data as 1 + B + 2r ρ Equation 24 b B (1 + B + r ρ ) 2 The straight-line equation allows the direct calculation of reserve (OGIP), once the slope and y- intercept is attained. OGIP = ( i y m ) (1 b) 1 + r ρb r ρ Equation 25 9 r ρ OGIP = [ 1 + ( 1 + r B )] ( 1 ) q ρ D gi 1 + r ρb Equation 26 i r ρ

21 10 Furthermore, Voelker (2004) was able to numerically demonstrate the ability of Arps hyperbolic equation to describe decline in a natural gas reservoir. Voelker presented a correlation that related Arps decline parameters to initial and boundary reservoir conditions, enabling the Arps hyperbolic equation to capture decline in a natural gas reservoir. The study was conducted for natural gas wells producing at a constant well flowing pressure. The results of the study concluded that the Arps hyperbolic exponent can be calculated explicitly prior to the analysis of production data. While the modern methods of decline curve analysis are suitable for determining OGIP in natural gas reservoirs, there are several advantages to using explicit methods. First, the ability to circumvent the requirement for iterations allows the determination of OGIP in a rapid manner. Next, explicit methods are substantially easier to implement and are computationally less intensive. Further attention to developing explicit methods for the analysis of natural gas reservoirs should be given. The focus of this work is to develop an explicit model that is capable of analyzing production data under the constant well flowing pressure constraint for the calculation of OGIP.

22 11 Chapter 2 Model Development The α-based Deliverability Equation for BDF Flow The proposed approach demonstrates the ability to map the Arps decline parameters to a rigorous, boundary dominated deliverability equation. Furthermore, it is also shown that the use of a straight-line analysis technique permits the explicit determination of OGIP for gas wells under constant drawdown. An established deliverability equation that describes the behavior of gas wells is the rigorous boundary dominated flow equation presented by Al-Hussainy and Ramey (1966). Al- Hussainy and Ramey (1966) introduced the rigorous boundary dominated flow equation and demonstrated that the gas flow rate can be related to the average gas pseudo-pressure during the boundary dominated flow period of a producing natural gas well. Al-Hussainy and Ramey s (1966) rigorous boundary dominated flow equation is πkht sc q gsc = P sc T [ln ( r e r ) 3 (m(p ) m(p wf )) w 4 ] Equation 27

23 12 Al-Hussainy also introduced the concept of gas pseudo-pressure, (Al-Hussainy et al,1966). Pseudo-pressure was introduced to acknowledge the strong dependency of gas compressibility and viscosity with respect to pressure. Pseudo-pressure as defined by Al- Hussainy is P m(p) = 2 P μ g Z dp 0 Equation 28 This study (Appendix A) demonstrates a good degree of linearity between pseudopressure and density data when plotted on a log-log scale. This linearity is captured through the fluid parameter α and is defined as d ln m(p) α = ( ) d ln ρ av Equation 29 The use of the α parameter allows for a bridge to dry gas material balance statements which can be written as m(p ) m(p i ) = ( ρ α ) ρ i Equation 30 As a result the α-based deliverability equation can be derived, (Appendix B). The α-based deliverability equation is: q gsc = q max g [( ρ α ) + r ρ m 1] i Equation 31 where r m represents drawdown in the reservoir and is defined as r m = m(p i) m(p wf ) m(p i ) Equation 32

24 13 In addition q g max is defined as q g max = πkht sc m(p i ) P sc T [ln ( r e r w ) 3 4 ] Equation 33 max The physical significance of q g is the maximum flow rate obtainable at the initial time (t=0) under a full drawdown condition. This is more clearly seen as at the initial time reservoir density and initial density are equivalent. In addition, under a full drawdown condition P wf = 0, equation 31 can be simplified to define q max g. The Al-Hussainy and Ramey (1966) rigorous boundary dominated flow equation and the α-based deliverability equation are both capable of capturing natural gas decline during the boundary dominated flow period. In addition, both deliverability equations are able to forecast gas production. Gas production forecasting using the deliverability equations involves the coupling of the material balance statement to calculate the average reservoir pressure or density. For a volumetric dry gas reservoir, the material balance statement can be written as: G p ρ = ρ i ( 1 OGIP ) Equation 34

25 14 Mapping the Arps Hyperbolic Decline parameters Having defined the α-based deliverability equation, this section demonstrates that it is possible to map the Al-Hussainy and Ramey (1966) rigorous deliverability equation to the Arps hyperbolic decline equation. The purpose of defining the α-based deliverability equation is to serve as a vehicle to map the rigorous deliverability equation to the Arps hyperbolic decline equation. The Arps decline parameter,q gi describes the gas flow rate at the initial time of boundary dominated flow. By evaluating the α-based deliverability equation, at initial conditions, we are able to derive an equivalent decline parameter q gi. (See Appendix B) max q gi = r m q g Equation 35 Similarly the initial Arps decline rate, D i can also be defined in terms of the α-based deliverability equation. Following the definition of D, equation 1, we can take the derivative of the α-based deliverability equation, with respect to time. This will yield an equivalent decline in terms of the α-based deliverability equation (See Appendix B): q giα D i = r m OGIP Equation 36 Having fully defined the initial decline in terms of the deliverability equation, it is now possible to define the Arps hyperbolic exponent in terms of the deliverability equation. Following the definition of the Arps exponent, equation 2, the initial Arps hyperbolic exponent can be derived as follows (See Appendix B): b i = r m(α 1) α Equation 37

26 For the full drawdown condition, where r m = 1, the Arps hyperbolic exponent can be predicted by equation 37 to be 15 (α 1) b i rm =1 = α Equation 38 Using equation 38 it is possible to calculate Arps hyperbolic exponent for ideal gasses. Appendix C demonstrates that for ideal gasses, α = 2. Therefore, under full drawdown production, the Arps hyperbolic exponent for ideal gasses can be predicted to be 0.5. Equation 38 can also be used for predicting the Arps decline exponent for liquids. Appendix C demonstrates that for slightly compressible liquids, α = 1. The Arps decline exponent for slightly compressible liquids can be predicted to be 0 for full drawdown production. For gas wells under full drawdown the α-based method predicts an initial Arps exponent between 0 and 0.5. Fetkovich also demonstrated the calculation of the Arps initial decline exponent from the backpressure equation. Fetkovich showed that gas wells under full drawdown experience an Arps decline exponent value between 0 and 0.5 (Fetkovich et. al, 1987). The calculation of the Arps initial exponent by the α-based method is in agreement with results presented by Fetkovich in For cases of less than full drawdown, the calculation of initial Arps exponent (equation 37) becomes time dependent. This implies that the well experiences a distinct period during which the initial Arps exponent can be assumed to be constant. This window is referred to as the hyperbolic window. Voelker (2004) indicates conclusions from his study that indicates that for low drawdown, the value of initial Arps exponent are lower and get lower as drawdown decreases. The results from the α-based calculation of initial Arps exponent show a behavior which is in agreement with the conclusions of Voelker (2004) for less than full drawdown

27 production. It is noted that for low drawdowns, the value of r m decreases and lower values of the Arps exponent are predicted 16 Explicit Reserves Analysis Using the α-based Deliverability Equation Having related Arps decline parameters to the rigorous deliverability equation, it is now possible to generate an equation capable of performing straight-line analysis of BDF production data and returning an estimate for initial gas in place. Ye and Ayala (2013) demonstrated that a straight-line analysis equation for the analysis of boundary dominated flow can be achieved from the cumulative production expression. The cumulative production equation is expressed as for hyperbolic decline is: G p (t) = 1 b q gi D i (1 b) (1 (q gsc(t) ) ) q gi Equation 39 Equation 39 can be re-written as a straight line equation by solving for q 1 b gsc : q 1 b gsc = q 1 b (1 b) gi D i b q gi G p (t) Equation 40 The decline parameter, D i has now been derived through the α-based deliverability equation and can be substituted into the above equation to yield: q 1 b gsc = q 1 b α(1 b) gi (1 r m OGIP G p(t)) Equation 41 Equation 41, can be used for the explicit straight-line analysis of natural gas reserves.

28 17 Extrapolating equation 41 to the x-axis will yield: [G p ] qgsc =0 = r m α(1 b) OGIP Equation 42 From where OGIP can be calculated as: OGIP = α(1 b) r m [G p ] qgsc =0 Equation 43 Additionally, Figure 1-1 illustrates the proposed straight-line analysis equation can yield OGIP 1 b directly from extrapolation to the x-axis when q gsc vs. G p (t) is plotted. Where G p (t) is defined as G p (t) = α(1 b) G r p (t). The proposed straight-line analysis equation is: m q 1 b gsc = q 1 b gi [1 G p (t) OGIP ] Equation 44 1 b q gsc Early Time Data Hyperbolic Late Time Data q 1 b gsc = q b gi [1 G p (t) OGIP ] OGIP G p (t)

29 Figure 1-1: Analysis of boundary dominated flow production data using the straight-line analysis technique. 18 It is important to place an emphasis on the applicability of the straight-line analysis equation. The straight-line analysis equation is applicable only to boundary dominated flow data that is contained within the hyperbolic window. The BDF regime, is a period of production where the boundaries of the reservoir have experienced a decline in pressure due to production. The hyperbolic window is initiated during the early times of the BDF regime. It is a production window where the Arps decline exponent can be considered a constant. The identification of the hyperbolic window for natural gas decline is made possible by use of the harmonic stem. Ye and Ayala (2012) presented an approach for the identification of the hyperbolic window using the Universal type curve. The equation for the Universal type curve is q b 1 Equation 45 Dd = 1 + t DAd b q Dd is the dimensional flow rate, and t DAd is the dimensional time. The implementation of the harmonic curve can be used to identify the hyperbolic window. Using the proposed method, the b dimensionless flow parameter q Dd follows: can be defined from the Arps hyperbolic decline equation as ( q b g 1 ) = q gi 1 + bd i t Equation 46 Where q Dd = q g q gi Equation 47

30 19 And t DAd = bd i t Equation 48 b The dimensional parameters q Dd and t DAd can be calculated by using the transformation parameters X and Y to obtain a match between production data and the harmonic stem. This process can be assisted using computers. By implementing a least squares solution, the transformation parameters can be used to translate production data to obtain the best match possible. The calculation of the dimensionless parameters is as follows: (q Dd ) b b = Y q gsc Equation 49 t DAd = X t Equation 50 Following a match between the production points and the harmonic stem the identification of hyperbolic window is possible. The figure 3-2 represents a match between the production points of a synthetic gas well and the harmonic stem.

31 20 Figure 3-2: Selection of boundary dominated flow data through matching with the harmonic stem The harmonic stem allows the identification of the start and end of the hyperbolic window. The start of the hyperbolic widow is realized when production points begin to lay on the harmonic stem. Additionally, the end of the hyperbolic window is found when the production points begin to deviate from the harmonic stem. In this study, the harmonic stem has been used to identify the start and end of the hyperbolic window.

32 21 Chapter 3 Results and Discussions The α-based deliverability equation has been derived and serves as a vehicle to map Arps decline parameters to Al-Hussainy and Ramey (1966) rigorous boundary dominated flow equation. Furthermore, the proposed methodology has presented the application of a straight-line analysis technique to calculate the OGIP of gas wells produced at a constant well flowing pressure. In this section an exploration into the performance of the α-based deliverability equation will be presented. In addition, the proposed straight-line analysis technique will also be tested for the calculation of OGIP for gas wells produced at a constant well flowing pressure. Case Studies 1 and 2 will compare the performance of the α-based deliverability equation and Al-Hussainy and Ramey (1966) rigorous boundary dominated flow equation to forecast boundary dominated flow of natural gas reservoirs. Case Study 1 will test the forecasting ability of both deliverability equations against changes in reservoir extent. Case Study 2 will investigate the forecasting ability of the deliverability equations in relation to different levels of reservoir drawdown. Next, Case Studies 3 and 4 will demonstrate the ability to use the mapped Arps parameters with the straight-line analysis technique to calculate OGIP. More specifically, Case Study 3 considers reservoirs of different sizes and Case Study 4 considers reservoir with different levels of drawdown.

33 22 Case Study 1 Case Study 1 considers a synthetic dry natural gas reservoir with a reservoir pressure of 5000 psia, a specific gravity 0f 0.55, a well flowing pressure of 100 psia, and a reservoir temperature of 200F. Additionally, the reservoir has a permeability of 0.01mD and 5% porosity. Case Study 1 contains three cases, Cases 1(a-c). For each case, the reservoir fluid properties are kept constant and only the reservoir extent is changed. For Case 1(a) the reservoir extent is considered to be 175 ft. The reservoir extent in Case 1(b) and Case 1(c) have been increased to 350 ft and 700 ft respectively. The reservoir and fluid properties for Case Study 1 can be found in Table 4-1. Table 4-1: Reservoir and fluid properties for Case Study 1, constant drawdown condition and varying reservoir extent. Permeability, k (md) 0.01 Porosity, ϕ (%) 5 Pay Zone Thickness, h (ft) 300 Specific Gravity, SG 0.55 Wellbore Radius, r w (ft) 0.25 Initial Reservoir Pressure, P i (psia) 5000 Initial Temperature, T (F) 200 Well Flowing Pressure, P wf (Psia) 100 Case 1(a) Reservoir Outer Radius, r e (ft) 175 Reported OGIP, (BCF) 0.37 Case 1(b)

34 23 Reservoir Outer Radius, r e (ft) 350 Reported OGIP, (BCF) 1.5 Case 1(c) Reservoir Outer Radius, r e (ft) 700 Reported OGIP, (BCF) 6.0 Case Study 1 considers three scenarios of varying reservoir extent, producing at a constant well flowing pressure of 100 psia. The explicit calculation of α and the Arps decline exponent, b is possible for Case Study 1(a-c) and can be calculated between initial and well flowing conditions as follows: α = ln m(ρ i) ln m(ρ wf ) ln ρ i ln ρ wf = 1.95 Equation 51 b ρ =ρi b i = r m(α 1) α = 0.48 Equation 52

35 24 Figure 4-1 demonstrates the boundary dominated flow using both of the deliverability equations for Case Study 1a. The results from the analytical calculations are compared to the numerical simulation result using an in-house simulator (in blue). The α-based deliverability equation (in red) shows a good degree of agreement in comparison to the Al-Hussainy and Ramey (1966) rigorous boundary dominated flow deliverability equation (in green). Figure 4-2 is the analytical calculations from the deliverability equations versus numerical simulation for Case Study 1b. The reservoir extent has increased to re = 350 ft. Both equations seem to be in close agreement despite an increase in reservoir extent. Figure 4-3 shows the results from Case Study 1c. The reservoir extent has been increased to re = 700 ft. Through the comparison of analytical results of both the rigorous deliverability equation as well as the α-based deliverability equation with results from simulation, it is noted that the hyperbolic decline in gas wells is captured by the deliverability equations for varying reservoir extent. Furthermore, the α-based deliverability equation compares closely with the rigorous deliverability equation for each case considered.

36 qgsc, scf/day Al-Hussainy Deliverability Alpha Based Deliverability time,days Figure 4-1: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based Deliverability Equation versus numerical simulation for Case1a.

37 qgsc, scf/day Al-Hussainy Deliverability Alpha Based Deliverability time,days Figure 4-2: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based Deliverability Equation versus numerical simulation for Case1b.

38 qgsc, scf/day Al-Hussainy Deliverability Alpha Based Deliverability time,days Figure 4-3: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based Deliverability Equation versus numerical simulation for Case1c.

39 28 Case Study 2 Case Study 2 is centered on comparing the performance of the α-based deliverability equation and Al-Hussainy and Ramey s (1966) rigorous deliverability equation for cases of different levels of reservoir drawdown. Case Study 2 entails three cases where the reservoir drawdown has been decreased in each case. Case Study 2 differs from Case Study 1, where the reservoir extent had been changed for each case. Case Study 2 considers a reservoir extent of 700 ft for all cases. The reservoir and fluid properties for Case Study 2 can be found in Table 4-2. Table 4-2: Reservoir and fluid properties for Case Study 2, constant reservoir extent and varying draw down condition. Case Study 2 Permeability, k (md) 0.01 Porosity, ϕ (%) 5 Pay Zone Thickness, h (ft) 300 Specific Gravity, SG 0.55 Wellbore Radius, r w (ft) 0.25 Initial Reservoir Pressure, P i (psia) 5000 Initial Temperature, T (F) 200 Reservoir Outer Radius, r e (ft) 700 Reported OGIP, (BCF) 6.0

40 29 Case P wf [Psia] r m a b c Case Study 2 considers three scenarios with different well flowing pressures and a reservoir extent of 700 ft. The calculation of α and the Arps decline exponent can be calculated for Case Study 2(a-c). It is important to note the trend originally identified by Voelker for the behavior of the initial decline exponent. Voelker concluded that for cases of decreasing reservoir drawdown, the value of the Arps initial decline exponent decreases as the reservoir drawdown decreases. This trend is noticed in Case Study 2, for each case of decreasing reservoir drawdown the value of the Arps initial decline exponent is decreasing. The results are calculated and plotted in Figure 4-4. Figure 4-4: Calculation of alpha and b for Case Study 2

41 30 Figure 4-5 demonstrates that the rigorous deliverability equation and the α-based deliverability equation compare closely to numerical simulation for case 2a. It is important to note that case 2a represents a scenario of high reservoir drawdown. Figure 4-6 represents case 2b, where the reservoir drawdown has decreased. A good agreement between the analytical calculations and numerical simulation is observed. Figure 4-7, shows the results from case 2c. The well flowing pressure for case 2c has been increased, a good agreement between the analytical deliverability equations and numerical simulation exists. Case Study 2 demonstrates the ability for the α-based deliverability equation to capture hyperbolic decline in gas wells for different well flowing pressures qgsc, scf/day Al-Hussainy Deliverability Alpha Based Deliverability time,days Figure 4-5: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based Deliverability Equation versus numerical simulation for Case2a.

42 qgsc, scf/day Al-Hussainy Deliverability Alpha Based Deliverability time,days Figure 4-6: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based Deliverability Equation versus numerical simulation for Case2b.

43 qgsc, scf/day Al-Hussainy Deliverability Alpha Based Deliverability time,days Figure 4-7: Al-Hussainy and Ramey (1966) Deliverability Equation and Alpha Based Deliverability Equation versus numerical simulation for Case2c. From case studies 1 and 2 the α-based deliverability equation has been tested against the Al-Hussainy rigorous boundary dominated flow equation for gas wells produced at constant drawdown. From these studies it has been observed that there is an excellent agreement between the α-based deliverability equation and Al-Hussainy and Ramey s (1966) rigorous boundary dominated flow equation. The studies are able to show that both equations are capable of forecasting boundary dominated flow of gas wells produced at constant drawdown.

44 33 Case Study 3 The proposed methodology demonstrates that it is possible to map the Arps decline parameters to Al-Hussainy and Ramey s (1966) rigorous boundary dominated flow equation through the α-based deliverability equation. Once the Arps decline parameters have been identified the method is able to implement a straight-line analysis technique to calculate the OGIP of gas wells produced at constant drawdown. Having established the ability of the α-based deliverability equation to capture the hyperbolic decline of natural gas wells produced at a constant well flowing pressure, it is now possible to investigate the performance of the straightline analysis technique to predict OGIP. In this Case Study, the straight-line analysis technique is used to predict the OGIP of a synthetic reservoir. Case Study 3 considers a total of 6 different scenarios, in each scenario the reservoir extent has been changed. For each case the well flowing pressure is held constant at 100 psia. Table 4-3 summarizes the reservoir and fluid properties for Case Study 3. Table 4-3: Reservoir and fluid properties for Case Study 3, constant drawdown condition and varying reservoir extent. Case Study 3 Permeability, k (md) 0.01 Porosity, ϕ (%) 5 Pay Zone Thickness, h (ft) 300 Specific Gravity, SG 0.55 Wellbore Radius, r w (ft) 0.25 Initial Reservoir Pressure, P i (psia) 5000

45 34 Initial Temperature, T (F) 200 Well Flowing Pressure, P wf (Psia) 100 Case 3(a) Reservoir Outer Radius, r e (ft) 175 Reported OGIP, (BCF) 0.37 Case 3(b) Reservoir Outer Radius, r e (ft) 350 Reported OGIP, (BCF) 1.5 Case 3(c) Reservoir Outer Radius, r e (ft) 700 Reported OGIP, (BCF) 6.0 Additional Cases (reservoir fluid properties maintained, r e has been changed as follows) Additional Case 3(d) Reservoir Outer Radius, r e (ft) 200 Reported OGIP, (BCF) 0.5 Additional Case 3(e) Reservoir Outer Radius, r e (ft) 500 Reported OGIP, (BCF) 3.10 Additional Case 3(f) Reservoir Outer Radius, r e (ft) 800 Reported OGIP, (BCF) 7.93

46 35 Case Study 3a considers a reservoir extent of 175 ft. Figure 4-8 demonstrates the straightline analysis applied to Case Study 3a. Next, Case Study 3b has an increase in reservoir extent to 350 ft. Figure 4-9 shows the straight-line analysis for Case Study 3b. Additionally, Case Study 3c has a reservoir extent of 700 ft. Figure 4-10 is the straight-line analysis applied to Case Study 3c. Case Study 3d has a reservoir extent of 200 ft. Figure 4-11 shows the straight-line analysis for Case Study 3d. Also, Case Study 3e has a reservoir extent of 500 ft. Figure 4-12 is the straight analysis applies to Case Study 3e Lastly, Case Study 3f has a reservoir extent of 800 ft. Figure 4-13 shows the straight-line analysis applied to Case Study 3f. For each Case Study 3(a-f ) the dashed blue lines shown in figures 4-(8-13) indicate the start and end of the hyperbolic window, and the blue arrow denotes the extrapolation of the straight-line to explicitly calculate OGIP. The straight-line analysis has been applied to data that is contained within the hyperbolic window. The production data that is included within the hyperbolic window for Case Study 3(a-f) was identified following a match between production data and the harmonic stem. The use of the harmonic stem to select production data that is found within the hyperbolic window for Case Study 3(a-f) can be found in Appendix D. Case Study 3 demonstrated the ability of the model to predict OGIP for different size synthetic reservoirs. It is noted that the model is able to accurately yield OGIP with reasonably low error. The greatest error in estimating OGIP for Case Study 3 was less than one percent. The model was able to accurately describe the OGIP of the synthetic reservoirs presented in Case Study 3.

47 Straight-line Analysis 36 qgsc ( 1-b), scfd ( 1-b) OGIP Gp*, scf x 10 8 Figure 4-8: Case 3a-re=175ft, straight line analysis using the alpha based straightline analysis approach Straight-line Analysis qgsc ( 1-b), scfd ( 1-b) OGIP Gp*, scf x 10 8 Figure 4-9: Case 3b-re=350ft, straight line analysis using the alpha based straight-line analysis approach

48 Straight-line Analysis qgsc ( 1-b), scfd ( 1-b) OGIP Gp*, scf x 10 9 Figure 4-10: Case 3c-re=700ft, straight line analysis using the alpha based straightline analysis approach Straight-line Analysis qgsc ( 1-b), scfd ( 1-b) OGIP Gp*, scf x 10 8 Figure 4-11: Case 3d-re=200ft, straight line analysis using the alpha based straightline analysis approach

49 qgsc ( 1-b), scfd ( 1-b) OGIP Gp*, scf x 10 9 Figure 4-12: Case 3e-re=500ft, straight line analysis using the alpha based straightline analysis approach Straight-line Analysis qgsc ( 1-b), scfd ( 1-b) OGIP Gp*, scf x 10 9 Figure 4-13: Case 3f-re=800ft, straight line analysis using the alpha based straightline analysis approach

50 39 Table 4-4: Results of Case Study 3 Case Reported Predicted Absolute difference OGIP[BCF] OGIP[BCF] % Case 3(a)- re=175 ft Figure 4-8 Case 3(b)-re=350ft Figure 4-9 Case 3( c)-re=700ft Figure 4-10 Case3(d)-re=200ft Figure 4-11 Case 3(e)-re=500ft Figure 4-12 Case 3(f)-re=800ft Figure 4-13

51 40 Case Study 4 In Case Study 4, the straight-line analysis technique is used to predict the OGIP of a synthetic natural gas reservoir. Case Study 4 is centered on investigating the ability of the straight-line analysis technique to calculate OGIP for natural gas reservoirs producing under different levels of reservoir drawdown. Case Study 4 includes a total of six separate cases where the reservoir drawdown is decreased. While Case Study 3 investigates the ability of the straightline analysis technique to calculate the OGIP of reservoirs of different extent, Case Study 4 considers a reservoir extent of 700 ft. for all cases. The reservoir and fluid properties for Case Study 4 can be found in Table 4-5.

52 41 Table 4-5: Reservoir and fluid properties for Case Study 4, constant reservoir extent and varying draw down condition. Case Study 4 Permeability, k (md) 0.01 Porosity, ϕ (%) 5 Pay Zone Thickness, h (ft) 300 Specific Gravity, SG 0.55 Wellbore Radius, r w (ft) 0.25 Initial Reservoir Pressure, P i (psia) 5000 Initial Temperature, T (F) 200 Reservoir Outer Radius, r e (ft) 700 Reported OGIP, (BCF) 6.0 P wf [Psia] r m, drawdown Case a b c d e F

53 42 Case Study 4a considers a well flowing pressure of 100 psia (r m = 0.99). Figure 4-14 demonstrates the straight-line analysis applied to Case Study 4a. Next, Case Study 4b has an increase well flowing pressure to 1600 psia (r m = 0.86). Figure 4-15 shows the straight-line analysis for Case Study 4b. Additionally, Case Study 4c has well flowing pressure of 2500 psia (r m = 0.69). Figure 4-16 is the straight-line analysis applied to Case Study 4c. Case Study 4d has a well flowing pressure of 3200 psia (r m = 0.52). Figure 4-17 shows the straight-line analysis for Case Study 4d. Also, Case Study 4e has a well flowing pressure of 3600 psia (r m = 0.41). Figure 4-18 is the straight analysis applied to Case Study 4e Lastly, Case Study 4f has a well flowing pressure of 4000 psia (r m = 0.30). Figure 4-19 shows the straight-line analysis applied to Case Study 4f. For each Case Study 4(a-f ) the dashed blue lines shown in figures 4-(14-19) indicate the start and end of the hyperbolic window, and the blue arrow denotes the extrapolation of the straight-line to explicitly calculate OGIP. The straight-line analysis has been applied to data that is contained within the hyperbolic window. The production data that is included within the hyperbolic window for Case Study 4(a-f) was identified following a match between production data and the harmonic stem. The use of the harmonic stem to select production data that is found within the hyperbolic window for Case Study 4(a-f) can be found in Appendix D. Case Study 4 demonstrates the ability of the model to predict OGIP for synthetic reservoirs with different reservoir drawdowns. It is noted that error in predicting OGIP increases as compared to Case Study 3, where a high reservoir drawdown is considered. This is due primarily to the length of the hyperbolic window and the amount of data points available for analysis. The loss of accuracy for cases of lower drawdown results from less data points available for analysis. This loss of accuracy is physically related to shorter hyperbolic windows for lower reservoir drawdown.