Production Rate Decline-Based Models for Oil Reservoir Performance Prediction in Niger Delta

Production Rate Decline-Based Models for Oil Reservoir Performance Prediction in Niger Delta

Anietie N. Okon, Ukpong U. Edem and Daniel T. Olagunju

Department of Chemical & Petroleum Engineering, University of Uyo, Uyo-AKS, Nigeria.

Abstract – In the oil and gas industry, reservoir performance analyses are established to facilitate the field development and planning strategies. One of the available tools to perform this analysis is production rate decline analysis. Thus, several models: Arps, Reciprocal and Quadratic model have been developed and fitted to handle this estimation in some oil producing fields in the world. In Niger Delta, no fitted production rate decline models are available in the public domain for the prediction of oil reservoir(s) production performance. In this course, the Arps model: Exponential, Harmonic, and Hyperbolic, Reciprocal and Quadratic models were fitted using multivariate analysis to predict the production history: production rate (q) and cumulative production (Np) of an oil field in Niger Delta. The fitted models have decline constants of 0.000353day-1, 0.000434day-1, 0.000332day-1, 0.000403day-1 and 0.000189day-1 for Exponential, Harmonic, Hyperbolic, Reciprocal and Quadratic, respectively. For the Hyperbolic model, the obtained decline exponent (b) is 0.5957. Also, the statistical validation of these fitted models resulted in absolute error, standard deviation and coefficient of determination of 0.0089, 0.4433 and 0.9991 for Exponential, -0.0011, 0.4455 and 0.9935

for Harmonic and 0.0005, 0.4460 and 0.9939 for Hyperbolic.

Additionally, the reciprocal and quadratic models have absolute error, standard deviation and coefficient of determination of -0.0136, 0.4452 and 0.9883, and 0.0001, 0.4460 and 0.9998, respectively. These statistical results indicate that the Quadratic and Exponential models are more prolific models than the other for predicting the production rate decline of the oil field. Therefore, the fitted Quadratic and Exponential models can be used as a quick and robust tool to predict the reservoir performance on the XYZ oil field in the Niger Delta.

Keyword: Production rate decline analysis, Oil reservoir, Arps models, Reciprocal model, Quadratic model, Niger Delta

1. INTRODUCTION

The early exploration of oil and gas in the Niger Delta region of Nigeria dates back to the early 1950s when the first commercial reserve was discovered at Oloibiri, in the present day Bayelsa State in 1956 (Okon, 2010). Since then, exploration and production of oil and gas activities have been on the increase; as more discoveries are made. In other words, several oil fields have been developed in the Niger Delta. It is worth noting that in oil field development plan, reserves have to be well established before the companys limited available resource is expended on the execution of the given project. This strategic decision is made to avoid engaging in a risky and unprofitable venture (Makinde et al., 2011). Therefore, one of the sole responsibilities of a petroleum engineer; especially

reservoir engineer is to estimate the recoverable reserves of a reservoir. Brons (1963) mentioned that choosing the reserves estimation method is critical for accurate forecast that are, in turn, vital for sound managerial planning. Thus, one of the approaches of estimating the reservoir recoverable reserves is the reservoir performance analysis using production rate decline analysis. Hook (2009) added that decline curve analysis (DCA) is the frequently used approach for recoverable reserves estimation as it is a function of rate of decline of petroleum extraction over a period of time. Therefore, this approach is based on the assumptions that the trend of production history of oil and/or gas reservoir(s) and factors causing the historical decline remain unchanged during the forecast period (Okon et al., 2017). Hence, these factors include both reservoir conditions and operating conditions. Ahmed and McKinney (2005) maintained that some of the reservoir factors that affect the decline rate are pressure depletion, number of producing wells, drive mechanism, reservoir characteristics, saturation changes and relative permeability. Also, Ahmed (2006) added that the operating conditions that influence the decline rate include: separator pressure, tubing size, choke setting, workovers, compression, operating hours and artificial lift. Although the production rate decline analysis comes with its limitations, the biggest advantage of this reservoir(s) recoverable reserves estimation method is that, it is virtually independent of the size and shape or the actual drive mechanism of the reservoir (Doublet et al., 1994). Thus, the detailed description of the reservoir or production data is not required to perform this production rate decline analysis. Arps in 1945 put together the earlier works by Arnold and Anderson (1908), Cutler (1924), Roeser (1925) and Miller (1942) to develop an all-inclusive empirical model for production rate decline analysis. This Arps equation as expanded in Table 1 categorized production rate decline curve into exponential, harmonic and hyperbolic declines. The hyperbolic equation is the universal Arps equation of which the exponential and harmonic declines are special cases (Makinde et al., 2011). Fetkovitch in 1980 presented type curve approach to analyze production rate decline data. The type curve consists of two segments, that is, transient and boundary dominated production periods. The transient portion comes from constant pressure type curve developed by Van Everdingen in 1949 while the boundary dominated portion is the same as Arps (1945) depletion stems (Okon et al., 2017). Further works by Blasingame and Lee (1988) and

Agarwal et al. (1999) are similar to Fetkovitchs type- curves for analysis of production data. The major difference is the introduction of flowing pressure data in the production rate to solve for hydrocarbon in-place analytically. Recently, Reese et al. (2007) and Johnson et al. (2009) presented reciprocal and quadratic models respectively, for production rate decline analysis. The reciprocal model assumes that flowing well bottom-hole pressure is approximately constant and was used to estimate hydrocarbon reserves using only rate-time production data. This model requires a plot of the reciprocal of production rate (1) against the cumulative

decline curves of various forms as mentioned can be used to create significant outlooks for hydrocarbon production of a single well or an entire field. However, it should be noted that in many field cases a single curve is not sufficient to obtain a good fit and it may be necessary to use a combination of curves to obtain good agreement (Haavardsson and Huseby, 2007). Though, the mentioned models have been tested and validated to be effective in some regions of the world, they are yet to be used as much as the Arps approach; especially in the Niger Delta. Then, Okon et al. (2017) fitted the production rate decline models for gas field in the Niger Delta and established the

production to production rate ratio

quadratic model as the most efficient and robust model for

( ) as presented in Table 1. Additionally, the quadratic model as developed by Johnson et al. (2009) is based on the Semi-analytical

formulation by Blasingame and Rushing (2005) and

Empirical formulation by Ilk et al. (2008). In essence,

analyzing the reservoir performance of the gas field. Therefore in this paper, the various production rate decline models were fitted and validated for use as quick and robust tools for oil reservoirsperformance predictions in the Niger Delta.

Table 1: Production Rate Decline Models

S/N Author(s) Models Production Rate (qt)

1. Exponential

   q q e Dit

   t i

2. Harmonic

   Arps (1945)

3. Hyperbolic

   qt

   qt

   qi

   1 Dit

   qi

   1

   1

4. Reese et al. (2007) Reciprocal

   1 Dibt b

   1 Di N p

   q q q q

   t i i t

   D2 2

5. Johnson et al. (2009) Quadratic

q q D N

i N

q

t i i p p i

where:

= Production Rate at time t, Stb/day

= Initial Production Rate, Stb/day t = time, Days

= Decline Constant, Day 1

b = Decline Exponent

= Cumulative Oil Production, Stb

1. MATERIALS AND METHODS

   2.1. Data Acquisition and Models Fitting

   The oil production data of the XYZ oil field in the Niger Delta for a period of about 12.32 years (i.e., 4500 days) were obtained from 25 wells. These data include: oil production rate (qt) and cumulative oil production (Np) of the XYZ oil field. The range of these oil production data is presented in Table 2. Multivariate analyses were performed based on the existing oil production rate decline models available in the literature, these include: Arps (i.e., Exponential, Harmonic and Hyperbolic), Reciprocal and

   Quadratic model to determine the decline constant () and decline exponent (b) – in terms of Hyperbolic model for the XYZ oil field in Niger Delta. Using the Microsoft Excel Solver, the generalized reduced gradient (GRG) iteration protocol was used to perform nonlinear regression to fit the mentioned production rate decline models to the field history production data. The obtained production rate decline constant () for the various models and decline exponent (b) for Hyperbolic model are presented in Table

2. Also, the fitted production rate decline and cumulative production models are presented in Table 4.

Table 2: Summary of Production data

Type of data Range

Production Rate (), Stb/Day 501.30 5998.90 Cumulative Production (), Stb 5998.90 14724688 Number of Wells 25

Period of Production (t), Day 4500

2.2 Models Validation

The various fitted models predictions were compared with the obtained field production data from the XYZ oil field. The parameters considered for comparison were field oil production rate () and fitted model predicted oil production rate () against time (), field cumulative oil production () versus fitted model predicted cumulative oil production (), and field cumulative

oil production () and fitted model predicted

mean square error (Erms), coefficient of determination (r2), standard deviation (SD) and normalized standard deviation (NSD). The respective mathematical equations of these statistical tools are expanded in Appendix A, and the results of the statistical analyses are presented in Table 5.

3. RESULTS AND DISCUSSION

The performed multivariate analysis resulted in different production rate decline constants (); as depicted in Table 3, for the various production rate decline models. This

cumulative oil production (

) against time (). These

result indicates that the production rate decline constant

obtained for the XYZ oil field in Niger Delta depends on

comparisons are presented in Log-Log plot owing to the extremity of the data range. In addition to these comparisons, statistical analyses were performed to validate the reliability of the fitted oil production rate decline models predicted values. The statistical methods used are the average error (Eavg), absolute error (Eabs), root

the production rate decline model. Therefore, establishing the decline constant () and the rate decline model that will accurately predict the production performance of the XYZ field is of essence. Hence, Table 4 present the fitted production rate decline models for evaluating the XYZ field performance.

Table 3: Decline Constants for the Fitted Production Rate Decline Models

S/N Constants Production Rate Decline Models

Table 4: Production Rate Decline and Cumulative Production Fitted Models

S/N Models Flow Rate (Qt) Cumulative Production (Np)

1. Exponential

   q q e0.000353t

   N p 2824.85qi qt

   t i t

2. Harmonic

   q qi

   N 1.5107 ln qi

   t p

   1 0.000434t

   q qi

   t

   qt

   i

   7 q0.4034

3. Hyperbolic

   t 1 0.000198t1.676

   N

   p

   t

   4.610

   q 1 t

   q0.4034

   i

   1 1 4.03104 N p

   N 2481.38q q

4. Reciprocal

   t i p p

   pt

   i t

   qt qi qi qt

   pt i t

5. Quadratic

   q q 3.07104 N

   5.971012 N 2

   N 3257.33q q

   Table 5: Statistical Validation Analysis

   	Exponential	Harmonic	Hyperbolic	Reciprocal	Quadratic
   1. Average Error (Eavg)	-0.0089	-0.0011	0.0005	-0.0136	0.0001
   2. Absolute Error (Eabs)	0.0272	0.0287	0.0221	0.0488	0.0001
   3. Root Mean Square Error (Erms)	0.0013	0.0011	0.0009	0.0039	0.0000
   4. Coefficient of Determination (r2)	0.9991	0.9935	0.9939	0.9883	0.9998
   5. Standard Deviation (SD)	0.4433	0.4455	0.4460	0.4452	0.4460
   6. Normalized Standard Deviation 0.2974		0.2986	0.3005	0.2986	0.2992

   S/N Validation Tools Production Rate Models

   (NSD)

   3.1. Comparison of Fitted Models Predictions

   As earlier alluded that nonlinear regression using generalized reduced gradient (GRG) iteration protocol was performed with the obtained field production data to determine the decline constants of the various rate decline models. The fitted production rate decline models predictions were compared with the obtained field production data. The results are presented in Figures 1 through 15 for the various production rate decline models. Additionally, the statistical validations of the fitted models are present in Table 4.

   1. Arps Models:

      The obtained results based on the Arps models are presented in Figures 1 through 9. The exponential approach for the decline production rate analysis is presented Figures

      1

      Production Rate (q); MStb/Day

      1 through 3. The comparison of the fitted models (as presented in Table 4) prediction with field production data in Figure 1 indicates an alignment of the predicted oil production rate with the actual field oil production rate data. Figure 2 also shows the comparison of the field cumulative oil production data with the model predicted cumulative oil production on cumulative oil production(Np) versus time (t) plot. The Figure depicts a close prediction of the field XYZ cumulative production data with the fitted model. Then, the degree of this close prediction between the field cumulative production and

      predicted cumulative production is evident in Figure 3 with the coefficient of determination (r2) of 0.9991. On the other hand, Figures 4 through 6 show the predictions of the harmonic approach rate decline model. The fitted models prediction based on the mentioned approach resulted in close prediction of the XYZ field production data, as the predicted data closely aligned with the field data. These alignments of the predicted data are observed in both the production rate time plot (Figure 4) and cumulative production time plot (Figure 5). In addition, the statistical evaluation of predicted cumulative production and field cumulative production resulted in coefficient of determination (r2) of 0.9935; as depicted in Figure 6.

      Also, Figures 7 through 9 present the comparison of the fitted models predictions based on hyperbolic approach with the field production data. Similarly, the obtained results for the production rate time and cumulative production time plots show close alignment of the predicted data and the field production data; with a coefficient of determination (r2) of 0.9939. In all, the fitted Arps models prediction of the XYZ field production performance is promising, as the different models: exponential, harmonic and hyperbolic model with decline constants of 0.000353/day, 0.000434/day and 0.000332/day respectively resulted in close prediction of the Niger Delta field production data.

      Field

      Model

      10

      1

      10

      100

      Time (t); Days

      1000

      10000

      0.1

      Figure 1: Field and Predicted Oil Production Rate vs. Time Plot (Exponential Model)

      Field

      Model

      100

      10

      1

      0.1

      0.01

      0.001

      1

      10

      100

      Time (t); Days

      1000

      10000

      8

      12

      Cumulative Production (Np); MMStb

      Field Cumulative Production

      (Npfield)

      Figure 2: Field and Predicted Cumulative Production vs. Time Plot (Exponential Model)

      16

      R² = 0.9991

      0

      0

      4

      8

      12

      16

      Predicted Cumulative Production (Npmodel)

      1

      10

      Model

      Field

      4

      Production Rate (q); MStb/Day

      Figure 3: Comparison of Field and Predicted Cumulative Oil Production (Exponential Model)

      1

      10

      100

      Time (t); Days

      1000

      10000

      0.1

      Figure 4: Field and Predicted Oil Production Rate vs. Time Plot (Harmonic Model)

      100

      10

      1

      0.1

      0.01

      0.001

      Field Model

      8

      12

      16

      10000

      1000

      100

      Time (t); Days

      10

      1

      Cumulative Production (Np); MMStb

      Field Cumulative Production

      (Npfield)

      Figure 5: Field and Predicted Cumulative Production vs. Time Plot (Harmonic Model)

      R² = 0.9935

      0

      0

      4

      8

      12

      16

      Predicted Cumulative Production (Npmodel)

      4

      Production Rate (q); MStb

      Figure 6: Comparison of Field and Predicted Cumulative Oil Production (Harmonic Model)

      Field

      Model

      10

      1

      0.1

      1

      10

      100

      Time (t); Days

      1000

      10000

      Figure 7: Field and Predicted Oil Production Rate vs. Time Plot (Hyperbolic Model)

      Field

      Model

      100

      10

      1

      0.1

      0.01

      0.001

      1

      10

      100

      Time (t); Days

      1000

      10000

      8

      12

      Field Cumulative Production

      (Npfield)

      Cumulative Production (Np); MMStb

      Figure 8: Field and Predicted Cumulative Production vs. Time Plot (Hyperbolic Model)

      16

      R² = 0.9939

      0

      0

      Predicted Cumulative Production (Npmodel)

      16

      12

      8

      4

      4

      Figure 9: Comparison of Field and Predicted Cumulative Oil Production (Hyperbolic Model)

   2. Reciprocal Model

      For the fitted production rate decline model based on reciprocal approach, Figures 10 through 12 depicts the obtained results for the comparison of the predicted and field production data. That is, production rate time relationship and cumulative production time relationship. The predicted cumulative production of the field XYZ from the fitted Reciprocal model resulted in a close alignment with the actual field XYZ cumulative production data (Figure 11). However, the predicted production rate in Figure 10 shows slight difference at the early year of production, but later aligns with the actual

      field production rate data. In this connection, Okon et al. (2017) maintained that this is attributed to the reciprocal nature of the modeled production rate data which restrict the flexibility of the model; since the reciprocated production rate data are returned to normal form to compare with the actual field data. This effect is also observed in the comparison of the predicted cumulative production () with the actual field cumulative production () in Figure 12. The disparity at the early predicted production data accounted for the obtained coefficient of determination (r2) of 0.9883; which is the least among the fitted production rate decline models.

      10

      Field

      Model

      10000

      1000

      100

      Time (t); Days

      10

      1

      0.1

      1

      Cumulative Production (Np); MMStb

      Production Rate (q); MMStb/Day

      Figure 10: Field and Predicted Oil Production Rate vs. Time Plot (Reciprocal Model)

      Field

      Model

      100

      10

      1

      0.1

      0.01

      0.001

      1

      10

      100

      Time (t); Days

      1000

      10000

      8

      12

      R² = 0.9883

      16

      Field Cumulative Production (Npfield)

      Figure 11: Field and Predicted Cumulative Production vs. Time Plot (Reciprocal Model)

      0

      4

      8

      12

      16

      Predicted Cumulative Production (Npmodel)

      0

      4

      Figure 12: Comparison of Field and Predicted Cumulative Oil Production (Reciprocal Model)

   3. Quadratic Model

   Figures 13 through 15 depict the fitted Quadratic model predictions of the XYZ field production data. Figure 13 presents the comparison of the predicted production rate

   models prediction. Interestingly, the fitted model predicted cumulative production aligned closely with the actual field cumulative production; as observed in Figure 14. Thus, the alignment of these data resulted in coefficient of

   (

   ) with the actual field production rate (

   ).

   determination (r2) of about 1.0 (i.e., 0.9998). Additionally,

   1

   Production Rate (q); MStb/Day

   From the Figure, there is a close alignment of the predicted and the actual production data; especially at the early production period of the XYZ field. The latter period production prediction shows slight difference between the predicted and actual data; as also noticed with the Arps

   the statistical evaluation of the fitted Quadratic model as presented in Table 5 shows that this model gives a better prediction of the XYZ field production performance than the other production rate decline models.

   Field

   Model

   10

   1

   10

   100

   Time (t); Days

   1000

   10000

   0.1

   Cumulative Production (Np);

   MMStb

   Figure 13: Field and Predicted Oil Production Rate vs. Time Plot (Quadratic Model)

   100

   10

   1

   0.1

   0.01

   0.001

   Field

   Model

   1

   10

   100

   Time (t); Days

   1000

   10000

   p>Figure 14: Field and Predicted Cumulative Production vs. Time Plot (Quadratic Model)

   16

   R² = 1

   0

   0

   4

   8

   12

   16

   Predicted Cumulative Production (Npmodel)

   4

   8

   12

   Field Cumulative Production

   (Npfield)

   Figure 15: Comparison of Field and Predicted Cumulative Oil Production (Quadratic Model)

   Finally, a comparison of all the fitted models predictions depicted in Figures 1-B and 2-B in Appendix B, indicate that the Arps and Quadratic models have about the same predictions of the field production data for both production rate and cumulative production. The Reciprocal model predictions are very close to the actual field production data at the later year of production than its predictions of the early period of production. Therefore, the fitted Exponential and Quadratic models can be used as a quick tool to predict the production performance of the XYZ oil field in the Niger Delta.

   4. CONCLUSION

   In the petroleum industry, reservoir performance analysis using production rate decline approach has centered on the traditional Arps models; even the recently propounded reciprocal and quadratic models are not left out. However, no available literature has fitted these models to evaluate their prediction capabilities of oil field(s) in the Niger Delta. Therefore, this paper compares the potential of the fitted production rate decline models based on production data obtained from the XYZ oil field in Niger Delta and the following conclusions were drawn:

   1. the established decline constants () of the various models: Exponential, Harmonic, Hyperbolic, Reciprocal and Quadratic are 0.000353day-1, 0.000434day-1, 0.000332day-1, 0.000403day-1 and 0.000189day-1 respectively for the XYZ oil field in the Niger Delta;

   2. the Arps models predicted the XYZ oil field performance with an absolute error, standard deviation and coefficient of determination of – 0.0089, 0.4433 and 0.9991 for Exponential, – 0.0011, 0.4455 and 0.9935 for Harmonic and 0.0005, 0.4460 and 0.9939 for Hyperbolic;

   3. the fitted Reciprocal model prediction of the oil field production performance with an absolute error, standard deviation and coefficient of determination of -0.0136, 0.4452 and 0.9883 respectively; and

   4. the Quadratic model accurately predicted the field production history with an absolute error, standard deviation and coefficient of determination of 0.0001, 0.4460 and 0.9998, respectively. In lieu of the established coefficient of determination, the quadratic and exponential models can be used as quick and robust tools to predict the XYZ oil field performance in the Niger Delta.

REFERENCES

1. Agarwal, R. G., Gardner, D. C., Kleinsteiber, S. W. and Fussell, D.

   D. (1999). Analysing Well Production Data Using Combined-Type- Curve and Decline-Curve Analysis Concepts. Society of Petroleum Engineers Reservoir Evaluation and Engineering Journals, pp. 478- 486.

2. Ahmed, T. (2006). Reservoir Engineering Handbook. Elsevier Incorporation, Oxford, United Kingdom.

3. Ahmed, T. and McKinney, P. D. (2005). Advanced reservoir Engineering. Gulf Professional Publishing, Elsevier, Oxford, United Kingdom.

4. Arnorld, R. and Anderson, R. (1908). Preliminary Report on Coalinga Oil District Fresno and Kings Countries. United State Geological Survey Bulletin. Vol 79, p. 357.

5. Arps, J. J. (1945). Analysis of Decline Curves. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers. Vol. 160, pp. 228-247.

6. Blasingame, T. A. and Lee, W. J. (1988). Variable-Rate Reservoir Limits Testing. Society of Petroleum Engineers paper presented at Permain Basin Oil and Gas Conference held in Midland, Texas, USA, March 13-14.

7. Blasingame, T. A. and Rushing, J. A. (2005). A Production-Based Method for Direct Estimation of Gas-in-place and Reserves. Paper presented at the Society of Petroleum Engineers, Eastern Regional Meeting, Morgantown, West Virginia. 14-16 September.

8. Brons, F. (1963). On the Use and Misuse of Production Decline Curves, Producers Monthly. Paper Presented at American Petroleum Institute. 801-39E, Denver, Colarado 16-18

9. Cutler, W. W. (1924). Estimation of Underground Oil Reserves by Well Production Curves. United State Bureau of Mines Bulletin 228.

10. Doublet, L. E., Pande, P. K., McCollom, T. J. and Blasingame, T. A. (1994). Decline Curve Analysis using Type Curve – Analysis of Oil Well Production Data using Material Balance Time: Application of Field Cases. Society of Petroleum Engineers paper presented at the International Petroleum Conference and Exhibition at Veracruz, Mexico, October 10-13.

11. Fetkovich, M. J. (1980). Decline Curve Analysis Using Type Curves. Journal of Petroleum Technology, Vol. 32, No. 6, pp. 1065- 1077.

12. Hook, M. (2009). Depletion and Decline Curve Analysis in Crude Oil Production. M.Sc. Thesis submitted to the Department of Physics and Astronomy, Uppsala University.

13. Haavardsson, N. F. and Huseby, A. B. (2007). Multisegment Production Profile Models A Tool for Enhanced Total Value Chain Analysis. Journal of Petroleum Science and Engineering, Vol. 58, No. 1, pp. 325-338.

14. Ilk, D., Perego, A. D., Rushing, J. A. and Blasingame, T. A. (2008). Exponential vs. Hyperbolic Decline in Tight Gas Sands Understanding the Origin and Implications for Reserve Estimates Using Arps' Decline Curves. Paper presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 21-24 September.

15. Johnson, N. L., Currie, S. M., Ilk, S. M. and Blasingame, T. A. (2009). A Simple Methodology for Direct Estimation of Gas-in- place and Reserves Using Rate-Time Data. Paper presented at the Society of Petroleum Engineers, Rocky Mountain Petroleum Technology Conference held in Denver, Colorado, USA, 1416 April 2009.

16. Makinde F. A, Orodu O. D., Ladipo A. O. and Anawe P. A. L. (2011). Cumulative Production Forecast of Oil Well Using Simplified Hyperbolic-Exponential Decline Models. Global Journal of Researches in Engineering, Vol.12 , No 2, pp.13-26.

17. Miller, H.C. (1942). Oil Reservoir Behavior Based upon Pressure- Production Data. United State Bureau of Mines Bulletin Vol. 3634.

18. Okon, A. N. (2010). Formulation of Water-Based Drilling Fluid using Local Clay. B.Eng. Project submitted to the Department of Chemical and Petroleum Engineering, University of Uyo. Unpublished.

19. Okon, A. N., Olagunju. D. T. and Akpabio, J. U. (2017). Rate Decline-Based Models for Gas Reservoir Performance Prediction in Niger Delta Region. British Journal of Applied Science and Technology, Vol. 19, No. 6, pp. 1-14.

20. Reese, P. D., Ilk, D. and Blasingame, T. A. (2007). Estimation of Reserves Using the Reciprocal Rate Method. Paper presented at the Society of Petroleum Engineers, Rocky Mountain Oil and Gas Technology Symposium held in Denver, Colorado, U.S.A., 1618 April.

21. Roeser, H. M. (1925). Determining the Constants of Oil- Production Decline Curves. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers. Vol. 71, pp.13-15.

22. Van Everdingen, A. F. and Hurst, W. (1949). The Application of the Laplace Transformation to Flow Problems in Reservoirs. Paper presented at American Institute of Mining, Metallurgical and Petroleum Engineers Annual meeting in San Francisco, USA, February 13-17.

APPENDIX A

The equations used to perform the statistical analysis of the fitted models prediction and field cumulative production:

1. Average Error:

   1 n N p N p

   Eavg

   field mod el

   n N

   1-A

   i1

   p field

2. Absolute Error:

   1 n N p N p

   Eabs

   field mod el

   n N

   2-A

   i1/p>

   p field

3. Root Mean Square Error:

   2

   1 n N p N p

   E field mod el

   3-A

   rms

   n i1

   N p field

4. Coefficient of Determination:

   n

   p p

   2

   N N

   field mod el

   n

   N N p

   r2 1 i1

   4-A

   i1

   2

   p field mod el

5. Standard Deviation:

   n N N 2 n N

   • N p

     2

     S 1

     p field pmod el

     • p field mod el

     5-A

     D n N N

     i1

     p field

     i1

     p field

6. Normalized Standard Deviation

2

1 n N p N p

NS 100

field mod el

6-A

D n 1

i1

N p field

where:

= Field Cumulative Production

= Predicted Cumulative Production n = Total Number Field Production Data

= Average Predicted Cumulative Production

APPENDIX B

10

Production Rate (q); MStb/day

Comparison of all the fitted models predictions with the actual field production data of the XYZ field in Niger Delta:

Field Data

Exponential Model

Harmonic Model

Hyperbolic Model Reciprocal Model Quadratic Model

1

10

100

Time (t); Days

1000

10000

0.1

1

Cumulative Production (Np); MMStb

Figure 1-B: Comparison of Field and Predicted Production Rate (All Models)

Field Data

Hyperbolic Model

Exponential Model

Reciprocal Model

Harmonic Model

Quadratic Model

100

10

1

0.1

0.01

0.001

1

10

100

Time (t); Days

1000

10000

Figure 2-B: Comparison of Field and Predicted Cumulative Production (All Models)