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Abstract A mathematical adaptation of the Vogel correlation was developed to obtain the drainage-area average pressure ( P ) and the skin factor (s) values from multi-stabilized-rate (flow-after-flow) tests data. A methodology is presented to smooth, validate and compute the data of these tests. The analysis of a theoretical example is presented to show the application of the methodology. The proposed graphic method Pwf vs. J contributes to verify the applicability of Vogel model to a specific data set. Introduction This paper presents a second method (a first one was previously presented by the author1) to analyze multi-stabilized-rate (flow-after-flow) test data with the main objective of estimating well-reservoir system properties. The method is based on the popular correlation of Vogel2. Conventionally, this correlation has been used for analyzing these kinds of tests with the final objective of determining the production capacity of wells. This utilization is named in the present paper as the “production approach” of the correlation. On the other hand, this paper presents the “reservoir approach” of the correlation among the current (transient-pressure) analysis techniques as an alternative and additional tool to determine the well-reservoir system properties in the special conditions of the multiphase reservoir flow.

At present, the most common method used for evaluating multiphase reservoir flow is the use of analytical single-phase flow models considering the Perrine3-Martin suggestions.

Although Vogel correlation is based only on simulated solution-gas-drive reservoir results, with a certain number of assumptions constrains, the daily engineering practice with this correlation has been verified satisfactory. A graphic

method is proposed to validate the acceptable fitting of a data set with the Vogel correlation model. Method Development Previous Theoretical Considerations. The graphical behavior of a Flow-After-Flow test run in an undersaturated reservoir is expected to be linear. The slope of this linear performance is denominated “productivity index”. In the literature has also been reported some field cases contrasting this linear rule. It is considered that these cases has been due to the presence of rate-dependent effects on the skin factor, drainage area and even on the average-pressure. These kinds of phenomena are not considered in this specific work.

On the other hand, the graphical behavior of these tests when run in solution-gas-drive reservoirs is expected to be rather curve, basically due to the increasingly restrictive effect of the gas phase to the oil flow specially near the wellbore when the pressure “drawdown” P -Pwf is increasing. In this case, the slope of the test graphical behavior is dependent of the drawdown value and is denominated “differential productivity index” described by:

( )( )wf

PP dP

dQJ wf

wf−= (1)

Muskat4 proposed the use of a constant reference slope value J0 computed at the asymptotic approximation of J(Pwf) to a pressure drawdown zero. Expressing this concept in terms of reservoir parameters with a pseudosteady state flow, a drainage-area static pressure P and a skin effect factor s present, we have:

( )( ) Pr

r

we Bk

srrhkJ

⋅

⋅+−⋅

⋅=

µ43ln2.1410 (2)

Some few attempts have been realized to describe the whole curve behavior of these tests under the saturation condition. The quantity and complexity of the variables involved offers a high difficulty to find simple analytical solutions meanwhile the reservoir numerical simulation offers only specific solutions to specific well-reservoir situations.

In 1969, Vogel presented a practical solution. He evaluated the reservoir numerical approximation of Weller5 under different solution-gas-drive reservoir conditions and found a

SPE 76719

Use of Vogel Equation to Estimate Drainage-Area Static Pressure and Skin Factor Raul A. Medina Parra, SPE, Innovaciones Medina C.A.

2 RAUL MEDINA SPE 76719

simple average correlation applicable to most of the studied cases:

( )

−

−=

2

8.02.01P

PP

PQQ wfwf

MAXPwf (3)

where QMAX and P are assumed constants. The utilized Weller reservoir model consisted of a

bounded circular homogeneous and isotropic reservoir producing in pseudosteady state regimen from a completely penetrating well in its center. The gas saturation gradients were negligible along the solution-gas-drive reservoir.

Vogel explained that the best match of the correlation (equation) was found in the early stage of the reservoir life and at the higher pressure drawdowns. Likewise, the worst matches were found with viscous crude, positive skin effect and initially saturated reservoir cases. These Vogel theoretical observations should be considered also in the application of the present method. The feasible presence of the previously mentioned rate-dependant effects in the test actual data should be also taken in account.

In spite of this some limitations, the Vogel correlation has been long verified as satisfactory in its field use since its publication. Additionally, in 1992 Wiggins6 presented an analytical deduction of this equation-correlation for the description of reservoir multiphase flow.

For the interest of this work, it is also utilized a relation defined by Standing7 between QMAX and J0:

8.10 PJQMAX

⋅= (4)

Mathematical Deduction of Present Method. To deduce the present method, we find first the derivative of the Vogel explicit function (Eq. 3), respect Pwf :

( )( )wf

wfP

wfMAX

wf

P JP

PP

QdP

dQ−=

+

−= 26.12.0

Then, solving for Pwf we have:

( ) PJQPP

wfPMAX

wf ⋅−⋅

⋅= 125.0

6.1

2

(5)

This means a linear relation, with a slope to denominate (A) and an intercept to denominate (B), which could be both determined from a Cartesian graph of Pwf vs. J(Pwf) . So:

( )125.0−

=BP (6)

and,

)(6.1

2

APQMAX ⋅

= (7)

Another useful expression can be obtained solving QMAX from Eq.5 and introducing it in Eq. 3 :

( )( )

−

−⋅

+

⋅=

2

8.02.016.12.0

PP

PP

PPPJ

Q wfwf

wf

PP

wf

wf

Arranging it conveniently:

( )

( ) ( )( )( ) ( )( )wfPwfP

PwfP

P

PQPJ

PQPJ

PJ

wfwf

wfwf

wf

28.0

2.0

0

2

2

+−

⋅+−

⋅=

And finding the (positive) root of this last second degree polynomial, we obtain an expression to determine P at each specific point data (Pwfi , Q’):

( ) ( )( )

( ) ( ) ( ) ( )

( )wf

wfwfwfwf

wf

wfwf

P

PPwfPwfP

P

PwfP

J

QQPJPJ

J

QPJP

⋅

+⋅⋅⋅+⋅⋅+

⋅

+=

10

16281

10

222

(8)↵

The skin value is derived from Eq. 2 and Eq. 4:

( ) 43Ln2.254

+−

⋅

⋅⋅

⋅⋅= we

Pr

r

MAXrr

Bk

QPhks

µ (9)

Recap of Method. The equations described above are used through the next method:

1º) Run a flow-after-flow test. This means to produce the oil well until finding a stabilized rate Q1 at the pseudosteady state bottomhole pressure Pwf,1. Then to change the rate until stabilizes in Q2 at Pwf,2 , and so on until x number of changes . The greater total x number of changes, the greater confidence of the method final results. Tubing head pressures information could be used to calculate respective required Pwf,i values if good correlation are available in the exploitation area of the testing well.

2º) Plot the Pwf,i vs. Qi data on a Cartesian plane and find the best fitting curve or equation Q’=f(Pwf) to the data. Determine the “smoothed” Q’i values on the curve or computed equation at each Pwf,i value. Determine also the respective Ji values as the slope of the tangent to the curve at each point (Pwf,i , Q’i) or its mathematical equivalent, the derivative of the function Q’(Pwf) respect Pwf at each mentioned point (Fig. 1).

3º) Plot the computed data Ji (abscissa axis) vs. the original Pwf,i data (ordinate axis) on a Cartesian graph. This proposed plot should draw a straight line if the data are consistent with Vogel model. Determine the slope (A) and the intercept (B) of the best straight line (Fig. 2).

4º) Finally obtain ( P ) through Eq. 6 and (s) through Eq. 8 and Eq. 9.

SPE 76719 USE OF VOGEL EQUATION TO ESTIMATE DRAINAGE-AREA STATIC PRESSURE AND SKIN FACTOR 3

0

500

1000

1500

2000

2500

3000

0 200 400 600 800 1000 1200 1400P wf

Q

(P w f ,i , Q i )

(P w f ,i , Q' i )Q' =f(P w f )

J (Pw f ) = - dQ' . dP w f

Fig. 1-Cartesian plot of Pwf,i vs. Qi and determination of the best fitting curve (Q’) . Computation of respective J and Q’ values for each Pwf,i .

900.000

1100.000

1300.000

1500.000

1700.000

1900.000

2100.000

0.700 0.800 0.900 1.000 1.100 1.200 1.300J

P wf

slope=(A )Intercept= (B )

Fig. 2- Proposed Cartesian (Pwf) vs. (J’) graph: determination of straight-line slope (A) and intercept (B).

Example of the Method Application A theoretical example is presented to show the application of the methodology described above. Table 1 shows the test data and the required reservoir information. The original test data was plotted on the Cartesian plane Pwf,i vs. Qi as shown in Fig. 3. Through this graph, the best-fit equation Q’=f(Pwf) was determined, and consequently the smoothed rates (Q’i ) and the differential-productivity-index values (Ji ) were obtained as shown in Table 2. The Fig. 4 shows the computed Ji and original Pwf data values in a Cartesian plot fitting a linear trend. Therefore, the method values, slope (A) and intercept (B) are determined.

TABLE 1 - DATA OF THE EXAMPLE.

i . Q

(STB/D) Pwf

(psi) . 1 1213 1700 2 1320 1600 3 1422 1500 4 1519 1400 5 1610 1300 6 1696 1200

• kkr= 40 md • h=20 ft • re=490 ft • rw=0.7 ft • µ=0.5 cp • Br=1.2 RB/STB

1,000

1,300

1,600

1,900

1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800Q(STB/D)

P wf(p

si)

Q '= -2,722E-04(P wf )2

-0,1769(P wf )+2300J = 0,0006(P wf ) +0,1769

Fig. 3- Cartesian plot of example source data. Determination of best fitting equation Q’=f(Pwf) and differential productivity index J(Pwf) functions.

TABLE 2 – COMPUTATION OF Q’i AND Ji VALUES.

From Fig. 3: Q’= -2.722E-04·(Pwf)2 -0.1769·(Pwf) +2300 J =(dQ/dPwf)= 5.444E-04·(Pwf) +0.1769

i. 1 2 3 4 5 6

Q’ (STB/D)

1213 1320 1422 1519 1610 1696

J (STB/D/psi)

1.1024 1.0479 0.9935 0.9391 0.8846 0.8302

-500

0

500

1000

1500

2000

2500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4J

P wf

slope:(A )= 1837Intercept:(B )= -325

Fig. 4- Method Cartesian (Pwf) vs. (J) graph of the example. Determination of straight-line slope (A) and intercept (B).

Finally, with the method values (A)=1837 and (B)= -325,

the sought values are determined: • From Eq.6: P = 2600 psi • From Eq.7: QMAX = 2300 STB/D • From Eq.9: s = 0.12

Conclusions A mathematical adaptation of the Vogel correlation was developed to obtain the drainage-area average pressure ( P ) and the skin factor (s) values from multi-stabilized-rate (flow-after-flow) tests data. A methodology is presented to (1°) smooth the field test data previously to analyze, (2°) validate

4 RAUL MEDINA SPE 76719

the consistency of the data with the Vogel equation model through the proposed graph Pwf vs. J , and (3°) compute the sought values. The analysis of a theoretical example is presented to show the application of the methodology.

The work lead to the following conclusions: 1. The proposed graph Pwf vs. J validates, in a non-

excusive way, the consistency of Multi-stabilized-rate (or Flow-after-flow) tests data with the Vogel equation-model.

2. The computation of the drainage-area average pressure ( P ) and the skin factor (s) values from these tests, through methods like the one presented in this paper, represents an additional tool to consider besides the conventional techniques of pressure-transient analysis. In the paper, this methodology is denominated the “reservoir approach” of the multi-stabilized-rate tests analysis.

Acknowledgments

Author acknowledges the benefits obtained from the SPE initiatives in the electronic communication media.

This work is dedicated to our father in the heaven. Nomenclature A = slope of the best straight line in the present

proposed graphic method Pwf vs. J, m2/L4t3, psi2/(STB/D)

B = intercept of the best straight line in the present proposed graphic method Pwf vs. J, m/Lt2, psi

Br = oil formation volume factor, RB/STB h = formation thickness, L, ft J(Pwf) = differential productivity index evaluated at Pwf,

L4t/m, (STB/D)/psi J0 = differential productivity index evaluated

asymptotically at Pwf = P , L4t/m, (STB/D)/ psi k = formation absolute permeability, L2, md kr = relative formation permeability to oil, fraction. P = reservoir drainage-area average pressure, m/Lt2,

psi Pwf = bottomhole flowing pressure, m/Lt2, psi Q = oil flowrate, L3/t, STB/D Q’ = oil flowrate computed from curve trend of flow-

after-flow data, L3/t, STB/D re = drainage area radius, L, ft rw = wellbore radius, L, ft s = skin effect factor, dimensionless µ = oil viscosity, m/Lt, cp Subscript i = index of flow-after-flow data References 1. Medina, R. A.: ‘“Use of Well Flowrate-Pressure Data to

Estimate Formation Drainage-Area Static Pressure,” paper SPE 72377 presented at the 2001 SPE Eastern Regional Meeting, Canton, Ohio, Oct. 17-19.

2. Vogel, J.V.: ‘“Inflow Performance Relationships for Solution-Gas Drive Well,” JPT (January 1968), 83-92.

3. Perrine, R. L.: ‘“Analysis of Pressure Buildup Curves,” Drill. and Prod. Prac., API (1956), 482-509.

4. Muskat, M.: Physical Principles of Oil Production, McGraw-Hill Book Company (1949), 341.

5. Weller, W. T.: ‘“Reservoir Performance During Two-Phase Flow,” J. Pet. Tech. (Feb., 1966) 240-246.

6. Wiggins, M. L., Russell, J. E. and Jennings, J. W.: ‘“Analytical Development of Vogel-type Inflow Performance Relationships,” SPE Journal (Dec., 1996) 355-362.

7. Standing, M. B.: ‘“Concerning the Calculation of Inflow Performance of Wells Producing by Solution Gas Drive,” J. Pet. Tech. (Sept., 1971) 1141-1142.

SI Metric Conversion Factors bbl x1.589 873 E-01 = m3 cp x 1.0* E-03 = Pa·s ft x 3.048* E-01 = m md x 9.869 233 E-04 = µm2 psi x 6.894 757 E-03 = Mpa *Conversion factors are exact.