SPE 135804

Direct Method of Estimating Average Reservoir Pressure for Flowing Oil and Gas Wells Ram G. Agarwal, SPE, Petrotel Inc.

Copyright 2010, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Florence, Italy, 19–22 September 2010. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract A knowledge of the average reservoir pressure ( p ) and its changes as a function of time or cumulative production is essential to determine the oil-in-place (OIP) or original gas-in-place (OGIP), to estimate reserves and to track and optimize reservoir performance. The common practice of determining p in moderate permeability reservoirs has been to run pressure buildup tests. In the current economic environment, buildup tests are almost non-existent except for very expensive exploratory wells. Moreover, time required for a pressure buildup test to reach p in low permeability reservoirs is prohibitively long. Fortunately, flowing pressures and rate data are continually collected from oil and gas wells. Data quality and quantity is usually good especially from wells installed with permanent pressure gauges. Such data for gas wells is currently being analyzed by assuming OGIP and estimating p required for calculating pseudo time. This is done in an iterative manner for using advanced decline curve analysis methods. The purpose of this paper is to discuss a new finding that will enable direct estimation of p using flowing pressures and rate data obtained from oil and gas wells during the pseudo steady-state flow period. In theory, pseudo steady-state flow requires that a well is produced at a constant rate. However, this limitation can be easily removed based on the work published in the SPE literature by this author and others whereby variable rate data can be converted to constant rate production data. The significance of the subject paper is that it will permit: a) direct determination of p using flowing wellbore pressures and rate data thus facilitating estimates of OGIP and OIP, b) estimation and/or validation of the value of the initial reservoir pressure ( )ip , which is normally suspect, and finally, c) enhancement or possible elimination of the current iterative process used for determining OGIP by advanced decline curve analysis methods. Introduction Estimation of oil-in-place, gas-in-place, and reserves is an integral part of reservoir development and management. This requires knowledge of initial reservoir pressure and average reservoir pressure. Initial reservoir pressure should be readily available from newly drilled wells. Unfortunately, a true value of initial reservoir pressure is rarely available. For new wells, initial reservoir pressure is often obtained from pressure buildup tests. The quality of such estimation depends on wellbore storage, the reservoir permeability and the duration of the well test. The same limitations apply to the estimation of average reservoir pressure. The problem is not too serious in case of moderate to high permeability reservoirs but it becomes practically impossible to estimate reservoir pressure in low permeability reservoirs. Moreover, average reservoir pressure values are needed as a function of production time or cumulative production. In the current economic environment, pressure buildup tests are rarely conducted. This poses a challenging situation for reservoir engineers to estimate OIP, OGIP, and reserves.

2 SPE 135804

The current trend in the industry is to utilize flowing well data or production data to estimate oil-in-place, gas-in-place, and reserves. Methods such as flowing material balance method and advanced decline curve analysis method are used. Fortunately, flowing pressure and rate data are available on a continuous basis especially from wells which have been installed with permanent pressure gauges. Purpose The purpose of this paper is to discuss a new finding to directly estimate average reservoir pressure ( p ) using flowing pressures and rate data collected from oil and gas wells during pseudo steady-state flow period. First, we will review background material and the pertinent transient and pseudo steady-state flow equations. Significance of these equations will be illustrated by means of figures and example problems. Second, it will be shown that the contacted oil and gas volumes and drainage areas can be estimated using flowing pressure as a function of time. Lastly, it will be demonstrated that the method proposed in this paper to calculate investigated reservoir volume and reservoir area is superior to the radius of investigation concept which has been commonly used in industry and often gives misleading results. Background Material for Oil Transient and Pseudo Steady-State Flow Conditions When a well is opened to flow, it is under a transient condition. It remains under this condition until the production from the well affects not only one or two boundaries but the total reservoir system. Then, the well is said to be under a pseudo steady-state (pss) flow condition or a boundary dominated flow condition. During the transient period, transient rate and pressure data is used to determine reservoir flow capacity and near wellbore condition (damage or improvement) denoted by skin effect. Such transient data for hydraulically fractured wells, horizontal wells, and other types of wells is used to derive parameters appropriate for such wells. During the pseudo steady-state period, the pss data is used to estimate oil and gas in place and reserves. This is described in papers by Agarwal et al. (1999) and Palacio and Blasingame (1993) and also in associated references cited in those papers.

Review of Pertinent Flow Equations for Oil Equations for radial flow systems during the transient and pseudo steady-state flow regimes have been widely published and are very well known in the petroleum literature. One example is the paper by van Everdingen and Hurst, which was published as back as 1949. However, these equations are repeated here for the ease of discussion and the continuity of the paper. Forms of equations as presented by Ramey and Cobb (1971) and also by Earlougher (1977) will be utilized here. These equations are of general nature. In presenting the following equations, it will be assumed that the well is located in the center of a square shaped reservoir of area, A. However, this technology will also be applicable to other types of closed systems. During the Transient Flow

( ) ( )[ ]⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 2

4ln5.0ln5.0wr

ADAtDAtDp

γ (1)

In equation (1) and elsewhere in this paper, the value of γ is 1.781. γ is the exponential of Euler’s constant whose value is equal to 0.5772. Substituting the value 1.781 for γ , we get,

( ) ( )[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2459.2ln5.0ln5.0w

DADAD rAttp (1a)

Equations (1) and (1a) are applicable when DAt is greater than 0.000025 and less than 0.1 for circular or square shaped reservoirs. Such values for DAt will be different for other shaped reservoirs and/or asymmetrical well location.

SPE 135804 3

Dimensionless Variables Dimensionless Pressure for single phase liquid,

( )

μqBpkhpD 2.141

Δ= (2)

( )wfi ppp −=Δ for drawdown data (2a) Dimensionless Time based on the area, A,

( ) Ac

kttit

DA μφ410637.2 −×= (3)

Dimensionless Time based on 2wr ,

( ) 2

410637.2wit

Dr rcktt

w μφ−×= (4)

Relationship between DAt and wDrt

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Artt w

DrDA w

2

(5)

Material Balance As shown by Ramey and Cobb (1971), for a single phase liquid in a closed reservoir, reservoir volume produced is equal to the expansion of initial reservoir fluids that is,

( )ppcAhtBq it −= φ24

615.5 (6)

In equation (6) and elsewhere in this paper, time, t is in hours. Rearranging parameters in equation (6), we get,

( ) ( )DA

iDADmb t

Bqppkhtp πμ

22.141

=−= (7)

And define it as a material balance dimensionless pressure. During the Pseudo Steady-State Flow

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

4ln5.02wA

DADAD rCAttp

γπ (8)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2459.2ln5.02wA

DADAD rCAttp π (8a)

Equations (8) and (8a) are applicable for 1.0≥DAt for circular or square shaped reservoirs with the well in the center.

4 SPE 135804

In equations (8) and (8a), the first term on the right hand side of both equations accounts for the pressure drop due to depletion (Material Balance) and the second term accounts for the pressure drop due to flow from the reservoir into the well (Darcy Equation). Time, t, in dimensionless time equations is in hours. The symbol AC denotes Dietz’s shape factor. Its value depends on the shape of the reservoir and the position of the well in reference to the reservoir boundaries. Its values are 30.8828 and 31.62 respectively for a well in the center of a square and circle. Values of AC for various other reservoir shapes and well locations are tabulated by Earlougher (1977). It also lists the values of

DAt at which pseudo steady-state equation such as equations (8) and (8a) are applicable. Such values of AC and DAt can also be generated for cases where published values are not available. Prime Derivatives and Log Derivative For this discussion, prime derivatives are those Dp or pΔ variables which are differentiated directly with respect to dimensionless time or real time. Log derivatives are those Dp or pΔ variables which are differentiated with respect to natural log of dimensionless time or real time. Each type of derivative provides useful insight into the behavior of transient and pseudo steady-state flow regimes. During transient radial flow, the prime derivative,

( ) ( ) 15.015.0' −=⎟⎟⎠

⎞⎜⎜⎝

⎛== DA

DADA

DDAD t

tdtdptp

p (9)

and the log derivative is given by,

( ) 5.0' =DAD tp (10)

During pseudo steady-state flow, the prime derivative is,

( ) π2' =DAD tpp

(11)

and the log derivative is given by,

( ) DADAD ttp π2' = (12)

Forms of equations (9) and (11) suggest that a log-log plot of the prime derivative, ( )DAD tpp' vs. DAt should result in a straight

line with a negative unit slope during the transient flow period. During the pseudo steady-state flow period, its value becomes constant, which is equal to π2 . Forms of equations (10) and (12) suggest that a log-log plot of the log derivative, ( )DAD tp ' vs. DAt results in a constant value equal to 0.5 during the transient radial flow period. The solution should result in a straight line with a positive unit slope during the pseudo steady-state flow period. Comparing the right hand side of the equation (12) with that of the material balance form of the equation (7), it should be noted that both equations have the same value. This suggests that during the pss flow period,

( ) ( ) DADADmbDAD ttptp π2' == (13)

This new finding forms the foundation of this paper. In this author’s knowledge, this observation is being reported for the first time in the petroleum literature. For a flowing well, pΔ is defined as ( )wfi pp − . Subtracting and adding p , we get,

( ) ( ) ( )wfiwfi pppppp −+−=− (14)

SPE 135804 5

In the dimensionless pressure form, equation (14) becomes,

( ) ( ) ( )DADDADmbDAD tptptp += (15)

Where,

( ) ( )μBq

ppkhtp wfiDAD 2.141

−= (15a)

( ) ( )μBq

ppkhtp iDADmb 2.141

−= (15b)

( ) ( )μBq

ppkhtp wfDAD 2.141

−= (15c)

During the pseudo steady-state flow period,

( ) ( ) ( )DADmbDADDAD tptptp −= (16)

Equation (16) is a rearrangement of equation (15) and,

( ) =DAD tp Constant (17)

Since values of wellbore flowing pressures, wfp , are known as a function of time, average reservoir pressure, p , can also be computed as a function of time. This should also enable the calculation and/or validation of the value of initial reservoir pressure,

ip , using the material balance identity shown in equation (15b). Values of average reservoir pressures, thus obtained, can be used to estimate OIP and OGIP. Graphical Presentations of Dimensionless Variables Application of the preceding concepts, which were discussed earlier, will be demonstrated using graphical plots of both dimensionless variables and real variables.

Figure 1: Dp and 'Dp vs DAt - For a well in the center of a square

Figure 1 shows a log-log plot of dimensionless pressure, Dp (shown in red) and the log derivative of dimensionless pressure, 'Dp

6 SPE 135804

(shown in dark blue) as a function of dimensionless time, DAt for a well in the center of a square. It is a familiar graph commonly used in pressure transient analysis. Characteristics of this plot are as follows. In the absence of wellbore storage effects, the derivative is flat during the transient radial flow period and it has a constant value, which is equal to 0.5. During the pss flow period, which starts at DAt equal to 0.1 for this case, the derivative, after a short transition, develops a positive unit slope line. This unit slope line also represents the material balance line based on ( )ppi − as was discussed earlier. One useful observation is that the log derivative unit slope line, which is also the material balance line, can be extrapolated backwards for times much less than the pss dimensionless time, which is equal to 0.1 in this case. This extrapolated line is also shown in Figure 1. On a Cartesian graph, this material balance line will go through zero at zero time and its slope will be equal to

π2 as expected. The dimensionless pressure line does not develop a unit slope until a value of DAt equal to 10. This occurs after almost two log cycles from the start of the pss flow period. The dimensionless pressure and log derivative solutions appear to merge together on the log-log graph at the value of DAt equal to 10 or slightly greater than 10. Although the two solutions namely Dp and 'Dp look alike after the merger on Figure 1, their values are different. This is so because they represent two different solutions. However, the difference between the two should become constant during the pss flow period. This aspect was also discussed earlier [see equations (16) and (17)]. It should be also noted that values of 'Dp and Dmbp are the same during the pss flow period and therefore, they can be used interchangeably.

Determine Average Pressure as a Function of Flowing Time Figure 2 is a semi-log plot of ( )'DDD ppp −= vs. DAt as shown below. It graphically shows the difference between Dp and 'Dp as a function of dimensionless time. During the pss flow period, the difference between the two solutions, Dp and 'Dp or Dmbp is equal to ( ) ( ) ( )wfiwfi pppppp −=−−− both in terms of dimensionless and real quantities. As shown in Figure 2, this difference is small at early times but it gradually increases and then becomes constant during the pss flow period with a value equal to π2 . This difference corresponds to the pressure drop due to Darcy flow from the reservoir into the well and is represented by ( )DAD tp as shown by equation (15c).

4

4.5

5

5.5

6

6.5

7

0.001 0.01 0.1 1 10 100

(pD

-pD' )

Dimensionless Time, tDA

(pD - pD') vs tDA

Figure 2: ( )'DD pp − vs. DAt - For a well in the center of a square

The preceding observation is significant because the pressure difference, ( )wfpp − can be determined from the difference of the

SPE 135804 7

two graphs or two known quantities namely ( ) ( )pppp iwfi −−− during the pss flow period. Since wellbore pressure, wfp is known as a function of time, this should enable the calculation of p also as a function of time. Graphical Characteristics of Prime Derivative Figure 3 is similar to Figure 1 except the prime derivative of the dimensionless pressure, '

pDp as a function of DAt has been

added to Figure 3. The prime derivative has been plotted in light blue. Its characteristics have been discussed earlier by means of equations and they can be visualized in Figure 3.

Figure 3: Dp , 'Dp & '

pDp vs. DAt - For a well in the center of a square

During the transient radial flow period, the prime derivative of the dimensionless pressure forms a negative unit slope line on a log-log plot. During the pss flow period, the value of the prime derivative, after a short transition period, becomes constant and it becomes equal to π2 . The transition from the transient flow to the pss flow takes place at an approximate value of 1.0=DAt . This transition period is similar to that of the log derivative of the dimensionless pressure, 'Dp for a well in the center of a square. Graphical Presentations Using Real Variables for Oil This will be illustrated by a synthetic example of a square shaped oil reservoir with a well in the center. Pertinent reservoir and well data is shown in Table 1. This well was produced for 1,000 hours at a constant rate, q, equal to 1,000 STB/D. Flowing pressures were generated as a function of time. Variables such as ( )wfi ppp −=Δ , ( )td

dpp ln' =Δ , prime derivative, dtdpp p =Δ ' , and ( )ppp imb −=Δ were

calculated and are plotted on log-log paper as a function of flowing time. Results are shown below in Figure 4. This graph is very similar to Figure 3. The main difference is that real variables are plotted in Figure 4 compared to Figure 3 where the same variables in dimensionless form have been plotted. The beauty of the log-log plot is that it retains the shape of the data plotted on the graph paper except that plotting scales may be different. Moreover, other characteristics of each variable are also retained. For example, 'pΔ vs. time graph (shown in dark blue) is flat and it has a constant value during the transient radial flow period. At a time of about 1 hour, the flow changes from transient to pseudo steady-state and the graph trends towards and follows a positive unit slope line. As discussed earlier, the positive unit slope line also represents mbpΔ which is equal to ( )ppi − . This ( )ppi − line (shown in dark blue) has been extrapolated backwards at early times and it represents material balance pressure drop during the transient flow period.

8 SPE 135804

Table 1: Oil Well Example (Well Located in Center of a Square Shaped Reservoir)

Reservoir and Well Properties

Initial Reservoir Pressure, psi 5000 Reservoir Temperature, °F 150

Porosity, fraction 0.2

Oil Saturation, fraction 0.8

Oil Viscosity, cp 0.5

Formation Volume Factor, Bo, res bbl/STB 1.2

Total System Compressibility, 1/psi 8.00E-06

Reservoir Permeability, k, md 200

Pay thickness, h, ft 150

Reservoir Area, ft2 1.00E+06

Wellbore Radius, ft 0.5

Oil-in-place, STB 3.56E+06

Production Rate, STB/D 1000

Producing Time, hours 1000 The prime derivative, dt

dpp p =Δ ' graph (shown in light blue) has a negative unit slope during the transient flow period and it

becomes flat with a constant value during the pss flow period.

0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100 1000

∆p, ∆

p', ∆

p mb,

dp/d

t

Flowing time, Hours

(dp/dt) vs Time(pi - pw) vs Time(pi-pbar) vs Time

t*dp/dt vs Time

Figure 4: pΔ , 'pΔ & 'ppΔ vs time - For a well in the center of a square

The pΔ equal to ( )wfi pp − graph has been plotted in red in Figure 4. As discussed earlier, in connection with the dimensionless

variables, this graph does not merge with the unit slope portion of the log derivative plot or the material balance plot, ( )ppi − shown in dark blue until after two log cycles (that is 100 hours) from the start of the pss flow period which takes place at about 1 hour. However, the difference between the two graphs ( pΔ and 'pΔ ), which represents ( )wfpp − , becomes constant starting at approximately 1 hour, which is the time for the beginning of the pss flow period. This has been shown in Figure 5.

SPE 135804 9

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10

Flowing Time, Hours

Pres

sure

Diff

eren

ces

and

[ t x

dp/

dt ]

(pbar - pwf) vs Time(pi - pwf) vs Time(pi - pbar) vs Time t x dp/dt vs Time

Figure 5: pΔ , 'pΔ & 'ppΔ vs time - For a well in the center of a square

Estimate Contacted Hydrocarbon Volume and Area The first paper on the reservoir limit test was published by Park Jones in 1956. Other papers have been also published in the petroleum literature; Earlougher (1977) has provided a good discussion on this topic in his monograph. Contacted Oil Volume For reservoir limit tests, the concept of prime derivative '

pDp has been used to make estimates for the original oil-in-place (OIP)

and the original gas-in-place (OGIP). During the pss flow period, the prime derivative becomes constant. This is shown by equation (11) and it has been reproduced below:

( ) == π2' DAD tpp

Constant (11)

Substituting the values of real variables in equation (11), we can derive equations for OIP and OGIP. Such an equation for estimating the contacted OIP as a function of time is shown below:

( )⎟⎠⎞⎜

⎝⎛

=

dtdpc

SqSTBOIP

t

o

615.5

23395.0 (18)

Where, q = Oil flow rate, STB/D

oS = Oil saturation, fraction

tc = System compressibility, 1/psi

dtdp = Slope of the pressure drawdown graph, psi/hour

It is assumed that the reservoir system is at a pss flow condition and the value of

dtdp has become constant. Inspection of Figure 3

reveals that the value of the prime derivative (see the light blue line) is very large at early times during the transient flow period

10 SPE 135804

but gradually decreases as a function of time until it becomes constant at the lowest value during the pss flow period. This observation suggests that if

dtdp values as a function of time are inserted in equation (18), it will provide the contacted oil

volume as a function of time. As expected, the contacted oil volume will be small at early times but will gradually become larger at later times. Finally, the contacted volume will become constant equal to OIP.

1.E+04

1.E+05

1.E+06

1.E+07

0.01 0.1 1 10 100

Flowing time, Hours

Con

tact

ed O

il Vo

lum

e (S

TB)

Oil Volume, STB vs Time

Figure 6: Oil Volume vs. Time - For a well in the center of a square

Figure 6 shows such a graph. This is a log-log plot of the contacted oil volume as a function of flowing or testing time. As expected, the image of this graph is the reverse of the '

pDp shown earlier in Figure 3. In the absence of faults or any other

boundaries, the early time data should form a unit slope line.

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

4.0E+06

0.01 0.1 1 10 100

Flowing time, Hours

Con

tact

ed O

il Vo

lum

e. S

TB

Oil Volume vs Time

Figure 7: Oil Volume vs. Time - For a well in the center of a square

Figure 7 is a semi-log plot of the data used in Figure 6. The contacted oil volume is plotted as a function of flowing time. It increases during transient flow and the oil volume becomes constant when the reservoir attains the pss flow condition.

SPE 135804 11

Contacted Reservoir Area Substituting the values of real variable in equation (11), we can derive equations for reservoir area, A. Such an equation for contacted reservoir area, A in square feet is shown below:

Reservoir Area, A (sq ft)

⎟⎠⎞⎜

⎝⎛

=

dtdpch

Bq

t

o

φ

23395.0 (19)

Where, q = Oil flow rate, STB/D

oB = Oil volume factor, Res Bbl/STB φ = Effective porosity, fraction h = Net oil pay, ft

tc = System compressibility at initial pressure, 1/psi

dtdp = Slope of the pressure drawdown graph, psi/hour

Although they are not shown here, graphs similar to Figure 4 and Figure 5 can be easily made where, on the y-axis, contacted oil volume is replaced by contacted reservoir area. Resultant graphs will not be the same but will be very similar. Background Material for Gas Review of Pertinent Equations for Gas Dimensionless Pressures

( ) ( ) ( )[ ]Tq

pmpmkhtp wfi

DAD 1422−

= (20a)

( ) ( ) ( )[ ]Tq

pmpmkhtp iDADmb 1422

−= (20b)

( ) ( ) ( )[ ]Tq

pmpmkhtp wfDAD 1422

−= (20c)

Equations (20a) through (20c) for gas wells are analogous to equations (15a) through (15c) presented earlier for single phase oil wells except pressures have been replaced by pseudo pressure, ( )pm introduced by Al-Hussainy et al. (1966). Other differences in dimensionless pressures for gas are due to flow units for gas wells, such as the flow rate, q in Mscf/D and reservoir temperature, T in degrees Rankine (°F + 460). Dimensionless Time

( ) Acktt

itDA μφ

410637.2 −×= (21)

This is analogous to equation (3) presented earlier for liquid systems and is applicable to gas wells during the transient flow period. However, the time term needs to be replaced by pseudo time, at during the pss flow period based on the average reservoir pressure, p . The concept of the pseudo time was originally introduced by Agarwal (1979) to correct the real time by re-evaluating viscosity and compressibility product ( )tcμ based on wellbore pressures. The concept of pseudo time was further extended by a number of authors for the pseudo steady-state flow of gas. The real time, for gas in pseudo steady-state flow, needs to be converted to pseudo

12 SPE 135804

time by taking into the account the variations in viscosity and compressibility product ( )tcμ based on the average reservoir pressure, p . Pseudo equivalent time, at developed by Palacio and Blasingame (1993) is shown below. It was obtained by extending the earlier work of Fraim and Wattenbarger (1987).

( ) ( ) ( )( ) ( ) ''1

0

dtpcp

tqctq

tt

giga ∫⎥

⎦

⎤⎢⎣

⎡=

μμ (22)

or,

( ) ( ) ( ) ( )[ ]pmpmpGzc

tqt i

i

iiiga −Δ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡=

21 μ (22a)

Graphical Presentations Using Real Variables for Gas This will be illustrated by a synthetic example of a square shaped gas reservoir with the well in the center. Pertinent reservoir and well data is shown in Table 2.

Table 2: Gas Well Example (Well Located in Center of a Square Shaped Reservoir)

Reservoir and Well Properties

Initial Reservoir Pressure, psi 5000 Reservoir Temperature, °F 150 Porosity, fraction 0.2

Gas Saturation, fraction 0.8

Initial Gas Viscosity, cp 0.03035

Formation Volume Factor, Bg, Surface vol/Res vol 303.07

Total System Compressibility, 1/psi 8.19E-05

Reservoir Permeability, k, md 200

Pay thickness, h, ft 150

Reservoir Area, ft2 1.00E+06

Wellbore Radius, ft 0.5 Gas-in-place, Bscf 7.274 Production Rate, MMscf/Day 100

Producing Time, hours 500 This well was produced for 500 hours at a constant rate of 100 MMscf/Day and it was followed by a 500 hour shut-in. Two cases are considered. Case 1 was not corrected for material balance (MB) but Case 2 included the material balance correction. It should be noted that Case 2 is a more realistic case than Case 1. The concept of pseudo pressure was used in both cases. Flowing and buildup pressures as a function of time are shown below in Figure 8 for both Case 1 and Case 2. Pressures for Case 1 are shown in dark blue. The trend of pressure decline is almost linear. The pressure at the end of flow period was 2170 psi. The final buildup pressure of about 2228 psi was reached after 500 hours of shut-in in this case. Pressures for Case 2 are shown in red. The trend of pressure decline is non-linear and it has a curvature which is concave upwards. The pressure at the end of flow period was 2941 psi. The final buildup pressure in this case after 500 hours was 2994 psi.

SPE 135804 13

Figure 8: Flowing and buildup pressures as function of time (Case 1 - not MB corrected, Case 2 - MB corrected) Average pressure calculated based on production of 2.083 Bscf gas out of 7.274 Bscf of the total gas contained in this synthetic reservoir is calculated as 2957 psi. This calculated average pressure is much closer to that obtained from Case 2. However, the static pressure reached in Case 1 is lower by about 766 psi compared to that of Case 2 or lower by 730 psi compared to calculated average pressure. These pressure difference values of 730 psi to 766 psi are not only significant but they are alarming. The major cause for such a large difference is that Case 1 does not correct for the material balance via use of changes in viscosity-compressibility product as a function of average reservoir pressure. It is normally done by using the pseudo time [see equations (22) and (22a)]. The main reason for discussing this aspect is to put a word of caution to those who use the analytical solutions or any well test software for gas wells where the option to correct for material balance is not available. It is possible that an engineer as a user of the PTA software may ignore to opt for the material balance correction option even though such an option may be available in certain software packages. Material balance correction is crucial for finite or closed gas systems. Numerical simulators automatically take the material balance into account. Flowing pressures were generated as a function of time and were converted into real gas pseudo pressure, m(p). Variables such as

( )pmΔ , ( )'pmΔ , prime derivative, ( ) 'ppmΔ , and ( )mbpmΔ were calculated and plotted on log-log paper as a function of flowing time. Results for Case 1 are shown below in Figure 9. Characteristics of Figure 9 are similar to those shown in Figure 4 for oil wells. The main difference is that real gas variables are plotted in Figure 9 compared to Figure 4 where the real oil variables were plotted.

Figure 9: ( )pmΔ , ( )'pmΔ , and ( )

dtpdm vs. time (Case 1)

14 SPE 135804

Results for Case 2 are shown in Figure 10 and they are plotted as a function of real time instead of pseudo time. It should be noted that pseudo time is the correct plotting variable for real gas cases in finite systems. This graph is very similar to Figure 9 and Figure 4 during the transient flow period. However, there is a noticeable difference between Figure 10 and Figures 4 and 9 at late times during the pss flow period. This kind of difference is evident in all solutions of the real gas variables which have been plotted and shown here in Figure 10.

Figure 10: ( )pmΔ , ( )'pmΔ , and ( )

dtpdm vs. time (Case 2)

For example, the ( )pmΔ variable vs. time graph (shown in red) does not develop a unit slope line and bends to the right at late times. ( )'pmΔ vs. time graph (shown in dark blue) develops a unit slope line during the early part of pss flow period but it also bends towards the right at late times. The partly developed unit slope line can be used to our advantage to locate ( )mbpmΔ line which represents ( ) ( )[ ]pmpm i − . This line can be extrapolated backwards into the transient flow period and forward into the late pss flow period. The prime derivative, ( )

dtpdm graph (shown in light blue) has a negative unit slope during the transient flow period and it becomes

flat with a constant value during the pss flow period. However, at late times this graph also bends downward. The characteristics related to the bending of the plotted solutions of above mentioned variables indicate that estimates of contacted gas volume and contacted reservoir area may be too optimistic, if conversion from real time to pseudo time is not done. Determine Average Pressure as a Function of Time Figure 10a is a semi-log graph of [ ( ) ( )wfpmpm − ] vs. flowing time. It graphically shows the difference between ( )pmΔ and

( ) 'pmΔ . These two variables were plotted respectively in red and dark blue in Figure 10. Case 2 with the material balance correction has been considered. In case of oil, the difference between the two solutions becomes constant during the pss flow period and it remains constant afterwards. This observation was discussed and demonstrated earlier by means of Figure 2. In case of real gas (Case 2), the difference between the two solutions increases going from the transient flow to pss flow condition and then becomes constant. However, in this case, the duration of this constant period lasts only for slightly more than one log cycle and then the difference increases significantly as shown in Figure 10a. This corresponds to the bending of the same two solutions on the right as was discussed in relation to Figure 10. The main reason for this behavior is that real time is being used instead of pseudo time for closed systems.

SPE 135804 15

Figure 10a: [ ( ) ( )wfpmpm − ] vs. Flowing time (Real Gas, Case 2)

However, it is possible to estimate gas-in-place and average reservoir pressures utilizing the data when the difference between the two solutions is constant. Estimation of gas-in-place and contacted reservoir area for this (Case 2) will be illustrated later by means of Figures 11 and 12. Although not shown here, average reservoir pressures, obtained from the constant difference period by converting pseudo pressure values to real pressures, can be used in constructing a p/z vs. cumulative production plot. Depending on the quality of the data, the graph can be extrapolated forward to tie up with the estimated OGIP value obtained and shown in Figures 11 and 12. Then, this p/z line can be extrapolated backwards to estimate the initial value of p/z and thus the value of initial reservoir pressure. This information can be of a great advantage for all published methods which require conversion from real time to pseudo time in the process of determining OGIP. They include advanced decline curve analysis methods published by Palacio and Blasingame (1993) and Agarwal et al. (1999). They also include publications related to flowing gas material balance by Mattar and McNeil (1998) and dynamic material balance method by Mattar and Anderson (2005) to name a few. It should be noted that this constant difference window, discussed earlier, can be larger or smaller for different real gas problems and the quality of the flowing pressures vs. time data can also be a factor. This aspect will be further investigated and reported in the future. Estimate Contacted Gas Volume and Reservoir Area Details for deriving the following equations were given earlier in this paper while discussing oil systems. Contacted Gas Volume Equation (23) for calculating the contacted gas volume is shown below:

( ) )()(

2873.83

ii

i

t

g

zp

dtpdmc

SqSCFOGIP

μ⎟⎠⎞⎜

⎝⎛

= (23)

Where, q = Gas flow rate, Mscf/Day

gS = Gas saturation, fraction

tc = System compressibility, 1/psi ( )

dtpdm = Slope of the m(p) drawdown plot, psi/hour

16 SPE 135804

It is assumed that the reservoir system is at pss flow condition and the value of ( )dt

pdm has become constant. Inspection of

Figure 3 reveals that the value of the prime derivative (see the light blue line) is very large at early times during the transient flow period but it gradually decreases as a function of time until it becomes constant at the lowest value during the pss flow period. This observation suggests that if ( )

dtpdm values as a function of time are inserted in equation (23), it will provide the contacted

gas volume as a function of time. As expected, the contacted gas volume will be small at early times but it will gradually become larger at later times. Finally, the contacted volume will become constant equal to OGIP for a short period of time.

Figure 11: Gas Volume vs. Time - For a well in the center of a square Figure 11 shows such a graph on log-log paper. The contacted gas volume has been plotted as a function of flowing or testing time. Data for Case 1 is plotted in pink and data for Case 2, which is a more realistic case, is shown in dark blue. They follow the trend of prime derivatives shown in Figures 9 and 10. These graphs are the reverse of the '

pDp shown earlier in Figures 9 and 10.

In the absence of any faults or other boundary effects, the early time data should form a unit slope line. The pss flow data shows a constant contacted true gas volume for Case 1 but the similar data for Case 2 shows a constant contacted true volume only from about 1 hour to 20 hours and then the contacted gas volume doubles up in value by the end of the test. This is not unexpected since Case 2 requires the use of pseudo time. In the absence of that, the calculated gas volume may be too optimistic unless the proper caution is exercised.

Figure 12: Gas Volume vs. Time - For a well in the center of a square

SPE 135804 17

Figure 12 is a semi-log plot of the same data as was shown in Figure 11. Same remarks apply to this graph as was discussed earlier for Figure 11. However, the effect of calculating twice the contacted true gas volume is clearly evident on this graph. Contacted Reservoir Area Substituting the values of real variables in equation (11), we can derive equations for reservoir area, A. Such an equation for contacted reservoir area, A, is shown below:

Reservoir Area, A (sq ft)⎟⎠⎞⎜

⎝⎛

=

dtpdmch

Tq

it)()(

3561.2

μφ (24)

Where, q = Gas flow rate, Mscf /Day T = Reservoir temperature, Deg R φ = Effective porosity, fraction h = Net oil pay, ft

tc = System compressibility, 1/psi ( )

dtpdm = Slope of m(pwf) vs. time graph, psi2/cp/hour

Although they are not shown here, graphs similar to Figure 11 and Figure 12 can be easily made where, on the y-axis, contacted gas volume is replaced by contacted reservoir area. Resultant graphs will not be same but will be very similar and similar conclusions will be applicable. Volume & Area of Investigation vs. Radius of Investigation Radius of investigation is a valuable concept and it has served our industry well for more than 50 years. The following equation has been presented by Earlougher (1977):

td c

tkrμφ

029.0= (25)

Other radius of investigation equations have also appeared in the literature but the main difference is that the multiplying constant in them is slightly different. This constant is equal to 0.029 in equation (25). Kuchuk (2009) has presented a comprehensive study on this topic and concludes that there is no identifiable radius of investigation from pressure diffusion. According to him, equation (25) gives conservative estimates for radius of investigation. Unfortunately, such equations are always used without giving any consideration to the shape or size of the reservoir. These types of equations often give unrealistic and misleading results for non-circular type reservoirs, circular reservoirs with single or multiple faults, channel type reservoirs and closed systems to name a few. It is believed that the concept of volume and area of investigation presented in this paper should provide realistic answers for radial and non-radial systems and they should be used instead of using the concept of the radius of investigation. Volume and area of investigation equations (18) and (19) are presented for liquid systems. Similar equations for the gas system are given by Equations (23) and (24). Summary and Conclusions

1. A new finding has been presented in this paper whereby average reservoir pressure can be directly estimated from the pressure vs. time data collected on flowing oil and gas wells. It is possible to estimate initial reservoir pressure in certain cases.

18 SPE 135804

2. The log derivative of pressure data (that is, the unit slope line on log-log paper) is also equal to the difference between average reservoir pressure and flowing well pressure during the pseudo steady-state flow period for oil systems.

3. Conclusion (2) also applies to gas systems but requires the use of pseudo time. If real time is used, the unit slope line can

be of a smaller time span. This will depend upon the magnitude of reservoir depletion; greater the depletion the smaller will be the size of this time span. This will limit the ability to calculate average reservoir pressure only during this time span.

4. It also has been shown by means of equations and figures that contacted hydrocarbon volume (OIP or OGIP) and

contacted reservoir area as function of time can be estimated by using flowing well pressure data.

5. For oil systems, the difference between the initial reservoir pressure and the average reservoir pressure vs. cumulative production or real time should have a linear relationship, if plotted on a Cartesian graph. For gas, the pseudo time should be used instead of real time.

6. Equations to calculate connected hydrocarbon volume and reservoir area presented here are superior to the radius of

drainage concept which is currently being used in the industry. The use of these equations is even more important in case of non-circular drainage areas and circular drainage areas with faults and boundaries. Although results presented in this paper mainly pertain to a well in the center of a square shaped reservoir, this technology is valid for other reservoir shapes and well locations.

7. It is suggested that the solution for the prime derivative should also be added to the log-log graph in both vendor and in-

house software packages, which are commonly used for pressure transient analysis.

8. Technology presented in this paper should be advantageous in the process required for converting real time to pseudo time for gas well analysis.

9. Caution should be exercised while analyzing or designing well tests for real gas wells using either analytical equations or

PTA software. It is essential to include the option to correct for material balance for finite reservoirs. Nomenclature A = drainage area, sq. ft. B = formation volume factor

tc = total system compressibility, 1/psi

AC = shape factor, dimensionless

iG = original gas-in-place, Mscf h = net pay, ft. k = effective permeability to oil or gas, md

( )pm = real gas pseudo pressure, psi2/cp ( )pmΔ = ( ) ( )wfi pmpm − , psi2/cp

( )pmΔ = ( ) ( )wfpmpm − , psi2/cp

( )mbpmΔ = ( ) ( )pmpm i − , psi2/cp

( )'pmΔ = ( )( )td

pdmln

, psi2/cp

( ) 'ppmΔ = ( )dt

pdm , psi2/cp/hour

ip = initial reservoir pressure, psi p = average reservoir pressure, psi

wfp = bottomhole flowing pressure, psi

pΔ = ( )wfi pp − , psi

SPE 135804 19

pΔ = ( )wfpp − , psi

'pΔ = ( )tddp

ln , psi/hour

mbpΔ = ( )ppi − , psi

'ppΔ = dtdp , psi/hour

Dp = dimensionless pressure q = flow rate, STB/D for oil & Mscf/Day for gas

wr = wellbore radius, ft.

gS = gas saturation, fraction

oS = oil saturation, fraction t = time, hours

at = pseudo time, hours

DAt = dimensionless time based on area, A

wDrt = dimensionless time based on wellbore radius, rw T = reservoir temperature, degrees Rankine

iz = z-factor at initial reservoir pressure Greek Letters γ = 1.781 (exponential of Euler’s constant = 0.5772) μ = gas viscosity, cp φ = hydrocarbon porosity, fraction π = 3.14159 Δ = difference Acknowledgments The author wishes to thank Arun Agarwal and Rahul Kumar of Petrotel for assistance provided during the preparation of this manuscript. Appreciation is also expressed to engineers of Reliance Industries, Mumbai, India for an opportunity to have thought provoking discussions with them. Thanks also go to Petrotel for encouragement and support for this work. References Agarwal, R. G.: “‘Real Gas Pseudo-Time’—A New Function for Pressure Buildup Analysis of MHF Gas Wells,” paper 8279 presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, Sept. 23-26. Agarwal, Ram G., Gardner, David C., Kleinsteiber, Stanley W., Fussell, Del D.: “Analyzing Well Production Data Using Combined-Type-Curve and Decline-Curve Analysis Concepts”, paper 57916 SPEJ (Oct. 1999), 478-486. Al-Hussainy, R., Ramey, Jr. H. J., and Crawford, P. B.: “The Flow of Real Gases Through Porous Media”, J. Pet. Tech. May (1966) 624-636. Earlougher, R. C.: Advances in Well Test Analysis: Monograph Vol. 5, SPE-AIME, 1977. Fraim, M. L. and Wattenbarger, R. A.: “Gas Reservoir Decline-Curve Analysis Using Type Curves with Real Gas Pseudo-pressures and Normalized Time”, SPEFE (December 1987) 620. Kuchuk, Fikri J.: “Radius of Investigation for Reserve Estimation from Pressure Transient Well Tests”, paper presented at the SPE Middle East Oil and Gas Show and Conference, Bahrain, Bahrain, March 15-18, 2009. Mattar, L. and Anderson, D: “Dynamic Material Balance (Oil or Gas-in-place without shut-ins)”, paper 2005-113 presented at 6th Canadian International Petroleum Conference, Calgary, Alberta, Canada, June 7-9, 2005.

20 SPE 135804

Mattar, L. and McNeil, R.: “The Flowing Gas Material Balance”, Journal of JCPT, vol. 37 #2, page, 1998. Palacio, J. C. and Blasingame, T. A..: “Decline Curve Analysis Using Type Curves: Analysis of Gas Well Production Data,” paper SPE 25909 presented at the 1993 Rocky Mountain Regional Meeting/ Low Permeability Reservoir Symposium and Exhibition, Denver, Colorado, 26-28 April. Park, Jones: “Reservoir Limit Test”, Oil and Gas J. (June 18, 1956) 184-196. Ramey, H. J. and Cobb, William M.: “A General Pressure Buildup Theory for a Well in a Closed Drainage Area”, Jour. Pet. Tech (Dec.1971), 1493. Van Everdingen, A. F. and Hurst, W.: “The Application of the Laplace Transformation to Flow Problems in Reservoirs”, Trans., AIME (1949) 186, 305-324. Appendix SI Metric Conversion Factors acre x 4.046 873 E+03 = m2 bbl x 1.589 873 E -01 = m3 cp x 1.0 E -03 = Pa·s ft x 3.048 E -01 = m psi x 6.894 757 E +00 = kPa ft3 x 2.831 685 E -02 = m3 md x 9.869 233 E -16 = m2 R x 5/9 = K