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Chapter 5 Oil Reservoir Performance Prediction (pp. 328-

342)

Three phases:

1) MBE to estimate the cumulative hydrocarbon production and ultimate oil recovery as

functions of declining reservoir pressure and increasing gas-oil ratio

2) To determine the recovery profile as a function of time using individual well performance

profile with declining reservoir pressure

3) To predict time-production.

5.1 Phase 1. Oil reservoir performance prediction methods

The MBE in its various mathematical forms as presented in Chapter 4 is designed to provide estimates of

the initial oil-in-place , size of the gas cap , and water influx . To use the MBE to predict the

reservoir future performance, two additional relations are required:

o Instantaneous GOR

o Equations to reservoir saturations

5.1.1 Instantaneous GOR

Three types of GORs:

o Instantaneous GOR

o Solution GOR: gas solubility

o Cumulative GOR:

 Instantaneous GOR

The produced GOR at any particular time: the ratio of the standard cubic feet of total

gas being produced at any time to the stock-tank barrels of oil being produced at that same

instant.

(

)(

)

(5.1)

Where:

Figure 5.1 Characteristics of solution gas drive reservoirs.

Point 1. When the reservoir pressure is above the bubble point pressure , there is no free gas

in the formation. and therefore:

The GOR remains constant at until the pressure reaches the bubble point pressure at point 2.

Point 2. As the reservoir pressure declines below , the gas begins to evolve from solution and

its saturation increases. However, this free gas cannot flow until the gas saturation reaches the

critical gas saturation at point 3.

From point 2 to point 3, the instantaneous GOR is described by a decreasing gas solubility, as:

Point 3. At this point, the free gas begins to flow with the oil and the values of GOR

progressively increase with the declining reservoir pressure to point 4. During this pressure

decline period, the GOR is described by Equation 5.1.

Point 4. At this point, the maximum GOR is reached due to the fact that the supply of gas has

reached a maximum and marks the beginning of the blow-down period to point 5.

Point 5. This point indicates that all the producible free gas has been produced and the GOR is

essentially equal to the gas solubility and continues to decline following the curve.

Figure 5.2 History of GOR and Rs for a solution gas drive reservoir.

 Solution GOR

( )

(5.2)

Where:

The cumulative gas produced is related to the instantaneous GOR and cumulative oil production by

the expression:

∫ ( )

(5.3)

Figure 5.3 Relationship between GOR and Gp.

The incremental cumulative gas produced, , between and is then given by:

∫ ( )

(5.4)

By using trapezoidal rule:

[ ( ) ( )

]( )

Or:

( )

∑( )

(5.5)

Example 5.1

The following production data is available on a depletion drive reservoir:

p, psi GOR, scf/STB Np, MMSTB

2925 1340 0

2600 1340 1.380

2400 1340 2.260

2100 1340 3.445

1800 1936 7.240

1500 3584 12.029

1200 6230 15.321

The initial reservoir pressure is 2925 psia with a bubble point pressure of 2100 psia. Calculate cumulative

gas produced Gp and cumulative GOR at each pressure.

Solution:

Construct the following table by applying Equations 5.2 and 5.5:

[ ( ) ( )

]( ) ( )

∑( )

5.1.2 Reservoir saturation equations and their adjustments

o Oil saturation adjustment in gravity drainage reservoirs

o Oil saturation adjustment due to water influx

o Oil saturation adjustment due to gas cap expansion

o Oil saturation adjustment for combination drive

o Oil saturation adjustment for shrinking gas cap

From Equation 4.1, assume no gas cap, the pore volume is:

Volume of the remaining oil ( ) , bbl

( )

(5.6)

Rearranging:

( )(

)

(5.7)

(5.8)

Example 5.2

A volumetric solution gas drive reservoir has an initial water saturation of 20%. The initial oil

formation volume factor is reported at 1.5 bbl/STB. When 10% of the initial oil was produced,

the value of decreased to 1.38. Calculate the oil saturation and gas saturation.

Solution

From Equations 5.6 and 5.7:

( )(

)

( )( )

The values of the relative permeability ratio

as a function of oil saturation can be generated

by using the actual field production as expressed in terms of , and data. The

recommended methodology involves the following steps:

Step 1. Calculate the oil and gas saturations.

( )(

)

Step 2. Using the actual field instantaneous GORs, solve relative permeability ratio.

( )(

)

Step 3. Plot

vs. on semi-log paper.

Oil saturation adjustment in gravity drainage reservoirs

Steps for adjusting Equation 5.6 to reflect the migration of gas to the top of the structure:

Step 1. Calculate the volume of the evolved gas that will migrate to the top of the formation to

form the secondary gas cap from the following relationship:

( ) [ ( ) ] [

( ) ]

Where:

( )

Step 2. Recalculate the volume of the evolved gas that will form the secondary gas cap from

following relationship:

( ) [ ]( )

Where:

( )

Step 3. Equating the two derived relationships and solving for secondary gas cap pore volume

gives:

( ) [ ( ) ] [

]

Step 4. Adjust Equation 5.6 to account for the migration of the evolved gas to the secondary gas

cap, to give:

( ) ( )

(

) ( )

(5.9

Oil saturation adjustment due to water influx

Figure 5.4 Oil saturation adjustment for water influx.

The proposed oil saturation adjustment methodology is illustrated in Figure 5.4 and described by

the following steps:

Step 1. Calculate the PV in the water-invaded region, as:

( ) ( )

( )

(5.10)

Where:

( )

Step 2. Calculate the oil volume in the water-invaded zone, or:

Volume of oil = ( )

(5.11)

Step 3. Adjust Equation 5.6 to account for the trapped oil by using Equations 5.10 and 5.11.

( ) [

]

(

) [

]

(5.12)

Oil saturation adjustment due to gas cap expansion

Figure 5.5 Oil saturation adjustment for gas cap expansion.

The oil saturation adjustment procedure is illustrated in Figure 5.5 and summarized below:

Step 1. Assuming no gas is produced from the gas cap, calculate the net expansion of the gas

cap, from:

Expansion of the gas cap = [( ) ]

(5.13)

Step 2. Calculate the PV of the gas-invaded zone, ( ) , by solving the following simple

material balance:

[( ) ] ( ) ( )

( ) [( ) ]

(5.13)

Where:

( )

Step 3. Calculate the volume of oil in the gas-invaded zone.

Oil volume = ( )

(5.14)

Step 4. Adjust Equation 5.6 to account for the trapped oil in the gas expansion zone by using

Equations 5.13 and 5.14, to give:

( ) [( ) ]

(

) [( ) ]

(5.15)

Oil saturation adjustment for combination drive

( ) [ (

)

( )

]

(

) [ (

)

]

(5.16)

Oil saturation adjustment for shrinking gas cap

Cole (1961): the magnitude of the loss of a substantial amount of oil may be quite large and

depends on:

 The area of the gas-oil contact

 The rate of gas cap shrinkage

 The relative permeability characteristics

 The vertical permeability

Expansion of the original gas cap = [( ) ]

Gas cap shrinkage = [( ) ]

From the volumetric equation:

[(

) ] ( )

(5.17)

Where:

Oil lost = [ (

)]

( )

(5.18)

Where:

Three methodologies that are widely used in the petroleum industry to perform a reservoir study are:

 The Tracy method

 The Muskat method

 The Tarner method

5.1.3 Undersaturated oil Reservoirs

The general material balance is expressed in Chapter 4 by Equation 4.15:

[ ( ) ] ( )

( ) ( ) [( ) ] ( )[( ) ( )]

For undersaturated oil reservoir with no fluid injection:



( ) [( ) ( )]

With:

Hawkins (1955) introduced the oil compressibility into the MBE to further simplify the

equation.

[( ) ( )]

[ (

)]

[

]

(5.19)

Effective compressibility:

(5.20)

(5.21)

( )

(5.22)

Expressed in a straight line:

[

]

(5.23)

Figure 5.6 Pressure voidage relationship.

Example 5.3

The following data is available on a volumetric undersaturated oil reservoir.

Estimate cumulative oil production when the reservoir pressure drops to 3500 psi. The oil

formation volume factor at 3500 psi is 1.414 bbl/STB.

Solution

Step 1. Determine the effective compressibility from Equation 5.20:

(

)( ) ( )(

)

Step 2. Estimate

( )

( )(

)(

)( )

5.1.4 Saturated oil Reservoirs

The general material balance is expressed in Chapter 4 by Equation 4.15:

[ ( ) ] ( )

( ) ( ) [( ) ] ( )[( ) ( )]

For saturated oil reservoir with no fluid injection:



 Expansion of rock and initial water can be neglected.

( ) ( ) ( )

(5.24)

All the techniques that are used to predict the future performance of a reservoir are based on

combining the appropriate MBE with the instantaneous GOR using the proper saturation

equation. The calculations are repeated at a series of assumed reservoir pressure drops.

( )(

)

( )(

)

There are several widely used techniques that were specifically developed to predict the

performance of solution gas drive reservoirs, including:

 The Tracy method

 The Muskat technique

 The Tarner method

Tarner method

Tarner (1944) suggested an iterative technique for predicting cumulative oil production and

cumulative gas production as a function of reservoir pressure.

Volumetric saturated oil reservoir:

Step 1. Select (assume) a future reservoir pressure below the initial (current) reservoir pressure

and obtain the necessary data. Assume that the cumulative oil production has

increased from to . (

@ )

Step 2. Estimate or guess the cumulative oil production at the selected (assumed) reservoir

pressure of step 1.

Step 3. Calculate the cumulative gas production by rearranging the MBE, i.e., Equation 5.24,

to give

[( )

] [ ]

(5.25)

( ) ( )

(5.26)

Step 4. Calculate the oil and gas saturations at the assumed cumulative oil production and the

selected reservoir pressure by applying Equations 5.7 and 5.8 respectively, or:

( )(

)

and:

where:

Step 5. Using the available relative permeability data, determine the relative permeability ratio

that corresponds to the calculated total liquid saturation of step 4 and compute the

instantaneous GOR at from Equation 5.1:

(

)(

)

Step 6. Calculate again the cumulative gas production at by applying Equation 5.5:

[

](

)

(5.27)

Step 7. The calculations as performed in step 3 and step 6 give two estimates for cumulative gas

produced at the assumed (future) pressure :

 as calculated from the MBE

 as calculated from the GOR equation

If the cumulative gas production as calculated from step 3 agrees with the value of step 6, the

assumed is correct. Otherwise, assume another value of and repeat steps 2 through 6.

Step 8. In order to simplify this iterative process, three values of can be assumes. When the

computed values of are plotted versus the assumed values of , the resulting two

curves will intersect. This intersection indicates that the cumulative oil and gas

production that will satisfy both equations.

Example 5.4

A saturated oil reservoir has a bubble point pressure of 2100 psi at 175 °F. The initial reservoir

pressure is 2400 psi. The following data summarizes the rock and fluid properties of the field:

Basic PVT data is as follows:

Relative permeability ratio:

Predict the cumulative oil and gas production at 2100, 1800, and 1500 psi.

Solution:

The required calculations will be performed under the following two different driving mechanisms:

 Initial pressure to bubble point: undersaturated, therefore, the MBE can be used directly

in cumulative production without restoring the iterative technique.

 Below bubble point pressure: saturated, and the Tarner method may be applied.

Oil recovery prediction from initial pressure to the bubble point pressure:

Step 1. The MBE for an undersaturated reservoir is given by Equation 4.47:

( )

Where:

[ ( )

]

̅̅̅

No water production,

( )

(5.28)

Step 2. Calculate the two expansion factors and for the pressure declines from the initial

reservoir pressure of 2400 psi to the bubble point pressure of 2100 psi:

[ ( )

]

[ ( )( ) ( )

] ( )

Step 3. Calculate the cumulative oil and gas production when the reservoir pressure declines

from 2400 to 2100 psi by applying Equation 5.28, to give:

( )

At or above the bubble point pressure, the producing GOR is equal to the gas solubility at

the bubble point and, therefore, the cumulative gas production is given by:

( )( )

Step 4. Determine the remaining oil-in-place at 2100 psi:

Remaining oil in place

the saturation pressure. That is:

Oil recovery prediction below the bubble point pressure:

Oil recovery prediction at 1800 psi with the following PVT properties:

Step 1. Assume that 1% of the bubble point oil will be produced when the reservoir pressure

drops 1800 psi. That is:

Calculate the corresponding cumulative gas by apply Equation 5.26:

( ) ( )

( ) ( )[ ( )( )]

Step 2. Calculate the oil saturation, to give:

( )(

)

( )(

)

Step 3. Determine the relative permeability ratio

from the tabulated data at total liquid

saturation of to give:

Step 4. Calculate the instantaneous GOR at 1800 psi by applying Equation 5.1

(

)(

)

( )(

)

Step 5. Solve again for the cumulative gas production by using the average GOR and applying

Equation 5.27 to yield:

[

](

)

[

]( )

Step 6. Since the cumulative gas production, as calculated by the two independent methods (step

1 to step 5), do not agree, the calculations must be repeated by assuming a different value

for and plotting results of the calculation. Repeated calculations converge at:

of bubble point oil and

of bubble point oil

Or:

( )

The cumulative oil and gas production as the pressure declines from the initial pressure to

the bubble point pressure is:

The actual cumulative recovery at 1800 psi is:

5.2 Phase 2: Oil well performance

5.2.1 Vertical oil well performance

Productivity index and IPR

The productivity index is the ratio of the total liquid flow rate to the pressure drawdown.

For a water-free oil production, the productivity index is given by:

̅̅̅

(5.29)

Where:

STB/day/psi

̅ ( )

Note: the productivity index is a valid measure of the well productivity potential only if the

well is flowing at pseudosteady-state conditions.

Figure 5.7 Productivity index during flow regimes.

( ̅̅̅ )

(5.30)

Figure 5.8 vs. relationship.

̅̅̅ (

)

(5.31)

Figure 5.8 IPR

̅̅̅

(5.32)

( ̅̅̅ )

[ ( ) ]

(5.33)

[ ( ) ]

(5.34)

Where:

STB/day/psi

The oil relative permeability:

[ ( ) ] (

)

(5.35)

Specific productivity index :

( ̅̅̅ )

(5.36)

Example 5.5

A productivity test was conducted on a well. The test results indicate that the well is capable of

producing at a stabilized flow rate of 110 STB/day and a bottom-hole flowing pressure of 900

psi. After shutting the well for 24 hours, the bottom0hole pressure reached a static value of 1300

psi. Calculate:

a) The productivity index b) The AOF

c) The oil flow rate at a bottom-hole flowing pressure of 600 psi d) The wellbore flowing pressure required to produce 250 STB/day.

Solution

a) Calculate J from Equation 5.29:

̅̅̅

b) Determine the AOF from Equation 5.32 ̅̅̅

Figure 5.9 IPR below

c) Solve for the oil flow rate by applying Equation 5.29

( ̅̅̅ )

( )

d) Solve for by using Equation 5.31:

̅̅̅ (

)

(

)

For Equation 5.35

[ ( ) ] (

)

Treating the term in the brackets as a constant , the above equation can be written in the

following form:

(

)

(5.37)

With the coefficient

[ ( ) ]

Figure 5.10 Effect of pressure on , and .

Figure 5.11 as a function of pressure.

Figure 5.12 Effect of reservoir pressure on IPR.

There are several empirical methods that are designed to predict the non-linear behavior of the

IPR for solution gas drive reservoirs. All the methods include the following two computational

steps:

 Using the stabilized flow test data, construct the IPR curve at the current average

reservoir pressure ̅̅̅.  Predict future IPRs as a function of average reservoir pressures.

1) the Vogel method 2) the Wiggins method 3) the Standing method 4) the Fetkovich method 5) the Klins and Clark method

Vogel method

Vogel (1968):

̅̅̅

( )

Where ( ) is the flow rate at zero wellbore pressure, i.e., the AOF.

( )

(

̅̅̅ ) (

̅̅̅ )

(5.38)

Where:

( )

Vogel’s methodology can be used to predict the IPR curve both for saturated oil reservoirs

and undersaturated oil reservoirs.

The computational procedure of applying the Vogel method in a saturated oil reservoir to

generate the IPR curve for a well with a stabilized flow data point, i.e., a recorded value at

, is summarized below:

Step 1. Using the stabilized flow data, i.e., and , calculate ( ) from Equation 5.38,

or:

( )

( ̅̅̅ ) (

̅̅̅ )

Step 2. Construct the IPR curve by assuming various values for and calculating the

corresponding by applying Equation 5.38:

( )

(

̅̅̅ ) (

̅̅̅ )

Or:

( ) [ (

̅̅̅ ) (

̅̅̅ )

]

Example 5.6

A well is producing from a saturated reservoir with an average reservoir pressure of 2500 psig.

Stabilized production test data indicates that the stabilized rate and wellbore pressure are 350

STB/day and 2000 psig, respectively. Calculate:

a) Calculate the oil flow rate at b) Calculate the oil flow rate assuming constant J. c) Construct the IPR by using the Vogel method and the constant productivity index

approach.

Solution

Step 1. Calculate ( ) :

( )

( ̅̅̅ ) (

̅̅̅ )

(

) (

)

Step 2. Calculate at by using Vogel’s equation:

( ) [ (

̅̅̅ ) (

̅̅̅ )

]

[ (

) (

)

]

b) Step 1. Apply Equation 5.29 to determine J:

̅̅̅

STB/day/psi

Step 2. Calculate :

( ̅̅̅ ) ( )

Assume several values for and calculate the corresponding :

Wiggins method

Wiggins (1993): proposed generalized correlations that are suitable for predicting the IPR during

three phase flow.

( ) [ (

̅̅̅ ) (

̅̅̅ )

]

(5.39)

( ) [ (

̅̅̅ ) (

̅̅̅ )

]

(5.40)

Where:

( )

( ) ( ) [ ( ̅̅̅) ( ̅̅̅)

( ( ̅̅̅) ( ̅̅̅)

)

]

(5.41)

( ) ( ) [ ( ̅̅̅) ( ̅̅̅)

( ( ̅̅̅) ( ̅̅̅)

)

]

(5.42)

Where:

( ̅̅̅) ( )

( ̅̅̅)

( )

Example 5.7

The information is given below:

Generate the current IPR data and predict future IPR when the reservoir pressure declines from

2500 to 2000 psig by using the Wiggins method.

Solution

Step 1. Using the stabilized flow test data, calculate the current maximum oil flow rate by

applying Equation 5.39:

( ) [ (

̅̅̅ ) (

̅̅̅ )

]

Solve for the present ( ) , to give:

( )

(

) (

)

Step 2. Generate the current IPR data by using the Wiggins method and compare the results with

those of Vogel. Results of the two methods are shown graphically in Figure 5.13.

Figure 5.13 IPR curves.

Step 3. Calculate future maximum oil flow rate by using Equation 5.41

( ) ( ) [ ( ̅̅̅) ( ̅̅̅)

( ( ̅̅̅) ( ̅̅̅)

)

]

[ (

) (

)

]

Step 4. Generate future IPR data by using Equation 5.39

( ) [ (

̅̅̅ ) (

̅̅̅ )

]

[ (

) (

)

]

Standing method

Standing (1970): rearrange Equation 5.38

( )

(

̅̅̅ )[ (

̅̅̅ )]

(5.43)

Productivity index as defined by Equation 5. 29, to yield:

( ) ̅̅̅

[ (

̅̅̅ )]

(5.44)

“zero drawdown” productivity index:

[

( ) ̅̅̅

]

(5.45)

[ (

̅̅̅ )]

(5.46)

( ̅̅̅ )

(5.47)

Combine Equation 5.45 with 5.43 to eliminate ( ) , to give:

[ ( ̅̅̅)

]{

( ̅̅̅)

[ ( ̅̅̅)

]

}

(5.48)

Where the subscript f refers to the future condition.

(

)

(

)

(5.49)

Where the subscript p refers to the present condition.

If the relative permeability data is not available, can be roughly estimated from:

[ ( ̅̅̅) ( ̅̅̅)

]

(5.50)

Standing’s methodology for predicting a future IPR is summarized in the following steps:

Step 1. Using the current time condition and the available flow test data, calculate ( ) from

Equation 5.38:

( )

( ̅̅̅ )[ (

̅̅̅ )]

Step 2. Calculate at the present condition by using Equation 5.45.

[

( ) ̅̅̅

]

Or from Equation 5.47:

( ̅̅̅ )

Step 3. Using fluid property, saturation, and relative permeability data, calculate both (

)

and (

) .

Step 4. Calculate by using Equation 5.49. Using Equation 5.50 if the oil relative permeability

data is not available.

(

)

(

)

Or:

[ ( ̅̅̅) ( ̅̅̅)

]

Step 5. Generate the future IPR by applying Equation 5.48:

[ ( ̅̅̅)

]{

( ̅̅̅)

[ ( ̅̅̅)

]

}

5.2.2 Horizontal oil well performance

Figure 5.14 Horizontal well drainage area.

Joshi (1991) proposed the following two methods for calculating the drainage area of a

horizontal well. Assuming that each end of the horizontal well is represented by a vertical well

that drains an area of a semicircle with a radius of b.

Method I

The drainage area of the horizontal well is represented by two semicircles of radius b at each

end and a rectangle, of dimensions in the center, the drainage area is:

( )

(5.51)

Where:

Method II

Assume the horizontal well drainage area is an ellipse and given by:

(5.52)

With:

(5.53)

Where a is the half major axis of an ellipse.

Average value for the drainage of the horizontal well:

√

(5.54)

Where:

Example 5.8

A 480 acre lease is to be developed by using 12 vertical wells. Assuming that each vertical well

would effectively drain 40 acres, calculate the possible number of either 1000 or 2000 ft long

horizontal wells that will drain the lease effectively.

Solution

Step 1. Calculate the drainage radius of the vertical well:

√ ( )

Step 2. Calculate the drainage area of the 1000 and 2000 ft long horizontal well using Joshi’s two

methods.

Method I:

For the 1000 ft horizontal well and using Equation 5.51:

( )

( )( ) ( )

For the 2000 ft horizontal well:

( )

( )( ) ( )

Method II:

For the 1000 ft horizontal well and using Equations 5.52 and 5.53:

( )( )

For the 2000 ft horizontal well:

( )( )

Step 3. Averaging the values from the two methods

For 1000ft:

For 2000ft:

Step 4. Calculate the number of horizontal wells 1000 ft long:

Step 5. Calculate the number of horizontal wells 2000 ft long:

5.2.3 Horizontal well productivity under steady-state flow The steady-state analytical solutions are the simplest form of horizontal well solutions. The

steady-state solution requires that the pressure at any point in the reservoir does not change with

time. The flow rate equation in a steady-state condition is represented by:

( )

(5.55)

Where:

STB/day

STB/day/psi

The productivity index of horizontal well :

There are several methods that are designed to predict the productivity index form the fluid and

reservoir properties. Some of these methods include:

 The Borisov method

 The Giger, Reiss, and Jourdan method

 The Joshi method

 The Benard and Dupuy method

Brisov method

Brisov (1984):

[ (

)

(

)]

(5.56)

Where:

STB/day/psi

Giger, Reiss, and Jourdan method

Giger et al. (1984): isotropic reservoir,

[(

) ( ) (

)]

(5.57)

Where:

√ [ ]

( )

(5.58)

To account for the reservoir anisotropy, the authors proposed the following relationships:

[(

) ( )

(

)]

(5.59)

With:

√

(5.60)

Joshi method

Joshi (1991): isotropic reservoirs:

[ ( )

(

)]

(5.61)

With:

√ (

)

( )

(5.62)

and a is half the major axis of the drainage ellipse and given by:

(

)[ √ ( )

]

(5.63)

For anisotropy reservoir:

[ ( )

(

)]

(5.64)

Where the parameters B and R are defined by Equations 5.60 and 5.62.

5.3 Phase 3: Relating reservoir performance to time

Figure 5.15 Cumulative production as a function of average reservoir pressure.

Figure 5.16 Overall field IPR at future average pressure.

The following methodology can be employed to correlate the predicted cumulative field

production with time :

Step 1. Plot the predicted cumulative oil production as a function of average reservoir

pressure as shown in Figure 5.15.

Step 2. Assume that the current reservoir pressure is with a current cumulative oil production

of ( ) and total field flow rate of ( )

Step 3. Select a future average reservoir pressure and determine the future cumulative oil

production from Figure 5.15.

Step 4. Using the selected future average reservoir pressure , construct the IPR curve for each

well in the field (Figure 1.16). Establish the total field IPR by taking the summation of

the flow rates of all wells at any time.

Step 5. Using the minimum bottom-hole flowing pressure ( ) , determine the total field

flow rate ( ) .

( ) ∑ ( )

Step 6. Calculate the average field production rate ( ̅̅̅̅ ) :

( ̅̅̅̅ ) ( ) ( )

Step7. Calculate the time required for the incremental oil production during the first

pressure drop interval, i.e., from to , by:

( ̅̅̅̅ )

Step 8. Repeat the above steps and calculate the total time t to reach an average reservoir

pressure p, by:

∑