A Semi-Analytical Approach for the Estimation of Reservesin a Dry-Gas Reservoir Using Rate-Time and Cumulative Production Data

Ibrahim Muhammad Buba

B.Eng., University of Bath (1999)M.Sc., Texas A&M University (2002)

Chair of Advisory Committee: Dr. Thomas A. Blasingame

ObjectivesThe objectives of the research being proposed are:

Development of a novel technique for the direct estimation of reserves (G) in a volumetric dry-gas reservoir using only production data. The direct solution proposed is expected to be in the form of a graphical solution to a generalized expression.

Derivation of a linear material balance expression from first principles to model the production profile of a volumetric dry-gas reservoir. This approach will require the use of dimensionless variables so as obtain a general expression which can be used to represent and/or analyze any volumetric dry-gas reservoir.

To investigate the possibility of a non-iterative material balance solution for the estimation of reserves in a dry-gas reservoir. This solution will avoid the need for calculations of supporting parameters or iterations, it is be expected to be a single independent calculation.

DeliverablesThe expected deliverables of the research are:

Presentation of a timesaving and straightforward technique for the estimation of ultimate recovery of naturally pressured dry-gas reservoirs using a material balance expression. The solution to this problem will be obtained as the root of the linearized graphical representation of the material balance expression.

Verification of the new method with existing methods. Such verification will consist of the accuracy and reliability of the new method in relation to existing methods.

A complete examination of the proposed method to data sensitivity and reliability using both field and numerical simulation data.

Present Status of the ProblemThe methods presently used in the estimation of original gas-in-place for a volumetric dry-gas reservoir may require prior knowledge of formation, well or fluid properties which can be inaccurate and yield an erroneous estimation. Some of the a priori information could be a statistical approximation, a weighted average value or an approximation based on historical performance. Wrong estimation of reserves will affect reservoir development, performance prediction and ultimate recovery. It is important for operators to have a reasonable estimate of original gas-in-place early on in the to improve the economical gains of such reservoirs.

Presently, calculation of original gas-in-place may require several iterations and/or secondary calculations of other reservoir or well parameters. These methods can be time consuming, complex and are susceptible to errors, as a wrong calculation of any of the other parameters will propagate the errors in the sequence of calculations. As mentioned earlier, this research presents a direct method for the estimation of original gas-in-place considerably reducing the secondary calculations requirements and iterations. The computation of the method being proposed also has a slim requirement for data usage.

The use of production data as a forecasting tool dates back to 1918 when Lewis and Beal 1 presented the consistent shape of the production decline curve as a mathematical tool which may be used to forecast future production. The authors observed the production decline on a Cartesian plot of rate against time has a power law function, with the coefficients calculated determined and a forecast of future production easily projected. Accuracy of early production data is necessary for this method, although the authors failed to realize that. Another flaw of this technique is the pressure stability of the reservoir in question. During early-time production, the reservoir pressure may not stabilize to represent the accurate drawdown potential. The Lewis and Beal observation when trying to linearize the power law function would require a considerable amount of tampering with the data, which will certainly yield erroneous projections for production.

The material balance method developed years after the volumetric method has become increasingly popular with most production data analysts. In 1941, Schilthius2, 3 presented a general form of material balance equation derived as a volumetric balance. Like the volumetric method, it was assumed that the reservoir pore volume remains unchanged or changes in a consistent manner with respect to reservoir pressure. The data required for this material balance equation are fluid production, reservoir temperature, reservoir pressure, reservoir fluid properties and core data. The material balance equation presented by Schilthius is given as

....................................................................................................................(1)

Where Gp is the cumulative gas production, G is the gas-in-place and Bgi and Bg are the initial and present time gas formation volume factors. The authors used the principle of conservation of mass as the basis for the expression. In volumetric terms, the remaining reserve is the initial reserve less the produced reserve.

The material balance equation was simplified and represented graphically, with the root of the expression as the potential ultimate cumulative production for the well without an external source of energy. In particular, the graph of p/z vs. Gp will be a straight line graph only requiring an extrapolation of the data points to the x-axis where the ultimate cumulative production for that well is estimated.

............................................................................................. (2)

The complexity associated with the material balance method for reserve estimation is linked to the calculation of average reservoir pressures. The average reservoir pressure is required for the calculation of pressure dependent parameters in the material balance equation. The average reservoir pressure will be the reference point for all parameters throughout the producing life of the reservoir, therefore its importance in the material balance method cannot be overstated.

The first comprehensive attempt to linearize the gas flow equation was by Agarwal4 in 1979. The author presented a pseudotime function for real gas, incorporating changes in gas properties with pressure change as a function of time. The pertinent changes in gas properties over the development period of the reservoir were associated with compressibility and viscosity. This would avoid the estimation of gas properties at one reference pressure as is the case with the p/z material balance approach for forecasting ultimate cumulative production. The authors approach was limited in its uses though, as results of this technique proved to be successful for hydraulically fractured wells only. The pseudotime function developed by Agarwal is given as

........................................................................................................(3)

2

In 1987, Fraim and Wattenbarger5 modified Agarwals’ pseudotime function by normalizing pseudopressure and pseudotime functions to account for the non-linear product of gas compressibility and viscosity, .

As expected, the normalized pseudopressure and pseudo-time functions linearized5 the gas diffusivity equation and allowed for the use of liquid flow solutions (Fetkovich type curve) for the analyses of gas production data. Although Fraim and Wattenbargers’ suggestion can be incorporated easily into decline-type curve analysis, it still requires the knowledge of average reservoir pressure for the normalized pseudotime function to calculate several gas properties. This is yet seen as a retrogressive step in the analyses of gas production data as gas properties are known to change somewhat drastically with large pressure changes.

Lee and Blasingame6 introduced a new concept which was quickly adapted in the analysis and interpretations of variable rate and variable pressure production data. Their paper was specifically for the prediction of drainage size area and reservoir shape from variable production rate. The solutions they proposed was not related to type-curve matching. It was clear from the conclusions that the idea would be useful in decline curve analysis. One of such is the possibility of estimating reservoir drainage area from the late-time data. Late-time data exhibit characteristics which suggest boundary dominated flow and its slope on a log-log graph can be applied to the Blasingame and Lee equations.

The idea of using rate versus cumulative production to forecast future production has been revisited several times since Lewis and Beal1 first observed the trend. Knowles7, 8 presented a new approach for linearizing the gas flow equation. Instead of using the constant parameter linearization as proposed by Carter9, Knowles introduced a straight-line linearization scheme in the form of a first order polynomial function. This resulted in the (p/z) 2 form of the stabilized flow equation coupling directly with the gas material balance equation to form analytical pressure-time and rate-time equations. Knowles dimensionless pressure-time and rate-time relations for pwf 0 are given as

........................................................................... (4)

...................................................(5)

The expressions above being functions of time, rate and pressure are not suitable for the determination of reserves in the forms represented. A dimensionless form of the cumulative production was defined7 and developed from both the pressure-time and rate-time relations. The dimensionless cumulative production relation, GpDd, is defined as the ratio of cumulative gas produced and gas reserves. This is represented mathematically as

............................................................................................................. (6)

3

Or in dimensionless pressure

..............................................................................................................................(7)

Recalling the expression for dimensionless pressure, pD when pwf 0 (Eq. 4) and substituting into Eq. 7 above gives

..................................................................(8)

Defining the term

Let ................................................................................(9)

Substituting Eq. 9 into Eq. 8 gives

.......................................................................................................................(10)

Recalling the dimensionless rate-time relation for pwf 0 (Eq. 5), and substituting Eq. 9 simplifies it to

.........................................................................................................(11)

Substituting Eq. 10 into Eq. 11 and rearranging gives

...................................................................(12)

Expanding Eq. 12 by substituting for all the dimensionless terms gives

...........................................(13)

The “decline” constant, Di , is defined as

..........................................................................................................(14)

4

Writing Eq. 13 in terms of the decline constant gives

.......................................................................................................(15)

Or writing Eq. 13 in arbitrary constants gives us

..................................................................................................................(16)

Eqs. 15 and 16 above are the final extensions of the “Knowles” gas flow equation referred to as the rate-cumulative production equation. This expression suggests that a Cartesian plot of qg versus Gp for which pwf 0 will yield a quadratic trend with the roots of the graph being the “movable” gas reserves. The above expression has no requirements for average reservoir pressure and other reservoir or fluid properties that are required for calculations in all other methods used in estimating total cumulative gas production. The rate-cumulative production relation will be the starting point for the derivation of the method being proposed for the estimation of gas reserves, G, in a volumetric gas reservoir.

ProcedureThe overall objective of this research is to present an analytical tool for the direct estimation of gas reserves. Such estimation will involve a graphical extrapolation of a function with respect to cumulative gas production. In order to achieve this, the rate-cumulative production equation, Eq. 15 will be manipulated arithmetically to a graphical form with a linear portion, which extrapolates to a point related to the gas reserves under the existing conditions. The final form will then be represented on a Cartesian plot with a special plotting function on the vertical axis, and cumulative gas production on the horizontal. In this work, several algebraic forms of Eq. 15 are presented all of which are arithmetically and graphically proven.

Derivation of a Direct Extrapolation Formula for Gas ReservesThree forms of the “quadratic” functions are proposed for the direct extrapolation method for estimation of gas reserves. All three expressions are derived from the governing quadratic equation (Eq. 15). The following assumptions are given for these derivations;

a. Reservoir being produced at a constant bottomhole pressure, (i.e., pwf = constant).b. The reservoir is flowing under pseudosteady state.

c. (Simplifying assumption for gas flow behaviour)

Recall the governing equation for the development of these formulae;

.......................................................................................................(15)

Where

..........................................................................................................(14)

5

By isolating the constant in the quadratic term in Eq. 15, we have:

Or

.....................................................................................................(17)

Given the dominance of the term, in a practical sense the term can often be assumed to be zero. In such a case (i.e ), Eq.15 becomes:

.........................................................................................................................(18)

Eq. 18 is identical in form to the result obtained for the case of a slightly compressible liquid, which validates at least conceptually, the liquid case as a subset of the gas case.

Comparing Eqs. 15 and 18 graphically, we have

Gas and Liquid Rate-Cumulative Relations

Gp, MSCF

(qg)

Gas

Rat

e, M

SCF/

D

GasLiquid

Gp,max, Eq. 18

Linear

Quadratic

2

2Gp

GDiDiGpqq gig

DiGpqq gig

(1) (2)

qgi

0

0

Fig. 1 Schematic behaviour of the “gas” and “liquid” forms of the “rate-cumulative production” relation.

Obviously the “liquid” form of the rate-cumulative relation (Eq.18), yields the most conservative estimate of the maximum reserves (Gp,max , Eq. 18 on Fig. 1). The issue is although this estimate will always be conservative (i.e low), the technique is both straight-forward and consistent.

6

Other issues remain

a. What is the physical significance of point (1) on Fig. 1 (i.e, qg = 0)? This would seem to be important (or least relevant), but the meaning is not clear.

b. What is the physical significance of point (2) on Fig.2 (i.e., )? It will evolve that this represents the maximum reserves for the case of the gas model (i.e. Eq. 15).

To resolve issue “b”, we will consider plots of qg versus Gp as well as versus Gp.Taking the derivative of the governing equation, Eq. 15 with respect to GP gives

.............................................................................................(19)

Plotting Eqs. 15 and 19 on separate plots, we have:

7

Gas Rate Versus Cumulative Gas Produced

Gp, MSCF

(qg)

Gas

Rat

e, M

SCF/

D

Gp, max (Eq.18)

Slope = Di

(qg)d,Gp=0

(G)

qgi

0

0

Gas Rate Derivative Function versus Cumulative Gas Produced

Gp, MSCF

(qg )

d,G

p =

dqg/

dGp

Slope = Di/G

Gp(qg)d,Gp=0

(G)

-Di

0

0

Figs. 2a and 2b – Schematic Plots for “gas” relation,

8

Setting Eq. 19 to zero, we obtain

Or

....................................................................................................................(20)

Using Fig. 2b and Eq. 20, we have established a direct technique to estimate gas reserves using only qg and GP data. Specifically, we have the following procedure:

1. Calculate using qg and Gp data

2. Construct a plot versus Gp as illustrated in Fig. 2b.

3. Extrapolate the straight-line trend of versus Gp to intersect

While this technique is both promising and straightforward, it is usually extremely difficult to obtain a meaningful estimate of the function due to erratic behaviour in the rate-time data. This issue leads us to consider the other formulations of this problem.

Specifically, consider the case of the rate-time integral, (qg)id, Gp. These functions are defined as:

..........................................................................................................(21)

And

.......................................................................................................(22)

Substituting Eq.15 into Eq.21, we obtain:

Which expands to

Or

...........................................................................................(23)

Substituting Eq. 23 into Eq. 22, we have

9

Or

.......................................................................................................(24)

Summary

.......................................................................................................(15)

.............................................................................................(19)

...........................................................................................(23)

.......................................................................................................(24)

A few options for estimating the gas reserves (G) using Eqs. 15, 19, 23 and 24 remain in particular, we can substitute Eq. 15 to obtain

.......................................................................................(25)

Solving for , Eq. 25 is arranged to yield

.............................................................................................(26)

Defining

................................................................................................................(27)

Substituting Eq. 27 into Eq. 26, we have

...........................................................................................................(28)

10

Rate-Time Integral Function versus Cumulative Gas Produced

Gp, MSCF

(qg )

i,Gp

- qg

/ Gp

Slope = 1/3(Di/G)

(Gp)yq,Gp=0

(a)

Intercept = Di/2

0

0

Fig. 3 Schematic plot for rate-time integral function versus cumulative gas produced.

11

Rate-Time Integral Function Versus Cumulative Gas Produced

Gp, MSCF

(qg )

i,Gp

- qg

/ Gp

Slope = 1/3(Di/G)

(Gp)yq,Gp=0

(a)

Intercept = Di/2

0

0

From Fig. 3, we have point (a) (i.e., (Gp)yq, Gp =0), we need to reconcile what (if any) representation this point has with gas reserves, G. Setting Eq. 28 equal to zero gives:

,

Or

...................................................................................................................(29)

Therefore, point (a) on Fig. 3 is

Similar to our work with qg and (qg)d,Gp on Figs.2, we can also use Eqs., 23 and 24 to generate a similar sequence of plots. Recalling

...........................................................................................(23)

.......................................................................................................(24)

Plotting Eqs. 23 and 24 on separate plots, we have:

Gas Rate-Integral Function versus Cumulative Gas Produced

Gp, MSCF

(qg)

Gas

Rat

e, M

SCF/

D

Slope = Di/2 (qg)d,Gp=0

qgi

(3/2G)

0

0

Gp,max - Linear extrapolation (qg)i,Gp

Fig. 4a Schematic plot for the Gas Relation-rate-integral function of the rate-cumulative production relation.

12

Gas Rate-Integral Difference Function versus Cumulative Gas Produced

Gp, MSCF

(qg )

id,G

p =

d(q g

)iGp/d

Gp

Slope = Di/3G

Gp(qg)id,Gp=0

-Di/2

(3/2G)

0

0

Fig. 4b Schematic plot for the Gas and integral difference functions of the rate-cumulative production relation.

From Eq. 24, setting equal to zero gives

...............................................................................................................(30)

This result is identical to Eq. 29

Using Figs. 4a and 4b we can establish an analysis procedure for the (qg)I,Gp and (qg)id,Gp functions as follows:

1. Calculate (qg)I,Gp and (qg)id,Gp using qg data

2. Construct a plot of (qg)id,Gp versus Gp (Fig. 4b)

3. Extrapolate the straight line portion of the data trend ((qg)id,Gp versus Gp to (qg)id,Gp=0. This

result is

It is important to note that the (qg)id,Gp versus Gp and versus Gp methods are mathematically identical, either technique should be equally applicable.

13

The final technique for estimating gas reserves, G is simply to use Eq. 15 as a regression model- where the parameters are established as follows:

1. Estimate qgi-using qg versus Gp. This should be straightforward provided transient data are not used.

2. Use as a base model and rearrange into the form

.............................................................................................................(31)

Or

..................................................................................................................(32)

Linear Regression Model

Gp, MSCF

(qg )

i - q

g / G

p

Slope = (Di/2)

(Gp){qgi - q/Gp}=0

(2G)

Intercept = Di

0

0

Fig. 5 Schematic plot for

From Fig. 5 estimates can be obtained for the “decline” constant, Di and the reserves G. Coupling these results with the qgi estimate we have essentially the data required to model qg as a function of Gp. Eq. 15 and the qi, Di and G estimates should be combined to simulate the data trend. Using MS-Excel or a similar computation/graphics software tool should be sufficient to obtain a refined analysis.

14

Auxiliary Functions

To this point, we have used the following auxiliary functions:

..........................................................................................................(21)

and

.......................................................................................................(22)

We can develop another auxiliary function by simply differentiating Eq. 22 again, which gives

....................................................................................................(33)

In addition to these functions we can also define a family of “double integral” functions based on Eq. 21. This family is given by:

...............................................................................................(34)

.....................................................................................................(35)

..................................................................................................(36)

Recalling the “single integral” results (i.e., combining Eqs. 15, 21, 22), we have:

....................................................(15)

.................................................(23)

....................................................(24)

Continuing

...........................................................(37)

For the “double integral” family, we have the following results

.....................................(38)

.........................................(39)

15

..................................................(40)

Substituting Eq. 37 into Eq. 24 gives

Or solving for the term, we have

................................................................................(41)

Substituting Eq. 37 and Eq. 41 into Eq. 24 gives us

Or

Solving for the qgi term, we obtain

......................................................(42)

Recalling Eq. 31, the rearrangement of Eq. 15 into a linear form,

.............................................................................................................(31)

Substituting Eq. 42 into Eq. 31, we obtain

...........................(43)

Or

Where

16

Setting y = 0, we have

Or

.........................................................................................................................(44)

The generalized expression for the normalized rate function (Eq. 44) is similar to that obtained for the “linear” regression model (Eq. 32). The graphical representation of the normalized rate function shown in Fig. 6 below also an exact replica of the “linear” regression model (Fig. 5).

Normalized Rate Function versus Cumulative Gas Produced

Gp, MSCF

{(qg )

iGp

- (q g

)id,G

p - (

qg)id

d,G

p(G

p-1/

2Gp^

2) - q

g }/

Gp

Slope = (Di/2)

(Gp)y=0

(2G)

Intercept = Di

0

0

Fig. 6 Schematic plot for Gp Normalized Rate functions

The calculation of the “single integral” auxiliary functions is straightforward. However, one may need to experiment with different schemes to calculate the integral and integral difference functions.

17

Summary of Direct Extrapolation Formulae for Gas Reserves (Graphical Solutions).Recall the governing rate-cumulative quadratic equation,

.......................................................................................................(15)

1. Rate-Derivative Function.

Or

....................................................................................................................(20)

2. Rate-Integral Function

...............................................................................(26)

Therefore,

....................................................................................................................(29)

3. Rate-Integral Difference Function

...........................................................................................(23)

.......................................................................................................(24)

At the root of the function,

.................................................................................................................(30)

4. “Linear” Regression Model

.............................................................................................................(31)

Hence

..................................................................................................................(32)

5. Auxiliary Functions

....................(43)

18

Then

.........................................................................................................................(44)

Validation of New MethodThe new method will be verified against existing methods for the estimation of ultimate cumulative production. Viability, acceptability and accuracy of the new method will be questioned by this exercise. A full field application case history will be carried out for the validation along with selected data from literature. Verification of the new method using numerical simulation data will also be carried out to ascertain the flexibility of such method. The reservoir model is that of a single-phase reservoir which does not exhibit water influx behavior and isothermal with no compressibility effects. The simple model applied for this exercise investigates the feasibility of the new method for estimation of reserves without reservoir complexities, which may be an extension of this research in the future.

Verification Using Synthetic Data

A gas production profile was numerically simulated with detailed knowledge of the value of total reserve for that particular production history. The data set generated represents a typical response from a volumetric gas reservoir producing at a constant bottomhole pressure. In particular, the simulation strictly followed the characteristics of the reservoir model listed earlier in the chapter. The mathematical models derived earlier are put to the test with this data set to ascertain their viability.

The parameters used to generate the production profile are;

Decline constant, Di = 0.221Gas reserve, G = 4.0 MSCF

The production profile was edited to exclude data points that may fall into transient flow, leaving only those representing pseudosteady state flow in the reservoir. The entire production profile of the numerically simulated gas production profile is shown in Fig. 7 below

19

Simulated Case Production ProfileGas Rate versus Cumulative Gas Produced

0.00E+00

4.00E+05

8.00E+05

1.20E+06

1.60E+06

0.00E+00 1.00E+06 2.00E+06 3.00E+06

Cumulative Gas Production, MSCF

Gas

Rat

e, M

SCF/

D

Raw Simulated Data

Edited Simulated Data

Fig. 7 Schematic plot for the simulated production profile showing raw and edited data sets

The edited portion of the production profile shown in Fig. 7 above is redrawn separately on a Cartesian plot and the parameters (coefficients) revealed by the equation of the plot. The edited profile follows the “rate-cumulative production quadratic equation”, Eq. 15, which is the basis for this exercise. The intercept on the y-axis is the taken as the initial gas rate as expected from the governing quadratic equation.

20

Simulated Case Production ProfileGas Rate versus Cumulative Gas Production - Edited Data

y = 3E-08x2 - 0.228x + 395788

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06 3.00E+06

Cumulative Gas Production, MSCF

Gas

Rat

e, M

SCF/

D

Fig. 8 Schematic plot showing the edited simulated production profile, its governing equation and parameters

The solutions according to each of the models listed are presented graphically using the edited data set above.

General Solution

Recall the general solution suggests the root of the production profile as the value of reserves. Although this solution can ever be duplicated by field data as required, it provides an invaluable insight into other possible solutions for the model. This is shown in Fig. 9 below

…………………………………………………(15)

21

Simulated Case - General Solution For Gas Reserves (Edited Data) Gas Rate-Cumulative Relations

-2.00E+05

0.00E+00

2.00E+05

4.00E+05

6.00E+05

0.00E+00 2.00E+06 4.00E+06 6.00E+06

Gp, MSCF

(qg)

Gas

Rat

e, M

SCF/

D

qgi

G

Fig. 9 Plot showing the gas reserves using the general solution of the quadratic equation on numerically simulated data

Rate-Derivative Function.

The rate-derivative function formula resolves the issue of “negative” production rates indicated by the general solution (Fig. 9). Excluding the negative data points, the governing quadratic equation data set is differentiated and plotted against the cumulative gas production. A linear relationship is noticed, thereby making extrapolation possible. The graphical solution and extrapolation are shown below;

Or

……………………………...……………...…….…...…(20)

22

Simulated Case - Rate Derivative Function (Edited Data)Rate Derivative Function versus Cumulative Gas Produced

0.00

0.10

0.20

0.30

0.40

0.00E+00 1.00E+06 2.00E+06 3.00E+06 4.00E+06 5.00E+06

Cumulative Gas Production, MSCF

dqg/

dGp

0,,

GpdqgpGdg G

GDiDiq

p

G 0,

GpdqgpG

G

Fig. 10 Plot showing the gas reserves using the rate-differential function model technique on numerically simulated data.

Rate-Integral Difference FunctionAnother solution is presented by integrating the production rates with respect to cumulative production and extrapolated as a function of cumulative production to the root of the data. For this purpose, the negative rates generated for the numerical solution are not included for this extrapolation exercise.

…………………...……………………(25)

………………….………………………….…….(26)

At the root of the function,

……………………………………………………..….(29)

23

Simulated Case - Rate Integral Difference Relation (Edited Data)Quadratic Plotting Function versus Cumulative Gas Produced

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.00E+00 2.00E+06 4.00E+06 6.00E+06 8.00E+06

Cumulative Gas Production, MSCF

Qua

drat

ic P

lotti

ng F

unct

ion

Gp = 3/2 G

2

, 62 ppgiGpig GG

DiGDiqq

pGpidg GG

DiDiq32,

GGGpidqgp 2

30,

Fig. 11 Plot showing the gas reserves using the rate-integral difference relation model technique on numerically simulated data.

“Linear” Regression ModelThe “linear” regression model is sensitive to the value of initial rate, qgi. Otherwise, this method presents the most consistent extrapolation to the solution of the “rate-cumulative production equation”, shown in Fig 12 below;

……….……..…………………...………....(31)

Hence

………………………………………………….(32)

24

Simulated Case - Regression Model (Edited Data)Regression Function versus Cumulative Gas Produced

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

0.00E+00 2.00E+06 4.00E+06 6.00E+06 8.00E+06 1.00E+07

Cumulative Gas Production, MSCF

(qgi

, - q

g)/G

p

Gp = 2G

pp

ggi GG

DiDiG

qq2

GGGpqqgip 2

0

Fig. 12 Plot showing the gas reserves using “linear” regression model technique.

The results obtained from the extrapolation of data points of each method indicate a great deal of accuracy of all the methods. In particular, those methods (“Linear” regression and rate-differential) which required very little data tampering, and manipulations proved to be more consistent.

Data RequirementsThe data required to solve the material balance equations are normally acquired during the producing life of the reservoir. Like all material balance methods, the method being proposed will also require ample production data. The uniqueness of this new technique is its limited data requirement. Rate-time and cumulative gas production are standard measured data in all producing wells, therefore no special consideration is required in obtaining data like well-tests for this method. More importantly, laboratory experiments or measurements will not be necessary in order to augment the data required.

Sensitivity AnalysisIn examining the sensitivity of the proposed method to field data, raw and edited data sets for a particular well will be used to calculate ultimate cumulative production. The edited data sets will be a reduced form of the raw data with the scattered points removed from the computation. Such scattering is largely due to human error in measurement or a faulty tool. The extrapolation to be carried out on the plot will require consistent data point as the case is with the p/z method. Unlike the p/z extrapolation, the proposed method with typically more data points requires the stem of the plot to be a perfect straight line for extrapolation to total cumulative production. This can only be achieved through extensive data editing. Data editing will be carried out from the exponential trend of gas production rate vs. cumulative gas production. The Fig. 13 below shows the edited data set overlapping the raw data. The raw data points excluded during computation (using edited data only) are shown as the scattered circular points.

25

Fig. 13 Production profile showing Raw and Edited data sets for a gas well

Field ExamplesField A, East Texas Well 5

Well 5 is a vertical gas well undergoing natural depletion at a fairly constant bottomhole pressure. The formation is recognized to be of low to very low permeability. Gas is produced from this well at a fairly constant bottomhole pressure and there are no indications of an aquifer or external driving force. Fig. 13 above shows the production profile for well 5, in which the edited data is used to for computational purposes. The “Linear” regression model being the most accurate of the models derived earlier is applied to the production data of this well. Fig. 14 below shows the resulting extrapolation;

26

Field A, Well 5 - Regression Model (Edited Data)Regression Function versus Cumulative Gas Produced

0.00E+00

5.00E-03

1.00E-02

1.50E-02

0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06

Cumulative Gas Production, MSCF

(qgi

, - q

g)/G

p

Gp = 2G

Fig. 14 Plot showing the gas reserves using “linear” regression model technique for well 5, Field A.

Ignoring the early-time (transient) data, qi was taken as 6,500 MSCF/D for the quadratic portion of the profile. The linear extrapolation on Fig. 14 shows well 5 to contain reserves of 900,000 MSCF.

West Virginia Reservoir, Well A

Well A is a vertical well which has been hydraulically fractured and is undergoing depletion. The formation is recognized to be of low to very low permeability. The original production data was reported by Fetkovich et. al10 and was later analyzed in Refs. 11, 12 and 13. Gas is produced from this well at a fairly constant bottomhole pressure. The well was flowing under a pseudosteady state regime when the original data reported was obtained. Summary of reservoir and fluid properties is given below

West Virginia Reservoir, Fluid Property and Production Data:Reservoir Properties

Average net pay thickness, h = 70 ftAverage Porosity, (fraction) = 0.06 (fraction)Average formation permeability, k* = 0.07 md

Fluid propertiesInitial gas formation volume factor, Bgi =7.1 x 10-4 RB/scf

Initial gas viscosity, gig = 0.0225 cpTotal compressibility, cti = 1.824 x 10-4 psi-1

Production parametersInitial reservoir pressure, pi = 4175 psiaBottomhole flowing pressure, pwf = 710 psiaInitial production, qi

# = 1820, psia

27

* From pressure transient test analysis# Ignoring early-time/ transient data

Application of the “linear” regression method to this well is shown in Fig. 15 below

Well A, West Virginia Reservoir - Regression ModelRegression Function versus Cumulative Gas Produced

0.00E+00

5.00E-04

1.00E-03

1.50E-03

0.00E+00 2.00E+06 4.00E+06 6.00E+06 8.00E+06

Gp, MSCF

Y =

(qg

- qi)/

Gp

Gp = 2G

Fig. 15 Plot showing the gas reserves using “linear” regression model technique for well A, West Virginia Reservoir.

Extrapolation of the data points of “Linear plotting function” versus cumulative gas production intersects the horizontal axis at Gp = 7.0 MMSCF, which translates to the value of 3.5 BSCF for OGIP of Well A.

Well A (West Virginia Reservoir) Analysis Summary

The results for the analysis of well A are given below in table 1. The value for OGIP obtained using the new semi-analytical method is the largest but not necessarily wrong as the well is reported to have produced 3.2 Bscf at economic limit. The new method is thus verified.

Method OGIP, BscfRef. 11 3.034Ref. 12 2.620Ref. 13 2.849This Work 3.500

Table 1 – Comparison of Results of OGIP Estimation for Well Co-A using different methods

28

DiscussionIn this report, we were able to present several direct methods for the estimation of gas reserves by solving the “cumulative production quadratic equation”, which is the resulting expression of coupling the gas flow and gas material balance equations. By observation, the solution to the numerically simulated model fell at the point of inflexion of the data set which then presented numerous avenues for analytical derivations consistent with real data. The graphical representations of the analytical solutions to the generalized expression are consistent and easily applicable to real data. Verification of the methods with numerically simulated data shows the sensitivity of this method with respect to data handling. In cases where the data is integrated, the integration technique being applied is very crucial and slight approximations may result in a distortion of the data and wrong estimates may result from such extrapolations. Nonetheless, all the graphical solutions will extrapolate within a reasonable percentage error. The field application of the new method showed its accuracy when compared to more demanding methods.

ConclusionsThe following conclusions can be made based on the results obtained from this work

The semi-analytical method developed is an alternative method for the estimation of original gas-in-place for low permeability and tight gas reservoirs using only production data. The successful application of the method and comparable results obtained with other techniques attests to the viability of this new semi-analytical method.

The most important aspect of this method is the type of reservoir, fluid and/or production data required. It has the leanest requirement of data type in the form of production history only. This reduces the time required to collect specific reservoir or well properties for the estimation of original gas-in-place.

The new method may not be applicable in early time due to data quantity, therefore more checks are required when more data is collected. The new method thus requires a decent amount of the reservoir to be depleted for accurate gas forecasting.

Another downside to this method is the quality of data required for the method. Data may require editing which is tailored to suit the method. Although this has a negligible effect on the results if the overwhelming production data profile is noticeable. In situations where data collection is poor and the production profile shows no consistent trend, application of this method may be erroneous.

An advantage of the new method over type curve analysis for reserve forecasting is the importance of only late-time or latest reported data. Type curve analysis is hindered with matching early and mid-term data as well as late-time data. This may affect its accuracy as early time data is usually the most unreliable.

This method is applicable only to harmonic and hyperbolic production decline type. Analysis of wells undergoing exponential decline are not possible using this semi-analytical method

Recommendations for Future WorkThe following recommendations are put forward for an extension into this novel research work

The coefficients of the new method can be determined using regression analysis so as produce a generalized equation similar to the Fetkovich apparent open flow (AOF) potential equation. This will save the need for a graphical extrapolation or calculations of the “quadratic plotting functions and make reserve forecasting possible in early times with limited production data.

Presently, abnormal reservoir conditions like water influx will offset the balance required for this method to work. Its’ application to such conditions will require adjustments in the development of the quadratic plotting function like the adjustments to the p/z method to accommodate such

29

conditions. The quadratic plotting function should be modified application to reservoirs with active aquifers, abnormally pressured reservoirs and other similar conditions. This will increase the acceptability of the method by field operators who experience a lot of these conditions.

Organization of the ResearchThe outline of the proposed research thesis is as follows:

Chapter I Introduction Research problem Research objectives Summary of results

Chapter II Literature Review Rigorous material balance methods (p/z versus Gp) Decline curve analysis Semi-Analytical analysis methods (Knowles approach)

Chapter III Development of a New Method for the Estimation of Original Gas-In-Place Development of "rate-cumulative production" relation Development of analysis methods using the "rate-cumulative production" relation Validation using synthetic data

Chapter IV Field Applications of a New Method for the Estimation of Original Gas-In-Place Various field examples taken from the petroleum literature Sensitivity analysis of new method – raw and edited data sets Case history of a full-field application (Field A) Practical considerations Chapter summary

Chapter V Summary and Conclusions Summary Conclusions Recommendations for future work

Nomenclature

References

Appendices

Appendix A Derivation of Rate-Time-Cumulative Production Relations for A Well Produced at A Constant Flowing Bottomhole Pressure (Liquid System)

Appendix B Derivation of Knowles Material Balance Expression (KMBE)

Appendix C Sensitivity Analysis – Comparison of Raw and Edited Data Appendix D Summary of Analysis for the Full-Field Case History (Field A)

30

References:

1. Lewis, J.O., and Beal, C.H.: “Some New Methods for Estimating the Future Productions of Oil Wells,” Trans.AIME (1918) 59, 492-525.

2. Lee, W.J., and Wattenbarger, R.A.: “Gas Reservoir Engineering,” SPE Textbook Series Volume 5, 1996, 230-255

3. Schilthius, R.J.: “Active Oil and Reservoir Energy,” Trans., AIME (1936) 118, 33-52

4. Agarwal, R.G.: “Real Gas Pseudo-Time - A New Function for Pressure Buildup Analysis of MHF Gas Wells,” paper SPE 8279 Presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, NV, Sept. 23 – 26

5. Fraim, M.L. and Wattenbarger, R.A..: “Gas Reservoir Decline Curve Analysis Using Type Curves with Real-Gas Pseudopressure and Normalized Time.” SPEFE (Dec. 1987) 671-682; Trans. AIME, 290

6. Lee, W.J. and Blasingame, T.A.: “Variable-Rate Reservoir Limits Testing,” paper SPE 15028 presented at the 1986 Permian Oil and Gas Recovery Conference of SPE, Midland, TX, March 13-14, 1986

7. Knowles S. R.: “Development and Verification of New Semi-Analytical Methods for the Analysis and Prediction of Gas Well Performance,” M.S. Thesis, Texas A & M University, College Station, TX. Dec 1996.

8. Ansah, J., Knowles, R.S., and Blasingame, T.A.: “A Semi-Analytic (p/z) Rate-Time Relation for the Analysis and Prediction of Gas Well Performance”, Paper SPE 35268 presented at the 1996 SPE Mid-Continent Gas Symposium, Amarillo, TX, Apr. 28-30.

9 Carter, R.D.: “Type Curves for Finite Radial and Linear Gas Flow Systems: Constant-Terminal Pressure Case,” SPEJ (Oct. 1985)

10 Fetkovich, M.J., et al: “Decline Curve Analysis Using Type Curves – Case Histories,” SPEFE (Dec. 1987) 637-656

11 Fraim, M.L. and Wattenbarger, R.A.: “Gas Reservoir Decline Curve Analysis Using Type Curves with Real-Gas Pseudopressure and Normalized Time.” SPEFE (Dec. 1987) 671-682; Trans. AIME, 290

12 Blasingame, T.A., McCray, T.C. and Lee, W.J.: “Decline Curve Analysis for Variable Pressure Drop/Variable Flowrate Systems,” Paper SPE 21513 presented at the 1991 SPE Gas Technology Symposium, Houston, TX, Jan. 23-24

13 Ansah, J., Knowles, R.S., and Blasingame, T.A.: “A Semi-Analytic (p/z) Rate-Time Relation for the Analysis and Prediction of Gas Well Performance”, Paper SPE 35268 presented at the 1996 SPE Mid-Continent Gas Symposium, Amarillo, TX, Apr. 28-30.

31

NOMENCLATURE

Variablesb = decline curve exponent, dimensionlessBgi = initial gas formation volume factor, RB/SCFBg = gas formation volume factor, RB/SCFC = coefficient of the flow equation, MSCF/D-psia2/cp cf = formation (rock) compressibility, psia-1

cg = gas compressibility, psia-1

ct = total system compressibility, psia-1

cw = water compressibility, psia-1

Di = initial decline rate constant, D-1 G = original-gas-in-place (OGIP), MSCF Gp = cumulative gas production, MSCF h = reservoir thickness, ft k = average permeability, md m(p) = real gas pseudopressure, psiaN = original-oil-in-place (OOIP), STB Np,mov = movable oil, STB

= average reservoir pressure, psia pwD = dimensionless wellbore pressurepi = initial reservoir pressure, psia pwf = flowing bottomhole pressure, psia qDd = dimensionless decline rate qg = gas flow rate, MSCF/D t = time, D t(p) = Pseudotime,z = gas compressibility factor

Subscriptsavg = averageb = base (reference) conditionsD = dimensionlessg = gaso = oili = initial condition p = pseudopressurewf = wellbore conditions

Greek Symbols = porosity, fraction = specific gravity (air = 1) = viscosity, cp

32