4
A Semi-Analytical Approach for the Estimation of Reserves
in 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
Objectives
The objectives of the research being proposed are:
l 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.
l 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.
l 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.
Deliverables
The expected deliverables of the research are:
l 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.
l 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.
l A complete examination of the proposed method to data sensitivity and reliability using both field and numerical simulation data.
Present Status of the Problem
The 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 Beal1 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)
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 ¹ 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)
Or in dimensionless pressure
. (7)
Recalling the expression for dimensionless pressure, pD when pwf ¹ 0 (Eq. 4) and substituting into Eq. 7 above gives
(8)
Defining the term a
Let (9)
Substituting Eq. 9 into Eq. 8 gives
(10)
Recalling the dimensionless rate-time relation for pwf ¹ 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)
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 ¹ 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.
Procedure
The 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 Reserves
Three 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)
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
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.
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:
Figs. 2a and 2b – Schematic Plots for “gas” relation,
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.