@article{Walsh1995,
abstract = {AbstractThis work completes the search for a generalized material balance equation. No restrictions are placed on reservoir fluid compositions. Unlike the conventional material balance equation, the new equation specifically accounts for volatilized-oil and, therefore, is uniquely applicable to the full range of fluids, including volatile-oils and gas-condensates. Equally important, the new material balance equation retains the same simplicity with which the conventional material balance equation has become known. The new, generalized material balance equation is featured here by showing how it is used to predict reservoir performance and estimate reserves in example volatile-oil and gas-condensate reservoirs. In comparison, the conventional material balance equation is shown to lead to erroneous results. The new, generalized material balance equation leads to an improved method of reservoir performance analysis.IntroductionMaterial balance calculations are a useful method of reservoir performance analysis. They are routinely used to estimate oil and gas reserves and predict future reservoir performance. Schilthuis(l), in 1936, was among the first to formulate and apply material balances. Inherent in his and others' use of material balances were the following assumptions: (1) at most, there are two hydrocarbon phases: oil and gas, (2) at most, there are two hydrocarbon components: stock-tank oil and separator (surface) gas, (3) the reservoir oil-phase consists of stock-tank oil and separator-gas, (4) the reservoir gas-phase consists of only separator-gas and no stock-tank oil, and (5) there are no compositional gradients within the system. Assumptions 1 and 2 define the popular two-hydrocarbon- component formulation. Assumption 3 effectively accounts for the effects of dissolved- or solution-gas. Assumption 4 ignores the possibility of volatilized-oil and. therefore, restricts application to black-oils and dry-gases and precludes application to volatile-oils, gas-condensates, or wet-gases. Volatilized-oil is the stock-tank oil content of the free reservoir gas-phase, Assumption 5 means that the system is effectively treated as a tank with no gradients; accordingly, Coats(2) and others have sometimes referred to this type of model as a tank or zero-dimensional model. Despite their apparent oversimplicity, early tank models found widespread use in reservoir engineering applications( 3-16).As time progressed, more sophisticated material balance models evolved, each striving for greater generality. Eventually, following the advent of digital computers, the material balance equations were discretized and the first generation of multi-dimensional, black-oil, finite-difference reservoir simulators were developed in the 1960's (17-19). These models, like their zero-dimensional counterparts, could effectively simulate black-oil and dry-gas reservoirs but could 1I0t model volatile-oil and gas-condensate reservoirs. To address this limitation, Cook et al 20 in 1974 introduced a volatile-oil, finite-difference simulator. Their formulation retained the simplicity of two hydrocarbon components, but allowed for volatilized-oil. Consequently, their formulation was applicable to the full range of reservoir fluids. The resulting formulation was applicable to the full range of reservoir fluids. Coats(21) later, in 1986, verified the applicability of the black- and volatile-oil approaches in reservoir simulation by showing agreement with a fully-compositional, equation-of-state (EOS) reservoir simulator.},
author = {Walsh, M P},
doi = {10.2118/95-01-07},
file = {:C$\backslash$:/Users/ftn60/OneDrive/Access/Material Balance/A Generalized Approach to Reservoir Material Balance Calculations .pdf:pdf},
issn = {0021-9487},
journal = {Journal of Canadian Petroleum Technology},
number = {01},
pages = {10},
publisher = {Petroleum Society of Canada},
title = {{A Generalized Approach to Reservoir Material Balance Calculations}},
url = {https://doi.org/10.2118/95-01-07},
volume = {34},
year = {1995}
}
@article{Walsh1994,
author = {Walsh, M. P. and Ansah, Joseph and Raghavan, Rajagopal},
doi = {10.2118/27728-MS},
file = {:C$\backslash$:/Users/ftn60/OneDrive/Access/Material Balance/The New, Generalized Material Balance as an Equation of a Straight Line- Part 2 - Applications to Saturated and Non-Volumetric Reservoirs .pdf:pdf},
journal = {Proceedings of the Permian Basin Oil {\&} Gas Recovery Conference},
pages = {859--865},
title = {{New, generalized material balance as an equation of a straight line: part 2- Applications to saturated and non-volumetric reservoirs}},
year = {1994}
}
@article{Pletcher2002a,
abstract = {SummaryExperience with material-balance data sets from the field and from simulation has revealed some procedures that can be used to improve analysis of both oil and gas reservoirs:Failure to account for a weak waterdrive can result in significant material-balance errors.The assertion of previous authors that weak waterdrive exhibits a negative slope on the Cole (gas) and Campbell (oil) plots has been confirmed. A weak waterdrive is much more unambiguous on these plots than on commonly used plots, such as the p/z plot for gas.A modified version of the Cole plot is proposed to account for formation compressibility.The reservoir drive indices are a useful tool for determining the correctness of the material-balance solution because they must sum to unity. The drive indices should never be normalized to sum to unity because this obscures their usefulness and leads to a false sense of security.A modified version of the Roach plot (for gas) is proposed that improves interpretation in some waterdrive situations.Material balance has not been replaced by reservoir simulation; rather, it is complementary to simulation and can provide valuable insights to reservoir performance that cannot be obtained by simulation.IntroductionClassical material balance is one of the fundamental tools of reservoir engineering. Many authors have addressed the difficult problem of solving the material balance in the presence of a waterdrive (Refs. 1 through 5 are just a few of the more significant ones). The emphasis in the literature has been on strong and moderate waterdrives. In this paper, examples of weak waterdrives are shown in which the effects on the material balance are significant. All aquifers studied here are of the "pot aquifer" type, which is time-independent.In gas reservoirs, the plot of p/z vs. cumulative gas production, Gp, is a widely accepted method for solving the gas material balance1 under depletion-drive conditions. Extrapolation of the plot to atmospheric pressure provides a reliable estimate of original gas in place (OGIP). If a waterdrive is present, the plot often appears to be linear, but the extrapolation will give an erroneously high value for OGIP. Many authors have addressed this problem (including those in Refs. 2 and 5 through 8), especially in cases of strong or moderate waterdrives. The p/z plot is actually more ambiguous in weak waterdrives than in strong or moderate ones.The Cole plot7,9 has proven to be a valuable diagnostic tool for distinguishing between depletion-drive gas reservoirs and those that are producing under a waterdrive. The analogous plot for oil reservoirs is the Campbell plot.10 The literature has emphasized strong and moderate waterdrives, the signature shapes of which are a positive slope and a hump-shaped curve, respectively, on these plots. Previous authors have recognized that weak waterdrives can produce negative slopes on these two diagnostic plots, but this author is not aware of any example plots in the literature. This paper shows examples, using simulation and actual field data, wherein a negative slope clearly reveals a weak waterdrive. These plots are much more diagnostic than the p/z plot. Once a weak waterdrive has been diagnosed, the appropriate steps can be taken in the material-balance equations to yield more accurate results.The Cole plot assumes that formation compressibility can be neglected, which is frequently the case with gas. However, in those reservoirs in which formation compressibility is significant, a modification to the Cole plot is presented that incorporates formation compressibility and gives more accurate results.The reservoir drive indices have been used to quantify the relative magnitude of the various energy sources active in a reservoir. It is shown here that the drive indices are also a useful diagnostic tool for determining the correctness of a materialbalance solution because they must sum to unity. If they do not sum to unity, a correct solution has not been obtained. In some commercial material-balance software, the drive indices are automatically normalized to sum to unity, which not only obscures their usefulness but also leads to the false impression of having achieved a correct solution.The Roach plot has been presented11 as a tool for solving the gas material balance when formation compressibility is unknown, with or without the presence of waterdrive. This paper shows that for waterdrives that fit the small pot aquifer model, incorporating cumulative water production into the x-axis plotting term improves the linearity of the Roach plot and gives more accurate values for OGIP.Finally, it is argued that even in those reservoirs for which a simulation study is performed, classical material-balance evaluation should be performed on a stand-alone basis. Simulation should not be viewed as a replacement for material balance because the latter can yield valuable insights that can be obscured during simulation. Performing a separate material balance study usually will improve overall reservoir understanding and enhance any subsequent simulation study. Material balance should be viewed as a complement to simulation, not as a competing approach.In this paper, formation compressibility, cf, is assumed to be constant and unchanging over the reservoir life under investigation. References are given for recommended methods to be used in those cases in which cf is variable.},
author = {Pletcher, J L},
doi = {10.2118/75354-PA},
file = {:C$\backslash$:/Users/ftn60/OneDrive/Access/Material Balance/Improvements to Reservoir Material-Balance Methods .pdf:pdf},
issn = {1094-6470},
journal = {SPE Reservoir Evaluation {\&} Engineering},
number = {01},
pages = {49--59},
publisher = {Society of Petroleum Engineers},
title = {{Improvements to Reservoir Material-Balance Methods}},
url = {https://doi.org/10.2118/75354-PA},
volume = {5},
year = {2002}
}
@article{Fetkovich1998,
abstract = {Summary This paper presents the derivation of a general gas material balance that has particular application to high-pressure gas reservoirs, [both normal pressured and overpressured (geopressured)]. Its main application is to calculate original gas in place and assist in calculating remaining recoverable reserves from pressure/production data.The form of the material-balance equation is (p/z)[1-ce(p)(pi-p)]=(p/Z)i(1-Gp/G), which includes a pressure-dependent cumulative effective compressibility term ce (p)that is defined in terms of the following reservoir parameters: pore compressibility, water compressibility, gas solubility, and total water associated with the gas reservoir volume. "Associated" water includes connate water, water within interbedded shales and nonpay reservoir rock, and any limited aquifer volume, cephysically represents the cumulative change in hydrocarbon pore volume (PV)calculated by compressibility effects and encroaching water.High pressure gas reservoirs typically have concave downward p/z vs. Gp plots which may result in serious overestimation of original gas in place and remaining recoverable reserves. The proposed form of the gas material balance equation provides a method to linearize the p/z vs. Gp plot, and thereby predict the true original gas in place. A method is suggested to determine initial gas in place by analyzing the behavior of cumulative effective compressibility backcalculated from pressure/production data. The ce(p) function determined by this procedure, or estimated from logs and geological maps when sufficient production data is not available, is then used to forecast pressure/cumulative behavior. Two field examples are provided showing the application of the material balance equation to high pressure gas reservoirs.IntroductionHigh pressure gas reservoirs experiencing depletion drive to have downward curving p/z vs. Gp behavior. Incorrect extrapolation of early depletion data may result in serious overestimation of original gas in place and remaining reserves.Bruns et al.1work in 1965 was a result of a field study conducted a large moderately overpressured gas reservoir in the Texas Gulf Coast area. Investments were made, and never needed, based on linear extrapolation of the early field p/z vs. Gpperformance to an apparent original gas in place that was later found to be overstated by about 200 Bscf. Fig. 5 in Ref. 1 (Run 20) shows the concave downward curvature typical for the pressure response of a limited external aquifer system that simulated the reservoir's response.This type of "limited" aquifer behavior, where pressure in the reservoir and aquifer are virtually equal, led to the derivativation of a general material balance for high pressure gas reservoirs (see Appendix, Ref. 2). The derivation includes pressure-dependent rock and water compressibility(with gas evolving from solution). All water and rock volumes associated with the reservoir and available for expansion, including a limited aquifer volume, were included in a cumulative effective compressibility term ce(p). Rock and water compressibilities were defined to account for cumulative changes in volume to be multiplied by the cumulative pressuare drop (pi-p); instantaneous compressibilities are not used at all. The final form of the material balance is similar to that published by Ramagost and Farshad,3 except that they considered ce, as a constant. The general gas material balance as presented in this paper defines a cumulative effective compressibility ce(p) as a function of pressure.Literature ReviewHarville and Hawkins4 and Hammerlindl5 attribute the concave dowaward shape of p/z vs. Gpcurves obtained in abnormally pressured gas reservoirs entirely to pore collapse and formation compaction. No definition of pore collapse is given in Ref. 4, but a plot of backcalculated PV change indicated a system compressibility change from 28´10-6 psi-1 at initial pressure to about 6´10-6 psi-1 at low pressures. This magnitude of PV change implies associated water volume. The decreasing "system"compressibility is expected for an overpressured reservoir with pressure-dependent PV compressibility, and based on results presented in this paper pore collapse is not a necessary condition for such behavior.The Anderson "L" reservoir performance presented by Duggan6 shows curved p/z vs. Gp field behavior which was primarily attributed to shale water influx with no evidence of reservoir pore compaction. The water influx drive mechanism was supported by the fact that several wells watered out. Wallace7 also concluded that shale water influx is an important drive mechanism in abnormally pressured gas reservoirs. Bass8 discounts shale water influx, and attributes curved p/zvs. Gp behavior to peripheral water influx from a limited aquifer and formation compaction treated with a constant PV compressibility cf. For a limited aquifer, Bass defines a term Fp as the ratio of peripheral water PV to the PV of gas-bearing rock.Roach9 and Ramagost and Farshad3 both use the term [1-ce(pi-p)] for geopressured and abnormally pressured gas reservoirs. Both authors consider ce a constant and they consider the Anderson "L"example.Bernard1 does not accept the rock collapse theory as the cause for overpressured p/z vs. Gp behavior, concluding that water influx is the basic drive mechanism. He also uses p/z[1-c(pi-p)] where c is a "catch-all"term for treating the effects of rock and water compressibility, a small steady-state acting aquifer, and steady state shale water influx. He further states that the term c is almost impossible to quantify in terms of reservoir properties.Begland and Whitehead,11 Prasad and Rogers,12 and Wang and Teasdale13all present studies of overpressured gas reservoirs based on computer models. Refs. 11 and 12 treat cfand cw as functions of pressure, including the effect of solution gas in the water. Extemal water sources are also included in Refs. 12 and 13. The differential forms of the material balance used in these references correctly apply instantaneous compressibility in a history-matching approach to determine initial gas in place. A direct plot of (p/z)[1-ce(pi-p)] vs. Gp was not made because the ce term had not been defined.Poston and Chen14 analyzed several abnormally pressured gas reservoirs, and recognized that calculated values of ce{\textgreater}30´10-6 psi-1 required to linearize the material-balance plot reflected the influence of water influx.Bourgoyne15 demonstrates that reasonable values of shale permeability and compressibilities treated as a function of pressure can be used to match abnormal gas reservoir performance behavior. He points out, however, that determining kand cf of the shale necessary for modeling this behavior is practically impossible.Ambastha16 uses Bourgoyne's general material-balance equation to develop a graphical matching technique based on a constant effective compressibility ce. The example given in that paper shows a lack of uniqueness in determining initial gas in place.},
author = {Fetkovich, Michael J and Reese, Dave E and Whitson, C H},
doi = {10.2118/22921-PA},
file = {:C$\backslash$:/Users/ftn60/AppData/Local/Mendeley Ltd./Mendeley Desktop/Downloaded/Fetkovich, Reese, Whitson - 1998 - Application of a General Material Balance for High-Pressure Gas Reservoirs (includes associated paper.pdf:pdf},
issn = {1086-055X},
journal = {SPE Journal},
number = {01},
pages = {3--13},
publisher = {Society of Petroleum Engineers},
title = {{Application of a General Material Balance for High-Pressure Gas Reservoirs (includes associated paper 51360)}},
url = {https://doi.org/10.2118/22921-PA},
volume = {3},
year = {1998}
}
@book{Walsh2003,
author = {Walsh, M. P. and Lake, Larry W},
edition = {1st},
editor = {Walsh, Mark P and Lake, Larry W},
isbn = {9780444506832},
pages = {652},
publisher = {Elsevier Ltd},
title = {{A Generalized Approach to Primary Hydrocarbon Recovery}},
url = {https://www.elsevier.com/books/a-generalized-approach-to-primary-hydrocarbon-recovery-of-petroleum-exploration-and-production/walsh/978-0-444-50683-2},
year = {2003}
}
@article{Walsh1994a,
author = {Walsh, M. P. and Ansah, Joseph and Raghavan, Rajagopal},
doi = {10.2118/27684-MS},
file = {:C$\backslash$:/Users/ftn60/OneDrive/Access/Material Balance/The New, Generalized Material Balance as an Equation of a Straight Line- Part 1 - Applications to Undersaturated, Volumetric Reservoirs .pdf:pdf},
journal = {Proceedings of the Permian Basin Oil {\&} Gas Recovery Conference},
pages = {549--564},
title = {{New, generalized material balance as an equation of a straight line: Part 1- Applications to undersaturated, volumetric reservoirs}},
year = {1994}
}
@book{Dake1983,
author = {Dake, L. P.},
isbn = {9780444418302},
pages = {462},
publisher = {Elsevier Science},
title = {{Fundamentals of Reservoir Engineering}},
year = {1983}
}
@book{Ahmed2004,
author = {Ahmed, Tarek and McKinney, Paul D.},
isbn = {9780750677332},
pages = {424},
publisher = {Gulf Professional Publishing},
title = {{Advanced Reservoir Engineering}},
year = {2004}
}
@book{IkokuChi1984,
author = {{Ikoku, Chi}, U.},
isbn = {0-471-89482-6},
pages = {503},
publisher = {John Wiley {\&} Sons, Ltd},
title = {{Natural Gas Reservoir Engineering}},
year = {1984}
}
