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SPE 71588 Evaluation of a Horizontal Gas-Condensate Well Using Numerical Pressure Transient Analysis

R.A. Harisch, SPE, Schlumberger, R.C. Bachman, SPE, Taurus Reservoir Solutions, P.J. Puchyr, SPE, SpaceTime Simulation Corp., G.W. Strashok, SPE

Copyright 2001, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September­3 October 2001. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

ABSTRACT This paper will describe the evaluation of a horizontal gascondensate well using numerical pressure transient analysis techniques. The reservoir contains near critical fluid which exhibits significant liquid dropout when subjected to depletion. The paper will describe the experimental work performed, its mathematical characterization and the matching of both a well test and long-term production data with an extended black-oil reservoir simulation model. The focus of the work was to determine how the multiphase flow effects impact test interpretation and how liquid dropout would affect the long-term production performance of the well. One of the important practical aspects was resolving production data at different separator conditions. The difficulties in characterizing near critical fluids and their impact on the test interpretation will also be reviewed. Introduction Gas-condensate systems are becoming increasingly important as higher pressure and temperature reservoirs are exploited. In many instances, the economic value of the condensate production far outweighs that of the gas. As a result, it is vital to accurately characterize the reservoir and reservoir fluids in order to maximize fluid recoveries and to accurately project future cash flows. This paper examines techniques to analyze gas-condensate systems using the extended black-oil model suggested by Coats1. To this end, a discussion of methods to convert laboratory PVT data (constant volume depletion and constant

composition expansion experiments) to an extended black-oil PVT model is provided. As the pressure in the reservoir declines in a typical gascondensate system, condensate drops out of solution, impeding the flow of gas due to relative permeability effects. At lower pressures, however, it is possible for some of the condensate to re-vaporize into the gas phase. This complex PVT behaviour requires the use of numerical rather than conventional analytical solutions to analyze well test behaviour and to predict future production performance. An extended production test on a horizontal well located in a gas-condensate system was used as a case study. Initially, a single-point well test was conducted with an extended shut-in period exceeding one month. Subsequently, the well was placed on production. Since permanent downhole gauges were installed in the well upon initial completion, it was possible to analyze the long-term production as part of the well test. Prior to the well test, a series of fluid samples were collected and constant volume depletion (CVD) and constant composition expansion (CCE) tests were conducted on the fluid samples. The laboratory experiments allowed derivation of extended black-oil PVT properties for use in the numerical modelling of the well test and long-term production history of the well. In general, well tests with their limited production times and, correspondingly, limited investigation depths into the reservoir often yield non-unique solutions. It was anticipated that the long-term production and pressure data available for the well would allow for a more unique reservoir characterization. The intent of the case study was to characterize the reservoir and reservoir fluid properties, initial fluid distributions, assess the impact of multiphase flow on the well test and to predict future production performance for the well. Of particular importance was the estimation of a long-term deliverability profile for purposes of nominating gas plant capacity and for project economic purposes. Background A conventional black oil model is represented by two hydrocarbon pseudo-components, oil and gas. The oil component can exist only in the oil phase, while gas can exist in both the gas and oil phases. Additionally, all PVT



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parameters including gas solubility in the oil (called the solution gas-oil ratio or Rs) can be expressed as functions of pressure alone. In an extended black oil model there are still two components. However, the oil component can now have solubility in the gas phase (in addition to gas solubility in the oil phase). This solubility is called the condensate-gas ratio (or rs), and again is a function of pressure alone. Coats has shown that this two-component PVT model yields results similar to more rigorous multi-component compositional models when only pressure depletion cases are considered. The relationship between compositional and black-oil properties is detailed in the Appendix, along with a discussion of black-oil property constraints required to achieve physically meaningful results. Coats also provided a technique for calculating the extended black oil PVT properties of the fluid to honor both the constant volume depletion data and the multi-stage flash experimental results. The calculation is complicated, however the basic idea is to set the solution oil gas ratio (rs) to the measured values from the CVD experiment accounting for separator conditions. Then a material balance on each step of the CVD experiment is performed, honoring the liquid dropout volumes and oil density. This allows direct calculation of the reciprocal gas formation volume factor (Eg), the oil formation volume factor (Bo), and the solution gas-oil ratio (Rs). An additional matching parameter is to ensure the calculated gas density matches measured values. The only issue with this approach is that dew point values for oil properties (Bo and Rs) are not well defined as no liquid is present. Standard extrapolation of sub-dew point properties to the dew point can lead to situations where the oil has a nonphysical negative compressibility. The Appendix provides a series of compressiblity equations for the extended black-oil treatment. Since the case study involved a horizontal well test, a review of horizontal well flow regimes is instructive. An idealized horizontal well test exhibits the log-log pressure derivative response illustrated in Fig. 1. An explanation of the flow regimes indicated in the figure follow2: Wellbore Storage ­ Wellbore storage is characterized by a unit slope on the log-log pressure derivative plot in earlytime. Conceptually, it is identical to wellbore storage for conventional vertical wells though, in general, the value is typically larger than for vertical wells as a consequence of the larger wellbore volume. Due to the large wellbore storage effect found in horizontal wells, it may mask the vertical radial flow regime. Vertical Radial Flow ­ Vertical radial flow is characterized by a zero slope on the log-log pressure derivative plot in early time, occurring just after wellbore storage. In this flow regime fully developed radial flow between the wellbore and the upper and lower boundaries of the reservoir occurs. This flow regime continues until the upper and lower reservoir boundaries are detected. Reservoir parameters that influence the duration of vertical radial flow include kv to kh ratio and formation thickness. Linear Flow ­ Linear flow is characterized by a half-slope on both the log-log pressure and log-log pressure derivative

plots. In this flow regime, the reservoir fluid streamlines become parallel to the upper and lower reservoir boundaries. During linear flow, a horizontal well behaves essentially as a vertical well in a narrow channel. This regime continues until the drainage area becomes much larger than the length of the horizontal well. Reservoir parameters that influence the duration of this flow regime include kh and horizontal well length. Horizontal Radial Flow ­ Horizontal radial flow is characterized by a zero slope on the log-log pressure derivative plot in late time, occurring after the linear flow period. This flow regime is similar to radial flow in a vertical well. The reservoir fluid streamlines become horizontal and are directed to the wellbore. The stabilization in the pressure derivative is dictated by the far field kh. Case Study - Introduction A field case study was analyzed using the techniques for conversion of the laboratory PVT experiments to extended black-oil PVT parameters. This case study consisted of an extended production test for a horizontal well in a gascondensate reservoir. The case study well was located within a carbonate reef. Geological analysis, supported by offset well drilling results, indicated the best portions of the reservoir had a significant quantity of dolomite. The surrounding limestone, however, was believed to have low permeability. Although the horizontal well was nearly 800m in length, a maximum of 140m of the well length was in the higher quality dolomitic reservoir. The remainder of the well was located in the poor quality limestone. The well test itself was a single-point test. A drawdown period of 109 hours in duration was followed by a shut-in period of about 850 hours. Due to the presence of permanent downhole pressure gauges, it was possible to analyze the subsequent long-term production period in conjunction with the well test using a single numerical reservoir model. One unique problem associated with this test was the production data was reported at separator conditions, rather than at standard conditions required for use in the numerical modelling. As a result, all reported fluid volumes were converted to standard conditions using the PVT characterization for the reservoir fluid. Due to the complexities of the fluid system, numerical well test analysis was required to allow the complex gascondensate PVT characterization developed for this well to be directly incorporated into well test analysis. Further, due to liquid dropout, the multiphase flow characteristics of the system were considered an important modelling characteristic. PVT Characterization Separator fluid samples from the horizontal well were retrieved and recombined to separator conditions. A series of

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experiments were performed to characterize the gascondensate fluid. These experiments consisted of: a compositional analysis of the fluid a constant composition experiment at the reservoir temperature of 103.7 oC · a constant volume experiment at reservoir temperature · a two stage flash separator experiment with the dew point fluid · measurement of the gas viscosity above the dew point pressure Following the experimental work, the fluid was mathematically characterized using the Peng-Robinson equation of state with 16 components. Oil viscosities were not directly measured but were calculated using the ChristensenFredenslund technique3. The composition of the recombined reservoir fluid is presented in Table 1, while Table 2 shows the fluid properties from the constant volume depletion experiment. Fig. 2 plots the liquid dropout versus declining pressure occurring throughout the experiment. A relatively small change in pressure (4,137 kPa) from the initial dew point pressure of 31,717 kPa resulted in significant amounts of liquid drop out in the reservoir (in excess of 32 percent). During the long term production history of the well liquid drop-out will be significant, and its proper effect on reservoir performance must be accounted for. Gas and oil (condensate) production data for the well was reported at a variety of conditions throughout the productive life of the well. This in turn led to various different liquid-gas ratios reported for the well over time. For our purposes, reported production data assumes the well undergoes a two stage separation process. The first stage's liquid is assumed to go to a second stage separator at standard conditions (Fig. 3). The first stage gas stream is assumed to contain no more liquid and its volume is simply corrected for pressure and temperature effects to get volumes at standard conditions. The total reported gas production includes non-hydrocarbon gases such as CO2, H2S and N2, which account for approximately 20 percent of the gas volume. Basic Data The goal of the history matching effort was to achieve an acceptable history match for both the well test and the extended production history using a single, cohesive numerical model. The model was run with specified gas production rates (historical field gas production rates corrected to standard conditions) and reservoir, well and fluid parameters were adjusted until the produced oil rate and bottomhole pressure matched observed field values. The single-point well test was analyzed first to determine general model characteristics such as reservoir permeability, effective horizontal well length, wellbore storage coefficient and skin factor. Numerical well test analysis software, incorporating an extended black-oil reservoir simulation model that explicitly modelled the complex PVT behaviour of · ·

the fluid system, was used to analyze the well test. The general model developed from the well test analysis was then incorporated into an extended black-oil simulator to history match the historical long-term production data. The laboratory CVD experiments were used to derive extended black oil PVT properties as detailed in the Background section of this paper. Separator conditions during the well test were 3,500 kPa and 35 oC. These calculations resulted in the PVT properties shown in Table 3 and illustrated in Fig. 4. Since the PVT representation consisted of a two pseudo-component system, a standard phase envelope versus gas mole fraction was constructed, as illustrated in Fig. 5. The initial composition of the fluid is the dew point entry at the 31,717 kPa. The dew and bubble point lines converge at the critical point. As can be seen from the phase envelope the fluid system is within a few mole percent of being classified as a volatile oil. At separator conditions, a solution gas-oil ratio (GOR) of 653.0 m3/m3 and a condensate-gas ratio (CGR) of 1.1101 m3/m3 at the dew point pressure of 31,717 kPa was obtained. Daily oil and gas rates for the long-term production period were recorded at conditions that did not represent standard conditions of either the gas or oil. Liquid rates from the first stage separator were metered and reported without regard to further flashing of this liquid to surface conditions. The gas volumes reported were metered values from the first stage separator, corrected to surface conditions using standard orifice correction factors. The actual field separator configuration is schematically illustrated in Fig. 6. Unfortunately, the numerical model required volumes input at standard conditions. As a result, it was necessary to convert the reported volumes to standard conditions using the PVT characterization. To compute actual surface volumes of gas and oil, gas volumes were first re-converted to volumes at first stage separator pressure and temperature. This was accomplished by inverting the pressure and temperature corrections of the standard orifice calculation. The oil and gas rates at first stage separator pressure and temperature were then known. This mixture was flashed using the known PVT properties of the fluid to give oil and gas rates at standard conditions. This last step must be accomplished using either a mathematically characterized EOS package or the extended black oil properties previously described. The flow rates, corrected to standard conditions as previously outlined, are shown in Table 4. Unfortunately, special core analysis was not available for this well. Therefore, relative permeability characteristics were estimated using typical values for a gas-condensate system. Of note was the low residual oil saturation in the presence of gas (Sorg), also referred to as the critical condensate saturation. A value of 10 percent was initially chosen for the residual oil saturation. In a conventional gas-oil-water system, the residual oil saturation is typically much higher, often in the range of 25 to 50 percent. Recent studies4, have suggested that low critical condensate saturations, ranging from 7 to 15 percent, are reasonable in gas-condensate systems. The low residual oil saturation is further supported by the history match.



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Other parameters input into the numerical well test model included: Porosity: 7 percent Initial water saturation, Swi: 25 percent Reservoir Temperature, Tres: 104 oC Dew point pressure, Pdew: 31,717 kPa Single-point Well Test Analysis The build-up pressure derivative response (Fig. 7) for the case study horizontal well is consistent with the idealized derivative response presented previously in Fig. 1, except that the case study well does not appear to enter horizontal radial flow in late-time. After a series of model runs, a good match between the observed and the model-generated pressures and oil production rates was obtained. The following summarizes the findings of the well test analysis: Horizontal permeability, kh: 0.75 md Vertical permeability, kv: 0.75 md Formation thickness, h: 25m Initial reservoir pressure, Pi: 33,800 kPa Well skin factor, S: -3.25 Effective well length, Lw: 125m Wellbore storage coefficient, cw: 1.8 x 10-4 m3/kPa Distance to boundaries: No boundaries Figs. 8 and 9 illustrate the cartesian pressure and log-log pressure derivative matches, respectively. Of the results presented above, the effective horizontal well length, formation thickness and skin require further explanation. Geological analysis indicated the effective horizontal well length (that which is within dolomite rather than limestone) was between approximately 70 and 140m, depending on the porosity cutoff value chosen. Therefore, the effective horizontal well length determined from the modelling is consistent with the geological interpretation. If the horizontal well length was reduced further in the model, then the pressure drawdown for a given gas production rate increased, resulting in a poor pressure match. One possible remedy would be to incorporate additional negative skin, however, the negative skin required to achieve a match with a shorter well was unrealistically large, given that the well was only subject to a foamed acid wash, rather than an acid fracture treatment. As a result, a reasonable compromise was struck between effective horizontal well length and the skin factor. The formation thickness required to achieve a history match was considerably lower than originally suggested by geological analysis. The geological interpretation indicated a formation thickness of between 50 and 75m compared to the 25m formation thickness determined from the well test. Increasing the formation thickness in the model resulted in a significantly delayed transition from the vertical radial flow to the linear flow regime, as illustrated conceptually in Fig. 10. As a result, in order to match the time at which the vertical radial flow regime ended, the formation thickness was reduced. It was possible to moderate the reduction in formation thickness with a corresponding increase in vertical

permeability, however, for this study, it was assumed that the vertical permeability would be no larger than the horizontal permeability (that is, a kv:kh ratio of 1:1). In order to assess the impact of multiphase effects on the well test pressure response, a dry gas numerical model was constructed for comparative purposes. The PVT properties for the dry gas model were chosen to mimic, as closely as possible within the limits of a single-phase PVT characterization, the gas-condensate PVT properties. Other model parameters, such as horizontal well length, reservoir thickness, permeabilities and the wellbore storage coefficient were identical to those of the gas-condensate numerical model. It was discovered it was possible to achieve an acceptable match between the pressure data and the model-generated pressure response for the dry gas model. The character of the pressure derivative response for the dry gas case was very similar to that of the gas-condensate case. This suggested the horizontal well flow regimes, governed by reservoir permeabilities and geometry, rather than multiphase flow effects, dominated the pressure response of the system during the well test. It is noted the BHP dropped below the dew point pressure of the fluid system for only a portion of the drawdown period of the single-point well test. As a result, a relatively small amount of condensate dropped out of solution in the reservoir. A review of the saturation profile for the gas-condensate model indicated a condensate saturation of 10 to 15 percent, extending out for a distance of only 3 meters. It is possible that multiphase flow effects may become more dominant in the pressure response of the system if the BHP were at a pressure much less than the dew point pressure for an extended period of time. Additional case studies with more severe and extended drawdowns are required to verify this postulation. Extended Production Period Analysis Following the test, the well was tied-in to a gas gathering system for continuous production. Thereafter, the well flowed at gas rates between approximately 40 and 60 103m3/d with corresponding condensate production rates of 40 m3/d, on average. One-month average daily production rates were input into the model. As noted previously, all volumes input into the model were converted from separator conditions to standard conditions using the PVT characterization developed for the well. A schedule of volumes used in the model is presented in Table 4. Although permanent downhole gauges were installed in the well, pressure data was recorded from these gauges sporadically. Despite the intermittent pressure data, a reasonably clear indication of the pressure trend was evident from the data. Most of the pressure points represent flowing BHP, although in some cases limited shut-in periods preceded the pressure measurement. The pressure history is presented on the history match plots (Figs. 11 and 12). A much lower value of residual oil saturation, Sorg, was used in this model than typically used in a conventional oilgas-water system. In addition to the published literature supporting this conclusion, the history match of the extended

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production period confirmed the requirement for a low Sorg. It was discovered that higher residual oil saturations resulted in low condensate recovery, relative to observed field data. At high Sorg, much of the condensate that dropped out of the gas was trapped in the reservoir as a residual saturation and, as a result, the model-generated condensate production at the well dropped dramatically, compared to the actual observed field production. The basic numerical model developed during the well test analysis portion of this study was used without modification for the initial history match of the extended production period. The most significant observation was that the well detected some form of reservoir boundary or permeability reduction away from the wellbore during the course of the extended production history. A review of the historical downhole gauge data suggests that the pressure declined significantly over the extended production period, beginning at approximately 33.8 MPa and declining to near 15 MPa by January 1998. When the model was run with an infinite-acting reservoir, however, the model calculated BHP did not match observed values from the downhole gauges as the pressure in the infinite-acting reservoir model did not show any significant decline. This led to the conclusion that the pressure decline was caused by either a limited reservoir or a composite reservoir in which the permeability away from the wellbore was very low (hereinafter referred to as the recharge case). Further history matching efforts using these two general models yielded successful matches between observed and model calculated bottom-hole pressure. It was discovered that the best history matches were obtained using a drainage area of 35.5 ha for the limited reservoir case and a composite permeability of 0.75 md near the wellbore (extending out over an area of 35.5 ha) with a far-field permeability of 0.001md. Since the rate of recharge from the tight rock in the recharge case was very low, both the recharge case and the limited reservoir case yielded similar pressure responses over the limited time period considered during the history match. However, in the long-term, significant recharge of the higher permeability rock in the vicinity of the wellbore is expected to occur. Thus, the long-term deliverability of the well will be substantially different, depending on which model is applied. Predictions In addition to providing a means to characterize the reservoir and fluid properties, the numerical model was constructed to provide the capability for predicting future deliverability for purposes of plant and gathering system nominations and project economics. Using the numerical models described previously, a series of performance predictions were created as follows: Case 1: The well was forecast to produce against a bottom hole pressure (BHP) of 5500 kPa (equivalent to a tubing head pressure of 1500 kPa). The limited drainage area reservoir model was used. Case 2: As in Case 1, except that the recharge reservoir model was used.

All prediction forecasts were run until January 1, 2018 or when the production volumes became uneconomic, whichever occurred first. Production rates of approximately 2.8 103m3/d (100 Mcfd) of gas and 0.5 m3/d (3 BCPD) of condensate were applied as economic limits. A summary of results for the predictive cases are presented in Table 5. Production plots for the predictive cases are contained in Figs. 13 and 14. Case 1 represented the limited reservoir predictive case. As expected with such a limited drainage area, the gas and condensate production rates dropped very quickly and the ultimate recovery was limited compared to Case 2 (approximately half that of the Case 2). Based on the geological interpretation and analysis of well test data from an offset well, this may not be the most realistic of the predictive cases. The current geological description has regions of moderate permeability and porosity (dolomitic reservoir) encased by poor quality (low permeability and porosity) limestone. While it is possible that the reservoir is of limited dimensions, the geological description is more consistent with the recharge case of a low outer region permeability. Case 2 represented the recharge model in which the region surrounding the wellbore contained rock of much higher permeability than the outer region. Like the limited reservoir model, the recharge model predicted gas and condensate production to decline sharply, however, the decline was far less severe than in the limited reservoir model. As the well was produced, the inner region depleted relatively quickly and gas from the tight, outer region slowly recharged the partially depleted inner region. As observed from Table 5, the quantity of fluid influxed into the inner region was small relative to the total volume originally in-place, but replaced a reasonable portion of the total produced gas and condensate. Conclusions 1. A properly conducted CVD experiment can be used to develop an extended black-oil PVT characterization suitable for numerical modelling. 2. Due to the complex PVT behaviour of a gas-condensate system, numerical rather than analytical techniques were required to more accurately model the reservoir. 3. It was vital to convert the reported volumes from separator conditions to standard conditions for modelling purposes. This required the use of the PVT characterization developed from the CVD experiments. 4. A low residual oil saturation (Sorg) was required to match the field observed data. This is consistent with other published data. 5. For this well test, with a limited drawdown period below the fluid dew point, multiphase effects appeared to have minimal impact on the pressure response of the system. Instead, horizontal well fluid flow regimes, driven by reservoir permeability and geometry, appeared dominant. 6. Long-term production and pressure data improved the quality of the model description. The long-term data indicated that either a limited reservoir or a low far-field permeabilty reservoir was present.



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7. A multiphase model was required for developing accurate performance predictions. A dry gas model cannot account for condensate production, which is of significant economic value. Nomenclature Bg= gas formation volume factor, res volume/stock tank volume, res m3/st m3 Bo= oil formation volume factor, res volume/stock tank volume, res m3/st m3 c= compressibility, kPa-1 cw= wellbore storage coefficient, m3/kPa Eg= gas expansion factor, st m3/res m3 h= formation thickness, m k= permeability, md Lw= effective horizontal well length, m Pdew= dew point pressure, kPa Pi= initial reservoir pressure, kPa Rs= solution gas-oil ratio, m3/m3 rs= solution oil-gas ratio, m3/m3 rw= wellbore radius, m S= skin factor, dimensionless Swi= initial water saturation, percent Tres= reservoir temperature, oC Subscripts h= horizontal v= vertical T= total o= oil g= gas References

1. 2. 3. 4. Coats, K.H.: "Simulation of Gas Condensate Performance", SPE 10512, SPE Symposium on Reservoir Simulation, Feb 1982. Lichtenberger, G., "Data Acquisition and Interpretation of Horizontal Well Pressure Transient Tests", Journal of Petroleum Technology, February 1994, 157. Christensen and Fredenslund, Chemical Engineering Science, Vol. 35, 1980, 871. Danesh, A., Henderson, G.D., Peden, J.M., "Experimental Investigation of Critical Condensate Saturation and Its Dependence on Interstitial Water Saturation in Water-Wet Rocks", SPE Reservoir Engineering, August 1991, 336. Dake, L.P., "Fundamentals of Reservoir Engineering", Elsevier Science Publishers B.V., 1978, 294.

parameters, and discusses the features that these parameters must have in order to ensure physical meaningfulness. Most notably, it will be shown that the critical pressure is determined entirely by the ratio parameters, and does not depend at all on the volume factors. In addition, in order to ensure the total compressibility remain finite as the critical point is approached, it is necessary for the quantities Bo - B g R s and B g - Bo rs to go to zero faster than 1 - rs Rs . This is because the total compressibility is Sg rs Bo - B g R s S o R s B g - Bo rs cT = - B g - - Bo - 1 - rs R s 1 - rs R s Bg Bo where the prime denotes the derivative with respect pressure.






Basic Notation The starting point for deriving expressions relating the black oil to compositional properties is to partition the total hydrocarbon mass into four parts. But first define the ratio of the stock tank density of gas to the stock tank density of oil as





Denote the mass of component "c" in phase "p" as m cp , so that mog is the mass of the oil component in the gas phase, and so on. Then the total mass of gas component and the total mass of oil component in the system are m gT = m gg + m go m oT = m og + m oo . ..........................(A-2)

Conversely, the total mass of each phase is mTg = m gg + m og mTo = m go + m oo . ..........................(A-3)


The solution gas-oil ratio is defined as the stock tank volume of gas which dissolves in a given stock tank volume of oil for varying pressure, i.e., m go Rs = gSTC m oo oSTC = 1 m go . m oo ..........................(A-4)

Appendix ­ Compositional Properties in the Black Oil Formulation In the reservoir simulation, the black oil properties are usually specified in the form of a table. Great care must be taken that the properties, as defined by these tables, are consistent and do not lead to non-physical results. It is not uncommon to define properties which lead to negative compressibility, which is not only physically unrealistic but also numerically troublesome. This is especially true if the simulation involves pressures near the critical pressure. This appendix describes the derivation of useful physical quantities in terms of the above four black oil

Similarly, the condensate-gas ratio is the stock tank volume of oil component dissolved (vaporized) in a specific stock tank volume of gas component, i.e.,

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m og rs = m og oSTC = . m gg m gg gSTC For the oil formation volume factor, consider a given volume of the oil phase at reservoir conditions. When that volume to taken to stock tank conditions, the volume generally will have shrunk due to the evolution of gas. The oil formation volume factor Bo is, by definition, the above volume at reservoir conditions divided by the volume at stock tank conditions. The thing to keep in mind here is that the composition has changed, and that the density, a phase property, includes that change in composition. So the oil formation volume factor is mTo oSTC + R s gSTC o Bo = = oSTC (1 + R s ) = . m oo o O oSTC ..........................(A-5)

yg = yo =

^ m gg ^ mTg ^ m og ^ mTg

= =

^ ^ + rs rs ^ + rs



The K values can now be expressed explicitly as functions of pressure, namely, Kg = 1 - rs R s ^R s + 1 = 1+ ^+r xg R s ^ + rs Rs s . y o rs ^R s + 1 ^ (1 - rs R s ) = = 1- Ko = xo ^ + rs ^ + rs yg =



) )




We note in passing that the inversion of equations (A-10) gives (A-6) Rs = rs R s = 1 1- Ko ^ K g - 1 Ko Kg


Similarly, the gas formation volume factor is the volume at reservoir conditions of a given amount of gas component (and associated oil component) relative to the volume at stock tank conditions of the same amount of gas component, i.e. gSTC m Tg gSTC r gSTC + rs oSTC .(A-7) 1 + s = Bg = = g m gg g g These latter expressions are the standard found in all references, but we re-derived here to show that the approach leads to expected results. Mole Fractions Letting the molecular weight of the gas and oil components be M g and M o , and using a caret to denote mass in moles, then the liquid mole fractions are defined xg = ^ m go ^ ^ m go + m oo = ^R s m oo ^R s m oo + m oo = ^R s ^R + 1


Critical Point It is a well-known property that at the critical point, the K values must be 1. From equations (A-10), it is clear that the K values become 1 when 1 - rs R s = 0


^ m oo m oo 1 xo = = = ^ ^ m go + m oo ^R s m oo + m oo ^R s + 1


so this provides a means of determining the critical pressure. The critical pressure is the solution to equation (A-11). Note that it depends only on the ratio's and not on the formation volume factors, and that at the critical pressure, the ratios must equal the reciprocals of each other. Furthermore, since 1 - rs R s is zero only at the critical point and is positive at stock tank conditions, it follows that it is positive for all pressures below the critical. It is a physical and computational requirement that at the critical point, the densities of the gas and oil phases must be equal. Equating the phase densities, and using equations (A-6) and (A-7) gives, after some re-arrangement

c Bg c Bo

where ^ = M o M g and the equations have been written out in agonizing detail to show how the unknown masses cancel, leaving only functions of pressure. In exactly analogous fashion, the vapor mole fractions are


+ rsc

c 1 + R s



c Rs

= rsc ,


where the superscript c has been added to denote being at the critical point, and the latter relations follow from equation (A11). Equations (A-11) and (A-12) place very strong restrictions on the black oil curves as they approach the critical point.



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Flash Calculation If the K values are known for a given total composition, then it is possible to determine the amount of material in each phase by doing a flash calculation. The general flash equation is

m gT m oT and zg =




z c (1 - K c ) =0, K c + (1 - K c ) f L


^ m gT ^ ^ m gT + m oT


^ . 1 + ^


where the summation is over all the components, and ^ ^ z c = m cT mTT is the total mole fraction of component c, and ^ ^ f L = mTo mTT is the liquid fraction. Substituting the above expressions for the K values and re-arranging gives 1- r R s s 1 - zg = ^ + r 1 + ^R s s ^ fL . .........(A-14)

Now an interesting identity expressing the amount of the oil component in the oil phase can be derived as follows. m oo = m oT - m og = m oT - = m oT = m oT rs m gg r - s m gT - m go rs m gT - R s m oo -

( (



This expression completely determines the phase behavior as a function of pressure. For constant composition expansion, the total mole fractions are fixed, and equation (A-14) shows how the liquid fraction changes with pressure. Conversely, f L can be fixed, and equation (A-14) used to determine the mole fraction in the gas phase as a function of pressure. On a plot of z g versus pressure, the curve obtained when f L is zero is the dew-point curve, and that obtained for f L equal to 1 is the bubble point curve. These two curves intersect at the pressure determined by equation (A-11), and they define the two-phase envelope for the fluid. An example of this is seen in Fig. 5. Phase Volumes The above flash calculation determines the mass in the liquid phase in terms of the total composition. In this section, the volume of each phase will be determined in a manner totally independent of the flash. This will be done starting from the phase density, and using the following definitions and identities. The amount of hydrocarbon in the system is specified by the volumes at stock tank conditions, i.e., m gT = gSTC V gSTC m oT = oSTC VoSTC . ........................(A-15)


Re-arranging and dividing through by the total mass of oil component gives m oo 1 - rs = , m oT 1 - rs R s and similarly m gg m gT = 1 - Rs . 1 - rs R s ........................(A-21) ........................(A-20)

Of course in equations (A-20) and (A-21), it is assumed that the pressure is not at the critical point. With these identities, the phase volumes can now be expressed in terms of the pressure functions, by starting simply with the phase density definition, Vg = = mTg g =

(m gg + rs m gg )B g

gSTC + rs oSTC . ...(A-22)

m gg B g gSTC 1 - Rs Bg 1 - rs R s

With this definition, it is implicitly assumed that the solution gas-oil ratio and the condensate-gas ratio are zero at stock tank conditions. Defining the stock tank volume ratio as = V gSTC VoSTC ...................................(A-16)

= V gSTC

The analogous steps for the oil phase give Vo = m oo Bo 1 - rs = VoSTC Bo . oSTC 1 - rs R s ...(A-23)

gives a convenient single parameter for the total composition of the hydrocarbon system in the volume representation. Using this and equation (A-1) gives

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The total hydrocarbon volume is, of course, just the sum of these two, which gives, after some re-arrangement, VT = V oSTC

( - R s )B g + (1 - rs )Bo

1 - rs R s

These equations can be used to solve for the volumes and saturation in a constant composition expansion, as Coates did in reference 1. Total Compressibility It is a physical and computational requirement that the total hydrocarbon compressibility be positive for the full range of pressures and saturations encountered in the simulation. This section derives an expression for the total compressibility in terms of the above quantities. It does so in a general two component formuation, and shows that the resulting expression reduces to the expected relation for the strictly black-oil ( rs = 0 ), and for the gas condensate ( R s = 0) cases. Compressibility is defined in general as c=- 1 V . V P ...................................(A-30)



This expression is entirely general. It contains constants which reflect the composition of the system, and explicitly known functions of pressure. Saturation The amount of each component in the system is arbitrary, and is defined by VoSTC and V gSTC . The amounts are arbitrary except that there should be enough of each component that both phases exist at reservoir conditions. In reservoir engineering terms, the oil saturation, S o , and the gas saturation, S g = 1 - S o , should both be non-zero at reservoir conditions, and it would preferable to express the equations in terms of saturations rather than the initial stock tank volumes. The saturation can be related to the initial composition as follows. The oil saturation by definition is S o = Vo VT . Using the above equations gives So =

(1 - rs )Bo ( - R s )B g + (1 - rs )Bo

Although it is tempting to use equation (A-29) to compute the total compressibility, that is not a good idea because the saturations are functions of pressure. To be sure, the saturation representation will be used, but only after doing the differentiation by substituting equation (A-24) in equation (A30). Doing the derivative and then replacing by the saturation is tedious but straightforward, with the result cT = - . Sg rs Bo - B g R s S o R s B g - Bo rs B - - Bo - g Bg 1 - rs R s 1 - rs R s Bo .........................................................(A-31)







This can be inverted, to express the stock tank volume ratio in terms of the saturation at any pressure, = S g Bo + S o B g R s S o B g + S g Bo rs , ........................(A-26)

where the prime denotes the derivative with respect to pressure. This is the primary result. Special Cases Since the above result is new, it would be prudent to ensure that it reduces to known results for the cases of strict black-oil, and gas condensate. In strict black-oil, the oil component does not vaporize at all, so rs = 0 and so is the derivative, in which case equation (A-31) becomes cT = - Sg So Bo - B g R s - B , g Bo Bg

an expression with marvelous symmetry. Because of the frequency with which certain combinations occur, note in passing that - Rs = and 1 - rs = S o B g (1 - rs R s ) S o B g + S g Bo rs . ...........(A-28) S g Bo (1 - rs R s ) S o B g + S g Bo rs ...........(A-27)



which is the conventional black-oil result, as seen in Dake [ref. 5], and Coats [ref. 1]. When gas does not dissolve in the oil, the gas condensate case, then R s and it's derivative are both zero, and equation (A-31) becomes cT = - Sg So Bo - B g - Bo rs , Bo Bb

With these relations, the total hydrocarbon volume is Bo B g Bo B g VT = VoSTC = V gSTC . S o B g + S g Bo rs S g Bo + S o B g R s .........................................................(A-29)



again the expected result for gas condensates.



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Discussion Based on the familiar saturation weighted average concept, the individual phase compressibilities can clearly be identified in equation (A-31) as cg = - Bo - B g R s B - rs g 1 - rs R s B g - B o rs 1 co = - Bo - R s Bo 1 - rs R s 1 Bg

Table 1 ­ Composition of Recombined Reservoir Fluid Mo1ecular Weight % Mole % Component Weight CO2 H2S N2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 mcyc-C5 BENZENE cycl-C6 C7 mcycl-C6 TOLUENE C8+ 44.010 34.080 28.013 16.043 30.070 44.097 58.124 58.124 72.151 72.151 86.200 84.160 78.110 82.150 100.200 98.190 92.140 3.375 9.745 2.903 23.022 4.540 3.925 1.454 2.964 1.705 1.836 3.159 0.644 0.228 0.535 3.365 1.160 0.965 34.475 2.971 11.078 4.015 55.595 5.850 3.449 0.969 1.976 0.916 0.986 1.420 0.296 0.113 0.252 1.301 0.458 0.406 7.949


One of the ultimate objectives was to ensure that the total compressibility was positive for all pressures and saturations expected in any simulation. Equation (A-31) makes it possible to compute the total compressibility in terms of the known functions of pressure, and thereby validate the PVT properties. Notice that if the compressibility of each phase is always positive, then the total compressibility must also be positive. However, this can be overly restrictive; it is possible for one of the phase compressibilities to go negative while the total compressibility remains positive. Notice also that as the pressure approaches critical pressure, the denominator of the transfer term in equations (A31) and (A-32) goes to zero. However, the requirement of equal density as the critical point makes the numerator of the term also zero. Care must be taken in specifying the PVT properties to ensure that the entire term remains finite at the critical pressure.

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Pressure (psia) 4600 4000 3250 2750 2000 1500 1000 562 (kPaa) 31717 27580 22409 18961 13790 10343 6895 3875

Table 2 ­ Constant Volume Depletion Data (@103.7 oC) Raw Experimental Data Vapor Volume Equilibrium Equilibrium Total Volume Displaced Liquid Volume Vapor Volume (cm3) 3 3 (cm3) (cm ) (cm ) 68.694 0.000 68.694 0.000 48.713 52.184 53.370 68.391 69.065 83.478 110.700 24.673 25.707 24.585 22.413 20.761 18.958 17.226 73.386 77.891 77.955 90.804 89.826 102.436 127.926 5.346 9.852 10.076 22.829 21.868 34.478 59.887

Displaced Vapor Density (g/cm3) 0.3291 0.2403 0.1883 0.1275 0.0935 0.0600 0.0324

Calculated Produced Gas Properties Pressure (psia) 4000 3250 2750 2000 1000 562 (kPaa) 27580 22409 18961 13790 6895 3875 Vapor Produced Per Step (gmol) (1) 0.0548 0.0837 0.0724 0.1188 0.0870 0.0757 (%) (1) 7.42 11.33 9.81 16.09 11.79 10.26 Cumulative Vapor Produced (gmol) (1) 0.0548 0.1384 0.2108 0.3296 0.5016 0.5774 (%) (1) 7.42 18.75 28.55 44.64 67.94 78.20 0.8421 0.8252 0.8250 0.8286 0.8542 0.9581 Vapor Z Factor Res Relative % Liquid (2) (Vol%) 36.26 37.78 36.22 32.97 27.90 25.32 Liquid Density (3) (g/cm3) 0.5096 0.5565 0.5868 0.6184 0.6655 0.6948



Table 3 ­ Black-oil PVT Properties Well test Separator Conditions

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Oil Stock Tank Density (kg/m3) Gas Density at standard conditions (kg/m3)

775.20 0.9782

Stage 1 Press Stage 1 Temp

3500 kPa 35 oC

Stage 2 Press Stage 2 Temp

101 kPa 15.6 oC

Pressure (kPa) 3875 6895 10343 13790 18961 22409 27580 31717 33000 35000

Solution GOR, Solution OGR, rs Rs (m3/103 m3) (m3/m3) 20.61 0.08808 51.05 103.33 151.13 159.43 331.29 445.68 653.03 717.34 817.58 0.01432 0.06694 0.08754 0.02838 0.34247 0.62305 1.11001 1.20000 1.30000

Oil Formation Volume Factor, Bo (res m3/st m3) 1.1447 1.2399 1.3769 1.4926 1.5868 1.9753 2.3767 2.6000 2.6200 2.6300

Gas expansion factor, Eg (st m3/res m3) 30.9610 60.6491 90.7686 121.8865 188.2627 193.2173 225.2276 250.0000 257.6826 269.6587

Oil viscosity (mPa-s) 0.892 0.620 0.425 0.304 0.196 0.150 0.100 0.060 0.048 0.028

Gas Viscosity (mPa-s) 0.01400 0.01510 0.01670 0.01880 0.02350 0.02820 0.03990 0.05670 0.06191 0.07003

Oil density (kg/m3) 694.80 665.50 636.40 618.40 586.80 556.50 509.60 543.84 563.70 598.84

Gas density (kg/m3) 32.40 60.00 93.50 127.50 188.30 240.30 329.10 459.67 491.77 535.53

Long-term Production Separator Conditions Oil Stock Tank Density (kg/m3) Gas Density at standard conditions (kg/m3) Pressure (kPa) 3875 6895 10343 13790 18961 22409 27580 31717 33000 35000 Solution GOR, Solution OGR, rs Rs (m3/m3) (m3/103 m3) 55.85 0.05159 92.04 145.32 195.94 316.63 354.34 477.09 600.00 638.12 697.54 0.00207 0.04385 0.06215 0.21739 0.29351 0.56306 1.02712 1.20000 1.30000 775.20 0.9782 Oil Formation Volume Factor, Bo (res m3/st m3) 1.1943 1.3001 1.4415 1.5635 1.8489 2.0158 2.4370 2.6000 2.6200 2.6300 Gas expansion factor, Eg (st m3/res m3) 31.8210 61.2368 92.3738 124.2237 164.2075 199.2984 232.6314 250.0000 255.3865 263.7832 Stage 1 Press Stage 1 Temp Oil viscosity (mPa-s) 0.892 0.620 0.425 0.304 0.196 0.150 0.100 0.060 0.048 0.028 9000 kPa 38 oC Gas Viscosity (mPa-s) 0.01400 0.01510 0.01670 0.01880 0.02350 0.02820 0.03990 0.05670 0.06191 0.07003 Stage 2 Press Stage 2 Temp Oil density (kg/m3) 694.80 665.50 636.40 618.40 586.80 556.50 509.60 523.89 534.12 554.19 101 kPa 15.6 oC Gas density (kg/m3) 32.40 60.00 93.50 127.50 188.30 240.30 329.10 443.60 487.39 523.86

Flow Period 1 2 3 4 5 6 7 8 9

Duration (hours) 0.917 17.500 26.000 65.000 863.750 468.250 744.000 192.000 528.000

Table 4 ­ Flow Period Details (All flow rates converted to standard conditions) Gas Rate Cond Rate Wtr Rate Flow Duration Gas Rate Cond Rate (103m3/d) (m3/d) (m3/d) Period (hours) (103m3/d) (m3/d) 117.530 62.417 0.524 10 744.000 45.253 42.598 100.527 113.776 80.237 0.000 57.523 51.859 0.000 34.530 112.800 127.500 93.000 0.000 77.659 52.496 0.000 33.522 1.083 0.822 0.491 0.000 0.000 0.000 0.000 0.000 11 12 13 14 15 16 17 18 720.000 744.000 744.000 720.000 744.000 720.000 744.000 552.000 42.749 41.565 44.923 42.426 35.300 53.779 50.929 50.810 35.930 40.102 59.177 58.481 40.217 55.881 45.682 40.475

Wtr Rate (m3/d) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

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Parameter Ultimate gas recovery Ultimate C5+ recovery Gas recovery factor C5+ recovery factor Gas influx into inner region at 01/01/2018 Well shut-in date

Table 5 ­ Prediction Case Summary Case 1 Case 2 (Limited reservoir) (Recharge case) 57.5 104.3 32.5 55.9 78.3 12.0 42.9 6.2 n/a 07/01/04 16.2 01/01/13

106m3 103m3 percent percent 106m3


Fig. 1 ­ Typical horizontal well flow regimes and their associated log-log pressure and log-log pressure derivative responses.

Fig. 2 ­ Liquid dropout curve derived from the laboratory CCE experiment.

Fig. 3 ­ Idealized separator configuration used in laboratory experiments and for conversion of measured field production volumes to standard conditons.



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Fig. 4 ­ Fluid PVT properties developed from CVD laboratory experiment.

Fig. 5 ­ Phase envelope for separator conditions of Psep = 3500 kPa, Tsep = 35 oC.

Fig. 6 ­ Actual field separator configuration. Measurement of fluids occur at separator conditions, rather than standard conditions.

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Fig. 7 ­ Observed pressure response (log-log pressure and pressure derivative plot) from the build-up portion of the singlepoint test.

Fig. 9 ­ Model generated pressure match (log-log pressure and pressure derivative) for the build-up portion of the well test. Model generated pressure response is indicated by solid line while field observed pressure response is denoted by symbols.

Fig. 8 ­ Model generated pressure match. Solid lines represent model generated parameters while symbols represent field observed pressures.

Fig. 10 ­ Effect of increasing formation thickness on the log-log pressure derivative. Increasing thickness results in delaying the onset of the linear flow period (indicated on the log-log derivative plot as a half-slope line).



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Fig. 11 - Long-term production history match for the limited reservoir case. Model generated results are presented as solid lines while observed data is presented as symbols.

Fig. 12 ­ Long-term production history match for the recharge case. Model generated results are presented as solid lines while observed data is presented as symbols.

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Fig. 13 ­ Production performance prediction for the limited reservoir case.

Fig. 14 ­ Production performance prediction for the recharge case.


Evaluation of a Horizontal Gas-Condensate Well Using Numerical Pressure Transient Analysis

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