SPE 94065

Estimation of Long Term Gas Condensate Well Productivity Using Pressure Transient Data R. Osorio, G. Stewart, A. Danesh, D. Therani and M. Jamiolahmady, Inst.of Petroleum Engineering, Heriot-Watt U.

Copyright 2005, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Europec/EAGE Annual Conference held in Madrid, Spain, 13-16 June 2005. 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 SPE, their officers, or members. 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 Conventional composite models can be used to calculate gas condensate well skin values including that due to condensate banking. However, the calculated skin varies in time as the fluid properties and gas and condensate fractional flow change, hence, making estimation of long term performance of a well unreliable. The dependency of gas and condensate mobility on the flow rate at near well bore conditions makes the problem more challenging. We have addressed this issue by developing an analytical method to use transient well test data to determine gas-condensate relative permeability, accounting for the rate effect, which makes future performance prediction a more reliable task. Synthetic well test data have been generated by a commercial compositional numerical simulator with and without the rate effect. The impact of including the rate effect in the simulated cases on well test analysis results has been evaluated. The results show that for the cases analysed that the derivative of pressure responses does not necessarily always exhibit a three-radial composite behaviour due to condensate banking, when the rate effect is present, though such behaviour could be exhibited in some cases. Our developed analytical methodology is based on a technique that allows estimating the reservoir pressure as a function of distance from the well, at the end of the drawdown period, using the bottom-hole pressure value before the shut-in. Once this reservoir pressure profile is available, the actual reservoir pressure gradient as a function of distance is determined, which can be used to calculate the values of gas and condensate relative permeability. Therefore, we show for the first time that it is possible to calculate from pressure transient data the near well gas and oil relative permeability values

accounting for the rate effect. The well must be shut in at the bottom of the hole to avoid any wellbore storage effect on pressure build-up. We also show that the size of the two-phase region can be predicted with excellent accuracy using an analytical approach. Introduction When bottom-hole flowing pressure drops below the dew point, a zone of high liquid saturation forms near the well bore. For a long time, engineers in oil industry considered that this high liquid saturation region always resulted in reduced gas relative permeability and lowered well deliverability causing in many cases a severe damage, which is often called skin due to condensate banking. It has been demonstrated by different studies, which focused on the measurement and correlation of gas and condensate relative permeabilities at near well bore conditions, that the rate effect as well as the negative inertial effect were both significant in many cases. However, the negative inertial effect over a wide range of conditions can be subordinated to the positive rate effect and therefore the gas relative permeability can increase with increasing velocity when significant quantity of condensate is present in the region near the wellbore. Danesh et al.[1] were first to report the improvement of relative permeability of condensing systems due to an increase in velocity as well as that caused by a reduction in interfacial tension. This flow behaviour, named as the positive coupling effect, was subsequently confirmed experimentally by other investigators [2-5]. The analysis of well tests on gas systems with retrograde condensation is based on either the two-zone radial composite model or the multiphase pseudo-pressure approach. However, analysis of well test in gas condensate reservoirs, either using composite models with single pseudo-pressures or multi-phase pseudo-pressure approaches, is still based on the wrong paradigm, which states that reduced gas relative permeability due to liquid saturation in the vicinity of the well bore is not improved by the rate effect, although there is evidence of cases where the rate effect concept has been used in reservoir simulation to match DST draw-down data from the Britannia and Cupiagua fields (Diamond et al.[6] and Salino[7]). At first

2 SPE 94065

glance, there would be good reasons to think that if the rate effect impacts the production matching process of a gas condensate well using reservoir simulation at the same time it should impact the well test interpretation using any of the existing standard approaches to calculate the skins, specially the skin due to condensate banking. The basis of the multiphase pseudo-pressure approach to interpret the well pressure responses in gas condensate reservoirs is the pressure integral that has been widely used in gas rate equations for a long time. This means that before multiphase pseudo-pressure functions were incorporated in the analysis of pressure transient data, some different types of pseudo-pressure integrals were used in order to evaluate well performance in gas condensate reservoirs. ODell and Miller[8] presented the first gas rate equation using a pseudo-pressure function to describe the effect of condensate blockage. Their equation is valid only if the zone with two-phase flow regime is small and the produced well stream is the original reservoir gas. Fussell[9] examined the performance of a well in a gas condensate reservoir and showed that the productivity of the well is much higher than the productivity calculated by the ODell and Miller[8] theory, which is unable to predict the saturation profile in the two-phase region correctly. Fussell[9] concluded that the model of ODell and Miller[8] might be used to predict sandface saturations, provided that the gas in the single-phase region is identical to the initial composition of the fluid. Jones and Raghavan[10-12] carried out different studies in order to build a framework to establish a coherent theory for analysing pressure transient behaviour (draw-down and build-up) in gas condensate reservoirs. They showed that draw-down pressure responses from retrograde gas condensate systems could be correlated with the classical liquid solution, if the pressures were transformed to appropriate two-phase reservoir pseudo-pressure. Raghavan et al.[13] presented analysis of several field and simulated cases using the steady-state pseudo-pressure and showed that their method works best when the reservoir pressure is much higher that the dew point pressure and the well bottom-hole pressure is much lower. These circumstances, as they pointed out, allow the formation of a stabilized bank of fluid with a very small transition zone around the well, in other words, this means that the steady-state theory used by Raghavan et al.[13] neglects a transition zone where oil saturation is below the critical oil saturation. Hence the steady-state approach models the flow in the reservoir in two regions: a near well bore region where condensate and gas are present (and mobile) and an outer region containing single-phase flow only. The clear advantage of using the steady-state concept is that saturations as function of radius at a given time in depletion do not need to be known. Fevang[14] developed two-phase pseudo-pressure using a pressure-saturation relationship computed separately after

defining three regions instead of two, that is, he accounted for the existence of a transition zone, where both oil and gas are present but only the gas is flowing. This approach is an extension of the pseudo-pressure method proposed by Evinger and Muskat[15] for solution gas drive wells. Xu and Lee[16] applied the Fevang[14] three-zone concept to gas condensate well test analysis and showed that the three-zone method is more accurate than the steady-state approach for evaluating both skin and reservoir flow capacity. There is no publication dealing with the issue of including the rate effect in the evaluation of multi-phase pseudo-pressure functions although Xu and Lee[16 and Roussenac[17] have recognised a technical vacuum for not using rate dependent gas and oil relative permeability curves to evaluate two-phase pseudo-pressure integrals. It is clear for us that any attempt to do so must sort out first the issue of calculating analytically the gas and condensate relative permeability values with rate effect required to evaluate the pseudo-pressure function. On composite models with single pseudo-pressures, there is only one publication by Gringarten et al.[18] dealing with the issue of including the rate effect on relative permeability functional forms in simulation of well tests of gas condensate reservoirs in order to generate synthetic responses to be interpreted using common approaches of determining the two-phase skin. The authors conclude that when capillary number effects are important, the pressure derivative should exhibit three stabilization periods. More recently a study using the two and three-radial composite model helped to gain a better understanding of Santa Barbara condensate reservoir behaviour when many well tests were interpreted by Briones et al.,[19]. The results obtained provide evidences of the existence of retrograde condensation and improvement of gas relative permeability around the wells. Four wells were presented where the near well bore zone with high capillary number was present. Unfortunately, the tests were not long enough in duration for the pressure derivative to exhibit the gas mobility at the outer zone. On muti-rate tests in gas condensates wells, to the best of our knowledge, there are only two publications dealing with issue. Raghavan et al.[13] analysed two actual multi-rate tests in gas condensate wells. Since no theoretical evaluations of multi-rate tests, from pressure transient analysis point of view, were available by that time, they conducted a number of simulations, without rate effect, using a compositional model to ensure that explanations they provided for the field cases were plausible. The second publication was that of Gringarten et al.[18]. They presented a simulated gas condensate multi-rate test with rate effect. The run consisted of 10 periods of alternating draw-downs and build-ups (1DD, 2PBU, 3DD, 4PBU, 9DD and 10 PBU) following a normal sequence, that is, a scheme based on increasing gas flow rate. Only the first draw-down (1DD) was analysed using single-phase pseudo-pressure in order to propose a three-zone radial composite model to incorporate rate effect in gas condensate well test

SPE 94065 3

analysis. Unlike Raghavan et al.[13] paper, this study did not present any interpretations for the subsequent build-up periods nor for the reverse sequence of the normal simulated multi-rate case even though variations of such production history were run with different rates, gas-oil relative permeability models and fluid compositions. Conventional Well Test Analysis Using Composite Models and Multiphase Pseudo-Pressure Functions From a general point of view, radial composite models were created to analyse systems where a discontinuity in reservoir properties (permeability for instance) can be specified at some radius from a well. Thus, the system is divided into a cylindrical inner region with the well at the centre and an infinite outer region. In homogenous gas condensate reservoirs, the physical nature of radial composite systems is usually connected with the two-phase conditions arising in the reservoir when the pressure in the vicinity of the wellbore falls below the dew point and retrograde liquid condenses out to give two-phase conditions near to the well. In this case, the reservoir pressure far from the well is above the dew point and the outer region is single-phase gas as it is shown in Figure 1. The applicability of the two-zone model to diagnose the presence of a condensate bank is based on the fact that any contrast in the gas mobility causes the pressure derivative curve to stabilize at a lower or higher level in a diagnostic log-log plot depending on whether the gas mobility is increased or decreased, respectively. Therefore, when the bottom-hole pressure drops below the dew point, a two-zone radial composite could be used to identify regions of condensate drop-out around the wellbore and of initial gas composition away from the well as it is shown in Figure 2. Using this methodology two skin values can be obtained, a mechanical skin from the higher plateau and the total skin from the lower plateau, hence the difference between those values is the skin due to condensate banking around the well. Two-zone radial composite models use a single phase pseudo-pressures function m(p), which includes fluid viscosity and compressibility factor, and is defined as:

dp)z)((

p)p(m = (1)

, where z and are the compressibility factor and viscosity of gas, respectively. The limits of integration are between a base pressure, usually taken as atmospheric pressure, and the pressure of interest. The value of the base pressure is arbitrary since only differences in pseudo pressure are considered. On the other hand, the two-phase pseudo-pressure function has been defined, by analogy, as:

dpkk)p(Pws

s,Pwf roo

org

g

g

+

= . (2)

Here is the relative permeability to phase m and the subscript o and g refer to the oil phase and gas phase, respectively.

rmk

When the two-phase pseudo-pressure function, given by Equation 2, is used for analysing well pressure responses, the two-zone radial composite behaviour is removed from the diagnostic log-log plot because the relative permeability curves yield homogeneous looking derivatives giving access to the mechanical skin only. Thus, the value of incorporating relative-permeability effects in the pseudo-pressure function is that skin due to condensate banking is already being included within the two-phase pseudo-pressure integral. To evaluate the two-phase pseudo-pressure integral it is necessary to express and as a function of pressure. For the purpose of transient pressure analysis this has been accomplished based on fluid saturation-pressure relationships corresponding to the steady-state flow. Such relationships are reported by Chopra and Carter

gk ok

[20] and also by Jones and Raghavan[11]. This theory states that the saturation pressure relationship for steady-state flow is given by,

go

og

rg

ro

VL

kk

= . (3)

Here L and V refer to the fraction of total moles of liquid and vapor, respectively. The left hand side of Equation 3 is function of saturation only and the right hand side, which can be calculated from a Constant Composition Expansion (CCE), is only function of pressure. Once the left hand side of Equation 3 is calculated then the values of and can be obtained from a single set of gas-oil relative permeability curves, which in this case do not account for the rate effect.

rgk rok

The steady-state flow also implies that the overall composition of the flowing mixture is the composition of the original reservoir gas. This means that there is no transition zone where oil saturation is below the critical oil saturation. Hence the steady-state approach models the flow in the reservoir into two regions: a near well bore region where condensate and gas are present (and mobile) and an outer region containing single-phase flow only. To include the third transition zone in the two-phase pseudo-pressure calculations another pressure-saturation relationship is computed separately in each of the three regions. This has been defined and used recently in well test analysis, however and involved in the two-phase pseudo-pressure calculations are still based on a single set of base gas-oil relative permeability curves without the rate effect. Details related to this different pressure-saturation

rgk rok

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relationship and its implementation in well test analysis of gas condensate reservoirs have been presented by Xu and Lee[16], Roussenac[17], Barrios[21] and Barrios and Stewart[22]. Single Well Simulation Models In order to evaluate the impact of including the rate effect in the well test analysis of gas condensate reservoirs, through the two-zone composite model approach and using the single-phase pseudo pressure, several well tests were simulated with the commercial compositional simulator Eclipse 300. A two-phase, one-dimensional model was built and different synthetic cases were simulated to model the behaviour of well pressure responses in gas condensate reservoirs. The purpose of using these single well models is to produce a data file which includes time, wet gas rate and bottom-hole pressure as variables to be imported into Pansystem, which is a commercial pressure transient analysis software, in order to apply the methodology explained previously. For both single-phase and two-phase conditions near the wellbore the simulation consisted of reproducing a well test in which the well is flowed at constant rate for some period (draw-down) and then closed during another period for a pressure restoration period (build-up). For all the predictions performed in this study, the reservoir is assumed to be a homogeneous porous medium of uniform thickness of 200 feet, where gravity and capillary pressure effects are negligible. Therefore the porosity and permeability data for each of the grid block is assumed to be the same. Hence any composite model identified in a log-log plot of pressure response is because of the existence of zones with different gas mobilities. It is also assumed that the well is shut in at the bottom of the hole and there is no wellbore storang effect present. The bases for the input reservoir properties were a limestone and sandstone cores with laboratory measurements. This means that porosity, permeability and gas-condensate relative permeability curves included in the single well model correspond to those measured in the laboratory for the specific case of zero water saturation. Limestone and sandstone absolute permeability values are 9.1 and 11 mD, respectively. Porosity values are 20.9% for limestone and 18.0% for sandstone. Figure 3 shows the gas-oil relative permeability curves of limestone and sandstone cores used in this study. The gas oil relative permeability curves correspond to those measured in the laboratory to the highest value of interfacial tension (IFT) and lowest velocity There are simulated well test cases accounting for three different compositions, as shown in Figure 4. Flowing gas rates from 5 MMscf/day to 40 MMscf/day before the build-up period were used in the simulations. The simulator calculates the fluid PVT properties using a 10-

component fluid and the three-parameter Peng-Robinson equation of state. Numerical simulation of well tests in gas condensate reservoirs should recognise the fact that the pressure transient information occurs during the first hours of the test and the near wellbore region is of great interest. The numerical simulation should therefore be performed on not only very small grid blocks around the well but also using very small time steps. The final model grid chosen consisted of one layer with 40-grid blocks in the radial direction, one grid in the angular direction and an outer radius of 5000 feet to ensure that no boundary effects are seen in the simulated well tests. Near the wellbore, the cells are small to simulate the gas-condensate near-wellbore behaviour accurately. The nearest cell to the wellbore has a size of 0.15 feet and the size of the other cells increases logarithmically away from the wellbore by a factor of 1.3 up to cell 40. After the grid size was optimised, the number of report steps and the time between report steps together with the right convergence criteria were set up by matching single-phase simulations to analytical solutions in which the bottom-hole pressure is above the dew point. However, in some cases, this approach is not sufficient and instabilities of pressure responses and production parameters are still observed when simulating two-phase flow cases. We used the analytically derived pressure derivative plots to check the validity of our simulations. This means that the optimum grid size, time steps between report steps and convergence criteria must be capable of eliminating relevant noises in the diagnostic log-log plot and at the same time the results of the analytical interpretation of the pressure derivative must fit the reservoir properties included in the single well model. In addition, an optimised model must ensure that production data during the draw down period (such as producing gas-oil ratio) do not oscillate and further tuning had to be made to accomplish this aim. We have seen that rigorously, both the time step tuning and convergence criteria should be done for each of the particular considered cases. Figures 5 and 6 show log-log plots of single-phase case and two-phase case, respectively, where the derivative response and the formation parameter calculated analytically match with those included in the simulator input data. An issue of particular importance in this study is the fact that the Velocity Dependent Relative Permeability option (VELDEP keyword in ECLIPSE 300) has been applied to calculate the gas and condensate relative permeabilities required for estimation of block to block flow. The experimental values of the exponents, together with the value of the base capillary number of both limestone and sandstone cores, were directly introduced to the simulation model. This means that the positive rate effect and the negative inertial

SPE 94065 5

effect on the relative permeabilities as a function of capillary number are being realistically incorporated in the simulation model or at least based on values of exponents measured using actual cores. Table 1 shows the exponents and constants for limestone and sandstone cores used in the simulation model. New Analytical Method In this section we will present theory and aspects of our technical strategy aimed at developing a comprehensive framework that allows inclusion of the rate effect in gas condensate well test data analysis. Our new analytical approach is based on the application of the probe radius concept of Peaceman[23] and Agarwal[24] multibank theory in gas condensate well test analysis. We have evaluated the application of probe radius concept to cases with different gas rates, rock properties and fluid systems. The first aim at using the probe radius is the calculation of rate dependent reservoir pressure gradient near the well bore from pressure transient analysis. More importantly, the second objective of using the probe radius, to interpret well test data, is the analytical calculation of rate dependent gas and oil relative permeability values from pressure transient data interpretation. We will show that we are able to accomplish this task near the well bore, that is, exactly where the steady-state assumption is valid and rate effect is important. This calculation has been carried out for cases with different flowing gas rates during the draw-down period before the build-up. The ability of calculating both reservoir pressure gradient and relative permeability values with rate effect is very important due to its potential application in a different number of ways in order to analyse productivity of a gas condensate well over its production life. Basics of Probe Radius Method The probe radius is the radius at which the steady-state flowing pressure (before shut-in) was equal to the current well pressure after the shut-in. Peaceman[23] used this parameter for calculating the pressure equivalent block radius in reservoir simulation with Cartesian co-ordinates. However, we have adopted the same concept for an entirely different purpose, i.e., for determination of the relative permeability, in the vicinity of the well within the reservoir, using the well test data. Peaceman[23] showed that the probe radius (Figure 7), r , is a function of shut-in time and through two independent mathematical derivations (using Bessel series and the Ei function) he verified that the probe radius and the shut-in time are related by the following equation:

445.02 =

rctk

t (4)

where, t is shut-in time, r is the probe radius, is porosity, is absolute permeability and total compressibility. k tc

From the probe radius concept, Equation 4, and replacing k/

by ___

T , the average total mobility ratio. The probe radius corresponding to each value of T can be estimated using the following equation:

t

T

ctr

=

___247.2

(5)

When pressure response data from a build-up test are analysed, shut-in time t is often changed to the equivalent time or the Agarwal equivalent time, which is defined by,

tttt

tp

pe +

= (6)

where is the effective producing time or flowing time at

constant rate (in draw-down test) and is the time elapsed from start of the transient test (in build-up test). A given pressure change,

ptt

p , that occurred at shut-it time, t , during a build-up test would have occurred at equivalent time,

et , in the test. An important parameter in Equation 5 is the average

mobility , which can be obtained from constant rate solutions of diffusivity equation commonly used by well test data analysis:

T

__

'62.70

___

phqB

T

= (7)

where is flow rate in scf/d, q B the gas formation volume factor in reservoir cf/scf and pay thickness in feet. h In Equation 7, 'p is the standard logarithm derivative of

p with respect to time. It can be obtained from the derivative of the pressure response on the diagnostic log-log plot and is given by,

( )etdpdp

=

ln' . (8)

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The mobility is almost constant for a single-phase flow, from practical point of view, if the calculation with Equation 7 is carried out using values from the outer radial flow regime on the diagnostic log-log plot. If values are picked up from either the condensate bank transition zone or

the inner radial flow regime, then is still a total mobility but we must refer to it as two-phase total mobility and it is not constant but changes with distance from the well bore.

T

___

'p'p

T

___

Analytical Calculation of Rate Dependent Gas and Oil Relative Permeability Values

The Analytical methodology we propose to obtain the gas and oil relative permeability values with rate effect has two different types of calculations. The first one is related to the rate dependent pressure gradient as a function of distance and the second one is the analytical calculation of the krg and kro with rate effect itself. We will show that these rate dependent gas and oil relative permeability values can be calculated based on the steady-state approach, which means that the following equations could be used for this purpose:

( )

=

rPrkh

xmk

g

gTrg

2

1 (9)

=

rPrkh

xmko

oTro

2 (10)

Where,

Tm : total mass rate. x : liquid mass fraction.

g , o : gas and condensate viscosities, respectively.

g , : gas and condensate densities, respectively. okh : permeability-thickness product.

rP

: rate dependent reservoir pressure gradient at radius r.

P : rate dependent reservoir pressure losses calculated based on a small r at some specific radius r.

Total mass rate can be calculated from gas and condensate flow rates measured during the actual well test. This total mass rate would correspond to that existing at the sand-face in the bottom hole. Fluid properties (densities, viscosities and liquid mass fraction) are obtained from a constant composition expansion (CCE) using an EoS and the produced gas composition based on the assumption of steady-state. The permeability-thickness product can be estimated by analysing just the outer zone from the diagnostic log-log plot of derivative of pressure (single phase flow) using single-phase pseudo-pressures. Rate dependent reservoir pressure gradient and rate dependent reservoir pressure losses would be available after applying probe radius concept. Fluid Properties Management We have verified that fluid properties (densities, viscosities and liquid mass fraction) required to calculate krg and kro values with rate effect, according to Equations 9 and 10, must be calculated from the constant composition expansion (CCE) test results using an EoS and the produced gas composition based on the assumption of steady-state flow in the immediate vicinity of the well. However, on the application of probe radius concept to gas condensate cases, the situation is different. It is not completely clear that required fluid properties (formation volume factor and total compressibility in Equation 5 and 7) must be obtained from the produced gas composition based on steady-state assumption. In fact, Peaceman[23] proved mathematically that it is possible to use the build-up curve to probe to a radius up to one-half that of an external boundary. This makes the probe radius a very robust and powerful concept, which is applicable both near the well bore and reservoir far away from the well. The external boundary in our single well models is located 5000 feet away from the well bore, which means that probe radius should still be valid in some part of the region with single-phase flow regime in our simulation runs. This means, as we will show later, that probe radius method is a practical concept to predict the size of a condensate bank from pressure transient data analysis. We have managed in two different ways the input related to fluid properties in order to extend the probe radius concept to gas condensate cases. This scheme of fluid input aims at modelling the phase behaviour related to different zones of the reservoir in the probe radius calculation, therefore, the objective here is to see how sensitive this concept is when it is applied to gas condensate systems. Firstly, we used original gas composition to calculate a constant formation volume factor at bottom-hole and constant total compressibility based on initial reservoir conditions. Then we obtained the same two parameters but now based on composition of produced fluid. We have verified that probe radius results in both cases are similar.

SPE 94065 7

Results and Discussion Analytical Calculation of Rate Dependent Gas and Oil Relative Permeability Values The procedure to convert bottom-hole pressures from build-up test into the reservoir pressure versus distance profile by using the probe radius concept could be summarized as follows: Synthetic bottom-hole pressure responses are generated

using a compositional simulator and different single well models depending on the specific case.

Pressure responses obtained from the build-up period are processed and analysed in pressure transient data analysis software.

Derivative values related to the build-up analysis are extracted from the well test data analysis.

Average total mobility values as a function of equivalent time are calculated by using Equation 7; therefore these total mobility values can also be expressed as a function of bottom-hole pressure.

Probe radius is estimated based on Equation 5. Bottom-hole pressure responses can be associated now to a

radius in the reservoir, hence a reservoir pressure versus distance is available from the above analytical approach.

The reservoir pressure profile calculated is compared with the reservoir pressure versus distance values given by the numerical simulator at the end of each draw-down period.

Figures 8 and 9 compare the rate dependent pressure gradient profiles calculated analytically through probe radius concept with those given by the single well models at the end of the eight-day draw-down period before the shut-in. As we pointed out previously, once the reservoir pressure as a function of distance is available, the reservoir pressure

gradient with rate effect, rP

, is also known and it is

possible to proceed with the calculation of the rate dependent krg and kro values using Equations 9 and 10. Figures 8 and 9 show results for limestone and sandstone cores, fluid system 1 and flowing gas rate of 30 MMscf/day, which has one of the lowest bottom-hole pressure values, at the end of the draw-down period, of all the cases we have analysed in our study. Therefore, these cases constitute a very good reference to show how accurate the probe radius concept can be in predicting the rate dependent pressure gradient in the immediate vicinity of a gas condensate well, which is the region where the assumption of steady-state flow is valid and therefore Equations 9 and 10 are applicable. Hence, if the rate

dependent rP

can be estimated with good accuracy from

probe radius method, we will show that the calculation of krg and kro with rate effect depends only on both the validity of the steady-state flow assumption and the management of fluid properties in Equations 9 and 10.

The estimation of rate dependent krg and kro using Equations 9 and 10 is quite sensitive to the composition of fluid used to determine fluid properties. Therefore, we have noted that, based on steady-state assumption, it is mandatory to use composition of produced fluid in order to obtain proper values of krg and kro near the well bore. As original fluid is richer than produced fluid, using fluid properties based on composition of original fluid in Equations 9 and 10 would lead us to overestimate values of kro and under-estimate krg. Rate dependent krg and kro can be calculated by our methodology presented above provided that steady-state-flow assumption is valid. In addition, we know that steady-state assumption has been a long standing, reasonable and sound approach to manage a number of issues near the well bore in gas condensate wells, which is precisely the place where rate effect is important. Figures 10 to 12 compare krg/kro ratio values calculated analytically based on steady-state theory with those given by the numerical single well models at the end of the eight-day draw-down period before the shut-in. Those charts correspond to limestone core, fluid system 1 and flowing gas rates of 10, 20 and 30 MMscf/day. The krg/kro ratio from steady-state theory was calculated based on composition of produced fluid and using Equation 3. Figures 10 to 12 confirm that there is a range of both reservoir pressure and krg/kro ratio values where the steady-state assumption is valid. It can also be observed that above some reservoir pressure value the curves shown in these charts do not match. In Figures 10 to 12, low pressures are related to a near well bore region and high pressures correspond to regions in the reservoir far away from the well. The higher the flowing gas rate the lower the bottom-hole pressure values at the end of the draw-down and the wider the range of reservoir pressure where the steady-state-flow assumption is valid. Figures 13 and 14 show results of rate dependent krg and kro distribution just prior to shutting the well in, calculated analytically using limestone core information, fluid system 1 and for a flowing gas rate of 20 MMscf/day. Composition from produced fluid was always used in order to calculate the required fluid properties in Equations 9 and 10. These charts show that both krg and kro with rate effect in the immediate vicinity of the well can be calculated analytically from pressure transient data with reasonable accuracy provided that steady-state-flow assumption is valid. It is also known that krg and kro can each be expressed directly as a function of the ratio krg/kro when both phases are mobile. We already showed that after applying our analytical methodology, krg and kro profiles as a function of distance from wellbore are determined. Therefore, if we pick up from those profiles values of krg and kro where the steady-state theory is valid, then we could obtain from pressure transient data analysis very useful relationships between rate dependent krg as function of krg/kro ratio at different rate and interfacial tension values, as it is shown in Figure 15. In this chart, IFT is calculated from a CCE run with composition of produced fluid, using an Equation of State and based on bottom-hole pressure at the end of the draw-down period.

8 SPE 94065

Now we show the validity of the probe radius concept in a realistic multi-rate test. Our joint industry research project[25,26], has addressed the issue of multi-rate testing, highlighting major differences between that of dry gas and gas condensate reservoir in order to develop insight into the multi-rate approach to test gas condensate wells. Scheme shown in Figure 16 is used with fluid system 1 and sandstone rock properties. This chart shows that total duration of the multi-rate test was 50 hours. We have defined certain criteria to establish the duration of the flowing and build-up periods. Firstly, the last build-up in any multi-rate test should be the longest. The aim here is to achieve radial flow in the reservoir and to calculate reservoir properties, which are necessary to perform calculations of probe radius in any build-up period. In addition, those reservoir properties are data that are conventionally is obtained from pressure transient analysis. Contrary to this, in other cases a build-up period is quite short and the objective in these scenarios is exclusively to acquire data to understand flow near the well bore, which is the region where the rate effect is most important. The main criterion related to the duration of the flowing periods is that a draw-down period must be as short as possible because their purpose is just that bottom-hole pressure drops below the dew point and two-phase flow regime takes place around the well bore under different flowing gas rates. However, we have also considered scenarios with a long drawdown period before a long build-up period, which is the most common scheme in single point gas condensate well tests. Summarizing, we have tried to take into account a number of possible schemes of flowing/build-up periods, in terms of test times, in order to evaluate the applicability of probe radius concept. However, we have kept total duration of the multi-rate test within the realistic limits of test times related to this type of operations in the field. Figure 17 compares rate dependent reservoir pressure profiles calculated analytically through probe radius concept with those given by the single well models at the end of each draw-down period before the shut-in based on multi-rate test shown in Figure 16. The results shown are related to the sandstone core and fluid type 1 (47% liquid drop out). We have simulated other multi-rate tests, all of them realistic from duration point of view, using other fluid systems and rock properties and the results are similar to those shown in Figure 17. From these results it can be concluded that it is possible for a build-up test in a gas condensate reservoir, to obtain from bottom-hole pressures at time the equivalent reservoir

pressure at radius r . This can be accomplished even for very short duration of drawdown and build-up periods and for a wide range of flowing gas rates. Additionally, it is not mandatory that flowing periods before shut-in be longer than build-up periods.

wsp t

rP

Multi-rate schemes with short test times would allow us to probe the region near the well bore for instance up to three times with three different flowing gas rates. Therefore, under steady-state conditions around the well bore, we could obtain from pressure transient data analysis krg and kro with rate effect up to some distance from the well. Alternatively, we could also define useful relationships between rate dependent krg as function of krg/kro ratio at different rate and interfacial tension values, as it was shown previously in Figure 15. Size of two-phase region (condensate bank size) It has been considered by many engineers that data related to size of condensate banks could be useful for workover operations in gas condensate wells. Figure 17 compares favorably reservoir pressure profiles calculated analytically with those given by the single well simulation models at the end of the drawdown period before the shut-in. Since we are interested only in the region around the well bore where the two-phase flow regime takes place, in all these charts reservoir pressures shown are related to distances in the r direction up to the specific radius where single-phase flow ends. We have run many different simulations and synthetic transient pressure data have been analysed using the extension of probe radius concept to gas condesante cases. In fact more than 200 simulated PBU tests have been interpreted in our study. The size of the two-phase region was predicted with excellent accuracy for the tested fluids. Size of condensate banks have been included in Tables 2 and 3 for some cases that we have simulated in order to compare the analytical results with the values obtained from single well models. In the simulated cases we have taken into account different rock types, gas flowing rates, skin values and type of drawdown before the shut-in. Results shown in Tables 2 and 3 are related to fluid type 1. Conclusions This study is the first step of a long-term project aimed at developing new and efficient techniques for estimating productivity of a gas condensate well over its production life by analysing pressure transient data. Important observations based on our results are as follows: It is possible for a pressure build-up well test in a gas condensate reservoir, including the rate effect, to derive the reservoir pressure , vs. radius r using the bottom-hole pressures, , vs. time after shut-in. Therefore, rate dependent reservoir pressure gradient can be known and based on steady-state-flow theory, gas and oil relative permeability values with rate effect can also be determined from pressure transient data analysis. This technique is what we call application of the probe radius method to gas condensate systems. Since gas-condensate relative permeability values are

rP

wsp

SPE 94065 9

rate dependent, skins due to condensate banking also depend on flow velocity near the wellbore. The applicability of the probe radius concept in gas condensate systems was proven for different flowing gas rates and different schemes of multi-rate tests. More importantly, it has been shown that probe radius method can work properly in cases with short flowing and build-up periods. Different rock types and fluid systems have been used to extend the probe radius concept to a variety of gas condensate scenarios. Size of the two-phase region (condensate bank) can be predicted with excellent accuracy from transient pressure data by applying the probe radius concept to gas condensate cases. Acknowledgements

The above study has been soponsored by UK Department of Trade and Industry, BP Exploration Operating Company Ltd., Gaz de France, Marathon Oil UK, Statoil A.S.A and Total Exploration UK plc, which is gratefully acknowledged. Schlumberger and Edinburgh Petroleum Services are thanked for the use of ECLIPSE300 and Pansystem, respectively. References 1. Danesh, A., Khazam, M., Henderson, G.M., Therani, D.H. and Peden, J.M.: "Gas Condensate Recovery Studies, DTI Improve Oil Recovery and Research Dissemination Seminar, London, June, 1994. 2. Henderson, G.M., Danesh, A., Therani, D.H., Peden, J.M.: The effect of velocity and interfacial tension on the relative permeability of gas condensate fluids in the wellbore region, presented at the 8th IOR European Symposium, Vienna, May 1995, proccedings page 201-208. 3. Henderson, G.M., Danesh, A., Therani, D.H., Peden, J.M.: Measurement and correlation of gas condensate relative permeability by the steady-state method, SPEJ, June 1996, 191-201. 4. Ali, J.K., McGauley, P.J. and Wilson, C.J.: The effects of high velocity flow and PVT changes near the wellbore on condensate well performance, SPE paper 38923, SPE Annual Technical Conference and Exhibition, 5-8 Oct., 1997, proccedings, page 823-838. 5. Blom, S.M.P., Hagoort, J. and Soetekouw, D.P.N.: Relative permeability near wellbore conditions, SPE 38935, SPE Annual Technical Conference and Exhibition, 5-8 Oct., 1997, proccedings, page 957-967. 6. Diamond, P.H., Pressney, R.A., Snyder, D.E. and Seligman, P.R.: Probabilistic prediction of well performance in a gas condensate reservoir, SPE paper 36894 presented at SPE European Petroleum Conference, October 1996, Milan, Italy. 7. Salino, P.: Gas condensate near-wellbore processes reconciling laboratory & field data, Internal Report RPT/054/98, bp, SPR-Reservoir Performance, 1998. 8. ODell, H.G. and Miller, R.N.: Succesfully cycling a low permeability, high-yield gas condensate reservoir, JPT (1967) 41-47; Trans., AIME, 240. 9. Fussel, D. D.: Single well performance predictions for gas condensate reservoirs, JPT (1973) 258-268; Trans., AIME, 255. 10. Jones, J. R.: Computation and analysis of single well responses for gas condensate systems, PhD Thesis, 1985, University of Tulsa, OK. 11. Jones, J.R. and Raghavan, R.: Interpretation of flowing well response in gas-condensate wells, SPE paper 14204, 1988.

12. Jones, J.R, Vo, D.T. and Raghavan, R.: Interpretation of pressure build-up responses in gas condensate wells, SPE paper 15535, 1989. 13. Raghavan, R., Chu, W.C, and Jones, J.R.: Practical considerations in the analysis of gas-condensate well tests, 1999, SPE Reservoir Eval. & Eng. 2, 288-295. 14. Fevang, O.: Gas Condensate Flow Behaviour and Sampling, PhD thesis, 1995, Norges Tekniske Hogskole. 15. Muskat, M.: Physical principles of oil production, McGraw-Hill Book Company, Inc., 1949. 16. Xu, S. and Lee, J.W.: Two-phase well test analysis of gas condensate reservoirs, SPE paper 56483, 1999. 17. Roussennac, B.: Gas Condensate well test analysis, Master of Science in Petroleum Engineering Thesis, 2001, Stanford University. 18. Gringarten, A.C., Al-Lamki, A. and Daungkaew, S.: Well Test Analysis in Gas-Condensate Reservoirs, 10th European Symposium on Improved Oil Recovery, August 1999, Brigthon, United Kingdom 19. Briones, M., Zambrano, J.A. and Zerpa, C.: Study of Gas-Condensate Well Productivity in Santa Barbara Field, October 2002, Venezuela, by Well Test Analysis, SPE paper 77538 presented at the SPE Annual Technical Conference, San Antonio, Texas. 20. Chopra, A. and Carter, R.: Proof of the two-phase steady-state theory for flow through porous media, SPE paper 14472, 1985. 21. Barrios, K.: Analysis of well test responses in gas condensate reservoirs, MPhil Thesis, 2002, Institute of Petroleum Engineering, Heriot-Watt University. 22. Barrios, K., Stewart, G. and Davies, D.: A novel methodology for analysis of well test responses in gas condensate reservoirs, SPE paper 81039, 2003. 23. Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation, SPE paper 6893, 1978. 24. Agarwal, R.G. and Yeh, N.S.: Pressure transient analysis of injection wells in reservoirs with multiple fluid banks, SPE paper 19775, 1989. 25. Progress Report 3, Gas Condensate Recovery Project, Institute of Petroleum Engineering, Heriot-Watt Unversity, January 2004. 26. Progress Report 4, Gas Condensate Recovery Project, Institute of Petroleum Engineering, Heriot-Watt Unversity, August 2004 Nomenclature B = Formation volume factor ct = Total compressibility p = Pressure q = Flow rate h = Thickness k = Absolute permeability kr = Relative permeability L = Liquid mole fraction mT = Total mass rate r = Probe radius tp = Effective producing time V = Vapor mole fraction x = Liquied mass fraction z = Compresibility factor = Density = Viscosity = Porosity

t = Shut-in time et = Equivalent time 'p = Rate dependent derivative of pressure

p = Rate dependent reservoir pressure losses

10 SPE 94065

T

__ = Averaged total mobility of gas and condensate.

T = Discrete total mobility m(p) = Single-phase pseudo-pressure Table 1. Exponents and parameters of the VDRP correlation for the cores used in the single well model.

Exponent or parameter Limestone Sandstone mg- exponent of residual gas saturation correlation ng- exponent of scale function correlation of gas relative permeability nc- exponent of scale function correlation of condensate relative permeability Ncb- Base Capillary number Inertia factor for dry gas phase (m-1) Two-phase inertia factor (F) C and D parameters to calculate the two-phase inertia factor

7.5 0.5100

0.013

3.81 E07 1.623 E10 2846.25

-0.3 and 1.2

2.0 0.2378

0.0694

1.41E-06 3.927 E9 548.52

-0.3 and 1.2

Table 2. Size of the region with two-phase flow regime given by numerical simulation and calculated analytically using probe radius method, at different rates.

Two-phase zone size /feet

Type of core Gas Rate /MMscf.day-1Simulation1 Analytical2

Limestone Limestone Limestone Sandstone Sandstone Sandstone

10 20 30 10 20 30

Between 53-67 Between 210-263 Between 70-77 Between 34- 43

Between 167-209 Between 329-413

65 243 72 43 200 332

1. This distance is based on the size the of two adjacent grid blocks in the single well model where two-phase flow begins (krg < 1 and liquid saturation is greater than zero).

2. Distance away from the well bore at which the reservoir pressure calculated by the probe radius method is equivalent to the dew point pressure.

Table 3. Size of the region with two-phase flow regime given by numerical simulation and calculated analytically using probe radius concept. Gas Rate of 10 MMscf/day

Two-phase zone size /feet Type

of core

Draw-down before shut-in

Mechanical Skin

Simulation1 Analytical2

Limestone Limestone Limestone Sandstone Sandstone Sandstone

Transient Pseudo-steady

Transient Transient

Pseudo-Steady Transient

0 0 6 0 0 6

Between 70-77 Between 175-191

Between 70-77 Between 45- 49

Between 122-133 Between 45-49

72 175 72 55 127 55

1. This distance is based on the size the of two adjacent grid blocks in the single well model where two-phase flow begins (krg < 1 and liquid saturation is greater than zero).

2. Distance away from the well bore at which the reservoir pressure calculated by the probe radius method is equivalent to the dew point pressure.

psat

pi

rw rsat re

p tp

pw

Gas Blockor

Liquid DropoutSingle PhasePressure Profile

Two PhasePressure Profile

Single Phase Region

p > psatTwo

PhaseRegion

Reservoir

Pressure

above

Saturation

Radius, r

p

psat

pi

rw rsat re

p tp

pw

Gas Blockor

Liquid DropoutSingle PhasePressure Profile

Two PhasePressure Profile

Single Phase Region

p > psatTwo

PhaseRegion

Reservoir

Pressure

above

Saturation

Radius, r

p

Figure 1. Radial composite behaviour near the wellbore in gas condensate reservoirs FFF

L o g - L o g D i a g n o s t i c a n d T y p e C u r v e

p

t e

t e

E T RW B S

M T R

s in g l ep h a s e

g a s

D P

D P

R a d ia l C o m p o s i t e

t w op h a s e

pw s

S e m i l o g P l o t

S e c o n dS t r a i g h t

L i n e

F i rs tS t r a ig h t

L in e

T o t a l( P s e u d o r a d i a l )

S k i n

T r a n s i t i o n

F ig 1 6 . 3 . 3

=

kk

1

1

=

kk

2

2

L o g - L o g D i a g n o s t i c a n d T y p e C u r v e

p

t e

t e

E T RW B S

M T R

s in g l ep h a s e

g a s

D P

D P

R a d ia l C o m p o s i t e

t w op h a s e

pw s

S e m i l o g P l o t

S e c o n dS t r a i g h t

L i n e

F i rs tS t r a ig h t

L in e

T o t a l( P s e u d o r a d i a l )

S k i n

T r a n s i t i o n

F ig 1 6 . 3 . 3

=

kk

1

1

=

kk

2

2

L o g - L o g D i a g n o s t i c a n d T y p e C u r v e

p

t e

t e

E T RW B S

M T R

s in g l ep h a s e

g a s

D P

D P

R a d ia l C o m p o s i t e

t w op h a s e

pw s

S e m i l o g P l o t

S e c o n dS t r a i g h t

L i n e

F i rs tS t r a ig h t

L in e

T o t a l( P s e u d o r a d i a l )

S k i n

T r a n s i t i o n

F ig 1 6 . 3 . 3

=

kk

1

1

=

kk

2

2

Figure 2. Composite model interpretation in gas condensate reservoirs.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00So

Krg - LimestoneKro - LimestoneKrg - SandstoneKro - Sandstone

Figure 3. Gas and condensate relative permeability curves of the limestone core at high IFT and low velocity (base curve).

SPE 94065 11

0

0.1

0.2

0.3

0.4

0.5

0 1000 2000 3000 4000 5000 6000Pressure /psia

Liqu

id S

atur

atio

n

Fluid System 1Fluid System 2Fluid System 3

Figure 4. Fluid systems used in this study.

Figure 5. Log-log plot of a single phase case.

Figure 6. Log-log plot of a two-phase case.

0tp

t

Pressure Build-Up in a Reservoir

t 1

t 2

t 3

t 4

t 5

p (t ,r )r p p1

p (t ,r )r p p2

p (t ,r )r p p3

p (t ,r )r p p4

p (t ,r )r p p5

pr

rrw

Reservoir pressure distributionat moment of shut-in, p (t )r p

Peaceman ProbeRadius Concept

Fig 2.5.11r

0tp

t

Pressure Build-Up in a Reservoir

t 1

t 2

t 3

t 4

t 5

p (t ,r )r p p1

p (t ,r )r p p2

p (t ,r )r p p3

p (t ,r )r p p4

p (t ,r )r p p5

pr

rrw

Reservoir pressure distributionat moment of shut-in, p (t )r p

Peaceman ProbeRadius Concept

Fig 2.5.11

0tp

t

Pressure Build-Up in a Reservoir

t 1

t 2

t 3

t 4

t 5

p (t ,r )r p p1

p (t ,r )r p p2

p (t ,r )r p p3

p (t ,r )r p p4

p (t ,r )r p p5

pr

rrw

Reservoir pressure distributionat moment of shut-in, p (t )r p

Peaceman ProbeRadius Concept

Fig 2.5.11

0tp

t

Pressure Build-Up in a Reservoir

t 1

t 2

t 3

t 4

t 5

p (t ,r )r p p1

p (t ,r )r p p2

p (t ,r )r p p3

p (t ,r )r p p4

p (t ,r )r p p5

pr

rrw

Reservoir pressure distributionat moment of shut-in, p (t )r p

Peaceman ProbeRadius Concept

Fig 2.5.11r

Figure 7. Ilustration of the Probe radius concept.

PVT from Original Fluid at Initial Reservoir Conditions

1

10

100

1000

10000

0 2 4 6 8

Distance /feet

Pres

sure

Gra

dien

t /p

si.ft

-1

10

Probe Radius

Simulation

Figure 8. Rate dependent reservoir pressure gradient profile at shut-in time. Used limestone core (k=9.1 mD) and fluid of 47% maximum liquid drop out. Gas rate of 30 MMscf/day.

PVT from Original Fluid at Initial Reservoir Conditions

1

10

100

1000

10000

0 2 4 6 8

Distance /feet

Pres

sure

Gra

dien

t /p

si.ft

-1

10

Probe Radius

Simulation

Figure 9. Rate dependent reservoir pressure gradient profile at shut-in time. Used sandstone core (k=11. mD) and fluid of 47% maximum liquid drop out. Gas rate of 30 MMscf/day.

12 SPE 94065

t = 8.2 days (Before PBU)

0

10

20

30

40

50

60

70

4900 5000 5100 5200 5300 5400 5500

Pressure /psia

k rg/k

ro

Simulation

Steady-State Theory - PVT from Produced Fluid

Figure 10. Rate dependent krg/kro ratio as a function of pressure at shut-in time. Used limestone core (k=9.1 mD), fluid of 47% maximum liquid drop out and gas rate of 10 MMscf/day.

t = 8.2 days (Before PBU)

0

10

20

30

40

50

4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500

Pressure /psia

k rg/k

ro

Simulation

Steady-State theory - PVT from Produced Fluid

Figure 11. Rate dependent krg/kro ratio as a function of pressure at shut-in time. Used limestone core (k=9.1 mD), fluid of 47% maximum liquid drop out and gas rate of 20 MMscf/day.

t = 8.2 days (Before PBU)

0

10

20

30

40

3700 3900 4100 4300 4500 4700 4900 5100 5300 5500

Pressure /psia

k rg/k

ro

Simulation

Steady-State Theory - PVT from Produced Fluid

Figure 12. Rate dependent krg/kro ratio as a function of pressure at shut-in time. Used limestone core (k=9.1 mD), fluid of 47% maximum liquid drop out and gas rate of 30 MMscf/day.

t = 8.2 days (Before PBU)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50 60 70

Distance /feet

k rg

Simulation

Probe Radius - Rate Dependent dP/dr

Figure 13. Rate dependent gas relative permeability as a function of distance at shut-in time. Used limestone core (k=9.1 mD), fluid of 47% maximum liquid drop out and gas rate of 20 MMscf/day.

t = 8.2 days (Before PBU)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50 60 70

Distance /feet

k ro

Simulation

Probe Radius - Rate Dependent dP/dr

Figure 14. Rate dependent oil relative permeability as a function of distance at shut-in time. Used limestone core (k=9.1 mD), fluid of 47% maximum liquid drop out and gas rate of 20 MMscf/day.

t = 8.2 days (Before PBU)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 1

krg/kro

k rg

10

10 MMscf/day - IFTwellbore=0.01 mNewton/mt20 MMscf/day - IFTwellbore=0.1 mNewton/mt30 MMscf/day - IFTwellbore=0.26 mNewton/mt

Figure 15. Rate dependent krg as function of krg/kro with rate effect at shut-in time. Used limestone core (k=9.1 mD), fluid of 47% maximum liquid drop out and gas rates of 10, 20 and 30 MMscf/day.

SPE 94065 13

MULTI-RATE TEST - FLUID SYSTEM 1

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 5Time/ hours

Rat

e/ M

Msc

f.day

-1

1 DD

2 DD

3 DD

1 PB

0

U 2 PBU 3 PBU

Figure 16. Multi-rate Test scheme for sandstone core. Used fluid of 47% maximum liquid drop out. Gas rates of 20, 30 and 40 MMscf/day.

3400

3700

4000

4300

4600

4900

5200

5500

0 10 20 30 40

Distance /feet

Pres

sure

/psia

50

Probe Radius

Simulation

Probe Radius

Simulation

Probe Radius

Simulation

Qg /Mmscfday-1

20

30

40

Figure 17. Reservoir pressure distribution at first, second and third shut-in times. Used sandstone core (k=11 mD) and fluid of 47% maximum liquid drop out. Gas rates of 20, 30 and 40 MMscf/day.