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A new fast approach for well production prediction in gas-condensate reservoirs
Accepted Manuscript A new fast approach for well production prediction in gas-condensate reservoirs Mahnaz Hekmatzadeh, Shahab Gerami PII:
S0920-4105(17)30812-4
DOI:
10.1016/j.petrol.2017.10.032
Reference:
PETROL 4355
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 8 May 2017 Revised Date:
20 September 2017
Accepted Date: 11 October 2017
Please cite this article as: Hekmatzadeh, M., Gerami, S., A new fast approach for well production prediction in gas-condensate reservoirs, Journal of Petroleum Science and Engineering (2017), doi: 10.1016/j.petrol.2017.10.032. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A New Fast Approach for Well Production Prediction in GasCondensate Reservoirs
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Mahnaz Hekmatzadeh*, Shahab Gerami
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IOR Research Institute, Tehran, Iran * [email protected]
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Prediction of gas-condensate well production is significant for sensible decision making for the reservoir development plan, infrastructure investment, and produced gas and condensate sales contract. Typically, time-consuming numerical simulation with fine grids or local grid refinement around the well is implemented in order to capture the effects of condensate blockage and predict gas-condensate well deliverability. In addition, in the commencement of the reservoir lifetime, the numerical simulation predicts erroneous results due to high uncertainty in reservoir data. A simple analytical model can estimate gas-condensate well production reasonable with a limited number of input data. On the other hand, uncertainty studies require repetitious usage of the analytical calculation. Therefore, a faster and more practical method for calculation of gas-condensates well flow rate has a precious value. In this paper, a fast analytical method is introduced for prediction of gas and condensate production profiles based on the two-phase pseudo-pressure integral and material balance equation. Proposed analytical model in contrast to previous ones, predicts the exact plateau time. As the result, it does not need iteration in each reservoir pressure step to compute the relevant flowing bottom-hole pressure during the constant gas production period. The model is expanded to take into account the high-velocity phenomena in near the wellbore region. It is also extended for the first time for different well geometries including vertical, deviated, horizontal, and hydraulicfractured wells. Additionally, the analytical model is validated using fine grid numerical simulation for a wide range of rock and fluid properties. The developed analytical model can be used as a fast engineering tool for evaluating the uncertainty in well and reservoir data to choose the best strategy for reservoir development.
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Abstract
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Keywords:
Gas-Condensate, Well Production Prediction, Analytical Model, High-Velocity Effects, Deviated Well.
1. Introduction
As long as the flowing bottom-hole pressure is above the dew-point in gas-condensate well, the production performance is the same as a dry gas well. When the flowing bottom-hole pressure falls below the dew point, condensate begins to drop out near the wellbore. This phenomenon is called “condensate blockage” that severely reduces effective gas permeability and gascondensate well productivity. On the other hand, high velocity and low interfacial tension (high 1
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capillary number) around the wellbore results in an increase in gas relative permeability which is named “positive coupling”. Meanwhile, high gas velocity in near wellbore region results in a reduction in gas relative permeability which is named “negative inertia”. The mutual interaction of positive coupling and negative inertia determines the gas-condensate well performance. Therefore, the condensate blockage and high-velocity phenomena should be modeled accurately in order to determine the exact well deliverability in gas-condensate reservoirs. Numerical compositional reservoir simulation is used to capture the complex behavior of gascondensate fluid around the wellbore and to predict the well deliverability correctly. But, coarse grid simulation cannot investigate the condensate blockage phenomena. Instead, fine grid simulation, especially around the wellbore, should be used which is time-consuming. Also, numerical compositional simulation requires details of fluid and rock properties e.g. tuned equation of state for PVT and distributed rock properties for each grid. Since accurate data of reservoir fluid and rock properties are not available at the beginning of the reservoir lifetime, simple analytical calculations are used for fast forecast of the gas-condensate wells' performance (Ayyalasomayajula, 2005). Analytical model has the advantage of assuming a homogeneous reservoir. It requires tabulated PVT properties versus pressure that could be generated from CVD experiments or correlations. Fevang and Whitson (1996) developed a two-phase pseudo-pressure integral for calculating gascondensate well deliverability by modifying the concept presented by Evinger and Muskat (1942). They introduced the idea of the presence of three regions around the gas-condensate well. The existence of these regions depends on the magnitudes of the flowing bottom-hole pressure and the average reservoir pressure. At the first region, which is the nearest to the wellbore, both phases (gas and condensate) are flowing (Region I). Whereas, at the second region, both gas and condensate phases are present, but only gas phase is flowing (Region II). At the third region, a single gas phase is present (Region III). Their integral requires tabulated pressure/volume/temperature (PVT) properties for each reservoir pressure step, relative permeability, and the producing gas-oil ratio (GOR). But, the values of GOR should be determined through numerical simulation. In addition, their developed equation does not contain the effect of high-velocity flow which has a major influence on the gas-condensate well productivity. Mott (2003) and Xiao and Muraikhi (2004) proposed novel analytical models to calculate gascondensate well production performance, using a material balance model for reservoir depletion and Fevang and Whitson (1996)’s two-phase pseudo-pressure for well inflow performance. Their methods rely on the fact that the increase in condensate saturation during pressure depletion in Region II is used at an extension of Region I. None of the methods need the numerical simulation which results to compute GOR, anymore. In contrast, they require defining a specified reservoir pressure step to calculate pseudo-pressure integral. As the matter of fact, iteration in each reservoir pressure step should be done to compute the relevant flowing bottomhole pressure during the constant gas production period. Whatever the plateau time is longer, the number of iterations increased that results in a longer analytical model run time. If larger reservoir pressure step is selected, the number of iteration decreased but the plateau time is computed erroneously. This disadvantage made these analytical models impractical for uncertainty studies. Al-Shawaf et al. (2014) demonstrated a simple method to predict IPR curves for condensate wells as a function of reservoir pressure using rock relative permeability and constant composition expansion (CCE) experiment data. The future IPR curve is also interpolated based
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on the current IPR curve if CCE data are available. Hence, the production profile and plateau time predictions are not exact and straightforward. In contrast to previous models, which have just been developed for vertical wells, proposed analytical model in this study is extended for different well geometries including deviated one. Most of the wells are deviated in offshore platforms in the gas-condensate reservoirs due to lack of possibility of vertical well drilling. Since nearly 20 percent of the world’s gas fields are located offshore, analytical modeling of deviated well in gas condensate reservoirs for uncertainty study is crucial. There are many literature studies for predicting productivity index for single phase flow in deviated wells including Cinco et al. (1975), Besson (1990), and Rogers et al. (1996). Cinco et al. (1975) claimed that their semi-analytical model could compute geometric skin factor for deviated wells in single phase gas reservoirs both for isotropic and non-isotropic media. Rogers et al. (1996) suggested another model and compare calculated skin factor with Cinco et al. (1975). Their results matched for isotropic cases but had a difference for non-isotropic reservoirs. Kyoon Choi et al. (2008) conducted a sensitivity analysis on available analytical models developed for calculating the productivity of deviated wells. They claimed that these correlations are inappropriate for extremely deviated wells; namely with a deviation angle of more than 75 degree. By using a sensitivity study, Ghahri (2010) claimed that the skin values obtained from Rogers et al. (1996) were under estimated and significantly different from those predicted by equations proposed by Besson (1990) and Cinco et al. (1975). According to this result, we use Cinco et al. (1975) model for computing geometric skin factor. In the current study, a new and fast approach to well production prediction in gas-condensate reservoirs is developed. The fundamental of this study is similar to Mott (2003). However, it has been modified as follows: - As the plateau time is one of the key parameters in gas and condensate sales contract and in selecting the reservoir development strategy, in this study, a new technique is proposed to determine the exact plateau time for both gas and condensate. - A new analytical model algorithm is proposed to calculate gas and condensate production profiles. This approach doesn’t require defining a pressure step and performing iteration in each pressure step during plateau time period for determining the following bottomhole pressure. Instead, the end of the plateau time is computed directly. Therefore, developed analytical model has the advantage of less run time which makes it suitable for studying the effect of data uncertainty on gas-condensate well deliverability. It also helps to selecting the best strategy for well completion especially in newly discovered fields with limited data and high uncertainty. - The proposed analytical method is extended for deviated wells in contrast to previous models that are just developed for vertical wells. Also, the model is extended for horizontal, and hydraulic-fractured wells with new method, as drill of these wells in low permeability gas-condensate reservoirs are more common due to condensate build-up. The model is validated for slant wells with different deviation angle, partially penetration wells, non-isotropic reservoirs, and horizontal and hydraulic-fractured wells with different length. - The model is expanded to take into account the high-velocity effect in near wellbore region including positive coupling and negative inertia with applying Jamialahmadi et al.
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(2009) correlation. This correlation in contrast to previous ones does not require experimentally determined constants. Consequently, a spreadsheet is developed based on the proposed model. The spreadsheet is also extended with stochastic programming to take into account the data uncertainty in well and reservoir. Therefore, developed spreadsheet provides an applied tool in development strategy determination in newly discovered reservoirs. After that, it is used for selecting development strategy in one of the Iranian offshore low permeability gas-condensate fields that will be reported in near future. The organization of the current study is as follows. First, we describe well inflow calculation method of different well geometries and the extension that is made to include the high-velocity flow effect. In the next section, the computational algorithm is described. Afterward, we present the model validation. At the last section of this paper, the summery is presented.
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2. Analytical Model for Well Inflow Calculation
Developed analytical model consists of the two-phase pseudo-pressure integral for flow equation term and a material balance equation for depletion term.
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Flow Equation Term
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Tabulated PVT properties for different pressure steps, a set of relative permeabilities from SCAL data, and reservoir and well data are used as an input for the analytical model to calculate gascondensate well productivity. Developed model applies two-phase pseudo-pressure integral for well inflow performance, as shown by Fevang and Whitson (1996);
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(1)
In which C is the productivity index that depends on the well and reservoir geometry. For vertical well, C can be determined from Eq. (2). 2π × 0.001127 × Kh r ln( e ) − 0.75 + s rw
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(2)
Pres and Pwf are average reservoir pressure and flowing bottom-hole pressure, respectively. P1 is the pressure in the interface between Region I and Region II, and Pdew is dew point pressure. The first integral, with integral limits Pdew to Pres, relates to Region III, in which only the gas phase is present. The second integral, with integral limits P1 to Pdew, relates to Region II in which condensate drop-out, but its saturation is less than critical condensate saturation. Then, it is immobile and only the gas phase is flowing. The third integral, with integral limits Pwf to P1, relates to Region I, near the wellbore region, in which both gas and condensate phases are flowing. In this integral, µ gBg, µ cBc, and Rs are determined from PVT table. One of the parameters, Pwf or qg, is known based on the well production strategy; constant bottom-hole pressure or constant flow rate, respectively. Pres is also determined from material balance equation that will be discussed later. krg in Region III is the only function of Swi and it is determined from SCAL table. krg in second integral (Region II) is a function of condensate saturation, So. So is evaluated for each pressure in Region II by Eq. (3). Then, krg in Region II is also finds out from SCAL data. S = r ( p ) × (1 − S ) (3) o v wi
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k rg k + R s rc dP µ c Bc µ g B g
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Pdew P1 Pres k rg k rg qg = C ∫ dP + ∫ dP + ∫ P µ g Bg µ g Bg P1 Pwf dew
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,
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−
(5)
Where qo,max is the maximum condensate flow rate at the interface Region I and II, So,1 is the average condensate saturation in Region I, which is estimated by arithmetic average of computed saturation at this region interfaces as is shown in Eq. (6); ,
=
1 ! r ( p w ) × (1 − S ) " + ! r ( p1 ) × (1 − S ) "$ v wi v wi 2
,
=
1 ! r ( p1 ) × (1 − S ) " + 0$ v wi 2
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So,2 is the average condensate saturation in Region II, which is calculated as is shown in Eq. (7) with calculating arithmetic average of interface saturations at Region II . (7)
PV is the pore volume in pressure contour map around a well that is dependent on the well geometry. For vertical well, it can be obtained from Eq. (8). ∆PV is also the increase pore volume of Region I growth size that can be calculated with knowing the previous PV of Region I. =π
ℎ 1 + '( + ' ∗
*
.∗
−
(,-
∅
(8)
Where r is the outer radius of Region I. In this study, r is estimated from Eq. (9) that is developed under the assumption of pseudo-steady state condition. Applying this equation implies acceptable error in P1 calculation and results in performing calculations faster. ln
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Where rv is vaporized condensate gas ratio. It is determined for each pressure by interpolating in tabulated PVT data. For calculating krg and krc in Region I, in the third part of the integral, combined flow in both condensate and gas phases should be considered as proposed by Fevang and Whitson (1996); krg 1 − rvf Rs µ g Bg = (4) krc rvf − rv µo Bo where rvf is the flowing condensate gas ratio of saturated gas at P1. After calculating the ratio krg/krc from Eq. (4), krg and krc are determined from relative permeability curves. These values should be updated for high velocity effect that is illustrated later. Finally, the only parameter that remains unknown in an evaluation of pseudo-pressure integral in Eq. (1) is P1. P1 is calculated according to Mott's concept as it is shown in Eq. (5).
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Gas production profile is divided into two parts; constant gas production and decline part. In constant gas production part, qg is known and equal to qg,cons. Then, Pwf should be determined which is required a trial procedure. Also, Eq. (5) is highly non-linear and finding P1 needs trial procedure, too. Therefore, finding Pwf and P1 in constant gas production based on the previous algorithms with two nested iteration procedures are very time-consuming. This circumstance makes previous methods inappropriate for uncertainty analysis with stochastic methods especially in spreadsheet programming. In this paper, an algorithm is developed to bypass first part in gas and condensate production profiles by finding two times, t1 and t2. t1 and t2 are the end of the plateau time in condensate and gas production profiles as is shown in Fig. 1 (A) and (B), respectively. Production profiles in Fig. 1 comprise the Mott and our method schematically. The number of iteration in Mott method depends on the plateau time duration and the defined pressure step. Whatever the plateau time is 5
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longer, the number of iterations increased that results in a longer analytical model run time. If larger reservoir pressure step is selected, the number of iteration decreased but the plateau time is computed erroneously. But, in our approach, just the last time in plateau production period is calculated. This method made the computed plateau time independent of either plateau time duration or the defined pressure step. For these times, t1 and t2, relevant Pwf is calculated. Therefore, qg can be computed directly and production profiles can be predicted without any iteration in pressure steps. As it is shown in Fig. 1 (C), three different slops exist in the curve of Pwf verses time. First slop relates to single-phase gas production. The end of the first slope, t1, shows the time at which Pwf equals to Pdew. Then, condensate starts to drop-out and Region II starts to growth. The size of Region I also increases with Region II growth. At this time, the qc declines, as it is shown in Fig. 1 (A) and (C). Therefore, second slop relates to two-phase flow of gas and condensate. In this part of the curve, Pwf is still greater than minimum bottom-hole pressure, Pwf,min, and gas flow rate is equal to qg,cons. At time t2, Pwf become equal to Pwf,min and gas flow rate declines. According to this fact, new algorithm is developed to compute exact plateau time for gas and condensate and to predict gas and condensate production profiles without need to define pressure step and perform iteration in each pressure step. This algorithm is illustrated in the following section.
(B)
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Fig. 1. Schematic of production profiles in gas-condensate well, (A) Gas flow rate, (B) Condensate flow rate, (C) Flowing bottom-hole pressure
Depletion Term
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The purpose of this section is to write the material balance to obtain Pres for a specified time, t. At Pi the total initial volume, Vi, can be calculated by Eq. (10); *
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Where PVi is the initial pore volume that depends on the reservoir geometry. The flow rate is known from the flow equation described in the previous section. For the first point (end of the plateau time, t1 or t2), qg is equal to, qg,cons. Then cumulative gas production, Ng,prod can be calculated by Eq. (11); ,9 :-
(11)
Total condensate production, Nc,prod (the area under the condensate production profile) in this time interval is equal to; 69,7( 8 = 6 ,7( 8 × @ < (12) − 8
=6
,7( 8
×
@
+
×
2
,9 :-
@
+
@
1 < <
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The material balance equation is written as Eq. (13); (13) * * − 6 ,7( 8 + 69,7( 8 = (, *:*: × (,= = , ℎ> 1 − * + '( + ' ∗ * .∗ (,- − * ? × (,For a known t, Nc,prod and Ng,prod can be computed. Then, by using Eq. (13), (,- can be obtained. High Velocity Effect in Near Wellbore Region
krg and krc, in Region I, are not only a function of saturation, but also a function of capillary number. Then, computed relative permeabilities from Eq. (4) are considered for the base condition, @( A and @(9A . After that, these parameters are correlated as a function of the capillary number via Jamialahmadi et al. (2009). This correlation selected because against previous correlations (i.e. Pope et al. (2000), Henderson et al. (2000), Whitson et al. (2003), etc.), it does not require experimentally determined constants. Eq. (14) computes gas relative permeability in miscible condition, @( , that is corrected BCDE for inertia effect, too. By using Eqs. (15) and (16), one can calculate the corrected immiscible gas relative permeability for negative inertia, @( F BCDE. @(
BCDE
` = @( 22 23 24 25
*
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@
Q S|U|VO RO
ab
c
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=
@
ab
c
+
@
db
c9
k EYZE =
,
[\]^
[\]^ L [\_^
@( A βρijD k@( 1+ fY μY e
=f
W
A
(14)
(15)
|u| n
p
(16)
o
Where β is the single-phase non-Darcy coefficient and subscript m refers to the miscible condition. ρavg is the weighted average density value by fg (Eq. (15)) at the base conditions. uT is total velocity and it is calculated by dividing the computed qg by the flow cross section area, Acs, which depend on the well geometry. Acs for vertical well is calculated by Eq. (17); 7
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(17)
krg = Yg (krgI)iner + (1− Yg) (krgm)iner
(18)
Finally, corrected krg for high-velocity effect in Region I can be obtained by using weighting factor, Yg, according to Eq. (18); The Yg equation can be found elsewhere (Jamialahmadi et al. (2009)).
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Different Well Geometries
The proposed method for the vertical well is also extended for different well geometries including deviated, horizontal, and hydraulic-fractured wells. The idea is to calculate the geometric skin and relevant apparent wellbore radius, rwa. After that, rw can be replaced by rwa in Eq. (2). In addition, the change in the shape of the pressure transient lines around each well is involved in the computational algorithm. For horizontal well, rwa is computed by Eq. (19); (Diyashev and Economides, 2006)
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:*
v + wv − x/2 { :* ℎ z+ |st x/2 x
:*
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•€ ••
„.… „.…
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}
(19)
(20)
(21)
(23)
3 ∗ A ∗ 43560 ∗ h 4v́
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(22) ,€ = √v́ ∗ b b is the minor axis of the drainage ellipse and is calculated with iterative use of Eqs. (23) and (24); b=~
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:* ℎ
where KH and Kv are horizontal and vertical permeabilities respectively. a is the half of the major axis of the drainage ellipse around the horizontal well and is calculated by Eq. (21); ƒ x ,€ v = u0.5 + 0.25 + ! " 2 x/2
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{ {
Where L is the horizontal well length and Iani is anisotropy index calculated by Eq. (20); {
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= st u
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(24)
where A is the reservoir drainage area. It also should be noted that in the horizontal well, the shape of the pressure transient lines is changed to elliptical (Ghahri (2010)). Then, for calculating PV and Acs, Eq. (25) and Eq. (26) are used instead of the Eq. (8) and Eq. (17) respectively. 4 = π v × b 1 + '( + ' ∗ 3 q9- = 4= v × b
*
.∗
−
*
∅
(25) (26)
Where P is the pressure of any contour-line around the wellbore. Effective wellbore radius for the hydraulic-fractured well is calculated according to Eq. (27); (Prats et al. (1962))
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= π v × b ℎ 1 + '( + ' ∗ q9- = = v + b ℎ
*
.∗
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*
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where Lf is the fracture length, xf is fracture half-length and xe is half of the reservoir length. As the shape of the pressure transient lines around the hydraulic-fractured well are changed to cylinder with ellipse cross-section (Ghahri (2010)), PV and Acs are calculated by, (28) (29) where a and b are the half of the major and minor axis of the drainage ellipse around the hydraulic fracture respectively, that can be calculated by; xŽ v=b+ 2 A ∗ 43560 b= =v
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(27)
(30) (31)
Geometric skin factor, ’ , for deviated well with deviation angle “ are calculated by Eq. (32); (Cinco et al. (1975)). But, it should be noted that Cinco correlation predict acceptable results for wells with deviation angle less than 75 degree (Ghahri, P., and M. Jamiolahmady (2012) and Suk Kyoon et al (2008)). Apparent wellbore radius to replace in Eq. (2) is computed by Eq. (34). (32) “ .„” “ .•”… ℎ– ’
= −! " 41
ℎ @€ ℎ– = ! " ~ @•
−! " 56
× s•a ! " 100
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(33)
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Developing a fast analytical model for computing the gas and condensate production profiles are the aim of this study. Most of the run-time in previous analytical models is spend in plateau time interval for computing the proper flowing bottom-hole pressure, Pwf, for a known average reservoir pressure, Pres. This disadvantage made them impractical for uncertainty analysis with stochastic methods that requires thousands run of a program. In this study, a new algorithm is introduced to bypass the constant gas production interval by calculating just the end of the plateau time. Based on our algorithm, the production profile is divided into two sections. The first section, is where the flow rate is constant and equal to qg,cons. The second section, where the Pwf is constant and equal to minimum flowing bottom-hole pressure, Pwf,min. In the first section, the shape of the gas production profile is uniform. Therefore, flow rate is known (qg,cons) but, the end of the plateau time and relavant Pres is unknown. Flowing bottomhole pressure at the end of the plateau time is also known and equal to Pwf,min. The strategy for this section is to find the end of the plateau time (t1 and t2 for condensate and gas production profiles respectively) and reduce the spend time in each reservoir pressure step to find relevant Pwf. To determine end of the plateau time following algorithm should be tracked.
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(34) = 2 —˜™ For deviated well, the shape of the pressure transient lines is changed to elliptical that is similar to horizontal well (Ghahri (2010)). Thus, for calculating PV and Acs, Eq. (25) and Eq. (26) could be used, respectively.
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Ž, *:
−
Ž,*:
(35)
k (P + P ) k k rg k rg rg rg res dew − + − µ B 2 µ B µ B µ B g g p res g g p dew g g Pdew g g =C k rg k rg k k (P1 − Pwf ) + R s rc − + R s rc + µ c B c P µ g B g µ c B c P 2 µ g B g 1 wf
(Pdew + P1 ) 2 P1
(36)
D. The trial values of Pres and P1 are replaced in Eq. (37) to obtain a trial value of plateau time interval, t2. min
(,- , 8,
−
1−
=∆
,
−
,
1
−
(37)
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Eq. (37) is written based on the Eq. (5). It reflects this fact that the amount of the condensate dropped in Region II from the beginning of the Region II growth to the end of the plateau time (left side of the Eq. (37)) is equal to the amount of condensate increase the Region I size in that specified time step (right side of the Eq. (37)); E. t2 is used in Eqs. (10) to (13) to compute the relevant Pres. F. Computed Pres in step E is compared with previous calculated Pres. The iteration is continued from step C until convergence in Pres is reached. Because of pseudo-steady state assumption, the pressure declines slowly and linearly, thus the convergence never failed. As the first guess comes from the dry gas equation, the convergence is fast. Normally, after three or four iteration Pres is obtained with accuracy around 0.1 psi. When the Region III is vanishes (Pres become less than Pdew and just Region I and Region II exist), there would be no fresh gas to support constant condensate drop in Region II. Therefore, the amount of condensate dropout in Region II is decreased with time and the pseudo-steady state assumption doesn’t valid anymore. In that condition the model’s accuracy decreased. But, it stills gives a good estimate of plateau time with precision around 10 psi.
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*
C. At the end of the gas production plateau time (t2), the flow rate and Pwf is known and equal to qg,cons and Pwf,min respectively. Then, Eq. (1) is written for the trial value of Pres obtained in the previous step to compute the trial value of P1. This is done by extension of Eq. (1) with trapezoidal method, as is shown in bellow equation;
q g ,cons
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=
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A. Using dry gas equation with Pi to obtain the first guess for bottom-hole pressure, Pwf,int. š› B. Assume pseudo-steady state condition in the reservoir, where šœ = •œž, and determine the first trial value for Pres by Eq. (35).
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To plot the condensate production profile, a similar procedure should be followed to obtain the condensate plateau time. The point is P1=Pdew at the end of the condensate plateau time. After that, Region II starts to improve and P1 becomes less than Pdew. Then, some of the condensate remains in the reservoir that causes a decrease in condensate flow rate. During condensate plateau time, gas flow rate is equal to qg,cons. Hence, to predict condensate production profile, t1 and Pres as unknown parameters should be determined. Suggested algorithm in this paper, described below; A. Assume Pres > Pdew .Solve Eq. (37) with assuming P1=Pdew and replacing t1 with t2 to obtain t1. B. Computed t1 is used in material balance equation, Eq. (13), to obtain the relevant Pres. 10
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By following aforementioned algorithm and without any iteration in each reservoir pressure step in the first section of the production profile, an exact amount of gas and condensate plateau times is determined. While, previous methods use two nested iteration for finding P1 and Pwf in each reservoir pressure step in plateau period. This approach results in a faster prediction of gas and condensate production profiles. In the second section, Pwf is equal to Pwf,min. then straight forward procedure of using Eq. (5) to calculate P1 and Eq. (1) to compute relevant flow rate should be followed. The flowchart algorithm is shown in Fig. 2.
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C. If computed Pres is greater than Pdew, the calculation is stopped. Otherwise, it shows that the Region III does not exist. Then, Eq. (37) is solved with the assumption of Pres < Pdew to obtain t1. D. Condensate flow rate in plateau time can be calculated by Eq. (38)
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Fig. 2. Flowchart algorithm for computing gas and condensate production profiles
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4. Analytical Model Validation and Discussion For validation of the developed model, the results are compared with a commercially available numerical compositional simulator, Eclipse 300 [Schlumberger-GeoQuest, 2010]. Various reservoir fluids (lean, intermediate and rich gas-condensate) and rock properties (different SCAL) in different well geometries (vertical, horizontal, hydraulic-fractured, and deviated wells with different deviation angles) with various production scenarios (different plateau rate and minimum bottom-hole pressures) are selected for model validation. Evaluation of horizontal and hydraulic-fractured wells are also repeated for several values of L and Lf . A three dimensional Cartesian reservoir model with a single layer and uniform porosity and permeability is considered as a base case. It is assumed that a fully-penetrated single producer well located at the center of the reservoir. Fig. 3 (A) shows the schematic of the reservoir with horizontal well model as an example. Local grid refinement (LGR) is made in near wellbore region in order to model condensate blockage more accurately in numerical simulation. Numerical simulation is repeated with refining the grid size to decrease the numerical error due to grid dimensions. Optimum grid size is selected such that the change in numerical simulation results is minimized. For deviated wells, the grid size is selected in a way that the well passed from the grid center as is shown in Fig. 3 (B). Assumed well and reservoir properties for numerical simulation are summarized in Table 1.
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(A) (B) Fig. 3. Schematic of the reservoir and well model, (A) Horizontal, (B) Deviated well
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Table 1. Well and reservoir properties that are selected for validation Unit Parameter Amount ∅
10
%
K
50
mD
Reservoir Dimension
6000 ×6000
ft
rw
'(
0.58
ft
'
4.25
(Psi-1) E-6
1.05
(Psi-1) E-6
Swi
10
%
h
200
ft
L
3200
ft
Lf
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The velocity dependent relative permeability keyword (Whitson Correlation with default parameters) and Forchheimer effect is also activated in numerical simulation for considering the effect of the capillary number and high velocity flow on relative permeability. Three multicomponent mixtures of gas condensate fluids with different condensate to gas ratio (CGR) are modeled. The key reservoir fluid properties are given in Table 2.
Rich Intermediate Lean
5480 3129 3535
38 21 3.1
Pi (Psia)
200 200 212
6000 3500 5500
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T (oF)
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Residual saturation (Somax, Somin, Sgmax, Sgmin) and Corey power (no , ng), kromax, krgmax and Swi for the construction of different relative permeability curves are reported in Table 3. SCAL 1 and SCAL 2 have different minimum condensate saturation. While, SCAL 1 and SCAL 3 differ in Corey power that results in different pore size distribution i.e. SCAL 3 has narrower pore size distribution than SCAL 1. Fig. 4 also shows the designed relative permeability curves that are used for comparison. Table 3. Residual saturation and Corey power for construction of gas and condensate relative permeability
Swi krgmax ng
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Somin Somax 0.15 0.35 0.35
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Pdew CGR Max Condensate (Psia) (bbl/Mscf) Drop out
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Table 2. Fluid properties that are selected for comparison
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Fig. 4. Relative permeability curves used for model validation
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Various production scenarios e.g. different plateau flow rate and minimum bottom-hole pressure, are also specified for model validation. After that, Gas and condensate production profiles 14
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computed by the analytical method are compared with the results of the numerical reservoir simulation. Totally, more than two hundred cases are designed for analytical model validation with numerical simulation. However, the average error for each well-geometry is reported, in Table 4.
Gas Production Profile
Average Error (%)
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Table 4. Computed error for the analytical results in comparison with the numerical reservoir simulation for different well geometries
Condensate Production Profile
Plateau Time
Decline section
Plateau Time
Decline section
1.84 2.23 1.46 4.1
2.03 3.7 2.1 5.3
2.05 2.84 1.27 2.53
2.2 3.6 2.67 3.23
Vertical Horizontal Hydraulic Fractured Deviated Well
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Based on the results of Table 4, the analytical model could predict numerical reservoir simulation with an average error less than 5%. In addition, average run time is less than a minute for developed analytical model, while the run time of each numerical simulation is approximately around ten minutes. This result indicates that developed model is a good engineering tool for gas and condensate production prediction. At the end, one case from predicted production profiles for each well type, with maximum observed average error, is reported here.
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4.1 Vertical Well Gas and condensate production profiles and flowing bottom-hole pressure plot for intermediate fluid and SCAL3 in the vertical well are shown in Fig. 5. Furthermore, the results of the developed model are compared with Mott’s method.
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(C) Fig. 5. Production profile for rich fluid and SCAL3 in the vertical well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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The average calculated error in plateau time and observed mean run time of our approach is also compared with Mott’s method in Table 5.
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Table 5. Comparison the results of our model with Mott’s method
Method
Average calculated error in plateau time (%)
Developed in this study Mott (2003)
1.84 7.4
6
It should be noted that the average calculated error in plateau time and observed mean run time in Mott’s method completely depend on the amount of the pressure step and the duration of the plateau time. The more the plateau time duration and pressure step is selected, the more the average computed error is observed. Anyway, our approach bypasses the plateau time period and just calculates the end of the plateau time. Because of that, it has the advantage of less run-time and exact prediction of the plateau time.
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Average run-time (sec) 35 150
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Flowing bottom-hole pressure, gas and condensate production profiles for lean fluid and SCAL1 as the case with maximum observed error in horizontal well cases are shown in Fig. 6.
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(C) Fig. 6. Production production profile for lean fluid and SCAL1 in the horizontal well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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Flowing bottom-hole pressure, gas and condensate production profiles for intermediate fluid and SCAL2 in the hydraulic-fractured well are shown in Fig. 7.
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(C) Fig. 7. Production production profile for intermediate fluid and SCAL2 in the hydraulic-fractured well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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Three cases for deviated well with deviation angle 15, 45, and 75 with maximum observed error are reported. Intermediate fluid and SCAL1 in the 15 degree slant well has the maximum error. Flowing bottom-hole pressure, gas and condensate production profiles are shown in Fig. 8.
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(C) Fig. 8. Gas production profile for intermediate fluid and SCAL1 in the 15 degree deviated well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
Lean fluid and SCAL2 in the 45 degree slant well has the maximum error. Flowing bottom-hole pressure, gas and condensate production profiles are shown in Fig. 9.
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(C) Fig. 9. Gas production profile for lean fluid and SCAL2 in the 45 degree deviated well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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Rich fluid and SCAL3 in the 75 degree slant well has the maximum error. Flowing bottom-hole pressure, gas and condensate production profiles are shown in Fig. 10.
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A fast analytical model is developed for prediction of gas and condensate production profiles based on the two-phase pressure integral and material balance equation. The model is extended for high-velocity flow phenomena and for different well geometries. The developed model does not need any iteration in each reservoir pressure step and just the end of the plateau time is computed directly and accurately. The model developed under the pseudo-steady state assumption. When the average reservoir pressure become less than dew point pressure, the assumption doesn’t valid anymore and the model’s accuracy decreased. But, it stills gives a good estimate of plateau time. This approach in conjugation with a stochastic programming provides an appropriate engineering tool for uncertainty studies and decision making for choosing the best reservoir development strategy. It also computes the exact gas and condensate plateau time interval which is a key parameter in gas and condensate sales contract.
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5. Conclusion
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1. Al-Shawaf, A., Kelkar, M., & Sharifi, M. (2014). A new method to predict the performance of gas-condensate reservoirs. SPE Reservoir Evaluation & Engineering, 17(02), 177-189. 2. Ayyalasomayajula, P. S., Silpngarmlers, N., & Kamath, J. (2005, January). Well deliverability predictions for a low permeability gas condensate reservoir. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. 3. Besson, J. (1990). Performance of slanted and horizontal wells on an anisotropic medium. In European Petroleum Conference. Society of Petroleum Engineers. 4. Cinco, H., F. G. Miller, and H. J. Ramey Jr. (1975). Unsteady-state pressure distribution created by a directionally drilled well. Journal of Petroleum Technology 27 (11), 1-392. 5. Diyashev, I. R., & Economides, M. J. (2006). The dimensionless productivity index as a general approach to well evaluation. SPE Production & Operations, 21(03), 394-401. 6. Evinger, H. H., & Muskat, M. (1942). Calculation of theoretical productivity factor. Transactions of the AIME, 146(01), 126-139. 7. Fevang, Ø. & Whitson, C. H. (1996). Modeling gas-condensate well deliverability. SPE Reservoir Engineering, 11(04), 221-230. 8. Ghahri, P. (2010). Modelling of Gas-condensate flow around horizontal and deviated wells and cleanup efficiency of hydraulically fractured wells (Doctoral dissertation, Heriot-Watt University). 9. Ghahri, P., and M. Jamiolahmady. (2012). A new, accurate and simple model for calculation of productivity of deviated and highly deviated well–Part I: Single-phase incompressible and compressible fluid." Fuel 97, 24-37. 10. Jamiolahmady, M., Sohrabi, M., Ireland, S., & Ghahri, P. (2009). A generalized correlation for predicting gas–condensate relative permeability at near wellbore conditions. Journal of Petroleum Science and Engineering, 66(3), 98-110. 11. Mott, R. (2003). Engineering Calculations of Gas-Condensate-Well Productivity. SPE Reservoir Evaluation & Engineering, 6(05), 298-306. 12. Pope, G. A., Wu, W., Narayanaswamy, G., Delshad, M., Sharma, M. M., & Wang, P. (2000). Modeling relative permeability effects in gas-condensate reservoirs with a new trapping model. SPE Reservoir Evaluation & Engineering, 3(02), 171-178. 13. Prats, M., Hazebroek, P., & Strickler, W. R. (1962). Effect of vertical fractures on reservoir behavior--compressible-fluid case. Society of Petroleum Engineers Journal, 2(02), 87-94. 14. Rogers, E. J., & Economides, M. J. (1996). The skin due to slant of deviated wells in permeability-anisotropic reservoirs. In International Conference on Horizontal Well Technology. Society of Petroleum Engineers. 15. Suk Kyoon, Choi, Liang-Biao Ouyang, and Wann-Sheng Huang. (2008). A comprehensive comparative study on analytical PI/IPR correlations. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. 16. Whitson, C. H., Fevang, Ø., & Sævareid, A. (2003). Gas condensate relative permeability for well calculations. Transport in porous media, 52(2), 279-311. 17. Xiao, J. J., & Al-Muraikhi, A. J. (2004, January). A new method for the determination of gas condensate well production performance. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers.
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References
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Nomenclature Half of the major axis of the drainage ellipse around the horizontal and hydraulic-fractured well Reservoir drainage area Flow cross section area Minor axis of the drainage ellipse around the horizontal and hydraulic-fractured wells Gas formation volume factor
Bc C '( ' h Iani K
Condensate formation volume factor Productivity index Rock compressibility factor Water compressibility factor Reservoir thickness Anisotropy index Permeability
krc
Condensate relative permeability
k rg
Gas relative permeability
KH Kv @(9A @( A @( F
Horizontal permeability Vertical permeability Measured condensate relative permeability in the base condition Measured gas relative permeability in the base condition Corrected immiscible gas relative permeability for negative inertia
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Gas relative permeability in miscible condition that is corrected for inertia effect too Condensate relative permeability at residual gas saturation Gas relative permeability at initial gas saturation Horizontal well length Fracture length in hydraulic-fractured well Condensate Corey power Gas Corey power Cumulative condensate production Cumulative gas production Pressure in the interface between Region I and Region II Dew point pressure Initial reservoir pressure Average reservoir pressure Flowing bottom-hole pressure Flowing bottom-hole pressure at Pi Minimum flowing bottom-hole pressure Pore volume in pressure contour map around a well that is dependent on the well geometry Initial pore volume that depends on the reservoir geometry pore volume growth of Region I Gas flow rate Gas plateau flow rate Condensate flow rate Outer radius of Region I Reservoir boundary radius Vaporized condensate gas ratio Wellbore radius
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kromax krgmax L Lf nc ng Nc,prod Ng,prod P1 Pdew Pi Pres Pwf Pwf,int Pwf,min PV PVi ∆PV qg qgcons qc r re rv rw
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uT Vi xe xf Yg β µ c
µg
Single-phase non-Darcy coefficient Condensate viscosity Gas viscosity
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Weighted average density Porosity
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Sc Sc,1 Sc,2 Swi Scmax Scmin Sgmax Sgmin ∆t
Apparent wellbore radius Solution Gas oil ratio Condensate saturation Average condensate saturation in Region I Average condensate saturation in Region II Initial water saturation Maximum condensate saturation Minimum condensate saturation Maximum gas saturation Minimum gas saturation Specified time interval Plateau time interval in condensate production profile Plateau time interval in gas production profile Total velocity Total initial volume Half of the reservoir length Fracture half-length Weighting factor for calculating velocity dependent relative permeability
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A fast analytical method is introduced for prediction of gas and condensate production profiles. Proposed model compute exact plateau time without any iteration in each reservoir pressure step during constant gas production period. This approach provides a practical tool for uncertainty study. The model is expanded to take into account the high-velocity phenomena in the near wellbore region. The proposed analytical method is extended for different well geometries including vertical, horizontal, hydraulic-fractured, and deviated wells.
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S0920-4105(17)30812-4
DOI:
10.1016/j.petrol.2017.10.032
Reference:
PETROL 4355
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 8 May 2017 Revised Date:
20 September 2017
Accepted Date: 11 October 2017
Please cite this article as: Hekmatzadeh, M., Gerami, S., A new fast approach for well production prediction in gas-condensate reservoirs, Journal of Petroleum Science and Engineering (2017), doi: 10.1016/j.petrol.2017.10.032. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A New Fast Approach for Well Production Prediction in GasCondensate Reservoirs
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Mahnaz Hekmatzadeh*, Shahab Gerami
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Prediction of gas-condensate well production is significant for sensible decision making for the reservoir development plan, infrastructure investment, and produced gas and condensate sales contract. Typically, time-consuming numerical simulation with fine grids or local grid refinement around the well is implemented in order to capture the effects of condensate blockage and predict gas-condensate well deliverability. In addition, in the commencement of the reservoir lifetime, the numerical simulation predicts erroneous results due to high uncertainty in reservoir data. A simple analytical model can estimate gas-condensate well production reasonable with a limited number of input data. On the other hand, uncertainty studies require repetitious usage of the analytical calculation. Therefore, a faster and more practical method for calculation of gas-condensates well flow rate has a precious value. In this paper, a fast analytical method is introduced for prediction of gas and condensate production profiles based on the two-phase pseudo-pressure integral and material balance equation. Proposed analytical model in contrast to previous ones, predicts the exact plateau time. As the result, it does not need iteration in each reservoir pressure step to compute the relevant flowing bottom-hole pressure during the constant gas production period. The model is expanded to take into account the high-velocity phenomena in near the wellbore region. It is also extended for the first time for different well geometries including vertical, deviated, horizontal, and hydraulicfractured wells. Additionally, the analytical model is validated using fine grid numerical simulation for a wide range of rock and fluid properties. The developed analytical model can be used as a fast engineering tool for evaluating the uncertainty in well and reservoir data to choose the best strategy for reservoir development.
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Abstract
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Keywords:
Gas-Condensate, Well Production Prediction, Analytical Model, High-Velocity Effects, Deviated Well.
1. Introduction
As long as the flowing bottom-hole pressure is above the dew-point in gas-condensate well, the production performance is the same as a dry gas well. When the flowing bottom-hole pressure falls below the dew point, condensate begins to drop out near the wellbore. This phenomenon is called “condensate blockage” that severely reduces effective gas permeability and gascondensate well productivity. On the other hand, high velocity and low interfacial tension (high 1
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capillary number) around the wellbore results in an increase in gas relative permeability which is named “positive coupling”. Meanwhile, high gas velocity in near wellbore region results in a reduction in gas relative permeability which is named “negative inertia”. The mutual interaction of positive coupling and negative inertia determines the gas-condensate well performance. Therefore, the condensate blockage and high-velocity phenomena should be modeled accurately in order to determine the exact well deliverability in gas-condensate reservoirs. Numerical compositional reservoir simulation is used to capture the complex behavior of gascondensate fluid around the wellbore and to predict the well deliverability correctly. But, coarse grid simulation cannot investigate the condensate blockage phenomena. Instead, fine grid simulation, especially around the wellbore, should be used which is time-consuming. Also, numerical compositional simulation requires details of fluid and rock properties e.g. tuned equation of state for PVT and distributed rock properties for each grid. Since accurate data of reservoir fluid and rock properties are not available at the beginning of the reservoir lifetime, simple analytical calculations are used for fast forecast of the gas-condensate wells' performance (Ayyalasomayajula, 2005). Analytical model has the advantage of assuming a homogeneous reservoir. It requires tabulated PVT properties versus pressure that could be generated from CVD experiments or correlations. Fevang and Whitson (1996) developed a two-phase pseudo-pressure integral for calculating gascondensate well deliverability by modifying the concept presented by Evinger and Muskat (1942). They introduced the idea of the presence of three regions around the gas-condensate well. The existence of these regions depends on the magnitudes of the flowing bottom-hole pressure and the average reservoir pressure. At the first region, which is the nearest to the wellbore, both phases (gas and condensate) are flowing (Region I). Whereas, at the second region, both gas and condensate phases are present, but only gas phase is flowing (Region II). At the third region, a single gas phase is present (Region III). Their integral requires tabulated pressure/volume/temperature (PVT) properties for each reservoir pressure step, relative permeability, and the producing gas-oil ratio (GOR). But, the values of GOR should be determined through numerical simulation. In addition, their developed equation does not contain the effect of high-velocity flow which has a major influence on the gas-condensate well productivity. Mott (2003) and Xiao and Muraikhi (2004) proposed novel analytical models to calculate gascondensate well production performance, using a material balance model for reservoir depletion and Fevang and Whitson (1996)’s two-phase pseudo-pressure for well inflow performance. Their methods rely on the fact that the increase in condensate saturation during pressure depletion in Region II is used at an extension of Region I. None of the methods need the numerical simulation which results to compute GOR, anymore. In contrast, they require defining a specified reservoir pressure step to calculate pseudo-pressure integral. As the matter of fact, iteration in each reservoir pressure step should be done to compute the relevant flowing bottomhole pressure during the constant gas production period. Whatever the plateau time is longer, the number of iterations increased that results in a longer analytical model run time. If larger reservoir pressure step is selected, the number of iteration decreased but the plateau time is computed erroneously. This disadvantage made these analytical models impractical for uncertainty studies. Al-Shawaf et al. (2014) demonstrated a simple method to predict IPR curves for condensate wells as a function of reservoir pressure using rock relative permeability and constant composition expansion (CCE) experiment data. The future IPR curve is also interpolated based
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on the current IPR curve if CCE data are available. Hence, the production profile and plateau time predictions are not exact and straightforward. In contrast to previous models, which have just been developed for vertical wells, proposed analytical model in this study is extended for different well geometries including deviated one. Most of the wells are deviated in offshore platforms in the gas-condensate reservoirs due to lack of possibility of vertical well drilling. Since nearly 20 percent of the world’s gas fields are located offshore, analytical modeling of deviated well in gas condensate reservoirs for uncertainty study is crucial. There are many literature studies for predicting productivity index for single phase flow in deviated wells including Cinco et al. (1975), Besson (1990), and Rogers et al. (1996). Cinco et al. (1975) claimed that their semi-analytical model could compute geometric skin factor for deviated wells in single phase gas reservoirs both for isotropic and non-isotropic media. Rogers et al. (1996) suggested another model and compare calculated skin factor with Cinco et al. (1975). Their results matched for isotropic cases but had a difference for non-isotropic reservoirs. Kyoon Choi et al. (2008) conducted a sensitivity analysis on available analytical models developed for calculating the productivity of deviated wells. They claimed that these correlations are inappropriate for extremely deviated wells; namely with a deviation angle of more than 75 degree. By using a sensitivity study, Ghahri (2010) claimed that the skin values obtained from Rogers et al. (1996) were under estimated and significantly different from those predicted by equations proposed by Besson (1990) and Cinco et al. (1975). According to this result, we use Cinco et al. (1975) model for computing geometric skin factor. In the current study, a new and fast approach to well production prediction in gas-condensate reservoirs is developed. The fundamental of this study is similar to Mott (2003). However, it has been modified as follows: - As the plateau time is one of the key parameters in gas and condensate sales contract and in selecting the reservoir development strategy, in this study, a new technique is proposed to determine the exact plateau time for both gas and condensate. - A new analytical model algorithm is proposed to calculate gas and condensate production profiles. This approach doesn’t require defining a pressure step and performing iteration in each pressure step during plateau time period for determining the following bottomhole pressure. Instead, the end of the plateau time is computed directly. Therefore, developed analytical model has the advantage of less run time which makes it suitable for studying the effect of data uncertainty on gas-condensate well deliverability. It also helps to selecting the best strategy for well completion especially in newly discovered fields with limited data and high uncertainty. - The proposed analytical method is extended for deviated wells in contrast to previous models that are just developed for vertical wells. Also, the model is extended for horizontal, and hydraulic-fractured wells with new method, as drill of these wells in low permeability gas-condensate reservoirs are more common due to condensate build-up. The model is validated for slant wells with different deviation angle, partially penetration wells, non-isotropic reservoirs, and horizontal and hydraulic-fractured wells with different length. - The model is expanded to take into account the high-velocity effect in near wellbore region including positive coupling and negative inertia with applying Jamialahmadi et al.
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(2009) correlation. This correlation in contrast to previous ones does not require experimentally determined constants. Consequently, a spreadsheet is developed based on the proposed model. The spreadsheet is also extended with stochastic programming to take into account the data uncertainty in well and reservoir. Therefore, developed spreadsheet provides an applied tool in development strategy determination in newly discovered reservoirs. After that, it is used for selecting development strategy in one of the Iranian offshore low permeability gas-condensate fields that will be reported in near future. The organization of the current study is as follows. First, we describe well inflow calculation method of different well geometries and the extension that is made to include the high-velocity flow effect. In the next section, the computational algorithm is described. Afterward, we present the model validation. At the last section of this paper, the summery is presented.
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2. Analytical Model for Well Inflow Calculation
Developed analytical model consists of the two-phase pseudo-pressure integral for flow equation term and a material balance equation for depletion term.
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Tabulated PVT properties for different pressure steps, a set of relative permeabilities from SCAL data, and reservoir and well data are used as an input for the analytical model to calculate gascondensate well productivity. Developed model applies two-phase pseudo-pressure integral for well inflow performance, as shown by Fevang and Whitson (1996);
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(1)
In which C is the productivity index that depends on the well and reservoir geometry. For vertical well, C can be determined from Eq. (2). 2π × 0.001127 × Kh r ln( e ) − 0.75 + s rw
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Pres and Pwf are average reservoir pressure and flowing bottom-hole pressure, respectively. P1 is the pressure in the interface between Region I and Region II, and Pdew is dew point pressure. The first integral, with integral limits Pdew to Pres, relates to Region III, in which only the gas phase is present. The second integral, with integral limits P1 to Pdew, relates to Region II in which condensate drop-out, but its saturation is less than critical condensate saturation. Then, it is immobile and only the gas phase is flowing. The third integral, with integral limits Pwf to P1, relates to Region I, near the wellbore region, in which both gas and condensate phases are flowing. In this integral, µ gBg, µ cBc, and Rs are determined from PVT table. One of the parameters, Pwf or qg, is known based on the well production strategy; constant bottom-hole pressure or constant flow rate, respectively. Pres is also determined from material balance equation that will be discussed later. krg in Region III is the only function of Swi and it is determined from SCAL table. krg in second integral (Region II) is a function of condensate saturation, So. So is evaluated for each pressure in Region II by Eq. (3). Then, krg in Region II is also finds out from SCAL data. S = r ( p ) × (1 − S ) (3) o v wi
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k rg k + R s rc dP µ c Bc µ g B g
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Pdew P1 Pres k rg k rg qg = C ∫ dP + ∫ dP + ∫ P µ g Bg µ g Bg P1 Pwf dew
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15 16 17 18
19 20 21
,
1
−
RI PT
−
(5)
Where qo,max is the maximum condensate flow rate at the interface Region I and II, So,1 is the average condensate saturation in Region I, which is estimated by arithmetic average of computed saturation at this region interfaces as is shown in Eq. (6); ,
=
1 ! r ( p w ) × (1 − S ) " + ! r ( p1 ) × (1 − S ) "$ v wi v wi 2
,
=
1 ! r ( p1 ) × (1 − S ) " + 0$ v wi 2
(6)
So,2 is the average condensate saturation in Region II, which is calculated as is shown in Eq. (7) with calculating arithmetic average of interface saturations at Region II . (7)
PV is the pore volume in pressure contour map around a well that is dependent on the well geometry. For vertical well, it can be obtained from Eq. (8). ∆PV is also the increase pore volume of Region I growth size that can be calculated with knowing the previous PV of Region I. =π
ℎ 1 + '( + ' ∗
*
.∗
−
(,-
∅
(8)
Where r is the outer radius of Region I. In this study, r is estimated from Eq. (9) that is developed under the assumption of pseudo-steady state condition. Applying this equation implies acceptable error in P1 calculation and results in performing calculations faster. ln
22 23 24 25 26 27 28 29 30 31 32 33
,
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13 14
∆ =∆
=
− 1 × ln 23 − 1
,
− ln
AC C
10 11 12
1−
SC
−
,
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5 6 7 8 9
Where rv is vaporized condensate gas ratio. It is determined for each pressure by interpolating in tabulated PVT data. For calculating krg and krc in Region I, in the third part of the integral, combined flow in both condensate and gas phases should be considered as proposed by Fevang and Whitson (1996); krg 1 − rvf Rs µ g Bg = (4) krc rvf − rv µo Bo where rvf is the flowing condensate gas ratio of saturated gas at P1. After calculating the ratio krg/krc from Eq. (4), krg and krc are determined from relative permeability curves. These values should be updated for high velocity effect that is illustrated later. Finally, the only parameter that remains unknown in an evaluation of pseudo-pressure integral in Eq. (1) is P1. P1 is calculated according to Mott's concept as it is shown in Eq. (5).
EP
1 2 3 4
+ ln
1
(9)
Gas production profile is divided into two parts; constant gas production and decline part. In constant gas production part, qg is known and equal to qg,cons. Then, Pwf should be determined which is required a trial procedure. Also, Eq. (5) is highly non-linear and finding P1 needs trial procedure, too. Therefore, finding Pwf and P1 in constant gas production based on the previous algorithms with two nested iteration procedures are very time-consuming. This circumstance makes previous methods inappropriate for uncertainty analysis with stochastic methods especially in spreadsheet programming. In this paper, an algorithm is developed to bypass first part in gas and condensate production profiles by finding two times, t1 and t2. t1 and t2 are the end of the plateau time in condensate and gas production profiles as is shown in Fig. 1 (A) and (B), respectively. Production profiles in Fig. 1 comprise the Mott and our method schematically. The number of iteration in Mott method depends on the plateau time duration and the defined pressure step. Whatever the plateau time is 5
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longer, the number of iterations increased that results in a longer analytical model run time. If larger reservoir pressure step is selected, the number of iteration decreased but the plateau time is computed erroneously. But, in our approach, just the last time in plateau production period is calculated. This method made the computed plateau time independent of either plateau time duration or the defined pressure step. For these times, t1 and t2, relevant Pwf is calculated. Therefore, qg can be computed directly and production profiles can be predicted without any iteration in pressure steps. As it is shown in Fig. 1 (C), three different slops exist in the curve of Pwf verses time. First slop relates to single-phase gas production. The end of the first slope, t1, shows the time at which Pwf equals to Pdew. Then, condensate starts to drop-out and Region II starts to growth. The size of Region I also increases with Region II growth. At this time, the qc declines, as it is shown in Fig. 1 (A) and (C). Therefore, second slop relates to two-phase flow of gas and condensate. In this part of the curve, Pwf is still greater than minimum bottom-hole pressure, Pwf,min, and gas flow rate is equal to qg,cons. At time t2, Pwf become equal to Pwf,min and gas flow rate declines. According to this fact, new algorithm is developed to compute exact plateau time for gas and condensate and to predict gas and condensate production profiles without need to define pressure step and perform iteration in each pressure step. This algorithm is illustrated in the following section.
(B)
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(C)
20 21 22
Fig. 1. Schematic of production profiles in gas-condensate well, (A) Gas flow rate, (B) Condensate flow rate, (C) Flowing bottom-hole pressure
Depletion Term
6
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The purpose of this section is to write the material balance to obtain Pres for a specified time, t. At Pi the total initial volume, Vi, can be calculated by Eq. (10); *
10 11 12 13 14 15 16 17 18 19 20 21
(10)
6
,7( 8
= ×
RI PT
Where PVi is the initial pore volume that depends on the reservoir geometry. The flow rate is known from the flow equation described in the previous section. For the first point (end of the plateau time, t1 or t2), qg is equal to, qg,cons. Then cumulative gas production, Ng,prod can be calculated by Eq. (11); ,9 :-
(11)
Total condensate production, Nc,prod (the area under the condensate production profile) in this time interval is equal to; 69,7( 8 = 6 ,7( 8 × @ < (12) − 8
=6
,7( 8
×
@
+
×
2
,9 :-
@
+
@
1 < <
SC
69,7(
The material balance equation is written as Eq. (13); (13) * * − 6 ,7( 8 + 69,7( 8 = (, *:*: × (,= = , ℎ> 1 − * + '( + ' ∗ * .∗ (,- − * ? × (,For a known t, Nc,prod and Ng,prod can be computed. Then, by using Eq. (13), (,- can be obtained. High Velocity Effect in Near Wellbore Region
krg and krc, in Region I, are not only a function of saturation, but also a function of capillary number. Then, computed relative permeabilities from Eq. (4) are considered for the base condition, @( A and @(9A . After that, these parameters are correlated as a function of the capillary number via Jamialahmadi et al. (2009). This correlation selected because against previous correlations (i.e. Pope et al. (2000), Henderson et al. (2000), Whitson et al. (2003), etc.), it does not require experimentally determined constants. Eq. (14) computes gas relative permeability in miscible condition, @( , that is corrected BCDE for inertia effect, too. By using Eqs. (15) and (16), one can calculate the corrected immiscible gas relative permeability for negative inertia, @( F BCDE. @(
BCDE
` = @( 22 23 24 25
*
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9
× 1−
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7 8
*
=G
EP
3 4 5 6
=
HIJKI
LMNO P
@
Q S|U|VO RO
ab
c
AC C
1 2
+
F BCDE
9
=
@
ab
c
+
@
db
c9
k EYZE =
,
[\]^
[\]^ L [\_^
@( A βρijD k@( 1+ fY μY e
=f
W
A
(14)
(15)
|u| n
p
(16)
o
Where β is the single-phase non-Darcy coefficient and subscript m refers to the miscible condition. ρavg is the weighted average density value by fg (Eq. (15)) at the base conditions. uT is total velocity and it is calculated by dividing the computed qg by the flow cross section area, Acs, which depend on the well geometry. Acs for vertical well is calculated by Eq. (17); 7
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(17)
krg = Yg (krgI)iner + (1− Yg) (krgm)iner
(18)
Finally, corrected krg for high-velocity effect in Region I can be obtained by using weighting factor, Yg, according to Eq. (18); The Yg equation can be found elsewhere (Jamialahmadi et al. (2009)).
RI PT
1 2 3
q9- = 2πrℎ
4 5
Different Well Geometries
The proposed method for the vertical well is also extended for different well geometries including deviated, horizontal, and hydraulic-fractured wells. The idea is to calculate the geometric skin and relevant apparent wellbore radius, rwa. After that, rw can be replaced by rwa in Eq. (2). In addition, the change in the shape of the pressure transient lines around each well is involved in the computational algorithm. For horizontal well, rwa is computed by Eq. (19); (Diyashev and Economides, 2006)
12
:*
v + wv − x/2 { :* ℎ z+ |st x/2 x
:*
= ~
•€ ••
„.… „.…
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z
x 2
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v́ = b +
}
(19)
(20)
(21)
(23)
3 ∗ A ∗ 43560 ∗ h 4v́
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•
20 21 22
+1
(22) ,€ = √v́ ∗ b b is the minor axis of the drainage ellipse and is calculated with iterative use of Eqs. (23) and (24); b=~
17 18 19
:* ℎ
where KH and Kv are horizontal and vertical permeabilities respectively. a is the half of the major axis of the drainage ellipse around the horizontal well and is calculated by Eq. (21); ƒ x ,€ v = u0.5 + 0.25 + ! " 2 x/2
15 16
{ {
Where L is the horizontal well length and Iani is anisotropy index calculated by Eq. (20); {
13 14
= st u
,
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ln
SC
6 7 8 9 10 11
(24)
where A is the reservoir drainage area. It also should be noted that in the horizontal well, the shape of the pressure transient lines is changed to elliptical (Ghahri (2010)). Then, for calculating PV and Acs, Eq. (25) and Eq. (26) are used instead of the Eq. (8) and Eq. (17) respectively. 4 = π v × b 1 + '( + ' ∗ 3 q9- = 4= v × b
*
.∗
−
*
∅
(25) (26)
Where P is the pressure of any contour-line around the wellbore. Effective wellbore radius for the hydraulic-fractured well is calculated according to Eq. (27); (Prats et al. (1962))
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=
6 7 8 9
= π v × b ℎ 1 + '( + ' ∗ q9- = = v + b ℎ
*
.∗
−
*
∅
RI PT
where Lf is the fracture length, xf is fracture half-length and xe is half of the reservoir length. As the shape of the pressure transient lines around the hydraulic-fractured well are changed to cylinder with ellipse cross-section (Ghahri (2010)), PV and Acs are calculated by, (28) (29) where a and b are the half of the major and minor axis of the drainage ellipse around the hydraulic fracture respectively, that can be calculated by; xŽ v=b+ 2 A ∗ 43560 b= =v
SC
4 5
(27)
(30) (31)
Geometric skin factor, ’ , for deviated well with deviation angle “ are calculated by Eq. (32); (Cinco et al. (1975)). But, it should be noted that Cinco correlation predict acceptable results for wells with deviation angle less than 75 degree (Ghahri, P., and M. Jamiolahmady (2012) and Suk Kyoon et al (2008)). Apparent wellbore radius to replace in Eq. (2) is computed by Eq. (34). (32) “ .„” “ .•”… ℎ– ’
= −! " 41
ℎ @€ ℎ– = ! " ~ @•
−! " 56
× s•a ! " 100
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xŽ •Ž `• ! " ≤ 0.3 4 •,
(33)
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Developing a fast analytical model for computing the gas and condensate production profiles are the aim of this study. Most of the run-time in previous analytical models is spend in plateau time interval for computing the proper flowing bottom-hole pressure, Pwf, for a known average reservoir pressure, Pres. This disadvantage made them impractical for uncertainty analysis with stochastic methods that requires thousands run of a program. In this study, a new algorithm is introduced to bypass the constant gas production interval by calculating just the end of the plateau time. Based on our algorithm, the production profile is divided into two sections. The first section, is where the flow rate is constant and equal to qg,cons. The second section, where the Pwf is constant and equal to minimum flowing bottom-hole pressure, Pwf,min. In the first section, the shape of the gas production profile is uniform. Therefore, flow rate is known (qg,cons) but, the end of the plateau time and relavant Pres is unknown. Flowing bottomhole pressure at the end of the plateau time is also known and equal to Pwf,min. The strategy for this section is to find the end of the plateau time (t1 and t2 for condensate and gas production profiles respectively) and reduce the spend time in each reservoir pressure step to find relevant Pwf. To determine end of the plateau time following algorithm should be tracked.
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10 11 12
(34) = 2 —˜™ For deviated well, the shape of the pressure transient lines is changed to elliptical that is similar to horizontal well (Ghahri (2010)). Thus, for calculating PV and Acs, Eq. (25) and Eq. (26) could be used, respectively.
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3. Computational Algorithm
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(,-
4 5 6 7
+
Ž, *:
−
Ž,*:
(35)
k (P + P ) k k rg k rg rg rg res dew − + − µ B 2 µ B µ B µ B g g p res g g p dew g g Pdew g g =C k rg k rg k k (P1 − Pwf ) + R s rc − + R s rc + µ c B c P µ g B g µ c B c P 2 µ g B g 1 wf
(Pdew + P1 ) 2 P1
(36)
D. The trial values of Pres and P1 are replaced in Eq. (37) to obtain a trial value of plateau time interval, t2. min
(,- , 8,
−
1−
=∆
,
−
,
1
−
(37)
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Eq. (37) is written based on the Eq. (5). It reflects this fact that the amount of the condensate dropped in Region II from the beginning of the Region II growth to the end of the plateau time (left side of the Eq. (37)) is equal to the amount of condensate increase the Region I size in that specified time step (right side of the Eq. (37)); E. t2 is used in Eqs. (10) to (13) to compute the relevant Pres. F. Computed Pres in step E is compared with previous calculated Pres. The iteration is continued from step C until convergence in Pres is reached. Because of pseudo-steady state assumption, the pressure declines slowly and linearly, thus the convergence never failed. As the first guess comes from the dry gas equation, the convergence is fast. Normally, after three or four iteration Pres is obtained with accuracy around 0.1 psi. When the Region III is vanishes (Pres become less than Pdew and just Region I and Region II exist), there would be no fresh gas to support constant condensate drop in Region II. Therefore, the amount of condensate dropout in Region II is decreased with time and the pseudo-steady state assumption doesn’t valid anymore. In that condition the model’s accuracy decreased. But, it stills gives a good estimate of plateau time with precision around 10 psi.
AC C
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
*
C. At the end of the gas production plateau time (t2), the flow rate and Pwf is known and equal to qg,cons and Pwf,min respectively. Then, Eq. (1) is written for the trial value of Pres obtained in the previous step to compute the trial value of P1. This is done by extension of Eq. (1) with trapezoidal method, as is shown in bellow equation;
q g ,cons
8 9
=
RI PT
3
SC
2
A. Using dry gas equation with Pi to obtain the first guess for bottom-hole pressure, Pwf,int. š› B. Assume pseudo-steady state condition in the reservoir, where šœ = •œž, and determine the first trial value for Pres by Eq. (35).
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1
To plot the condensate production profile, a similar procedure should be followed to obtain the condensate plateau time. The point is P1=Pdew at the end of the condensate plateau time. After that, Region II starts to improve and P1 becomes less than Pdew. Then, some of the condensate remains in the reservoir that causes a decrease in condensate flow rate. During condensate plateau time, gas flow rate is equal to qg,cons. Hence, to predict condensate production profile, t1 and Pres as unknown parameters should be determined. Suggested algorithm in this paper, described below; A. Assume Pres > Pdew .Solve Eq. (37) with assuming P1=Pdew and replacing t1 with t2 to obtain t1. B. Computed t1 is used in material balance equation, Eq. (13), to obtain the relevant Pres. 10
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9
=
×
(38)
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By following aforementioned algorithm and without any iteration in each reservoir pressure step in the first section of the production profile, an exact amount of gas and condensate plateau times is determined. While, previous methods use two nested iteration for finding P1 and Pwf in each reservoir pressure step in plateau period. This approach results in a faster prediction of gas and condensate production profiles. In the second section, Pwf is equal to Pwf,min. then straight forward procedure of using Eq. (5) to calculate P1 and Eq. (1) to compute relevant flow rate should be followed. The flowchart algorithm is shown in Fig. 2.
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C. If computed Pres is greater than Pdew, the calculation is stopped. Otherwise, it shows that the Region III does not exist. Then, Eq. (37) is solved with the assumption of Pres < Pdew to obtain t1. D. Condensate flow rate in plateau time can be calculated by Eq. (38)
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11
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1 2
Fig. 2. Flowchart algorithm for computing gas and condensate production profiles
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4. Analytical Model Validation and Discussion For validation of the developed model, the results are compared with a commercially available numerical compositional simulator, Eclipse 300 [Schlumberger-GeoQuest, 2010]. Various reservoir fluids (lean, intermediate and rich gas-condensate) and rock properties (different SCAL) in different well geometries (vertical, horizontal, hydraulic-fractured, and deviated wells with different deviation angles) with various production scenarios (different plateau rate and minimum bottom-hole pressures) are selected for model validation. Evaluation of horizontal and hydraulic-fractured wells are also repeated for several values of L and Lf . A three dimensional Cartesian reservoir model with a single layer and uniform porosity and permeability is considered as a base case. It is assumed that a fully-penetrated single producer well located at the center of the reservoir. Fig. 3 (A) shows the schematic of the reservoir with horizontal well model as an example. Local grid refinement (LGR) is made in near wellbore region in order to model condensate blockage more accurately in numerical simulation. Numerical simulation is repeated with refining the grid size to decrease the numerical error due to grid dimensions. Optimum grid size is selected such that the change in numerical simulation results is minimized. For deviated wells, the grid size is selected in a way that the well passed from the grid center as is shown in Fig. 3 (B). Assumed well and reservoir properties for numerical simulation are summarized in Table 1.
20
(A) (B) Fig. 3. Schematic of the reservoir and well model, (A) Horizontal, (B) Deviated well
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Table 1. Well and reservoir properties that are selected for validation Unit Parameter Amount ∅
10
%
K
50
mD
Reservoir Dimension
6000 ×6000
ft
rw
'(
0.58
ft
'
4.25
(Psi-1) E-6
1.05
(Psi-1) E-6
Swi
10
%
h
200
ft
L
3200
ft
Lf
400
ft
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21
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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The velocity dependent relative permeability keyword (Whitson Correlation with default parameters) and Forchheimer effect is also activated in numerical simulation for considering the effect of the capillary number and high velocity flow on relative permeability. Three multicomponent mixtures of gas condensate fluids with different condensate to gas ratio (CGR) are modeled. The key reservoir fluid properties are given in Table 2.
Rich Intermediate Lean
5480 3129 3535
38 21 3.1
Pi (Psia)
200 200 212
6000 3500 5500
SC
166 71 32.7
T (oF)
1−
−
−
−
−
*:
*:
*:
−
*:
:]
* :¡
(39)
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•( = •(
(40)
Residual saturation (Somax, Somin, Sgmax, Sgmin) and Corey power (no , ng), kromax, krgmax and Swi for the construction of different relative permeability curves are reported in Table 3. SCAL 1 and SCAL 2 have different minimum condensate saturation. While, SCAL 1 and SCAL 3 differ in Corey power that results in different pore size distribution i.e. SCAL 3 has narrower pore size distribution than SCAL 1. Fig. 4 also shows the designed relative permeability curves that are used for comparison. Table 3. Residual saturation and Corey power for construction of gas and condensate relative permeability
Swi krgmax ng
0.1
0.6
EP
Rock Type SCAL 1 SCAL 2 SCAL 3
1.5 1.5 3
Sgmin
0.3
Sgmax kromax no 0.9
0.3
1.5 1.5 3
Somin Somax 0.15 0.35 0.35
0.6
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27 28
Pdew CGR Max Condensate (Psia) (bbl/Mscf) Drop out
Different Condensate and gas relative permeabilities are constructed using Eq. (39) and Eq. (40) (Brooks and Corey (1964)). •( = •(
17 18 19 20 21 22 23 24 25 26
Fluid
RI PT
Table 2. Fluid properties that are selected for comparison
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29 30
Fig. 4. Relative permeability curves used for model validation
31 32
Various production scenarios e.g. different plateau flow rate and minimum bottom-hole pressure, are also specified for model validation. After that, Gas and condensate production profiles 14
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computed by the analytical method are compared with the results of the numerical reservoir simulation. Totally, more than two hundred cases are designed for analytical model validation with numerical simulation. However, the average error for each well-geometry is reported, in Table 4.
Gas Production Profile
Average Error (%)
RI PT
Table 4. Computed error for the analytical results in comparison with the numerical reservoir simulation for different well geometries
Condensate Production Profile
Plateau Time
Decline section
Plateau Time
Decline section
1.84 2.23 1.46 4.1
2.03 3.7 2.1 5.3
2.05 2.84 1.27 2.53
2.2 3.6 2.67 3.23
Vertical Horizontal Hydraulic Fractured Deviated Well
SC
1 2 3 4 5 6 7
Based on the results of Table 4, the analytical model could predict numerical reservoir simulation with an average error less than 5%. In addition, average run time is less than a minute for developed analytical model, while the run time of each numerical simulation is approximately around ten minutes. This result indicates that developed model is a good engineering tool for gas and condensate production prediction. At the end, one case from predicted production profiles for each well type, with maximum observed average error, is reported here.
17 18 19 20 21
4.1 Vertical Well Gas and condensate production profiles and flowing bottom-hole pressure plot for intermediate fluid and SCAL3 in the vertical well are shown in Fig. 5. Furthermore, the results of the developed model are compared with Mott’s method.
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(A)
(B)
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(C) Fig. 5. Production profile for rich fluid and SCAL3 in the vertical well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
3 4
The average calculated error in plateau time and observed mean run time of our approach is also compared with Mott’s method in Table 5.
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Table 5. Comparison the results of our model with Mott’s method
Method
Average calculated error in plateau time (%)
Developed in this study Mott (2003)
1.84 7.4
6
It should be noted that the average calculated error in plateau time and observed mean run time in Mott’s method completely depend on the amount of the pressure step and the duration of the plateau time. The more the plateau time duration and pressure step is selected, the more the average computed error is observed. Anyway, our approach bypasses the plateau time period and just calculates the end of the plateau time. Because of that, it has the advantage of less run-time and exact prediction of the plateau time.
TE D
7 8 9 10 11 12
Average run-time (sec) 35 150
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4.2 Horizontal Well
15 16
Flowing bottom-hole pressure, gas and condensate production profiles for lean fluid and SCAL1 as the case with maximum observed error in horizontal well cases are shown in Fig. 6.
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(C) Fig. 6. Production production profile for lean fluid and SCAL1 in the horizontal well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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1 2
(B)
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(A)
3
4.3 Hydraulic-Fractured Well
5 6
Flowing bottom-hole pressure, gas and condensate production profiles for intermediate fluid and SCAL2 in the hydraulic-fractured well are shown in Fig. 7.
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(C) Fig. 7. Production production profile for intermediate fluid and SCAL2 in the hydraulic-fractured well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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4.4 Slant Well
2 3 4
Three cases for deviated well with deviation angle 15, 45, and 75 with maximum observed error are reported. Intermediate fluid and SCAL1 in the 15 degree slant well has the maximum error. Flowing bottom-hole pressure, gas and condensate production profiles are shown in Fig. 8.
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(C) Fig. 8. Gas production profile for intermediate fluid and SCAL1 in the 15 degree deviated well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
Lean fluid and SCAL2 in the 45 degree slant well has the maximum error. Flowing bottom-hole pressure, gas and condensate production profiles are shown in Fig. 9.
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6 7
(B)
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(A)
(A)
(B)
18
(C) Fig. 9. Gas production profile for lean fluid and SCAL2 in the 45 degree deviated well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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3
Rich fluid and SCAL3 in the 75 degree slant well has the maximum error. Flowing bottom-hole pressure, gas and condensate production profiles are shown in Fig. 10.
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(C) Fig. 10. Gas production profile for rich fluid and SCAL3 in the 75 degree deviated well, (A) Gas production profile, (B) Condensate production profile, (c) flowing bottom-hole pressure plot
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A fast analytical model is developed for prediction of gas and condensate production profiles based on the two-phase pressure integral and material balance equation. The model is extended for high-velocity flow phenomena and for different well geometries. The developed model does not need any iteration in each reservoir pressure step and just the end of the plateau time is computed directly and accurately. The model developed under the pseudo-steady state assumption. When the average reservoir pressure become less than dew point pressure, the assumption doesn’t valid anymore and the model’s accuracy decreased. But, it stills gives a good estimate of plateau time. This approach in conjugation with a stochastic programming provides an appropriate engineering tool for uncertainty studies and decision making for choosing the best reservoir development strategy. It also computes the exact gas and condensate plateau time interval which is a key parameter in gas and condensate sales contract.
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2 3 4 5 6 7 8 9 10 11 12
5. Conclusion
SC
1
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1. Al-Shawaf, A., Kelkar, M., & Sharifi, M. (2014). A new method to predict the performance of gas-condensate reservoirs. SPE Reservoir Evaluation & Engineering, 17(02), 177-189. 2. Ayyalasomayajula, P. S., Silpngarmlers, N., & Kamath, J. (2005, January). Well deliverability predictions for a low permeability gas condensate reservoir. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. 3. Besson, J. (1990). Performance of slanted and horizontal wells on an anisotropic medium. In European Petroleum Conference. Society of Petroleum Engineers. 4. Cinco, H., F. G. Miller, and H. J. Ramey Jr. (1975). Unsteady-state pressure distribution created by a directionally drilled well. Journal of Petroleum Technology 27 (11), 1-392. 5. Diyashev, I. R., & Economides, M. J. (2006). The dimensionless productivity index as a general approach to well evaluation. SPE Production & Operations, 21(03), 394-401. 6. Evinger, H. H., & Muskat, M. (1942). Calculation of theoretical productivity factor. Transactions of the AIME, 146(01), 126-139. 7. Fevang, Ø. & Whitson, C. H. (1996). Modeling gas-condensate well deliverability. SPE Reservoir Engineering, 11(04), 221-230. 8. Ghahri, P. (2010). Modelling of Gas-condensate flow around horizontal and deviated wells and cleanup efficiency of hydraulically fractured wells (Doctoral dissertation, Heriot-Watt University). 9. Ghahri, P., and M. Jamiolahmady. (2012). A new, accurate and simple model for calculation of productivity of deviated and highly deviated well–Part I: Single-phase incompressible and compressible fluid." Fuel 97, 24-37. 10. Jamiolahmady, M., Sohrabi, M., Ireland, S., & Ghahri, P. (2009). A generalized correlation for predicting gas–condensate relative permeability at near wellbore conditions. Journal of Petroleum Science and Engineering, 66(3), 98-110. 11. Mott, R. (2003). Engineering Calculations of Gas-Condensate-Well Productivity. SPE Reservoir Evaluation & Engineering, 6(05), 298-306. 12. Pope, G. A., Wu, W., Narayanaswamy, G., Delshad, M., Sharma, M. M., & Wang, P. (2000). Modeling relative permeability effects in gas-condensate reservoirs with a new trapping model. SPE Reservoir Evaluation & Engineering, 3(02), 171-178. 13. Prats, M., Hazebroek, P., & Strickler, W. R. (1962). Effect of vertical fractures on reservoir behavior--compressible-fluid case. Society of Petroleum Engineers Journal, 2(02), 87-94. 14. Rogers, E. J., & Economides, M. J. (1996). The skin due to slant of deviated wells in permeability-anisotropic reservoirs. In International Conference on Horizontal Well Technology. Society of Petroleum Engineers. 15. Suk Kyoon, Choi, Liang-Biao Ouyang, and Wann-Sheng Huang. (2008). A comprehensive comparative study on analytical PI/IPR correlations. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. 16. Whitson, C. H., Fevang, Ø., & Sævareid, A. (2003). Gas condensate relative permeability for well calculations. Transport in porous media, 52(2), 279-311. 17. Xiao, J. J., & Al-Muraikhi, A. J. (2004, January). A new method for the determination of gas condensate well production performance. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers.
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References
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Nomenclature Half of the major axis of the drainage ellipse around the horizontal and hydraulic-fractured well Reservoir drainage area Flow cross section area Minor axis of the drainage ellipse around the horizontal and hydraulic-fractured wells Gas formation volume factor
Bc C '( ' h Iani K
Condensate formation volume factor Productivity index Rock compressibility factor Water compressibility factor Reservoir thickness Anisotropy index Permeability
krc
Condensate relative permeability
k rg
Gas relative permeability
KH Kv @(9A @( A @( F
Horizontal permeability Vertical permeability Measured condensate relative permeability in the base condition Measured gas relative permeability in the base condition Corrected immiscible gas relative permeability for negative inertia
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Gas relative permeability in miscible condition that is corrected for inertia effect too Condensate relative permeability at residual gas saturation Gas relative permeability at initial gas saturation Horizontal well length Fracture length in hydraulic-fractured well Condensate Corey power Gas Corey power Cumulative condensate production Cumulative gas production Pressure in the interface between Region I and Region II Dew point pressure Initial reservoir pressure Average reservoir pressure Flowing bottom-hole pressure Flowing bottom-hole pressure at Pi Minimum flowing bottom-hole pressure Pore volume in pressure contour map around a well that is dependent on the well geometry Initial pore volume that depends on the reservoir geometry pore volume growth of Region I Gas flow rate Gas plateau flow rate Condensate flow rate Outer radius of Region I Reservoir boundary radius Vaporized condensate gas ratio Wellbore radius
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kromax krgmax L Lf nc ng Nc,prod Ng,prod P1 Pdew Pi Pres Pwf Pwf,int Pwf,min PV PVi ∆PV qg qgcons qc r re rv rw
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uT Vi xe xf Yg β µ c
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Single-phase non-Darcy coefficient Condensate viscosity Gas viscosity
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Weighted average density Porosity
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Sc Sc,1 Sc,2 Swi Scmax Scmin Sgmax Sgmin ∆t
Apparent wellbore radius Solution Gas oil ratio Condensate saturation Average condensate saturation in Region I Average condensate saturation in Region II Initial water saturation Maximum condensate saturation Minimum condensate saturation Maximum gas saturation Minimum gas saturation Specified time interval Plateau time interval in condensate production profile Plateau time interval in gas production profile Total velocity Total initial volume Half of the reservoir length Fracture half-length Weighting factor for calculating velocity dependent relative permeability
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A fast analytical method is introduced for prediction of gas and condensate production profiles. Proposed model compute exact plateau time without any iteration in each reservoir pressure step during constant gas production period. This approach provides a practical tool for uncertainty study. The model is expanded to take into account the high-velocity phenomena in the near wellbore region. The proposed analytical method is extended for different well geometries including vertical, horizontal, hydraulic-fractured, and deviated wells.
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