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New technique for determining rate dependent skin factor & non-Darcy coefficient
Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
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New technique for determining rate dependent skin factor & nonDarcy coefficient Salam Al-Rbeawi Middle East Technical University, North Cyprus Campus, Mersin 10, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 June 2016 Received in revised form 8 September 2016 Accepted 11 September 2016 Available online 13 September 2016
A non-Darcy flow develops in porous media when the velocity of reservoir fluids becomes extremely high because of the continuous narrowing of the cross section area of the flow and the convergence of flow streamlines. Therefore, the inertial effect increases significantly and the total pressure drop, required by fluids to move from the outer drainage area towards the wellbore increases significantly due to the extra pressure drop caused by the non-Darcy flow. The extra pressure drop is described by the Forchheimer equation in which the deviation from Darcy's law is proportional to the inertial factor which in turn is a function of porous media characteristics such as permeability and porosity. This paper introduces a new technique for estimating rate dependent skin factors (DQsc) and nonDarcy flow coefficients (D). The new technique uses dimensionless pressure for steady state flow to determine these two parameters. A set of plots has been developed for the term (DQsc) based on reservoir configurations i.e. reservoir boundaries (2xe & 2ye), wellbore length, and anisotropy. All the plots have been developed based on the fact that the rate dependent skin factor represents the difference between the total pressure drop and the pressure drop caused by Darcy flow only when non-Darcy coefficient equals to zero. Dimensionless pressure drops for horizontal wells producing from finite acting reservoirs have been calculated for steady state flow when the impact of dimensionless time can be eliminated. The calculated pressure drops have been used in reservoir performance models to estimate rate dependent skin factors and non-Darcy coefficients. A point of special interest in this study is the ability to estimate these two parameters without the need for experimental work or the need to use currently proposed empirical models for calculating them. This new technique requires knowing reservoir configurations (reservoir width and length), the length of the wellbore, and the height of the formation in addition to the anisotropy. The study indicates that rate dependent skin factors and non-Darcy coefficients have the minimum values when the reservoirs having square-shape drainage area. It has also been found that both parameters have great impact on wells with short wellbore length. Additionally, isotropic formations and symmetrical wells perform better than anisotropic formations and asymmetrical wells in terms of extra pressure drop caused by non-Darcy flow. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Great attention has been given to non-Darcy flow beginning in the early 1900s when Forchheimer pointed out for the first time that Darcy's law is not applicable for high velocity gas flow in porous media without considering the inertial impact on pressure drop. For this reason, he stated that Darcy's law could be written in the following form (Forcheimer, 1901):
hm i dP ¼ v þ rbv2 dr k
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jngse.2016.09.028 1875-5100/© 2016 Elsevier B.V. All rights reserved.
(1)
where (b) is the inertial factor which has numerous empirical models tabulated in the literature (Economides et al., 2013). The pressure drop in gas reservoirs, caused by fluid flow, may have two trends: the first one is caused by Darcy flow when the velocity of reservoir fluid is low and the cross section area of flow is large enough for Darcy flow (Laminar flow approach) to be the dominant flow regime. The second is caused by non-Darcy flow when the cross section area of flow decreases gradually and flow streamlines come close together in such a way that the velocity increases to the turbulent condition limit. In this case, excessive pressure might be needed to overcome the inertial forces resulted from the convergence of flow streamlines. For horizontal wells, the extra pressure drop caused by non-Darcy flow, sometimes called choked flow, characterizes the flow regimes that typically result
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Fig. 1. Horizontal wellbore in bounded formation.
from change in the cross section area of flow and converging flow streamlines. This could happen for both early radial flow regimes observed during early production when reservoir fluids move radially in the vertical plane towards the wellbore or pseudo-radial flow regime, observed during late time production when reservoir fluids move radially in the horizontal plane towards the wellbore. However, it could happen also for linear flow regimes if the cross section area of flow is not large enough and therefore the developed velocity is so high that Darcy's law is no longer be applicable. The non-Darcy flow accompanies pseudo-radial flow is characterized and controlled by reservoir boundaries and the horizontal permeability while the non-Darcy flow accompanies early radial flow is characterized and controlled by formation thickness and vertical permeability. Spivey et al. (2004) stated that more than 90% of the pressure drop within a distance of (0.5 ft) from wellbore could be caused by the non-Darcy flow developed by completion technique such as gravel-packed for vertical wells and slotted liner for horizontal wells. The non-Darcy flow coefficient for pseudoradial flow when reservoir fluid flows in the outer drainage area and radially converges to the wellbore, is given by (Zeng and Zhao, 2008):
D ¼ 2:22*1015
bgg k 1 1 hm rw re
(2)
while the non-Darcy flow coefficient for early-radial flow near the wellbore, is given by:
D ¼ 2:22*1015
ðbs bÞgg k 1 1 rw rs hm
(3)
where: (b) is the inertial factor at the outer drainage area of the reservoir and (bs) is the inertial factor for the zone close to the wellbore or the damaged zone extended from the wellbore to radius of (rs). To characterize non-Darcy flow, the rate dependent skin factor (DQsc) should be determined. This term represents the dimensionless pressure drop and depends on the non-Darcy flow coefficient (D). Several models have been presented in the literature to estimate this coefficient. One of them was given by Ramey (1965):
D ¼ 2:22*1015
bgg k hmrw
(4)
It can be seen that the non-Darcy coefficient is a function of the petrophysical properties of the formation, such as permeability and porosity, inertial factor (b) which in turn is a function of permeability and porosity as well as reservoir fluid properties such as
viscosity and specific gravity, in addition to the formation thickness and wellbore radius. A method of estimating the inertial factor has long been known from experimental studies (Barree and Conway, 2004, 2009). Based on these studies, several mathematical models have been proposed for this factor (Economides et al., 2013). Many researchers have suggested using gas well tests such as flow after flow tests, isochronal, and modified isochronal tests to predict the value of non-Darcy coefficient (Ramey, 1965), (Kelker, 2000), and (Brar and Aziz, 1978). Experimental studies have also been conducted to estimate non-Darcy coefficients especially for the pressure drop across the completion system (Nguyen, 1986). Pressure drawdown and build up tests have also been used to determine the rate dependent skin factor (Zeng and Zhao, 2008). Spivey et al., 2004, suggested using sand face flow rate instead of surface gas flow rate for buildup test analysis to eliminate wellbore storage effects, while Kim and Kang, 1994, introduced a semianalytical model for determining the non-Darcy flow coefficient from single rate gas well pressure transient tests. In this technique, pressure profiles and flow regimes could be derived from pressure and pressure derivative curves. Camacho-V et al., 1996, stated that pressure transient analysis could be used to characterize the nonDarcy coefficient and described it as the Reynold number. Therefore, well test analysis can be an excellent tool for characterizing these two parameters. However, the existence of different flow regimes, especially for horizontal wells, could cause some difficulties for estimating non-Darcy coefficient from well test analysis. In this paper, the main focus will be on estimating rate dependent skin factors and non-Darcy flow coefficients using steady state pressure profiles and flow regimes. According to Matthews 1986 and Babu and Odeh 1989, the total pressure drop resulting from the depletion process of a horizontal well producing from finite acting reservoir could be separated into two terms. The first one represents the pseudo-steady state pressure drop that is a function of time and drainage area wherein reservoir pressure declines from its initial to average value. The second is the pressure drop at steady state when time does not have a significant impact on the pressure profile at any point in the reservoir. Therefore, dimensionless pressure can be simulated by several models and calculated for different reservoir configurations, horizontal wellbore lengths and anisotropy. Using the calculated dimensionless pressure, the rate dependent skin factor and non-Darcy flow coefficient can be determined. 1.1. Mathematical models for pressure profile Consider a rectangular-shape formation with two side boundaries having the dimensions (2xe & 2ye) and height of (h) as shown
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in Fig. 1. The formation can be considered either a conventional or an unconventional gas reservoir in case of pseudo-steady state flow has already been reached. It is depleted by a horizontal wellbore of length (2Lw). The formation is homogenous and anisotropic where permeability in the vertical direction (kV) is not the same as the permeability in the horizontal direction (kH). The horizontal wellbore could be totally extended in the drainage area or it could be partially extended in the X-direction. Formation porosity ð∅Þ is considered constant. The pressure drop at the wellbore of the above mentioned horizontal well can be determined by the solution of the diffusivity equation either by the instantaneous line source solution presented by different researches (Gringarten and Ramey, 1973; Daviau et al., 1988) or by Laplace transformation presented by Ozkan 1988. Mathematically, dimensionless pressure drop for horizontal wells extending in bounded reservoirs can be obtained applying line source solution for diffusivity equation. This solution is well documented in the literature such as the one introduced by Ozkan 1988 using Laplace domain, which is:
PPD ¼ PPD1 þ PPD2 þ PPD3 þ PPD4
(5)
These first two pressure items represent the solutions for fully penetrating vertical fractures while the second two pressure terms represent the solutions satisfy the boundary conditions in the diffusivity equation. More details for these four models are given by Eqs. (A-3), (A-4), (A-5), and (A-6) in the Appendix-A. To consider wellbore storage effect, skin factor, and non-Darcy flow pressure drop, dimensionless wellbore pressure can be written as follows:
PPwD ¼ L 1 PPwD þ PPND 0
1
SPPD þ s B C ¼ L 1 @ A þ PPND S þ CD S2 SPPD þ s
(6)
where (PPND) is the pressure drop caused by non-Darcy flow and defined by:
PPND ¼ DQsc
(7)
The non-Darcy flow coefficient in this model is represented by:
D¼C
bgg kH hmLw
(8)
Therefore. Eq. (7) becomes:
PPND ¼
IND LD Iani
(9)
where: (IND) is the inertial number, representing the ratio of the inertial force to the viscous force, and mathematically formulated as:
IND ¼
C rg kH Qsc
mL2w
(10)
(LD) is the dimensionless wellbore length and defined as:
sffiffiffiffiffiffi Lw kV LD ¼ h kH And the anisotropy (Iani) is defined as:
Iani
sffiffiffiffiffiffi kV ¼ kH
(12)
pffiffiffiffiffiffiffiffiffiffi where (kV ¼ kz) is the vertical permeability and ðkH ¼ kx ky Þ is the horizontal permeability. For late time production, the pressure drop caused by Darcy flow only, given by Eq. (5), consists of two parts. The first part is the pseudo-steady state behavior when the reservoir pressure declines with time by a constant rate. The second part is the pressure drop caused by fluid flow in the steady state period when pressure decline is no longer affected by the production time and reservoir boundaries have the greatest impact on pressure drop only (Matthews, 1986; Babu and Odeh, 1989). Therefore, Eq. (6) can be written as:
PPwD ¼ PPwDP þ PPwDS þ PPND
(13)
where the term (PPwDP) is pressure drop for pseudo-steady state caused by Darcy flow and given by:
PPwDP ¼ 2ptDA
(14)
and the term (PPwDS) is the pressure drop for steady state caused by Darcy flow and given by:
p PPwDS ¼ xeD yeD PPDx þ PPDy þ PPDz þ PPDxy þ PPDxz þ PPDyz 2 þ PPDxyz (15) The pressure terms in the right hand side of the above model are defined in the Appendix-A. The steady state pressure drop caused by Darcy flow is characterized by a straight line of unit slope for both dimensionless pressure and dimensional pressure derivative curves. Typical pressure behaviors and flow regimes for horizontal wells in finite acting reservoirs having small and large drainage areas, are shown in Figs. 2 and 3 respectively. Obviously a horizontal wellbore, acting in large drainage area (LD ¼ 10 and xeD ¼ yeD ¼ 0.1), responses to the depletion process differently over time than do horizontal wellbores acting in small drainage area (LD ¼ 10 and xeD ¼ yeD ¼ 0.5). The difference between the two cases could be seen in both the pressure profile and flow regimes. The pseudoradial flow regime is developed for large drainage areas but it is not observed for small drainage areas as there is enough time for reservoir fluids to move radially in the horizontal plane towards the wellbore in the first case. The radial movement of reservoir fluids means that the cross section area of flow decreases as the fluids approach the wellbore. This could lead to the convergence of flow streamlines and cause more lost energy due to the friction force or inertial impact. However, it does not mean there is no lost energy if there is no pseudo-radial flow. In general, the lost energy could potentially be observed for linear flow also when the cross section area of flow is not enough to demonstrate Darcy flow. The amount of the lost energy depends on the flow rate of reservoir fluids and the shrinkage in the cross section area of flow in addition to other rock and fluid properties. 1.2. Mathematical models for non-Darcy flow
(11)
Several models have been presented in the literature for steady state performance of reservoirs depleted by horizontal wells. One of these models was introduced by Joshi 1988 and developed later by several researchers (Frick and Economides, 1993) to include non-
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Fig. 2. Wellbore storage and skin effect for horizontal well acting in finite reservoir, small drainage area (xeD ¼ yeD ¼ 0.5).
Fig. 3. Wellbore storage and skin effect for horizontal well acting in finite reservoir, large drainage area (xeD ¼ yeD ¼ 0.1).
Darcy flow coefficient for gas reservoir performance. The most commonly used model is (Joshi, 1988):
Qsc
kH h PPe PPwf ¼ 3 1 2 0 pffiffiffiffiffiffiffiffiffiffiffi
aþ a2 L2w A Iani h Iani h 4 @ þ 2Lw ln rw ð1þIani Þ þ s þ DQsc 5 1422 T ln Lw (16) In dimensionless form, Eq. (12) can be written as:
PPwD
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 Iani þs ¼ ln aD þ a2D 1 þ ani ln 2LD rwD ðIani þ 1Þ þ DQsc
the change in reservoir fluid properties such as compressibility and viscosity with production time. The pressure drop caused by the depletion process of gas reservoirs, given by Eq. (17), can be classified into two types: The first one represents the pressure drop due to Darcy flow when reservoir fluids move from the outer drainage area towards the horizontal wellbore. This one can be mathematically formulated as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 Iani þs PPwDL ¼ ln aD þ a2D 1 þ ani ln 2LD rwD ðIani þ 1Þ (18)
(17)
The right hand side of this model represents the impact of reservoir configuration, wellbore length, and anisotropy as wells as normal skin factor and rate dependent skin factor while the lift hand side is the pseudo-pressure at the wellbore which considers
The second term is the pressure drop caused by the convergence of flow streamlines as the cross section area of flow narrows close to the wellbore. The convergence of flow streamlines causes excessive pressure drop due to the growing inertial forces. As a result, Darcy law may not accurately describe this pressure drop and Forchheimer model can be adequately used for this purpose. It is worthy noting that non-Darcy flow is commonly used to describe all types of flow where Darcy law is not applicable. The pressure drop caused by Forchheimer flow or non-Darcy flow can be written as:
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PPwDT ¼
2 Iani ðDQsc Þ 2LD
(19) 3
Equating the two terms given by Eqs. (18) and (19) with the calculated pressure drop by applying Eqs. (9) and (15) yields a new model for the rate dependent skin factor:
DQsc
" qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2LD ¼ 2 PPwDS þ PPNDP ln aD þ a2D 1 Iani
# I2 Iani þs ani ln 2LD rwD ðIani þ 1Þ
(20)
One point should be emphasized. The skin factor in the above model represents all types of skin factors but not the rate dependent skin factor. The skin factor might be considered for short horizontal wellbore or for the cases of anisotropy greater than one. However, for other cases it can be considered equal to (zero). In the above mentioned models, the parameter (aD) is defined as:
4
5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5 aD ¼ 0:5 þ 0:25 þ ðrehD Þ4
(21)
and:
rehD
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ¼ pxeD yeD
(22)
1.3. Results' analysis The wellbore dimensionless pressure drops by non- Darcy flow and Darcy flow have been calculated using Eqs. (9) and (15) respectively for different wellbore lengths, reservoir configurations, rate dependent skin factors, inertial number and different anisotropies. The skin factor has also been included in some cases to show its impact on rate dependent skin factor. The followings are on the trends or behaviors obtained from the results: 1 All cases demonstrate the high values for the rate dependent skin factor for rectangular-shape drainage areas but much less so for square-shape drainage areas as shown in Figs. 4 and 5. This fact indicates that linear flow regimes may have non-Darcy flow conditions if the cross section area of flow is small enough for this flow to be developed. Even though, there is no convergence for the flow streamlines in linear flow regimes as the area of flow is constant, the friction forces resulting within fluid layers and between fluid and rock are the reasons for the high value of rate dependent skin factor. For square-shape drainage areas, the cross section area of flow is large enough to eliminate, to some degree, the impact of the non-Darcy flow conditions and reduce the inertial factor. 2 Fig. 6 shows the behavior of rate dependent skin factors for different wellbore lengths acting in square drainage areas. The minimum value for this factor is produced in large square drainage areas or in reservoirs with large width to length ratios to the horizontal wellbore (the left side of the plot xeD ¼ yeD ¼ 0.2). The maximum value for this factor most often occurs when (xeD ¼ yeD ¼ 0.6). The rate dependent skin factors decrease slightly for drainage areas having a small ratio between reservoir boundaries and wellbore (the right side of the plot xeD ¼ yeD < 0.6). The large drainage area or the large ratio of the reservoir boundaries and wellbore is not recommended, even though it has the minimum rate dependent skin factor, because the wellbore length normalized rate dependent skin factor
6
7
shows high lost energy due to the turbulence when reservoir fluids approach the wellbore as shown in Fig. 7. The rate dependent skin factor changes linearly with wellbore length as shown in Fig. 8. This is due to the fact the total lost energy due to non-Darcy flow conditions is the product of the lost energy per wellbore length times the wellbore length. However, the change in the rate dependent skin factor with the change in the drainage area is not significantly recognized as shown in Fig. 9. For short wellbore lengths, the rate dependent skin factor is almost constant no matter the reservoir configurations, while it changes slightly with the reservoir drainage area for long wellbores. For all cases, two trends for rate dependent skin factor can be seen according to reservoir width or side boundary normal to the wellbore (yeD). The first trend exhibits a decrease in the rate dependent skin factor for all values of reservoir length parallel to the wellbore (xeD) for the cases (yeD < 0.5), while the second trend shows increasing values for this factor for all (yeD > 0.5) as shown in Fig. 10. Fig. 11 shows the impact of anisotropy on the rate dependent skin factor. It ca n be seen that high anisotropy (Iani > 1.0) maximizes the influence of non-Darcy flow. Physically this is explained by the fact that the anisotropy represents the index for how much the vertical permeability of porous media is greater than the horizontal permeability. For steady state conditions, pseudoredial flow and boundary-dominated flow are expected to be developed. For both of them the horizontal permeability is the controlling parameter rather than the vertical one, as reservoir fluids flow radially or linearly in the horizontal plane toward the wellbore. Therefore, high anisotropy means small horizontal permeability which in turn means narrow flow passages in the porous media definitely cause increased velocity of flow to the limits suitable for non-Darcy flow to occur. Skin factor effects are shown in Fig. 12. As the rate dependent skin factor is calculated for steady state flow, the other types of skin factors, such as mechanical skin factors and skin factors due to damage sections, do not have significant impact on rate dependent skin factor as they do not a have significant impact on the total pressure drop resulting from fluid flow towards the wellbore. The inertial number (IND) has a considerable effect on the rate dependent skin factor. This number refers to the size of inertial forces that resist fluid flow in the porous media. As it is given in Eq. (10), this number is a function of the flow rate or fluid superficial velocity in the porous media. Increasing the velocity causes a significant increase in the friction forces acting in the opposite direction of the flow, which in turn requires extra pressure to enable the reservoir fluid to reach the wellbore. Fig. 13 demonstrates the effect of the inertial number in rate dependent skin factors.
1.4. Assymmetricity impact Horizontal well assymmetricity has a significant impact on rate the dependent skin factor as shown in Figs. 14, 15 and 16. This factor increases as the well location is not centralized in the formation either in the vertical direction or in the horizontal plane. For the cases where the horizontal well is located close to the boundaries, non-Darcy flow might be developed and the inertial forces increased so that the rate dependent skin factor and non-Darcy flow coefficient exhibit increases. This could be caused by the rapid change in the cross section area of flow for the part of reservoir between the horizontal wellbore and the closer boundary.
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
Fig. 4. Rate dependent skin factor for (LD ¼ 64).
Fig. 5. Rate dependent skin factor for (LD ¼ 32).
Fig. 6. Rate dependent skin factor for square drainage area.
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Fig. 7. Wellbore length normalized rate dependent skin factor.
Fig. 8. Rate dependent skin factor for wellbore length.
Fig. 9. Rate dependent skin factor for drainage area.
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Fig. 10. Rate dependent skin factor for (LD ¼ 8).
Fig. 11. Rate dependent skin factor for different anisotropies.
Fig. 12. Rate dependent skin factor for different skin factors.
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1.5. Productivity index The dimensionless productivity index for steady state flow, controlled by non-Darcy flow, can be written as follows:
JnD ¼ ¼
Qsc PPe PPwf
1.6. Application
C qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Iani 2 ln aD þ aD 1 þ 2LD ln rwD ðIIani þ s þ DQ sc ani þ1Þ (23)
while the dimensionless productivity index without considering non-Darcy can be written as:
JD ¼ ¼
Qsc PPe PPwf
C qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I þ s ln aD þ a2D 1 þ 2LaniD ln rwD ðIIani þ1Þ ani
(24)
kH h 1; 422 T
The rate dependent skin factor and non-Darcy flow coefficient can be determined by using the proposed technique and by knowing reservoir configurations(xeD & yeD), horizontal wellbore length, and the formation anisotropy. In this new technique, there is no need to calculate the inertial factor (b). Also there is no need for well tests and stabilized flow rates or experimental studies to estimate the rate dependent skin factor and non-Darcy coefficient. The specifications given by Hashemi et al., 2006, are shown in Table 1 and Fig. 19. The model that has been proposed above has been used to calculate the rate dependent skin factor as follows: 1 The following parameters have been determined:
Lw LD ¼ h
where:
C¼
is greater than for short wellbores, while the ratio of non-Darcy flow to Darcy flow pressure drops for long horizontal wellbores is less than long wellbores.
(25)
The ratio between the two dimensionless productivity indexes
Iani
sffiffiffiffiffiffi kV ¼ ¼ 0:46 kH
is:
JnD ¼ JD
1 1þ
I2 ani 2LD
pffiffiffiffiffiffiffiffiffi 2
ln aD þ
xeD ¼
Lw ¼ 0:116 xe
yeD ¼
Lw ¼ 0:165 ye
xwD ¼
xw ¼ 0:785 xe
ywD ¼
yw ¼ 0:282 ye
zwD ¼
zw ¼ 0:7 h
(26) ðDQsc Þ
I2
ani ln aD 1 þ2L D
Iani rwD ðIani þ1Þ
þs
The productivity index ratio decreases with the increase of the drainage area of the porous media as shown in Fig. 17. This may be explained as the Darcy flow is the dominant for a large drainage area while non-Darcy flow is dominant for a small drainage area. This fact is confirmed by Fig. 18 wherein the ratio of pressure drop caused by non-Darcy flow to the pressure drop caused by Darcy flow is in the minimum limit for small drainage areas and reaches the maximum limit for large drainage areas. It is also important to note that the productivity index ratio for long horizontal wellbores
sffiffiffiffiffiffi kV ¼ 0:788 kH
Fig. 13. Rate dependent skin factor for different inertial number.
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
Fig. 14. Rate dependent skin factor for assymmetricity.
Fig. 15. Rate dependent skin factor for assymmetricity.
Fig. 16. Rate dependent skin factor for assymmetricity.
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Fig. 17. Productivity index ratio for square drainage area.
Fig. 18. Non-Darcy flow/Darcy flow pressure drop ratio. Table 1 Well specifications given by Hashemi et al., 2006. 3495 psi 410 ft 2,35 md 0,5 md 0.25 ft 120 ft
Initial reservoir pressure Horizontal wellbore length kH kV rw h
2 The simulator has been used to calculate the rate dependent skin factor based on the proposed technique. The results have been plotted in Fig. 20. 3 The rate dependent skin factor has been determined based on (xeD ¼ 0.116 and yeD ¼ 0.165):
DQsc ¼ 119 Therefore for ðQsc ¼ 10; 000 MScf =Day Þ; non-Darcy flow coefficient is:
value for rate dependent skin factor and non-Darcy flow coefficient. The dimensionless productivity index for the well given by Hashemi et al., 2006, has been calculated for both Darcy and nonDarcy flow and plotted as shown in Fig. 19. The following results have been obtained:
JD ¼ 0:325 JnD ¼ 0:008 JnD ¼ 0:24% JD It is apparent that the non-Darcy flow is not developed very well for this case and the productivity index of the well is not affected by the rate dependent skin factor. Knowing that:
kH h ¼ 3:42*104 1422T
D ¼ 0; 0119 Day=MScf
C¼
The application of the currently used technique; i.e. by calculating the inertial factor (b), and using Eq. (4) has given insignificant
then:
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Fig. 19. Rate dependent skin factor for the well given by Hashemi et al., 2006.
Fig. 20. Productivity index for the well given by Hashemi et al., 2006.
MScf J ¼ 1:14 D
. psi cp 2
2. Conclusions 1 The rate dependent skin factor and non-Darcy flow coefficient are two characterizing parameters for non-Darcy flow conditions that could be developed either in the vicinity of the horizontal wellbore during early radial flow or in the outer drainage area when pseudo-radial flow regimes be the dominant flow. 2 These two parameters have significant impact on the pressure profile of reservoirs especially for short wellbore lengths or small drainage areas. 3 The new proposed techniques suggest calculating these two parameters by knowing reservoir configurations, wellbore lengths and formation anisotropy. Therefore, there is no need for analytically calculating or experimentally measuring the inertial factor (b). 4 The minimum value for the rate dependent skin factor is obtained for square drainage area. 5 Formation anisotropy and well assymmetricity may provide the environment for developing non-Darcy flow and increasing the rate dependent skin factor.
6 The rate dependent skin factor has a significant negative contribution for horizontal well productivity index. It causes an increase in pressure drop as the non-Darcy flow conditions become the dominant conditions when reservoir fluid approaches the wellbore and the cross section area of flow becomes small and in turn leads to the convergence of flow streamlines.
2.1. Dimensionless parameters
Lw LD ¼ h PD ¼
sffiffiffiffiffiffi kV kH
kH hDP 1422 Qsc T
Lw rD ¼ h rwD ¼
sffiffiffiffiffiffi kV kH
rw h
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tD ¼
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
kH t ∅mct L2w
xwD ¼
xw xe
ywD ¼
y ye
zwD ¼
zw h
xeD ¼
Lw xe
yeD ¼
Lw ye
xD ¼
x xw Lw
yD ¼
y yw Lw
z zw zD ¼ Lw zD ¼
sffiffiffiffiffiffi kH kV
z zw ¼ zD LD h
kV Lw PPwf PPe PPwD PPwDL PPwDT Qsc r rw re S s T t tD v x y z xw yw zw xe ye
u r rg l
4
m h gg
Vertical Permeability, md Half-horizontal wellbore, ft Wellbore pseudo pressure, psi2 =cp Reservoir pseudo pressure, psi2 =cp Dimensionless wellbore pseudo pressure Dimensionless wellbore pseudo pressure for laminar flow Dimensionless wellbore pseudo pressure for non-Darcy flow Flow rate, MScf/D Radius, ft Wellbore radius, ft Drainage area radius, ft Laplace domain operator Skin factor Temperature, R Time, hr Dimensionless time Velocity, ft/sec Any point in the X-direction in the reservoir Any point in the Y-direction in the reservoir Any point in the Z-direction in the reservoir The X-coordinate of production point The Y- coordinate of production point The Z-coordinate of production point Distance to reservoir side boundary, ft Distance to reservoir side boundary, ft Storativity Density, Ib=ft 3 Gas density, Ib=ft 3 Interporosıty Porosity Viscosity, cp Diffusivity coefficient, ft2/sec Gas specific gravity
Nomenclatures Appendix-A B h CD ct dP DP k kH
Oil formation volume factor, RB/STB Formation height, ft Dimensionless wellbore storage coefficient Total compressibility, psi1 Pressure derivative, psi Pressure difference, psi Permeability, md Horizontal Permeability, md
Pressure drop caused by horizontal wells depleting finite reservoirs can be obtained using line source solution. This solution is mathematically derived by integrating the point source solution 0 from ðxwD 1Þ to ðxwD þ 1Þ with respect to ðxwD Þ. The point source solution in finite reservoirs having rectangular drainage area is given by (Ozkan, 1988):
(
pffiffiffi
pffiffiffi ) 3 2 cosh u ðyeD jyD ywD jÞ þcosh u ðyeD ðyD ywD ÞÞ
pffiffiffi pffiffiffi þ usinh u yeD 7 6 7 6 qffiffiffiffiffiffiffiffiffiffiffiffi 7 6 p ffiffiffiffiffiffiffiffiffiffiffiffi 8 9 7 6 2 2 7 6 P < = 0 ðy ÞÞ ðy cosh jy y jÞ þcosh ðy y uþa uþa eD D wD eD D wD xwD xD 7 6 ∞ 7 62 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi cos np cos np þ 7 6 n¼1 2 2 : ; x x eD eD uþa sinh uþa yeD 7 6 7 p 6 7 6 q ffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffi 8 9 P PD ¼ 7 6 7 xeD s 6 ∞
(A-1)
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
1057
The integration of the above mentioned model gives the line source solution:
PPD ¼ PPD1 þ PPD2 þ PPD3 þ PPD4
(A-2)
PDz ¼
2
∞ X 1
p2 L2D l¼1 l2
cosðlpzwD ÞcosðlpðzD LD þ zwD ÞÞ
(A-15)
The four separated parts are (Ozkan, 1988):
PD1 ¼
PD2
pffiffiffi
pffiffiffi u ðyeD jyD ywD jÞ þ cosh u ðyeD ðyD ywD ÞÞ
pffiffiffi pffiffiffi usinh u yeD
p cosh SxeD
(A-3)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ∞ u þ a2 ðyeD ðyD ywD ÞÞ u þ a2 ðyeD jyD ywD jÞ þ cosh 2X 1 np xwD xD 4cosh 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin cos np cos np ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s n¼1 n xeD xeD xeD u þ a2 y u þ a2 sinh
(A-4)
eD
PD3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ∞ cosh u þ b2 ðyeD jyD ywD jÞ þ cosh u þ b2 ðyeD ðyD ywD ÞÞ 2p X 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cosðlpzwD ÞcosðlpzD Þ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi SxeD u þ b2 y u þ b2 sinh l¼1
PD4 ¼
a¼
∞ ∞ X 8X 1 np x x sin cos np wD cos np D cosðlpzwD ÞcosðlpzD Þ S n xeD xeD xeD n¼1 l¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosh u þ c2 ðyeD jyD ywD jÞ þ cosh u þ c2 ðyeD ðyD ywD ÞÞ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ a2 yeD u þ c2 sinh
(A-7)
b ¼ lpLD
∞ X ∞ x 32 X 1
sin np eD 2 p3 xeD n¼1 m¼1 n n2 x2eD þ m2 y2eD x n p y ðx x þ xwD Þ cos mp wD cos np wD cos 2 2 D eD 2 mp cos ðyD yeD þ ywD Þ 2
(A-8) np xeD
(A-16) (A-9) x sin np eD 2 eD n¼1 l¼1 n n2 x2 þ 4l2 L2 eD D x np ðx x þ xwD Þ cosðlpzwD Þ cos np wD cos 2 2 D eD cosðlpðzD LD þ zwD ÞÞ
PDxz ¼ u ¼ Se2s for homogenous formation
(A-10)
u ¼ Sf ðSÞe2s for non homogenous formation ðdual porosity permeability porous mediaÞ (A-11) f ðSÞ ¼
PDx ¼
swð1 wÞ þ l sð1 wÞ þ l
(A-12)
32
∞ X ∞ X
p3 x
1
(A-17)
1 y cos mp wD 2 2 2 2 2 m¼1 l¼1 m yeD þ 4l LD mp ðyD y þ yÞ cosðlpzwD Þ cosðlpðzD LD þ zwD ÞÞ cos 2
PDyz ¼
∞ X ∞ 16 X
p2
(A-18)
∞ x x n p 16 X 1 ðx x sin np eD cos np wD cos 3 3 3 2 2 2 D eD p xeD n¼1 n þ xwD Þ
∞ X ∞ X ∞ 64 X 1 p3 xeD n¼1 m¼1 l¼1 n n2 x2 þ m2 y2 þ 4lL2 eD eD D x x n p ðxD xeD þ xwD Þ sin np eD cos np wD cos 2 2 2 ywD mp cos ðyD yeD þ ywD Þ cosðlpzwD Þ cos mp 2 2 cosðlpðzD LD þ zwD ÞÞ
PDxyz ¼ (A-13)
PDy ¼
(A-6)
PDxy ¼
np xeD
c ¼ lpLD þ
(A-5)
eD
∞ 8 X 1 ywD mp cos ðy cos m p y þ y Þ D eD wD 2 2 p2 x2eD m¼1 m2
(A-14)
(A-19)
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References Babu, D.K., Odeh, A.S., 1989, November 1. Productivity of a Horizontal Well (Includes Associated Papers 20306, 20307, 20394, 20403, 20799, 21307, 21610, 21611, 21623, 21624, 25295, 25408, 26262, 26281, 31025, and 31035). Society of Petroleum Engineers. http://dx.doi.org/10.2118/18298-PA. Barree, R.D., Conway, M.W., 2004, January 1. Beyond Beta Factors: A Complete Model for Darcy, Forchheimer, and Trans-forchheimer Flow in Porous Media. Society of Petroleum Engineers. http://dx.doi.org/10.2118/89325-MS. Barree, R.D., Conway, M., 2009, May 1. Multiphase Non-Darcy Flow in Proppant Packs. SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/ 109561-PA. Brar, G.S., Aziz, K., 1978, February 1. Analysis of Modified Isochronal Tests to Predict the Stabilized Deliverability Potential of Gas Wells without Using Stabilized Flow Data (Includes Associated Papers 12933, 16320 and 16391). SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/6134-PA. Camacho-V, R., Vasquez-C, M., Roldan-C, J., Samaniego-V, F., Macias-C, L., 1996, December 1. New Results on Transient Well Tests Analysis Considering Nonlaminar Flow in the Reservoir. SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/26180-PA. Daviau, F., Mouronval, G., Bourdarot, G., Curutchet, P., 1988, December 1. Pressure Analysis for Horizontal Wells. SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/14251-PA. Economides, M.J., et al., 2013. Petroleum Production System, second ed. Prentice Hall, Westford, MA, USA. Frick, T.P., Economides, M.J., 1993, February 1. Horizontal Well Damage Characterization and Removal. SPE Production & Facilities. Society of Petroleum Engineers. http://dx.doi.org/10.2118/21795-PA. Frocheimer, P., 1901. Wasserbewguang durch boden. ZVDI 45, 1781. Gringarten, A.C., Ramey, H.J., 1973, October 1. The Use of Source and Green's
Functions in Solving Unsteady-flow Problems in Reservoirs. SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/3818-PA. Hashemi, A., Nicolas, L., Gringarten, A.C., 2006, February 1. Well Test Analysis of Horizontal Wells in Gas-Condensate Reservoirs. SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/89905-PA. Joshi, S.D., 1988, June 1. Augmentation of Well Productivity with Slant and Horizontal Wells (Includes Associated Papers 24547 and 25308). SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/15375-PA. Kelkar, M.G., 2000, April 1. Estimation of Turbulence Coefficient Based on Field Observations. Society of Petroleum Engineers. http://dx.doi.org/10.2118/62488PA. Kim, J., Kang, J.M., 1994, January 1. A semi-analytical approach. In: Determining Non-Darcy Flow Coefficient from a Single Rate Gas Well Pressure Transient Test. Society of Petroleum Engineers. Matthews, C.S., 1986, April 1. Transient, Semisteady-State, and Steady-State Flow. Society of Petroleum Engineers. http://dx.doi.org/10.2118/15278-PA. Nguyen, T.V., 1986, January 1. Experimental Study of Non-Darcy Flow Through Perforations. Society of Petroleum Engineers. http://dx.doi.org/10.2118/15473MS. Ozkan, E., 1988. Performance of Horizontal Wells. PhD dissertation. University of Tulsa, Tulsa, Oklahoma, USA. Ramey, H.J., 1965. Non-Darcy flow and wellbore storage effects in pressure build-up and drawdown of gas wells. JPT 17 (2), 223e233. http://dx.doi.org/10.2118/ 1058-PA. Trans. AIME, 234. SPE 1058-PA. Spivey, J.P., Brown, K.G., Sawyer, W.K., Frantz, J.H., 2004, August 1. Estimating NonDarcy Flow Coefficient from Buildup-Test Data with Wellbore Storage. SPE Journal. Society of Petroleum Engineers. http://dx.doi.org/10.2118/88939-PA. Zeng, F., Zhao, G., 2008, April 1. Semianalytical Model for Reservoirs with Forchheimer's Non-Darcy Flow. SPE Reservoir Evaluation & Engineering. Society of Petroleum Engineers. http://dx.doi.org/10.2118/100540-PA.
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
New technique for determining rate dependent skin factor & nonDarcy coefficient Salam Al-Rbeawi Middle East Technical University, North Cyprus Campus, Mersin 10, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 June 2016 Received in revised form 8 September 2016 Accepted 11 September 2016 Available online 13 September 2016
A non-Darcy flow develops in porous media when the velocity of reservoir fluids becomes extremely high because of the continuous narrowing of the cross section area of the flow and the convergence of flow streamlines. Therefore, the inertial effect increases significantly and the total pressure drop, required by fluids to move from the outer drainage area towards the wellbore increases significantly due to the extra pressure drop caused by the non-Darcy flow. The extra pressure drop is described by the Forchheimer equation in which the deviation from Darcy's law is proportional to the inertial factor which in turn is a function of porous media characteristics such as permeability and porosity. This paper introduces a new technique for estimating rate dependent skin factors (DQsc) and nonDarcy flow coefficients (D). The new technique uses dimensionless pressure for steady state flow to determine these two parameters. A set of plots has been developed for the term (DQsc) based on reservoir configurations i.e. reservoir boundaries (2xe & 2ye), wellbore length, and anisotropy. All the plots have been developed based on the fact that the rate dependent skin factor represents the difference between the total pressure drop and the pressure drop caused by Darcy flow only when non-Darcy coefficient equals to zero. Dimensionless pressure drops for horizontal wells producing from finite acting reservoirs have been calculated for steady state flow when the impact of dimensionless time can be eliminated. The calculated pressure drops have been used in reservoir performance models to estimate rate dependent skin factors and non-Darcy coefficients. A point of special interest in this study is the ability to estimate these two parameters without the need for experimental work or the need to use currently proposed empirical models for calculating them. This new technique requires knowing reservoir configurations (reservoir width and length), the length of the wellbore, and the height of the formation in addition to the anisotropy. The study indicates that rate dependent skin factors and non-Darcy coefficients have the minimum values when the reservoirs having square-shape drainage area. It has also been found that both parameters have great impact on wells with short wellbore length. Additionally, isotropic formations and symmetrical wells perform better than anisotropic formations and asymmetrical wells in terms of extra pressure drop caused by non-Darcy flow. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Great attention has been given to non-Darcy flow beginning in the early 1900s when Forchheimer pointed out for the first time that Darcy's law is not applicable for high velocity gas flow in porous media without considering the inertial impact on pressure drop. For this reason, he stated that Darcy's law could be written in the following form (Forcheimer, 1901):
hm i dP ¼ v þ rbv2 dr k
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jngse.2016.09.028 1875-5100/© 2016 Elsevier B.V. All rights reserved.
(1)
where (b) is the inertial factor which has numerous empirical models tabulated in the literature (Economides et al., 2013). The pressure drop in gas reservoirs, caused by fluid flow, may have two trends: the first one is caused by Darcy flow when the velocity of reservoir fluid is low and the cross section area of flow is large enough for Darcy flow (Laminar flow approach) to be the dominant flow regime. The second is caused by non-Darcy flow when the cross section area of flow decreases gradually and flow streamlines come close together in such a way that the velocity increases to the turbulent condition limit. In this case, excessive pressure might be needed to overcome the inertial forces resulted from the convergence of flow streamlines. For horizontal wells, the extra pressure drop caused by non-Darcy flow, sometimes called choked flow, characterizes the flow regimes that typically result
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
1045
Fig. 1. Horizontal wellbore in bounded formation.
from change in the cross section area of flow and converging flow streamlines. This could happen for both early radial flow regimes observed during early production when reservoir fluids move radially in the vertical plane towards the wellbore or pseudo-radial flow regime, observed during late time production when reservoir fluids move radially in the horizontal plane towards the wellbore. However, it could happen also for linear flow regimes if the cross section area of flow is not large enough and therefore the developed velocity is so high that Darcy's law is no longer be applicable. The non-Darcy flow accompanies pseudo-radial flow is characterized and controlled by reservoir boundaries and the horizontal permeability while the non-Darcy flow accompanies early radial flow is characterized and controlled by formation thickness and vertical permeability. Spivey et al. (2004) stated that more than 90% of the pressure drop within a distance of (0.5 ft) from wellbore could be caused by the non-Darcy flow developed by completion technique such as gravel-packed for vertical wells and slotted liner for horizontal wells. The non-Darcy flow coefficient for pseudoradial flow when reservoir fluid flows in the outer drainage area and radially converges to the wellbore, is given by (Zeng and Zhao, 2008):
D ¼ 2:22*1015
bgg k 1 1 hm rw re
(2)
while the non-Darcy flow coefficient for early-radial flow near the wellbore, is given by:
D ¼ 2:22*1015
ðbs bÞgg k 1 1 rw rs hm
(3)
where: (b) is the inertial factor at the outer drainage area of the reservoir and (bs) is the inertial factor for the zone close to the wellbore or the damaged zone extended from the wellbore to radius of (rs). To characterize non-Darcy flow, the rate dependent skin factor (DQsc) should be determined. This term represents the dimensionless pressure drop and depends on the non-Darcy flow coefficient (D). Several models have been presented in the literature to estimate this coefficient. One of them was given by Ramey (1965):
D ¼ 2:22*1015
bgg k hmrw
(4)
It can be seen that the non-Darcy coefficient is a function of the petrophysical properties of the formation, such as permeability and porosity, inertial factor (b) which in turn is a function of permeability and porosity as well as reservoir fluid properties such as
viscosity and specific gravity, in addition to the formation thickness and wellbore radius. A method of estimating the inertial factor has long been known from experimental studies (Barree and Conway, 2004, 2009). Based on these studies, several mathematical models have been proposed for this factor (Economides et al., 2013). Many researchers have suggested using gas well tests such as flow after flow tests, isochronal, and modified isochronal tests to predict the value of non-Darcy coefficient (Ramey, 1965), (Kelker, 2000), and (Brar and Aziz, 1978). Experimental studies have also been conducted to estimate non-Darcy coefficients especially for the pressure drop across the completion system (Nguyen, 1986). Pressure drawdown and build up tests have also been used to determine the rate dependent skin factor (Zeng and Zhao, 2008). Spivey et al., 2004, suggested using sand face flow rate instead of surface gas flow rate for buildup test analysis to eliminate wellbore storage effects, while Kim and Kang, 1994, introduced a semianalytical model for determining the non-Darcy flow coefficient from single rate gas well pressure transient tests. In this technique, pressure profiles and flow regimes could be derived from pressure and pressure derivative curves. Camacho-V et al., 1996, stated that pressure transient analysis could be used to characterize the nonDarcy coefficient and described it as the Reynold number. Therefore, well test analysis can be an excellent tool for characterizing these two parameters. However, the existence of different flow regimes, especially for horizontal wells, could cause some difficulties for estimating non-Darcy coefficient from well test analysis. In this paper, the main focus will be on estimating rate dependent skin factors and non-Darcy flow coefficients using steady state pressure profiles and flow regimes. According to Matthews 1986 and Babu and Odeh 1989, the total pressure drop resulting from the depletion process of a horizontal well producing from finite acting reservoir could be separated into two terms. The first one represents the pseudo-steady state pressure drop that is a function of time and drainage area wherein reservoir pressure declines from its initial to average value. The second is the pressure drop at steady state when time does not have a significant impact on the pressure profile at any point in the reservoir. Therefore, dimensionless pressure can be simulated by several models and calculated for different reservoir configurations, horizontal wellbore lengths and anisotropy. Using the calculated dimensionless pressure, the rate dependent skin factor and non-Darcy flow coefficient can be determined. 1.1. Mathematical models for pressure profile Consider a rectangular-shape formation with two side boundaries having the dimensions (2xe & 2ye) and height of (h) as shown
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S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
in Fig. 1. The formation can be considered either a conventional or an unconventional gas reservoir in case of pseudo-steady state flow has already been reached. It is depleted by a horizontal wellbore of length (2Lw). The formation is homogenous and anisotropic where permeability in the vertical direction (kV) is not the same as the permeability in the horizontal direction (kH). The horizontal wellbore could be totally extended in the drainage area or it could be partially extended in the X-direction. Formation porosity ð∅Þ is considered constant. The pressure drop at the wellbore of the above mentioned horizontal well can be determined by the solution of the diffusivity equation either by the instantaneous line source solution presented by different researches (Gringarten and Ramey, 1973; Daviau et al., 1988) or by Laplace transformation presented by Ozkan 1988. Mathematically, dimensionless pressure drop for horizontal wells extending in bounded reservoirs can be obtained applying line source solution for diffusivity equation. This solution is well documented in the literature such as the one introduced by Ozkan 1988 using Laplace domain, which is:
PPD ¼ PPD1 þ PPD2 þ PPD3 þ PPD4
(5)
These first two pressure items represent the solutions for fully penetrating vertical fractures while the second two pressure terms represent the solutions satisfy the boundary conditions in the diffusivity equation. More details for these four models are given by Eqs. (A-3), (A-4), (A-5), and (A-6) in the Appendix-A. To consider wellbore storage effect, skin factor, and non-Darcy flow pressure drop, dimensionless wellbore pressure can be written as follows:
PPwD ¼ L 1 PPwD þ PPND 0
1
SPPD þ s B C ¼ L 1 @ A þ PPND S þ CD S2 SPPD þ s
(6)
where (PPND) is the pressure drop caused by non-Darcy flow and defined by:
PPND ¼ DQsc
(7)
The non-Darcy flow coefficient in this model is represented by:
D¼C
bgg kH hmLw
(8)
Therefore. Eq. (7) becomes:
PPND ¼
IND LD Iani
(9)
where: (IND) is the inertial number, representing the ratio of the inertial force to the viscous force, and mathematically formulated as:
IND ¼
C rg kH Qsc
mL2w
(10)
(LD) is the dimensionless wellbore length and defined as:
sffiffiffiffiffiffi Lw kV LD ¼ h kH And the anisotropy (Iani) is defined as:
Iani
sffiffiffiffiffiffi kV ¼ kH
(12)
pffiffiffiffiffiffiffiffiffiffi where (kV ¼ kz) is the vertical permeability and ðkH ¼ kx ky Þ is the horizontal permeability. For late time production, the pressure drop caused by Darcy flow only, given by Eq. (5), consists of two parts. The first part is the pseudo-steady state behavior when the reservoir pressure declines with time by a constant rate. The second part is the pressure drop caused by fluid flow in the steady state period when pressure decline is no longer affected by the production time and reservoir boundaries have the greatest impact on pressure drop only (Matthews, 1986; Babu and Odeh, 1989). Therefore, Eq. (6) can be written as:
PPwD ¼ PPwDP þ PPwDS þ PPND
(13)
where the term (PPwDP) is pressure drop for pseudo-steady state caused by Darcy flow and given by:
PPwDP ¼ 2ptDA
(14)
and the term (PPwDS) is the pressure drop for steady state caused by Darcy flow and given by:
p PPwDS ¼ xeD yeD PPDx þ PPDy þ PPDz þ PPDxy þ PPDxz þ PPDyz 2 þ PPDxyz (15) The pressure terms in the right hand side of the above model are defined in the Appendix-A. The steady state pressure drop caused by Darcy flow is characterized by a straight line of unit slope for both dimensionless pressure and dimensional pressure derivative curves. Typical pressure behaviors and flow regimes for horizontal wells in finite acting reservoirs having small and large drainage areas, are shown in Figs. 2 and 3 respectively. Obviously a horizontal wellbore, acting in large drainage area (LD ¼ 10 and xeD ¼ yeD ¼ 0.1), responses to the depletion process differently over time than do horizontal wellbores acting in small drainage area (LD ¼ 10 and xeD ¼ yeD ¼ 0.5). The difference between the two cases could be seen in both the pressure profile and flow regimes. The pseudoradial flow regime is developed for large drainage areas but it is not observed for small drainage areas as there is enough time for reservoir fluids to move radially in the horizontal plane towards the wellbore in the first case. The radial movement of reservoir fluids means that the cross section area of flow decreases as the fluids approach the wellbore. This could lead to the convergence of flow streamlines and cause more lost energy due to the friction force or inertial impact. However, it does not mean there is no lost energy if there is no pseudo-radial flow. In general, the lost energy could potentially be observed for linear flow also when the cross section area of flow is not enough to demonstrate Darcy flow. The amount of the lost energy depends on the flow rate of reservoir fluids and the shrinkage in the cross section area of flow in addition to other rock and fluid properties. 1.2. Mathematical models for non-Darcy flow
(11)
Several models have been presented in the literature for steady state performance of reservoirs depleted by horizontal wells. One of these models was introduced by Joshi 1988 and developed later by several researchers (Frick and Economides, 1993) to include non-
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
1047
Fig. 2. Wellbore storage and skin effect for horizontal well acting in finite reservoir, small drainage area (xeD ¼ yeD ¼ 0.5).
Fig. 3. Wellbore storage and skin effect for horizontal well acting in finite reservoir, large drainage area (xeD ¼ yeD ¼ 0.1).
Darcy flow coefficient for gas reservoir performance. The most commonly used model is (Joshi, 1988):
Qsc
kH h PPe PPwf ¼ 3 1 2 0 pffiffiffiffiffiffiffiffiffiffiffi
aþ a2 L2w A Iani h Iani h 4 @ þ 2Lw ln rw ð1þIani Þ þ s þ DQsc 5 1422 T ln Lw (16) In dimensionless form, Eq. (12) can be written as:
PPwD
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 Iani þs ¼ ln aD þ a2D 1 þ ani ln 2LD rwD ðIani þ 1Þ þ DQsc
the change in reservoir fluid properties such as compressibility and viscosity with production time. The pressure drop caused by the depletion process of gas reservoirs, given by Eq. (17), can be classified into two types: The first one represents the pressure drop due to Darcy flow when reservoir fluids move from the outer drainage area towards the horizontal wellbore. This one can be mathematically formulated as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 Iani þs PPwDL ¼ ln aD þ a2D 1 þ ani ln 2LD rwD ðIani þ 1Þ (18)
(17)
The right hand side of this model represents the impact of reservoir configuration, wellbore length, and anisotropy as wells as normal skin factor and rate dependent skin factor while the lift hand side is the pseudo-pressure at the wellbore which considers
The second term is the pressure drop caused by the convergence of flow streamlines as the cross section area of flow narrows close to the wellbore. The convergence of flow streamlines causes excessive pressure drop due to the growing inertial forces. As a result, Darcy law may not accurately describe this pressure drop and Forchheimer model can be adequately used for this purpose. It is worthy noting that non-Darcy flow is commonly used to describe all types of flow where Darcy law is not applicable. The pressure drop caused by Forchheimer flow or non-Darcy flow can be written as:
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S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
PPwDT ¼
2 Iani ðDQsc Þ 2LD
(19) 3
Equating the two terms given by Eqs. (18) and (19) with the calculated pressure drop by applying Eqs. (9) and (15) yields a new model for the rate dependent skin factor:
DQsc
" qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2LD ¼ 2 PPwDS þ PPNDP ln aD þ a2D 1 Iani
# I2 Iani þs ani ln 2LD rwD ðIani þ 1Þ
(20)
One point should be emphasized. The skin factor in the above model represents all types of skin factors but not the rate dependent skin factor. The skin factor might be considered for short horizontal wellbore or for the cases of anisotropy greater than one. However, for other cases it can be considered equal to (zero). In the above mentioned models, the parameter (aD) is defined as:
4
5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5 aD ¼ 0:5 þ 0:25 þ ðrehD Þ4
(21)
and:
rehD
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ¼ pxeD yeD
(22)
1.3. Results' analysis The wellbore dimensionless pressure drops by non- Darcy flow and Darcy flow have been calculated using Eqs. (9) and (15) respectively for different wellbore lengths, reservoir configurations, rate dependent skin factors, inertial number and different anisotropies. The skin factor has also been included in some cases to show its impact on rate dependent skin factor. The followings are on the trends or behaviors obtained from the results: 1 All cases demonstrate the high values for the rate dependent skin factor for rectangular-shape drainage areas but much less so for square-shape drainage areas as shown in Figs. 4 and 5. This fact indicates that linear flow regimes may have non-Darcy flow conditions if the cross section area of flow is small enough for this flow to be developed. Even though, there is no convergence for the flow streamlines in linear flow regimes as the area of flow is constant, the friction forces resulting within fluid layers and between fluid and rock are the reasons for the high value of rate dependent skin factor. For square-shape drainage areas, the cross section area of flow is large enough to eliminate, to some degree, the impact of the non-Darcy flow conditions and reduce the inertial factor. 2 Fig. 6 shows the behavior of rate dependent skin factors for different wellbore lengths acting in square drainage areas. The minimum value for this factor is produced in large square drainage areas or in reservoirs with large width to length ratios to the horizontal wellbore (the left side of the plot xeD ¼ yeD ¼ 0.2). The maximum value for this factor most often occurs when (xeD ¼ yeD ¼ 0.6). The rate dependent skin factors decrease slightly for drainage areas having a small ratio between reservoir boundaries and wellbore (the right side of the plot xeD ¼ yeD < 0.6). The large drainage area or the large ratio of the reservoir boundaries and wellbore is not recommended, even though it has the minimum rate dependent skin factor, because the wellbore length normalized rate dependent skin factor
6
7
shows high lost energy due to the turbulence when reservoir fluids approach the wellbore as shown in Fig. 7. The rate dependent skin factor changes linearly with wellbore length as shown in Fig. 8. This is due to the fact the total lost energy due to non-Darcy flow conditions is the product of the lost energy per wellbore length times the wellbore length. However, the change in the rate dependent skin factor with the change in the drainage area is not significantly recognized as shown in Fig. 9. For short wellbore lengths, the rate dependent skin factor is almost constant no matter the reservoir configurations, while it changes slightly with the reservoir drainage area for long wellbores. For all cases, two trends for rate dependent skin factor can be seen according to reservoir width or side boundary normal to the wellbore (yeD). The first trend exhibits a decrease in the rate dependent skin factor for all values of reservoir length parallel to the wellbore (xeD) for the cases (yeD < 0.5), while the second trend shows increasing values for this factor for all (yeD > 0.5) as shown in Fig. 10. Fig. 11 shows the impact of anisotropy on the rate dependent skin factor. It ca n be seen that high anisotropy (Iani > 1.0) maximizes the influence of non-Darcy flow. Physically this is explained by the fact that the anisotropy represents the index for how much the vertical permeability of porous media is greater than the horizontal permeability. For steady state conditions, pseudoredial flow and boundary-dominated flow are expected to be developed. For both of them the horizontal permeability is the controlling parameter rather than the vertical one, as reservoir fluids flow radially or linearly in the horizontal plane toward the wellbore. Therefore, high anisotropy means small horizontal permeability which in turn means narrow flow passages in the porous media definitely cause increased velocity of flow to the limits suitable for non-Darcy flow to occur. Skin factor effects are shown in Fig. 12. As the rate dependent skin factor is calculated for steady state flow, the other types of skin factors, such as mechanical skin factors and skin factors due to damage sections, do not have significant impact on rate dependent skin factor as they do not a have significant impact on the total pressure drop resulting from fluid flow towards the wellbore. The inertial number (IND) has a considerable effect on the rate dependent skin factor. This number refers to the size of inertial forces that resist fluid flow in the porous media. As it is given in Eq. (10), this number is a function of the flow rate or fluid superficial velocity in the porous media. Increasing the velocity causes a significant increase in the friction forces acting in the opposite direction of the flow, which in turn requires extra pressure to enable the reservoir fluid to reach the wellbore. Fig. 13 demonstrates the effect of the inertial number in rate dependent skin factors.
1.4. Assymmetricity impact Horizontal well assymmetricity has a significant impact on rate the dependent skin factor as shown in Figs. 14, 15 and 16. This factor increases as the well location is not centralized in the formation either in the vertical direction or in the horizontal plane. For the cases where the horizontal well is located close to the boundaries, non-Darcy flow might be developed and the inertial forces increased so that the rate dependent skin factor and non-Darcy flow coefficient exhibit increases. This could be caused by the rapid change in the cross section area of flow for the part of reservoir between the horizontal wellbore and the closer boundary.
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
Fig. 4. Rate dependent skin factor for (LD ¼ 64).
Fig. 5. Rate dependent skin factor for (LD ¼ 32).
Fig. 6. Rate dependent skin factor for square drainage area.
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Fig. 7. Wellbore length normalized rate dependent skin factor.
Fig. 8. Rate dependent skin factor for wellbore length.
Fig. 9. Rate dependent skin factor for drainage area.
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Fig. 10. Rate dependent skin factor for (LD ¼ 8).
Fig. 11. Rate dependent skin factor for different anisotropies.
Fig. 12. Rate dependent skin factor for different skin factors.
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1.5. Productivity index The dimensionless productivity index for steady state flow, controlled by non-Darcy flow, can be written as follows:
JnD ¼ ¼
Qsc PPe PPwf
1.6. Application
C qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Iani 2 ln aD þ aD 1 þ 2LD ln rwD ðIIani þ s þ DQ sc ani þ1Þ (23)
while the dimensionless productivity index without considering non-Darcy can be written as:
JD ¼ ¼
Qsc PPe PPwf
C qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I þ s ln aD þ a2D 1 þ 2LaniD ln rwD ðIIani þ1Þ ani
(24)
kH h 1; 422 T
The rate dependent skin factor and non-Darcy flow coefficient can be determined by using the proposed technique and by knowing reservoir configurations(xeD & yeD), horizontal wellbore length, and the formation anisotropy. In this new technique, there is no need to calculate the inertial factor (b). Also there is no need for well tests and stabilized flow rates or experimental studies to estimate the rate dependent skin factor and non-Darcy coefficient. The specifications given by Hashemi et al., 2006, are shown in Table 1 and Fig. 19. The model that has been proposed above has been used to calculate the rate dependent skin factor as follows: 1 The following parameters have been determined:
Lw LD ¼ h
where:
C¼
is greater than for short wellbores, while the ratio of non-Darcy flow to Darcy flow pressure drops for long horizontal wellbores is less than long wellbores.
(25)
The ratio between the two dimensionless productivity indexes
Iani
sffiffiffiffiffiffi kV ¼ ¼ 0:46 kH
is:
JnD ¼ JD
1 1þ
I2 ani 2LD
pffiffiffiffiffiffiffiffiffi 2
ln aD þ
xeD ¼
Lw ¼ 0:116 xe
yeD ¼
Lw ¼ 0:165 ye
xwD ¼
xw ¼ 0:785 xe
ywD ¼
yw ¼ 0:282 ye
zwD ¼
zw ¼ 0:7 h
(26) ðDQsc Þ
I2
ani ln aD 1 þ2L D
Iani rwD ðIani þ1Þ
þs
The productivity index ratio decreases with the increase of the drainage area of the porous media as shown in Fig. 17. This may be explained as the Darcy flow is the dominant for a large drainage area while non-Darcy flow is dominant for a small drainage area. This fact is confirmed by Fig. 18 wherein the ratio of pressure drop caused by non-Darcy flow to the pressure drop caused by Darcy flow is in the minimum limit for small drainage areas and reaches the maximum limit for large drainage areas. It is also important to note that the productivity index ratio for long horizontal wellbores
sffiffiffiffiffiffi kV ¼ 0:788 kH
Fig. 13. Rate dependent skin factor for different inertial number.
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
Fig. 14. Rate dependent skin factor for assymmetricity.
Fig. 15. Rate dependent skin factor for assymmetricity.
Fig. 16. Rate dependent skin factor for assymmetricity.
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Fig. 17. Productivity index ratio for square drainage area.
Fig. 18. Non-Darcy flow/Darcy flow pressure drop ratio. Table 1 Well specifications given by Hashemi et al., 2006. 3495 psi 410 ft 2,35 md 0,5 md 0.25 ft 120 ft
Initial reservoir pressure Horizontal wellbore length kH kV rw h
2 The simulator has been used to calculate the rate dependent skin factor based on the proposed technique. The results have been plotted in Fig. 20. 3 The rate dependent skin factor has been determined based on (xeD ¼ 0.116 and yeD ¼ 0.165):
DQsc ¼ 119 Therefore for ðQsc ¼ 10; 000 MScf =Day Þ; non-Darcy flow coefficient is:
value for rate dependent skin factor and non-Darcy flow coefficient. The dimensionless productivity index for the well given by Hashemi et al., 2006, has been calculated for both Darcy and nonDarcy flow and plotted as shown in Fig. 19. The following results have been obtained:
JD ¼ 0:325 JnD ¼ 0:008 JnD ¼ 0:24% JD It is apparent that the non-Darcy flow is not developed very well for this case and the productivity index of the well is not affected by the rate dependent skin factor. Knowing that:
kH h ¼ 3:42*104 1422T
D ¼ 0; 0119 Day=MScf
C¼
The application of the currently used technique; i.e. by calculating the inertial factor (b), and using Eq. (4) has given insignificant
then:
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Fig. 19. Rate dependent skin factor for the well given by Hashemi et al., 2006.
Fig. 20. Productivity index for the well given by Hashemi et al., 2006.
MScf J ¼ 1:14 D
. psi cp 2
2. Conclusions 1 The rate dependent skin factor and non-Darcy flow coefficient are two characterizing parameters for non-Darcy flow conditions that could be developed either in the vicinity of the horizontal wellbore during early radial flow or in the outer drainage area when pseudo-radial flow regimes be the dominant flow. 2 These two parameters have significant impact on the pressure profile of reservoirs especially for short wellbore lengths or small drainage areas. 3 The new proposed techniques suggest calculating these two parameters by knowing reservoir configurations, wellbore lengths and formation anisotropy. Therefore, there is no need for analytically calculating or experimentally measuring the inertial factor (b). 4 The minimum value for the rate dependent skin factor is obtained for square drainage area. 5 Formation anisotropy and well assymmetricity may provide the environment for developing non-Darcy flow and increasing the rate dependent skin factor.
6 The rate dependent skin factor has a significant negative contribution for horizontal well productivity index. It causes an increase in pressure drop as the non-Darcy flow conditions become the dominant conditions when reservoir fluid approaches the wellbore and the cross section area of flow becomes small and in turn leads to the convergence of flow streamlines.
2.1. Dimensionless parameters
Lw LD ¼ h PD ¼
sffiffiffiffiffiffi kV kH
kH hDP 1422 Qsc T
Lw rD ¼ h rwD ¼
sffiffiffiffiffiffi kV kH
rw h
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tD ¼
S. Al-Rbeawi / Journal of Natural Gas Science and Engineering 35 (2016) 1044e1058
kH t ∅mct L2w
xwD ¼
xw xe
ywD ¼
y ye
zwD ¼
zw h
xeD ¼
Lw xe
yeD ¼
Lw ye
xD ¼
x xw Lw
yD ¼
y yw Lw
z zw zD ¼ Lw zD ¼
sffiffiffiffiffiffi kH kV
z zw ¼ zD LD h
kV Lw PPwf PPe PPwD PPwDL PPwDT Qsc r rw re S s T t tD v x y z xw yw zw xe ye
u r rg l
4
m h gg
Vertical Permeability, md Half-horizontal wellbore, ft Wellbore pseudo pressure, psi2 =cp Reservoir pseudo pressure, psi2 =cp Dimensionless wellbore pseudo pressure Dimensionless wellbore pseudo pressure for laminar flow Dimensionless wellbore pseudo pressure for non-Darcy flow Flow rate, MScf/D Radius, ft Wellbore radius, ft Drainage area radius, ft Laplace domain operator Skin factor Temperature, R Time, hr Dimensionless time Velocity, ft/sec Any point in the X-direction in the reservoir Any point in the Y-direction in the reservoir Any point in the Z-direction in the reservoir The X-coordinate of production point The Y- coordinate of production point The Z-coordinate of production point Distance to reservoir side boundary, ft Distance to reservoir side boundary, ft Storativity Density, Ib=ft 3 Gas density, Ib=ft 3 Interporosıty Porosity Viscosity, cp Diffusivity coefficient, ft2/sec Gas specific gravity
Nomenclatures Appendix-A B h CD ct dP DP k kH
Oil formation volume factor, RB/STB Formation height, ft Dimensionless wellbore storage coefficient Total compressibility, psi1 Pressure derivative, psi Pressure difference, psi Permeability, md Horizontal Permeability, md
Pressure drop caused by horizontal wells depleting finite reservoirs can be obtained using line source solution. This solution is mathematically derived by integrating the point source solution 0 from ðxwD 1Þ to ðxwD þ 1Þ with respect to ðxwD Þ. The point source solution in finite reservoirs having rectangular drainage area is given by (Ozkan, 1988):
(
pffiffiffi
pffiffiffi ) 3 2 cosh u ðyeD jyD ywD jÞ þcosh u ðyeD ðyD ywD ÞÞ
pffiffiffi pffiffiffi þ usinh u yeD 7 6 7 6 qffiffiffiffiffiffiffiffiffiffiffiffi 7 6 p ffiffiffiffiffiffiffiffiffiffiffiffi 8 9 7 6 2 2 7 6 P < = 0 ðy ÞÞ ðy cosh jy y jÞ þcosh ðy y uþa uþa eD D wD eD D wD xwD xD 7 6 ∞ 7 62 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi cos np cos np þ 7 6 n¼1 2 2 : ; x x eD eD uþa sinh uþa yeD 7 6 7 p 6 7 6 q ffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffi 8 9 P PD ¼ 7 6 7 xeD s 6 ∞
(A-1)
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The integration of the above mentioned model gives the line source solution:
PPD ¼ PPD1 þ PPD2 þ PPD3 þ PPD4
(A-2)
PDz ¼
2
∞ X 1
p2 L2D l¼1 l2
cosðlpzwD ÞcosðlpðzD LD þ zwD ÞÞ
(A-15)
The four separated parts are (Ozkan, 1988):
PD1 ¼
PD2
pffiffiffi
pffiffiffi u ðyeD jyD ywD jÞ þ cosh u ðyeD ðyD ywD ÞÞ
pffiffiffi pffiffiffi usinh u yeD
p cosh SxeD
(A-3)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ∞ u þ a2 ðyeD ðyD ywD ÞÞ u þ a2 ðyeD jyD ywD jÞ þ cosh 2X 1 np xwD xD 4cosh 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin cos np cos np ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s n¼1 n xeD xeD xeD u þ a2 y u þ a2 sinh
(A-4)
eD
PD3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ∞ cosh u þ b2 ðyeD jyD ywD jÞ þ cosh u þ b2 ðyeD ðyD ywD ÞÞ 2p X 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cosðlpzwD ÞcosðlpzD Þ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi SxeD u þ b2 y u þ b2 sinh l¼1
PD4 ¼
a¼
∞ ∞ X 8X 1 np x x sin cos np wD cos np D cosðlpzwD ÞcosðlpzD Þ S n xeD xeD xeD n¼1 l¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosh u þ c2 ðyeD jyD ywD jÞ þ cosh u þ c2 ðyeD ðyD ywD ÞÞ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ a2 yeD u þ c2 sinh
(A-7)
b ¼ lpLD
∞ X ∞ x 32 X 1
sin np eD 2 p3 xeD n¼1 m¼1 n n2 x2eD þ m2 y2eD x n p y ðx x þ xwD Þ cos mp wD cos np wD cos 2 2 D eD 2 mp cos ðyD yeD þ ywD Þ 2
(A-8) np xeD
(A-16) (A-9) x sin np eD 2 eD n¼1 l¼1 n n2 x2 þ 4l2 L2 eD D x np ðx x þ xwD Þ cosðlpzwD Þ cos np wD cos 2 2 D eD cosðlpðzD LD þ zwD ÞÞ
PDxz ¼ u ¼ Se2s for homogenous formation
(A-10)
u ¼ Sf ðSÞe2s for non homogenous formation ðdual porosity permeability porous mediaÞ (A-11) f ðSÞ ¼
PDx ¼
swð1 wÞ þ l sð1 wÞ þ l
(A-12)
32
∞ X ∞ X
p3 x
1
(A-17)
1 y cos mp wD 2 2 2 2 2 m¼1 l¼1 m yeD þ 4l LD mp ðyD y þ yÞ cosðlpzwD Þ cosðlpðzD LD þ zwD ÞÞ cos 2
PDyz ¼
∞ X ∞ 16 X
p2
(A-18)
∞ x x n p 16 X 1 ðx x sin np eD cos np wD cos 3 3 3 2 2 2 D eD p xeD n¼1 n þ xwD Þ
∞ X ∞ X ∞ 64 X 1 p3 xeD n¼1 m¼1 l¼1 n n2 x2 þ m2 y2 þ 4lL2 eD eD D x x n p ðxD xeD þ xwD Þ sin np eD cos np wD cos 2 2 2 ywD mp cos ðyD yeD þ ywD Þ cosðlpzwD Þ cos mp 2 2 cosðlpðzD LD þ zwD ÞÞ
PDxyz ¼ (A-13)
PDy ¼
(A-6)
PDxy ¼
np xeD
c ¼ lpLD þ
(A-5)
eD
∞ 8 X 1 ywD mp cos ðy cos m p y þ y Þ D eD wD 2 2 p2 x2eD m¼1 m2
(A-14)
(A-19)
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