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Pseudo-steady state analysis in fractured tight oil reservoirs
Journal of Petroleum Science and Engineering 129 (2015) 40–47
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Pseudo-steady state analysis in fractured tight oil reservoirs Obinna Daniel Ezulike n, Ebrahim Ghanbari, Shahab Siddiqui, Hassan Dehghanpour Petroleum Engineering, University of Alberta, Edmonton, AB, Canada T6G 2W2
art ic l e i nf o
a b s t r a c t
Article history: Received 3 April 2014 Accepted 9 January 2015 Available online 19 January 2015
The log–log plots of normalized rate versus material balance time of two fractured horizontal wells completed in the Cardium and Bakken formations show three distinct regions of quarter, half and unit slopes. The first two regions can be interpreted by the existing semi-analytical models for bi-linear and linear transient flow. This study develops a pseudo-steady state (PSS) model for linear dual porosity reservoirs that describes the third region (unit slope). The application of this PSS model on field production data from two tight oil wells yields reasonable estimates of hydraulic fracture half-length and average matrix permeability. Also, this paper demonstrates how the proposed PSS analysis complements the transient linear analysis commonly used in industry. & 2015 Elsevier B.V. All rights reserved.
Keywords: Pseudo-steady state flow Tight reservoirs Estimate fracture half-length and average matrix permeability Quick analysis tool
1. Introduction Unconventional (tight oil, tight gas, shale gas and coal-bed methane) resources have emerged as important sources of energy supply in the United States and Canada (Franz and Jochen, 2005). These resources have low matrix permeability and require large reservoir contact area per well to achieve economic production (Ning et al., 1993). A combination of horizontal drilling and hydraulic fracturing stimulation provides the required reservoir contact area to justify economic production from these tight reservoirs (Wang et al., 2008; Medeiros et al., 2010). In contrast to conventional vertical wells, fractured horizontal wells demonstrate a complex rate and pressure transient behavior. Analyzing the production data of various fractured tight reservoirs indicates the development of extended linear flow regimes. Linear flow may occur when a square geometry reservoir drains into hydraulic fractures or a tight layer drains into a high permeability layer (Wattenbarger, 2007; Bello and Wattenbarger, 2010; Clarkson and Pedersen, 2010; Al-Ahmadi et al., 2010). El-Banbi (1998) presented a linear dual porosity model to describe flow in various naturally and artificially fractured reservoirs. Bello (2009) extended this linear dual porosity model to describe gas depletion in hydraulically fractured shale gas reservoirs. He identified five possible flow regimes, and developed approximate analytical solutions for the transient linear and bilinear regions, excluding the fifth pseudo-steady depletion region. n
Corresponding author. E-mail addresses: [email protected] (O. Daniel Ezulike), [email protected] (E. Ghanbari), [email protected] (S. Siddiqui), [email protected] (H. Dehghanpour). http://dx.doi.org/10.1016/j.petrol.2015.01.009 0920-4105/& 2015 Elsevier B.V. All rights reserved.
Similar models have been developed to describe transient flow through more complicated systems composed of hydraulic fractures, reactivated natural fractures and matrix blocks (Ozkan et al., 2010; Al-Ahmadi, Wattenbarger, 2011; Dehghanpour and Shirdel, 2011; Ali et al., 2013). Although linear transient flow regime is dominant in most of the multifractured horizontal wells, recent studies indicate the occurrence of a pseudo-steady state (PSS) flow regime after late linear transient region (Clarkson and Pedersen, 2010, 2011;Kabir et al., 2011; Samandarli et al., 2011; Song et al., 2011). During PSS (boundary dominated) flow, virtual no-flow boundaries develop between adjacent hydraulic fractures and the average pressure in the stimulated reservoir volume drops with time at a constant rate. However, due to the existence of possible fluid influx beyond the fracture tip, the change in pressure is not quite linear with time. As a result the derivative plot may show a slope which is slightly less than one (Samandarli et al., 2012). Most of the existing analytical models capture the transient regions (Bello and Wattenbarger, 2010; Ali et al., 2013) and there is a need for an approximate analytical model for quick analysis of PSS data observed in hydraulically fractured tight oil reservoirs. In this paper, we combine the continuity and linear diffusivity equations and arrive at a linear relationship between rate normalized pressure (RNP) and material balance time (MBT) for the flow region 5 from the linear dual porosity model (Bello, 2009). This paper applies the linear relationship to analyze the PSS data of two fractured horizontal wells completed in Cardium and Bakken Formations. Also, it shows how the combined analysis of production data related to linear transient and pseudo steady state regions gives reasonable estimates of average matrix permeability and fracture half length.
O. Daniel Ezulike et al. / Journal of Petroleum Science and Engineering 129 (2015) 40–47
Nomenclature Acw B c h k Lf Np Nf P qo Qo MBT Vm Xe
well-face cross-sectional area to flow, m2 formation volume factor, m3 =s m3 1 compressibility of rock, kpa reservoir thickness, m permeability, m2 fracture spacing, m cumulative oil production, STB number of fractures pressure, kPa oil rate produced from one fracture, m3/s daily oil rate, STB=day material balance time, day matrix control volume, m3 length of horizontal well, m
ye φ ρ μ
41
fracture half-length, m porosity density; kg=m3 viscosity, cP
Subscript i m f o wf t r
initial matrix fracture oil bottom-hole flowing total rock
2. Field data
3. Model geometry and assumptions
Figs. 1 and 2 show the log–log plot of normalized rate versus material balance time for two tight oil wells completed in Cardium and Bakken formations. The data show three distinct flow regions. Region 1 with an approximate negative quarter slope represents bi-linear transient flow (Xu et al., 2012; Bello, 2009). Region 2, with a negative half slope, represents linear transient flow, and is the dominant flow regime. Although material balance time successfully eliminates the effect of variable-rate production on field data (Medeiros et al., 2010), its application for analyzing transient flow regimes has not been proved mathematically. However, it has been shown recently (Samandarli et al., 2012) that bi-linear and linear transient regions keep their characteristic slopes when plotted versus material balance time (as shown in Figs. 1 and 2). This paper focuses on Region 3 (negative unit slope). Based on previous works (Song et al., 2011; Clarkson and Pedersen, 2011; Song and Ehlig-Economides, 2011; Samandarli et al., 2012), we hypothesize that this region represents PSS flow due to the pressure interference between the consecutive hydraulic fractures, and insufficient fluid influx from the formation beyond the fracture tip.
Fig. 3 and 4 shows the conceptual linear dual porosity model which comprises hydraulic fractures and matrix blocks. The key assumptions include (i) equally spaced hydraulic fractures, (ii) which are perpendicular to the horizontal well, (iii) this well is located at the center of rectangular drainage area, (iv) simultaneous occurrence of pressure interference at the center of all matrix blocks, (v) single phase flow perpendicular to fractures (vi) negligible flow from matrix to well, (vii) negligible stress, temperature and secondary fracture effects on reservoir depletion.
Fig. 1. Log–log plot of daily normalised oil rate versus material balance time for Cardium well (flow regions are based on constant pressure draw-down assumption).
4. Mathematical solution The PSS model for analyzing flow region 5 from linear dual porosity model (Bello, 2009) is derived by combining material balance and linear diffusivity equations. The control volume (Vm) shown in Fig. 5, represents a fraction of reservoir volume feeding one fracture. When the pressure interference occurs between adjacent fractures, the no-flow boundaries appear at the distance of Lf =2 from the fracture faces (Song et al., 2011), where Lf is the fracture spacing. Applying mass balance on V m , the change in
Fig. 2. Log–log plot of daily normalised oil rate versus material balance time for Bakken well (flow regions are based on constant pressure draw-down assumption).
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Fig. 3. Plan view of reservoir showing hydraulic fractures connected to the horizontal well. The direction of flow is from the matrix to the fractures and from the fractures to the well (arrows indicate the flow direction).
2003; Mattar and Anderson, 2003; Palacio and Blasingame, 1993). Integrating, and applying the boundary conditions, the matrix pressure as a function of space during the PSS flow becomes: μo q μo qo Lf P m ðxÞ ¼ x þ Pf ð3Þ o x2 þ km 2 Vm km V m 2 Eq. (3) represents pressure profile in the matrix blocks where P f denotes fluid pressure in the fracture. Averaging the pressure over Vm results (Fig. 5) in average matrix pressure: Pm ¼
μBo qo Lf 2 þP f 12 km V m
ð4Þ
The derivation details are given in Appendix B. 4.2. Average matrix pressure
Fig. 4. Front view of reservoir showing hydraulic fractures connected to the horizontal well qt.
matrix pressure with respect to time is given by dP m qo ¼ dt V m φm C t
ð1Þ
Eq. (1) is the well-known material balance equation and detailed derivations are presented in Appendix A. 4.1. Linear diffusivity equation Diffusivity equation for linear flow from the matrix towards the fracture is given by 2 d Pm φm μo ct dP m ð2Þ ¼ km dt dx2 dP m =dt in Eq. (2) is constant during pseudo-steady depletion (Fig. 6) and can be replaced by Eq. (1). A similar approach has been used to solve pressure in space during pseudo-steady depletion of conventional oil or gas reservoirs into vertical wells (Lee et al.,
The primary drive mechanism during PSS flow is oil expansion due to pressure depletion in the entire reservoir volume (Vtm). Therefore, total oil (qt) production rate is proportional to oil compressibility and the change in pressure with respect to time (Eq. C-1). Appendix C shows that combining Eq. (4) and Eq. C-3 results in the following linear relationship between fracture pressure and cumulative oil prodution: Pi Pf ¼
μBo L2f qt Bo Np þ ðV t φC t Þm 12km V tm
ð5Þ
During PSS flow, the pressure drop in highly-conductive hydraulic fractures is assumed to be negligible (Bello and Wattenbarger, 2008). Therefore, fracture pressure (Pf ) can be replaced by wellbore flowing pressure (Pwf). After substituting for total matrix volume (Vtm), Eq. 5 becomes: P i P wf μBo Lf Np Bo ¼ð Þ þð Þ qt ð2ye hN f Lf Þðφct Þm qt 24 km ye hN f
ð6Þ
Eq. (6) is analogous to the linear pressure drop relationship developed for pseudo-steady state flow in vertical wells (Lee et al., 2003; Mattar and Anderson, 2003; Palacio and Blasingame, 1993). In petroleum literature, ðPi P wf =qt Þ and ðN p =qt Þ are defined as rate normalized pressure (RNP) and material balance time (MBT), respectively (Palacio and Blasingame, 1993). Eq. (6) can be
O. Daniel Ezulike et al. / Journal of Petroleum Science and Engineering 129 (2015) 40–47
simplified as: RNP ¼ mss MBT þ bss
ð7Þ
where mss and bss are the slope and intercept from the best linear fit of RNP versus MBT data on a Cartesian plot: mss ¼
Bo 2 X e h ye ðφct Þm
ð8Þ
where Xe ENf Lf and stimulated reservoir volume (SRV)E 2Xeyeh The intercept is given by bss ¼
μBo Lf 24 km ye hN f
ð9Þ
Using the slope of RNP versus MBT, we can calculate ye and km.
43
1998; Clarkson and Pedersen, 2011) in 18 stages of hydraulic fracturing and horizontal length of 1370 m. This well has been producing for 2 years and the reservoir pressure is above bubble point. Obtained horizontal and vertical permeabilities from well testing analysis are 1.34 md and 9.03 10 4 md, respectively. Negligible vertical permeability compared with horizontal permeability justifies the assumption of horizontal flow towards vertical hydraulic fractures. Table 1 shows the pertinent features of this reservoir. Fig. 7 shows the linear relation between RNP and MBT for this well: RNP ¼ 0:058 MBT þ0:35 Using Eq. (8) and the line slope, we calculate ye: ye ¼ 243 m
5. Field data analysis The proposed model leads to a simple analytical procedure for analyzing PSS regions observed in the production data of fracture tight oil reservoirs:
Using Eq. (9) and the line intercept, we calculate average matrix permeability: km ¼ 173:7 mD These calculations considered the appropriate conversion factors.
1. Make rate normalised pressure (RNP) against material balance time (MBT) production data plot on log–log scale. 2. Identify various flow regions. PSS flow region should be recognized by a unit slope. 3. Plot RNP versus MBT of the data points of PSS region on a Cartesian graph. 4. Record the line slope, mss and the line intercept, bss. 5. Calculate ye by substituting mss in ye ¼
Bo 2 mss X e h ðφct Þm
6. Calculate the average permeability of the matrix by substituting bss in km ¼
5.2. Bakken well: This well was completed in Bakken formation (Hlidek and Rieb, 2011; Clarkson and Pedersen, 2011; Alcoser et al., 2012) in 16 stages of hydraulic fracturing and horizontal length of 1707 m. This well has been producing for 3 years and the reservoir pressure is above bubble point. Table 2 shows the pertinent features of this reservoir. Fig. 8 shows the linear relation between RNP and MBT for well: RNP ¼ 0:0521 MBT þ 8:0981 By using the line slope and intercept relationship from Eqs. (8 and 9), ye and km are given by ye ¼ 80 m km ¼ 0:15 mD These calculations considered the appropriate conversion factors.
μBo Lf 24bss ye hN f
We demonstrate the application of the presented procedure in calculating ye and km of two wells completed in Cardium and Bakken formations: 5.1. Cardium well This well was completed in Cardium formation (Justen, 1957; MacKenzie, 1975; Purvis and Bober, 1979; Krasey and Fawcett,
Table 1 Reservoir and fluid properties of well in Cardium formation. Reservoir type Dominant flow Multiphase flow in reservoir Reservoir pressure (Pi) Flowing BHP (Pwf) Number of stages of fractures Length of horizontal well (Χe) Permeability—horizontal(khor) Permeability—vertical (kver) Reservoir thickness (h) Matrix porosity (φm ) Oil formation volume factor (Bo) Oil viscosity (μo) Total compressibility (ct) (assumed)
Homogeneous Oil phase No 15575 kPa 7413 kPa 18 1370 m 1.34 mD 9.03 10 4 mD 7m 0.108 1.221 m3/s m3 1.13 cP 3.91 10-5 kPa 1
Fig. 5. Scheme of the selected control volume showing a hydraulic fracture surrounded by the matrix blocks. Two no-flow boundaries are virtually created at the center of adjacent fractures at distance of 0.5 Lf from the fracture.
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Table 2 Reservoir and fluid properties of well in Bakken formation. Reservoir type Dominant flow Multiphase flow in reservoir Reservoir pressure (Pi) Flowing BHP (Pwf) Number of stages of fractures Length of horizontal well (Χe) Reservoir thickness (h) Matrix porosity (φm ) Oil formation volume factor (Bo) Oil viscosity (μo) Total compressibility (ct) (assumed)
Fig. 6. The pressure profile in the matrix during the boundary dominated flow. The rate of pressure drop with respect to time remains constant at pseudo-steady state conditions.
Unknown Oil phase No 46884 kPa 4826 kPa 16 1707 m 5.8 m 0.09 1.329 m3/s m3 0.5643 cP 4.02 10 5 kPa 1
Table 3 Reservoir parameters of Cardium and Bakken wells obtained from pseudo-steady state flow analysis. Cardium well
Matrix permeability Stimulated reservoir volume Matrix permeability Stimulated reservoir volume
Bakken well
173.7 mD 4.66MM m3 0.15 mD 1.58MM m3
Fig. 7. Specialized plot of RNP versus MBT for analyzing boundary dominated flow of the well completed in Cardium formation.
6. Discussion of results 6.1. Comparative analysis of PSS and Transient regions Table 4 compares the fracture half-lengths (ye ) for the Bakken well estimated by applying (i) the model proposed in this paper for analyzing PSS region and (ii) the dual porosity model proposed by Bello (2009) for analyzing linear transient (LT) region. The detailed calculations of the ye for LT region of Cardium and Bakken wells are presented in Ezulike and Dehghanpour (2013). The ye calculated using LT model from the Bakken well is higher than that calculated from PSS model. Eq. 9 shows that, ye from PSS analysis does not depend on the matrix permeability (km), while ye from LT analysis is inversely proportional to the square root of permeability. Therefore, LT analysis is expected to overestimate ye due to ignorance of secondary fractures in the dual porosity model, especially when laboratory-measured plug permeability is used as the input matrix permeability. However, PSS analysis should give a more reasonable estimate of ye because it does not depend on km as it is mainly based on a volumetric analysis (tank model). km from intercept analysis of PSS model should be higher than plug permeability because it is an average matrix
Fig. 8. Specialized plot of RNP versus MBT for analyzing boundary dominated flow of the well completed in Bakken formation.
Table 4 Hydraulic fracture half-length from LT and PSS analyses on production data from Bakken well. Analysis equation Estimated half-length (m) PSS LT
80 176
permeability that accounts for possible secondary fractures. Therefore, substitution of km from PSS analysis into the dual porosity analysis equation for LT analysis (Bello, 2009) should give a more reasonable estimate of ye . To put the above discussion into a practical framework we propose the following complementary analyses.
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1- Calculate ye from the slope of the PSS analysis match. Calculate average km by substituting ye into the intercept from the same match. Recalculate ye from the slope of LT analysis match using the km from the PSS match. Compare the two ye values from PSS and LT analysis. 2- Calculate ye from the slope of the PSS analysis match. Calculate average km by substituting ye into the intercept from the same match. Recalculate km from the slope of LT analysis using the ye from the PSS analysis. Compare the two km values from PSS and LT analysis.
45
Application of the proposed model on the pseudo-steady flow region of production data from two fractured tight oil wells yielded reasonable fracture half-length and matrix permeability estimates.
Acknowledgements The authors are grateful to Natural Sciences and Engineering Research Council of Canada and Vanier Canada for funding this study.
6.2. Comparing the analysis results of Bakken and Cardium wells Table 3 shows the summary of our analysis results for Cardium and Bakken wells. The estimated average matrix permeability from the Bakken well is near the upper end value of 0.5 mD as reported by Clarkson and Pedersen (2011). Although the typical average matrix permeability in the Cardium Formation is o 10 mD (Hoch et al., 2003), the overestimation (173.7 mD) from PSS analysis on the Cardium well data could be explained by the assumption of negligible secondary fracture effects. The active secondary fractures (higher permeability) for this Cardium well have significant effects when lumped with matrix. Therefore, this suggests that the Cardium well’s hydraulic fractures have a greater connection with active secondary fractures compared to those of the Bakken well. The same reason cab explain why the half-length estimate from PSS analysis (243 m) is higher than that from LT analysis (168 m) for the Cardium case. The perfect linear relation between RNP and MBT of Cardium well and the scatter observed in the Bakken well plots can be explained by reviewing well and formation characteristics. The horizontal permeability of the Cardium formation is much higher than the vertical one. Also, the geological reports show that the formation thins out in the southern parts of the Cardium well. Therefore, Cardium well would be a good candidate for our analysis that assumes fluid flow is linear toward the fracture faces and there is negligible influx beyond the fracture tip. On the other hand, there are some unknown parameters for Bakken formation that results in some uncertainty in the analysis results. As it can be seen in Table 2, we do not have clear information about geological characteristics and permeabilities in horizontal and vertical directions. Therefore, flow may not necessarily be linear and PSS (boundary dominated). Moreover, most of the horizontal wells in Bakken formation are completed open hole while those in Cardium formation are completed cased hole (Hlidek and Rieb, 2011; Clarkson and Pedersen, 2011). Therefore, Bakken well may benefit from additional inflow from formation directly to the wellbore.
7. Summary A methodology for analyzing late-time production data of lowpermeability oil reservoirs has been proposed which incorporates flow-regime analysis. Average matrix permeability and fracture half-length can be calculated using the production data and proposed model. Following are the main conclusions drawn by this study:
Three flow regimes (transient bi-linear, transient linear and
pseudo-steady state boundary) are observed in the Cardium and Bakken wells studied here. A model is derived for pseudo-steady state flow analysis using a linear dual porosity model and some simplifying assumptions. Average fracture half-length and matrix permeability can be estimated using the slope and intercept from this model.
Appendix A. Derivation of pressure change with respect to time for matrix Starting from material balance equation we have Mass in–Mass out¼ Accumulation in the matrix Assuming that the dominant flow direction is horizontal and towards the fracture surfaces (Ozkan et al., 2009; Wattenbarger, 2007) and considering an under saturated reservoir with single phase and slightly compressible fluid we can write qo ρo Δtj x þ Δx qo ρo Δtj x ¼ ρo φm V m j t
þ Δt ρo φm V m jt
ðA 1Þ
For unconventional reservoirs with low matrix permeabilities, the contribution of the reservoir beyond the stimulated volume is usually negligible (Medeiros et al., 2008). By neglecting contribution of reservoir beyond SRV we can write 0 qo ρo ¼
ρo φm V m j t
þ Δt ρo φm V m jt
Δt
ðA 2Þ
where subscripts o and m denote oil and matrix, respectively. Taking limits as Δt approaches to zero in Eq. (A-2) gives qo ρo ¼
d ρ φ Vm dt o m
qo ρo d ρ φ ¼ dt o m Vm Using chain rule we can write qo ρo dP m d d φm ¼ ρo þρo φm dP m dP m Vm dt Dividing both sides by ϕm and ρo gives qo dP m 1 d 1 d ¼ ρo þ φm φm dP m V m φm dt ρo dP m
ðA 3Þ ðA 4Þ
ðA 5Þ
ðA 6Þ
qo dP m ¼ ðc o þ c r Þ V m φm dt
ðA 7Þ
qo dP m ¼ ðc t Þ V m φm dt
ðA 8Þ
dP m qo ¼ dt V m φm C t
ðA 9Þ
where V m is the control matrix volume given by V m ¼ 2ye h Nf Lf
ðA 10Þ
Appendix B. Derivation of pressure profile in matrix block Considering linear diffusivity equation in the control volume (Vm in Fig. 5), we can write 2 d Pm φm μo ct dP m ðB 1Þ ¼ km dt dx2
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O. Daniel Ezulike et al. / Journal of Petroleum Science and Engineering 129 (2015) 40–47
During pseudo-steady depletion, dP m =dt can be replaced by Eq. (A-9) 2 d Pm φm μ o c t qo ¼ ðB 2Þ km V m φm c t dx2
Also, since the integral on the left hand side corresponds to the cumulative production, we can write,
Integrating both sides gives dP m μo q ¼ o x þ C1 dx km Vm
Replacing P m from Eq. (B-12) and accounting for the total flow rate from all fractures and total matrix volume (Vtm = Nf Vm and qt¼Nf qo)
ðB 3Þ
Applying the first boundary condition At ¼
Pi Pm ¼
Pi Pf ¼
Bo N p ðV t φC t Þm
μBo L2f qt Bo Np þ ðV t φC t Þm 12km V tm
ðC 3Þ
ðC 4Þ
Eq. (C-4) can be used for constant rate only. To extend this equation for variable rate and pressure, we divide it by qt , as follow:
Lf dP m ; ¼0 2 dx
C1 is given by μo qo Lf C1 ¼ km V m 2
ðB 4Þ
μBo L2f Pi Pf Np Bo ¼ð Þ þð Þ qt ðV t φct Þm qt 12km V tm
ðC 5Þ
In which V m ¼ 2ye hN f Lf Substituting C1 from Eq. (B-4) into Eq. (B-3), we obtain dP m μo q μo qo Lf ¼ ðB 5Þ o xþ dx km Vm km V m 2 Z
Integrating both sides of Eq. (B-5) Z Z μo q μo qo Lf dx dP m ¼ o xdx þ km Vm km V m 2
μo Pm ¼ km
q o x2 þ 2 Vm
ðB 6Þ
μ o qo L f x þ C2 km V m 2
ðB 7Þ
And the second boundary condition is At x ¼ 0, P m ¼ P f By applying second boundary condition to Eq. (B-7), we obtain: C2 ¼ P f
ðB 8Þ
By replacing obtained value for C2 into Eq. (B-7), we arrive at: μo q μo qo Lf x þ Pf o x2 þ ðB 9Þ Pm ¼ km 2 Vm km V m 2 Average pressure of the control volume V m is given by R L2f
P m dV m R L2f 0 dV m
Pm ¼
0
dV m ¼ 2 ye h dx
ðB 10Þ
ðB 11Þ
Replacing P m from Eq. (B-9) and using differential form of control volume, the average matrix pressure becomes: Pm ¼
μBo qo Lf 2 þ Pf 12 km V m
ðB 12Þ
Appendix C. Derivation of final solution for boundary dominated region From the compressibility definition it can be shown that (Palacio and Blasingame, 1993): qt ¼
AφhC t dp Bo dt
ðC 1Þ
Since oil is the only phase flowing in the total reservoir volume (Vtm), C t is assumed to be constant. Thus the integration of Eq. (C1) produces Z t Z ðV t φC t Þm p qt dt ¼ dp ðC 2Þ Bo 0 pi
Pi Pf μBo Lf Np Bo Þ ¼ð Þ þð qt ð2ye hN f Lf Þ ðφct Þm qt 12 km ð2ye hN f Þ
ðC 6Þ
High permeability hydraulic fractures are assumed to have negligible pressure drop during PSS flow (Bello and Wattenbarger, 2008) therefore we can replace fracture pressure (P f ) with flowing bottom hole pressure (P wf ). P i P wf μBo Lf Np Bo Þ ¼ð Þ þð qt ð2ye hN f Lf Þ ðφct Þm qt 12 km ð2ye hNf Þ
ðC 7Þ
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Ozkan, E., Brown, M., Raghavan, R., Kazemi, H. 2009.Comparison of fractured horizontal-well performance in conventional and unconventional reservoirs. Paper SPE121290, presented at SPE Western Regional Meeting, San Jose, California, 24–26 March. http://dx.doi.org/10.2118/121290-MS. Ozkan, E., Raghavan, R., Apaydin, O.G. 2010.Modeling of fluid transfer from shale matrix to fracture network. In: Paper SPE 134830 Presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy, 19-22 September. http:// dx.doi.org/10.2118//134830-MS. Palacio, J.C., Blasingame, T.A. 1993. Dec line-curve analysis using type curves— analysis of gas well production data. In: Oral Presentation Given at the Rocky Mountain Regional/Low Permeability Reservoirs Symposium and Exhibition, Denver, Colorado, 12-14 April.10.2118/25909-MS. Purvis, R.A., Bober, W.G., 1979. A reserves review of the Pembina Cardium oil pool. J. Can. Pet. Technol. 18 (3), 20–34. http://dx.doi.org/10.2118/79-03-02 (PETSOC79-03-02). Samandarli, O., Al-Ahmadi, H., Wattenbarger, R.A. 2011.A new method for history matching and forecasting shale gas reservoir production performance with a dual porosity model. In: Paper SPE-144335-MS Presented at the SPE North American Unconventional Gas Conference and Exhibition, The Woodlands, Texas, USA, 14-16 June. http://dx.doi.org/10.2118/144335-MS. Samandarli, O., Valbuena, E., Ehlig-Economides, C. 2012. Production data analysis in unconventional reservoirs with rate-normalized pressure (RNP): theory, methodology, and applications. In: Paper SPE 155614 Presented at Americas Unconventional Resources Conference, Pittsburgh, Pennsylvania USA, 5-7 June. http://dx.doi.org/10.2118/155614-MS. Song, B., Economides, M.J., Ehlig-Economides, C. 2011.Design of multiple transverse fracture horizontal wells in shale gas reservoirs. In: Paper SPE-140555-MS Presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA, 24-26 January. http://dx.doi.org/10.2118/140555-MS. Song, B., Ehlig-Economides, C. 2011. Rate-normalized pressure analysis for determination of shale gas well performance. In: Paper SPE-144031-MS Presented at the SPE North American Unconventional Gas Conference and Exhibition, The Woodlands, Texas, USA, 14–16 June. http://dx.doi.org/10.2118/144031-MS. Wattenbarger, R.A., 2007. Some reservoir performance aspects of unconventional gas production. In: Private Conference Presentation, Lake Louise, AB, Canada. Wang, Y., Holditch, S.A., McVay, D.A. 2008. Simulation of Gel Damage on FractureFluid Cleanup and Long-Term Recovery in Tight-Gas Reservoirs. Paper SPE 117444 Presented at Eastern Regional/AAPG Eastern Section Joint Meeting, Pittsburgh, Pennsylvania, USA, 11–15 October. http://dx.doi.org/10.2118/ 117444-MS. Xu, B., Haghighi, M., Cooke, D. Li, X. 2012. Production data analysis in Eagle Ford shale gas reservoir. In: Paper SPE 153072 Presented at SPE/EAGE European Unconventional Resources Conference and Exhibition, Vienna, Austria, 20–22 March. http://dx.doi.org/10.2118/153072-MS.
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Pseudo-steady state analysis in fractured tight oil reservoirs Obinna Daniel Ezulike n, Ebrahim Ghanbari, Shahab Siddiqui, Hassan Dehghanpour Petroleum Engineering, University of Alberta, Edmonton, AB, Canada T6G 2W2
art ic l e i nf o
a b s t r a c t
Article history: Received 3 April 2014 Accepted 9 January 2015 Available online 19 January 2015
The log–log plots of normalized rate versus material balance time of two fractured horizontal wells completed in the Cardium and Bakken formations show three distinct regions of quarter, half and unit slopes. The first two regions can be interpreted by the existing semi-analytical models for bi-linear and linear transient flow. This study develops a pseudo-steady state (PSS) model for linear dual porosity reservoirs that describes the third region (unit slope). The application of this PSS model on field production data from two tight oil wells yields reasonable estimates of hydraulic fracture half-length and average matrix permeability. Also, this paper demonstrates how the proposed PSS analysis complements the transient linear analysis commonly used in industry. & 2015 Elsevier B.V. All rights reserved.
Keywords: Pseudo-steady state flow Tight reservoirs Estimate fracture half-length and average matrix permeability Quick analysis tool
1. Introduction Unconventional (tight oil, tight gas, shale gas and coal-bed methane) resources have emerged as important sources of energy supply in the United States and Canada (Franz and Jochen, 2005). These resources have low matrix permeability and require large reservoir contact area per well to achieve economic production (Ning et al., 1993). A combination of horizontal drilling and hydraulic fracturing stimulation provides the required reservoir contact area to justify economic production from these tight reservoirs (Wang et al., 2008; Medeiros et al., 2010). In contrast to conventional vertical wells, fractured horizontal wells demonstrate a complex rate and pressure transient behavior. Analyzing the production data of various fractured tight reservoirs indicates the development of extended linear flow regimes. Linear flow may occur when a square geometry reservoir drains into hydraulic fractures or a tight layer drains into a high permeability layer (Wattenbarger, 2007; Bello and Wattenbarger, 2010; Clarkson and Pedersen, 2010; Al-Ahmadi et al., 2010). El-Banbi (1998) presented a linear dual porosity model to describe flow in various naturally and artificially fractured reservoirs. Bello (2009) extended this linear dual porosity model to describe gas depletion in hydraulically fractured shale gas reservoirs. He identified five possible flow regimes, and developed approximate analytical solutions for the transient linear and bilinear regions, excluding the fifth pseudo-steady depletion region. n
Corresponding author. E-mail addresses: [email protected] (O. Daniel Ezulike), [email protected] (E. Ghanbari), [email protected] (S. Siddiqui), [email protected] (H. Dehghanpour). http://dx.doi.org/10.1016/j.petrol.2015.01.009 0920-4105/& 2015 Elsevier B.V. All rights reserved.
Similar models have been developed to describe transient flow through more complicated systems composed of hydraulic fractures, reactivated natural fractures and matrix blocks (Ozkan et al., 2010; Al-Ahmadi, Wattenbarger, 2011; Dehghanpour and Shirdel, 2011; Ali et al., 2013). Although linear transient flow regime is dominant in most of the multifractured horizontal wells, recent studies indicate the occurrence of a pseudo-steady state (PSS) flow regime after late linear transient region (Clarkson and Pedersen, 2010, 2011;Kabir et al., 2011; Samandarli et al., 2011; Song et al., 2011). During PSS (boundary dominated) flow, virtual no-flow boundaries develop between adjacent hydraulic fractures and the average pressure in the stimulated reservoir volume drops with time at a constant rate. However, due to the existence of possible fluid influx beyond the fracture tip, the change in pressure is not quite linear with time. As a result the derivative plot may show a slope which is slightly less than one (Samandarli et al., 2012). Most of the existing analytical models capture the transient regions (Bello and Wattenbarger, 2010; Ali et al., 2013) and there is a need for an approximate analytical model for quick analysis of PSS data observed in hydraulically fractured tight oil reservoirs. In this paper, we combine the continuity and linear diffusivity equations and arrive at a linear relationship between rate normalized pressure (RNP) and material balance time (MBT) for the flow region 5 from the linear dual porosity model (Bello, 2009). This paper applies the linear relationship to analyze the PSS data of two fractured horizontal wells completed in Cardium and Bakken Formations. Also, it shows how the combined analysis of production data related to linear transient and pseudo steady state regions gives reasonable estimates of average matrix permeability and fracture half length.
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Nomenclature Acw B c h k Lf Np Nf P qo Qo MBT Vm Xe
well-face cross-sectional area to flow, m2 formation volume factor, m3 =s m3 1 compressibility of rock, kpa reservoir thickness, m permeability, m2 fracture spacing, m cumulative oil production, STB number of fractures pressure, kPa oil rate produced from one fracture, m3/s daily oil rate, STB=day material balance time, day matrix control volume, m3 length of horizontal well, m
ye φ ρ μ
41
fracture half-length, m porosity density; kg=m3 viscosity, cP
Subscript i m f o wf t r
initial matrix fracture oil bottom-hole flowing total rock
2. Field data
3. Model geometry and assumptions
Figs. 1 and 2 show the log–log plot of normalized rate versus material balance time for two tight oil wells completed in Cardium and Bakken formations. The data show three distinct flow regions. Region 1 with an approximate negative quarter slope represents bi-linear transient flow (Xu et al., 2012; Bello, 2009). Region 2, with a negative half slope, represents linear transient flow, and is the dominant flow regime. Although material balance time successfully eliminates the effect of variable-rate production on field data (Medeiros et al., 2010), its application for analyzing transient flow regimes has not been proved mathematically. However, it has been shown recently (Samandarli et al., 2012) that bi-linear and linear transient regions keep their characteristic slopes when plotted versus material balance time (as shown in Figs. 1 and 2). This paper focuses on Region 3 (negative unit slope). Based on previous works (Song et al., 2011; Clarkson and Pedersen, 2011; Song and Ehlig-Economides, 2011; Samandarli et al., 2012), we hypothesize that this region represents PSS flow due to the pressure interference between the consecutive hydraulic fractures, and insufficient fluid influx from the formation beyond the fracture tip.
Fig. 3 and 4 shows the conceptual linear dual porosity model which comprises hydraulic fractures and matrix blocks. The key assumptions include (i) equally spaced hydraulic fractures, (ii) which are perpendicular to the horizontal well, (iii) this well is located at the center of rectangular drainage area, (iv) simultaneous occurrence of pressure interference at the center of all matrix blocks, (v) single phase flow perpendicular to fractures (vi) negligible flow from matrix to well, (vii) negligible stress, temperature and secondary fracture effects on reservoir depletion.
Fig. 1. Log–log plot of daily normalised oil rate versus material balance time for Cardium well (flow regions are based on constant pressure draw-down assumption).
4. Mathematical solution The PSS model for analyzing flow region 5 from linear dual porosity model (Bello, 2009) is derived by combining material balance and linear diffusivity equations. The control volume (Vm) shown in Fig. 5, represents a fraction of reservoir volume feeding one fracture. When the pressure interference occurs between adjacent fractures, the no-flow boundaries appear at the distance of Lf =2 from the fracture faces (Song et al., 2011), where Lf is the fracture spacing. Applying mass balance on V m , the change in
Fig. 2. Log–log plot of daily normalised oil rate versus material balance time for Bakken well (flow regions are based on constant pressure draw-down assumption).
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O. Daniel Ezulike et al. / Journal of Petroleum Science and Engineering 129 (2015) 40–47
Fig. 3. Plan view of reservoir showing hydraulic fractures connected to the horizontal well. The direction of flow is from the matrix to the fractures and from the fractures to the well (arrows indicate the flow direction).
2003; Mattar and Anderson, 2003; Palacio and Blasingame, 1993). Integrating, and applying the boundary conditions, the matrix pressure as a function of space during the PSS flow becomes: μo q μo qo Lf P m ðxÞ ¼ x þ Pf ð3Þ o x2 þ km 2 Vm km V m 2 Eq. (3) represents pressure profile in the matrix blocks where P f denotes fluid pressure in the fracture. Averaging the pressure over Vm results (Fig. 5) in average matrix pressure: Pm ¼
μBo qo Lf 2 þP f 12 km V m
ð4Þ
The derivation details are given in Appendix B. 4.2. Average matrix pressure
Fig. 4. Front view of reservoir showing hydraulic fractures connected to the horizontal well qt.
matrix pressure with respect to time is given by dP m qo ¼ dt V m φm C t
ð1Þ
Eq. (1) is the well-known material balance equation and detailed derivations are presented in Appendix A. 4.1. Linear diffusivity equation Diffusivity equation for linear flow from the matrix towards the fracture is given by 2 d Pm φm μo ct dP m ð2Þ ¼ km dt dx2 dP m =dt in Eq. (2) is constant during pseudo-steady depletion (Fig. 6) and can be replaced by Eq. (1). A similar approach has been used to solve pressure in space during pseudo-steady depletion of conventional oil or gas reservoirs into vertical wells (Lee et al.,
The primary drive mechanism during PSS flow is oil expansion due to pressure depletion in the entire reservoir volume (Vtm). Therefore, total oil (qt) production rate is proportional to oil compressibility and the change in pressure with respect to time (Eq. C-1). Appendix C shows that combining Eq. (4) and Eq. C-3 results in the following linear relationship between fracture pressure and cumulative oil prodution: Pi Pf ¼
μBo L2f qt Bo Np þ ðV t φC t Þm 12km V tm
ð5Þ
During PSS flow, the pressure drop in highly-conductive hydraulic fractures is assumed to be negligible (Bello and Wattenbarger, 2008). Therefore, fracture pressure (Pf ) can be replaced by wellbore flowing pressure (Pwf). After substituting for total matrix volume (Vtm), Eq. 5 becomes: P i P wf μBo Lf Np Bo ¼ð Þ þð Þ qt ð2ye hN f Lf Þðφct Þm qt 24 km ye hN f
ð6Þ
Eq. (6) is analogous to the linear pressure drop relationship developed for pseudo-steady state flow in vertical wells (Lee et al., 2003; Mattar and Anderson, 2003; Palacio and Blasingame, 1993). In petroleum literature, ðPi P wf =qt Þ and ðN p =qt Þ are defined as rate normalized pressure (RNP) and material balance time (MBT), respectively (Palacio and Blasingame, 1993). Eq. (6) can be
O. Daniel Ezulike et al. / Journal of Petroleum Science and Engineering 129 (2015) 40–47
simplified as: RNP ¼ mss MBT þ bss
ð7Þ
where mss and bss are the slope and intercept from the best linear fit of RNP versus MBT data on a Cartesian plot: mss ¼
Bo 2 X e h ye ðφct Þm
ð8Þ
where Xe ENf Lf and stimulated reservoir volume (SRV)E 2Xeyeh The intercept is given by bss ¼
μBo Lf 24 km ye hN f
ð9Þ
Using the slope of RNP versus MBT, we can calculate ye and km.
43
1998; Clarkson and Pedersen, 2011) in 18 stages of hydraulic fracturing and horizontal length of 1370 m. This well has been producing for 2 years and the reservoir pressure is above bubble point. Obtained horizontal and vertical permeabilities from well testing analysis are 1.34 md and 9.03 10 4 md, respectively. Negligible vertical permeability compared with horizontal permeability justifies the assumption of horizontal flow towards vertical hydraulic fractures. Table 1 shows the pertinent features of this reservoir. Fig. 7 shows the linear relation between RNP and MBT for this well: RNP ¼ 0:058 MBT þ0:35 Using Eq. (8) and the line slope, we calculate ye: ye ¼ 243 m
5. Field data analysis The proposed model leads to a simple analytical procedure for analyzing PSS regions observed in the production data of fracture tight oil reservoirs:
Using Eq. (9) and the line intercept, we calculate average matrix permeability: km ¼ 173:7 mD These calculations considered the appropriate conversion factors.
1. Make rate normalised pressure (RNP) against material balance time (MBT) production data plot on log–log scale. 2. Identify various flow regions. PSS flow region should be recognized by a unit slope. 3. Plot RNP versus MBT of the data points of PSS region on a Cartesian graph. 4. Record the line slope, mss and the line intercept, bss. 5. Calculate ye by substituting mss in ye ¼
Bo 2 mss X e h ðφct Þm
6. Calculate the average permeability of the matrix by substituting bss in km ¼
5.2. Bakken well: This well was completed in Bakken formation (Hlidek and Rieb, 2011; Clarkson and Pedersen, 2011; Alcoser et al., 2012) in 16 stages of hydraulic fracturing and horizontal length of 1707 m. This well has been producing for 3 years and the reservoir pressure is above bubble point. Table 2 shows the pertinent features of this reservoir. Fig. 8 shows the linear relation between RNP and MBT for well: RNP ¼ 0:0521 MBT þ 8:0981 By using the line slope and intercept relationship from Eqs. (8 and 9), ye and km are given by ye ¼ 80 m km ¼ 0:15 mD These calculations considered the appropriate conversion factors.
μBo Lf 24bss ye hN f
We demonstrate the application of the presented procedure in calculating ye and km of two wells completed in Cardium and Bakken formations: 5.1. Cardium well This well was completed in Cardium formation (Justen, 1957; MacKenzie, 1975; Purvis and Bober, 1979; Krasey and Fawcett,
Table 1 Reservoir and fluid properties of well in Cardium formation. Reservoir type Dominant flow Multiphase flow in reservoir Reservoir pressure (Pi) Flowing BHP (Pwf) Number of stages of fractures Length of horizontal well (Χe) Permeability—horizontal(khor) Permeability—vertical (kver) Reservoir thickness (h) Matrix porosity (φm ) Oil formation volume factor (Bo) Oil viscosity (μo) Total compressibility (ct) (assumed)
Homogeneous Oil phase No 15575 kPa 7413 kPa 18 1370 m 1.34 mD 9.03 10 4 mD 7m 0.108 1.221 m3/s m3 1.13 cP 3.91 10-5 kPa 1
Fig. 5. Scheme of the selected control volume showing a hydraulic fracture surrounded by the matrix blocks. Two no-flow boundaries are virtually created at the center of adjacent fractures at distance of 0.5 Lf from the fracture.
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Table 2 Reservoir and fluid properties of well in Bakken formation. Reservoir type Dominant flow Multiphase flow in reservoir Reservoir pressure (Pi) Flowing BHP (Pwf) Number of stages of fractures Length of horizontal well (Χe) Reservoir thickness (h) Matrix porosity (φm ) Oil formation volume factor (Bo) Oil viscosity (μo) Total compressibility (ct) (assumed)
Fig. 6. The pressure profile in the matrix during the boundary dominated flow. The rate of pressure drop with respect to time remains constant at pseudo-steady state conditions.
Unknown Oil phase No 46884 kPa 4826 kPa 16 1707 m 5.8 m 0.09 1.329 m3/s m3 0.5643 cP 4.02 10 5 kPa 1
Table 3 Reservoir parameters of Cardium and Bakken wells obtained from pseudo-steady state flow analysis. Cardium well
Matrix permeability Stimulated reservoir volume Matrix permeability Stimulated reservoir volume
Bakken well
173.7 mD 4.66MM m3 0.15 mD 1.58MM m3
Fig. 7. Specialized plot of RNP versus MBT for analyzing boundary dominated flow of the well completed in Cardium formation.
6. Discussion of results 6.1. Comparative analysis of PSS and Transient regions Table 4 compares the fracture half-lengths (ye ) for the Bakken well estimated by applying (i) the model proposed in this paper for analyzing PSS region and (ii) the dual porosity model proposed by Bello (2009) for analyzing linear transient (LT) region. The detailed calculations of the ye for LT region of Cardium and Bakken wells are presented in Ezulike and Dehghanpour (2013). The ye calculated using LT model from the Bakken well is higher than that calculated from PSS model. Eq. 9 shows that, ye from PSS analysis does not depend on the matrix permeability (km), while ye from LT analysis is inversely proportional to the square root of permeability. Therefore, LT analysis is expected to overestimate ye due to ignorance of secondary fractures in the dual porosity model, especially when laboratory-measured plug permeability is used as the input matrix permeability. However, PSS analysis should give a more reasonable estimate of ye because it does not depend on km as it is mainly based on a volumetric analysis (tank model). km from intercept analysis of PSS model should be higher than plug permeability because it is an average matrix
Fig. 8. Specialized plot of RNP versus MBT for analyzing boundary dominated flow of the well completed in Bakken formation.
Table 4 Hydraulic fracture half-length from LT and PSS analyses on production data from Bakken well. Analysis equation Estimated half-length (m) PSS LT
80 176
permeability that accounts for possible secondary fractures. Therefore, substitution of km from PSS analysis into the dual porosity analysis equation for LT analysis (Bello, 2009) should give a more reasonable estimate of ye . To put the above discussion into a practical framework we propose the following complementary analyses.
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1- Calculate ye from the slope of the PSS analysis match. Calculate average km by substituting ye into the intercept from the same match. Recalculate ye from the slope of LT analysis match using the km from the PSS match. Compare the two ye values from PSS and LT analysis. 2- Calculate ye from the slope of the PSS analysis match. Calculate average km by substituting ye into the intercept from the same match. Recalculate km from the slope of LT analysis using the ye from the PSS analysis. Compare the two km values from PSS and LT analysis.
45
Application of the proposed model on the pseudo-steady flow region of production data from two fractured tight oil wells yielded reasonable fracture half-length and matrix permeability estimates.
Acknowledgements The authors are grateful to Natural Sciences and Engineering Research Council of Canada and Vanier Canada for funding this study.
6.2. Comparing the analysis results of Bakken and Cardium wells Table 3 shows the summary of our analysis results for Cardium and Bakken wells. The estimated average matrix permeability from the Bakken well is near the upper end value of 0.5 mD as reported by Clarkson and Pedersen (2011). Although the typical average matrix permeability in the Cardium Formation is o 10 mD (Hoch et al., 2003), the overestimation (173.7 mD) from PSS analysis on the Cardium well data could be explained by the assumption of negligible secondary fracture effects. The active secondary fractures (higher permeability) for this Cardium well have significant effects when lumped with matrix. Therefore, this suggests that the Cardium well’s hydraulic fractures have a greater connection with active secondary fractures compared to those of the Bakken well. The same reason cab explain why the half-length estimate from PSS analysis (243 m) is higher than that from LT analysis (168 m) for the Cardium case. The perfect linear relation between RNP and MBT of Cardium well and the scatter observed in the Bakken well plots can be explained by reviewing well and formation characteristics. The horizontal permeability of the Cardium formation is much higher than the vertical one. Also, the geological reports show that the formation thins out in the southern parts of the Cardium well. Therefore, Cardium well would be a good candidate for our analysis that assumes fluid flow is linear toward the fracture faces and there is negligible influx beyond the fracture tip. On the other hand, there are some unknown parameters for Bakken formation that results in some uncertainty in the analysis results. As it can be seen in Table 2, we do not have clear information about geological characteristics and permeabilities in horizontal and vertical directions. Therefore, flow may not necessarily be linear and PSS (boundary dominated). Moreover, most of the horizontal wells in Bakken formation are completed open hole while those in Cardium formation are completed cased hole (Hlidek and Rieb, 2011; Clarkson and Pedersen, 2011). Therefore, Bakken well may benefit from additional inflow from formation directly to the wellbore.
7. Summary A methodology for analyzing late-time production data of lowpermeability oil reservoirs has been proposed which incorporates flow-regime analysis. Average matrix permeability and fracture half-length can be calculated using the production data and proposed model. Following are the main conclusions drawn by this study:
Three flow regimes (transient bi-linear, transient linear and
pseudo-steady state boundary) are observed in the Cardium and Bakken wells studied here. A model is derived for pseudo-steady state flow analysis using a linear dual porosity model and some simplifying assumptions. Average fracture half-length and matrix permeability can be estimated using the slope and intercept from this model.
Appendix A. Derivation of pressure change with respect to time for matrix Starting from material balance equation we have Mass in–Mass out¼ Accumulation in the matrix Assuming that the dominant flow direction is horizontal and towards the fracture surfaces (Ozkan et al., 2009; Wattenbarger, 2007) and considering an under saturated reservoir with single phase and slightly compressible fluid we can write qo ρo Δtj x þ Δx qo ρo Δtj x ¼ ρo φm V m j t
þ Δt ρo φm V m jt
ðA 1Þ
For unconventional reservoirs with low matrix permeabilities, the contribution of the reservoir beyond the stimulated volume is usually negligible (Medeiros et al., 2008). By neglecting contribution of reservoir beyond SRV we can write 0 qo ρo ¼
ρo φm V m j t
þ Δt ρo φm V m jt
Δt
ðA 2Þ
where subscripts o and m denote oil and matrix, respectively. Taking limits as Δt approaches to zero in Eq. (A-2) gives qo ρo ¼
d ρ φ Vm dt o m
qo ρo d ρ φ ¼ dt o m Vm Using chain rule we can write qo ρo dP m d d φm ¼ ρo þρo φm dP m dP m Vm dt Dividing both sides by ϕm and ρo gives qo dP m 1 d 1 d ¼ ρo þ φm φm dP m V m φm dt ρo dP m
ðA 3Þ ðA 4Þ
ðA 5Þ
ðA 6Þ
qo dP m ¼ ðc o þ c r Þ V m φm dt
ðA 7Þ
qo dP m ¼ ðc t Þ V m φm dt
ðA 8Þ
dP m qo ¼ dt V m φm C t
ðA 9Þ
where V m is the control matrix volume given by V m ¼ 2ye h Nf Lf
ðA 10Þ
Appendix B. Derivation of pressure profile in matrix block Considering linear diffusivity equation in the control volume (Vm in Fig. 5), we can write 2 d Pm φm μo ct dP m ðB 1Þ ¼ km dt dx2
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O. Daniel Ezulike et al. / Journal of Petroleum Science and Engineering 129 (2015) 40–47
During pseudo-steady depletion, dP m =dt can be replaced by Eq. (A-9) 2 d Pm φm μ o c t qo ¼ ðB 2Þ km V m φm c t dx2
Also, since the integral on the left hand side corresponds to the cumulative production, we can write,
Integrating both sides gives dP m μo q ¼ o x þ C1 dx km Vm
Replacing P m from Eq. (B-12) and accounting for the total flow rate from all fractures and total matrix volume (Vtm = Nf Vm and qt¼Nf qo)
ðB 3Þ
Applying the first boundary condition At ¼
Pi Pm ¼
Pi Pf ¼
Bo N p ðV t φC t Þm
μBo L2f qt Bo Np þ ðV t φC t Þm 12km V tm
ðC 3Þ
ðC 4Þ
Eq. (C-4) can be used for constant rate only. To extend this equation for variable rate and pressure, we divide it by qt , as follow:
Lf dP m ; ¼0 2 dx
C1 is given by μo qo Lf C1 ¼ km V m 2
ðB 4Þ
μBo L2f Pi Pf Np Bo ¼ð Þ þð Þ qt ðV t φct Þm qt 12km V tm
ðC 5Þ
In which V m ¼ 2ye hN f Lf Substituting C1 from Eq. (B-4) into Eq. (B-3), we obtain dP m μo q μo qo Lf ¼ ðB 5Þ o xþ dx km Vm km V m 2 Z
Integrating both sides of Eq. (B-5) Z Z μo q μo qo Lf dx dP m ¼ o xdx þ km Vm km V m 2
μo Pm ¼ km
q o x2 þ 2 Vm
ðB 6Þ
μ o qo L f x þ C2 km V m 2
ðB 7Þ
And the second boundary condition is At x ¼ 0, P m ¼ P f By applying second boundary condition to Eq. (B-7), we obtain: C2 ¼ P f
ðB 8Þ
By replacing obtained value for C2 into Eq. (B-7), we arrive at: μo q μo qo Lf x þ Pf o x2 þ ðB 9Þ Pm ¼ km 2 Vm km V m 2 Average pressure of the control volume V m is given by R L2f
P m dV m R L2f 0 dV m
Pm ¼
0
dV m ¼ 2 ye h dx
ðB 10Þ
ðB 11Þ
Replacing P m from Eq. (B-9) and using differential form of control volume, the average matrix pressure becomes: Pm ¼
μBo qo Lf 2 þ Pf 12 km V m
ðB 12Þ
Appendix C. Derivation of final solution for boundary dominated region From the compressibility definition it can be shown that (Palacio and Blasingame, 1993): qt ¼
AφhC t dp Bo dt
ðC 1Þ
Since oil is the only phase flowing in the total reservoir volume (Vtm), C t is assumed to be constant. Thus the integration of Eq. (C1) produces Z t Z ðV t φC t Þm p qt dt ¼ dp ðC 2Þ Bo 0 pi
Pi Pf μBo Lf Np Bo Þ ¼ð Þ þð qt ð2ye hN f Lf Þ ðφct Þm qt 12 km ð2ye hN f Þ
ðC 6Þ
High permeability hydraulic fractures are assumed to have negligible pressure drop during PSS flow (Bello and Wattenbarger, 2008) therefore we can replace fracture pressure (P f ) with flowing bottom hole pressure (P wf ). P i P wf μBo Lf Np Bo Þ ¼ð Þ þð qt ð2ye hN f Lf Þ ðφct Þm qt 12 km ð2ye hNf Þ
ðC 7Þ
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