Transient and Pseudo-Steady-State Inflow Performance Relationships for Multiphase Flow in Fractured Unconventional Reservoirs
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Abstract
The objective of this paper is developing new methodology for constructing the inflow performance relationships (IPRs) of unconventional reservoirs experiencing multiphase flow. The motivation is eliminating the uncertainties of using single-phase flow IPRs and approaching realistic representation and simulation to reservoir pressure–flow rate relationships throughout the entire life of production. Several analytical models for the pressure drop and decline rate as wells productivity index of two wellbore conditions, constant Sandface flow rate and constant wellbore pressure, are presented in this study. Several deterministic models are also proposed in this study for multiphase reservoir total mobility and compressibility using multi-regression analysis of PVT data and relative permeability curves of different reservoir fluids. These deterministic models are coupled with the analytical models of pressure drop, decline rate, and productivity index to construct the pressure–flow rate relationships (IPRs) during transient and pseudo-steady-state production time. Transient IPRs are generated for early-time hydraulic fracture linear flow regime and intermediate-time bilinear and trilinear flow regimes, while steady-state IPRs are generated for pseudo-steady-state flow regime in case of constant Sandface flow rate and boundary-dominated flow regime in case of constant wellbore pressure. The outcomes of this study are as follows: (1) introducing the impact of multiphase flow to the IPRs of unconventional reservoirs; (2) developing deterministic models for reservoir total mobility and compressibility using multi-regression analysis of PVT data and relative permeability curves; (3) developing analytical models for different flow regimes that could be developed during the entire production life of reservoirs; (4) predicting transient and steady-state IPRs of multiphase flow for different wellbore conditions. The study has pointed out: (1) Multiphase flow conditions have significant impact on reservoir IPRs. (2) Multiphase reservoir total mobility and compressibility exhibit significant change with reservoir pressure. (3) Constant Sandface flow rate may demonstrate IPR better than constant wellbore pressure. (4) Late production time is not affected by multiphase flow conditions similar to transient state flow at early and intermediate production time.
Keywords
Multiphase flow Unconventional reservoirs Hydraulic fracturing Fractured formations Inflow performance relationshipList of symbols
- \( B_{\text{g}} \)
\( {\text{Gas}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( B_{\text{g}} ' \)
\( {\text{Derivative}}\;{\text{of}}\;{\text{gas}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( B_{\text{o}} \)
\( {\text{Oil}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( B_{\text{o}} ' \)
\( {\text{Derivative}}\;{\text{of}}\;{\text{oil}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( B_{\text{t}} \)
\( {\text{Total}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( B_{\text{w}} \)
\( {\text{Water}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( B_{\text{w}}^{{\prime }} \)
\( {\text{Derivative}}\;{\text{of}}\;{\text{water}}\;{\text{formation}}\;{\text{volume}}\;{\text{factor}} \)
- \( c_{\text{AFq}} \)
\( {\text{Shape}}\;{\text{factor}}\;{\text{for}}\;{\text{constant}}\;{\text{Sandface}}\;{\text{flow}}\;{\text{rate}}\;{\text{approach}} \)
- \( c_{\text{AFP}} \)
\( {\text{Shape}}\;{\text{factor}}\;{\text{for}}\;{\text{constant}}\;{\text{wellbore}}\;{\text{pressure}}\;{\text{approach}} \)
- \( c_{\text{F}} \)
\( {\text{Reservoir}}\;{\text{fluid}}\;{\text{total}}\;{\text{compressibility,}}\;{\text{psi}}^{ - 1} \)
- \( c_{\text{g}} \)
\( {\text{Gas - phase}}\;{\text{compressibility}},\,{\text{psi}}^{ - 1} \)
- \( c_{\text{o}} \)
\( {\text{Oil - phase}}\;{\text{compressibility}},\,{\text{psi}}^{ - 1} \)
- \( c_{\text{w}} \)
\( {\text{Water - phase}}\;{\text{compressibility}},\,{\text{psi}}^{ - 1} \)
- \( \left( {c_{\text{t}} } \right)_{\text{mp}} \)
\( {\text{Multiphase}}\;{\text{reservoir}}\;{\text{total}}\;{\text{compressibility}},\,{\text{psi}}^{ - 1} \)
- \( F_{\text{CD}} \)
\( {\text{Hydraulic}}\;{\text{fracture}}\;{\text{conductivity,}}\,{\text{dimensionless}} \)
- \( J_{\text{DP}} \)
\( {\text{Productivity}}\;{\text{index}}\;{\text{of}}\;{\text{constant}}\;{\text{wellbore}}\;{\text{pressure,}}\,{\text{dimensionless}} \)
- \( J_{\text{Dq}} \)
\( {\text{Productivity}}\;{\text{index}}\;{\text{of}}\;{\text{constant}}\;{\text{Sandface}}\;{\text{flow}}\;{\text{rate,}}\,{\text{dimensionless}} \)
- \( h \)
\( {\text{Formation thickness,}}\;{\text{ft}} \)
- \( k_{\text{i}} \)
\( {\text{Induced}}\;{\text{matrix}}\;{\text{permeability,}}\,{\text{md}} \)
- \( k_{{\rm m}} \)
\( {\text{Matrix}}\;{\text{permeability,}}\,{\text{md}} \)
- \( \left( {k /\mu } \right)_{\text{mp}} \)
\( {\text{Multiphase}}\;{\text{reservoir}}\;{\text{total}}\;{\text{mobility,}}\,{\text{md/cp}} \)
- \( P \)
\( {\text{Pressure,}}\,{\text{psi}} \)
- \( P_{\text{b}} \)
\( {\text{Bubble}}\;{\text{point}}\;{\text{pressure,}}\,{\text{psi}} \)
- \( \Delta P_{\text{wf}} \)
\( {\text{Wellbore}}\;{\text{pressure}}\;{\text{drop,}}\,{\text{psi}} \)
- \( P_{\text{D}} \)
\( {\text{Pressure}}\;{\text{drop,}}\,{\text{dimensionless}} \)
- \( P_{\text{Di}} \)
\( {\text{Initial}}\; {\text{reservoir}}\;{\text{pressure,}}\,{\text{dimensionless}} \)
- \( P_{\text{wD}} \)
Wellbore pressure drop, dimensionless
- \( t_{\text{D}} xP_{\text{D}}^{{\prime }} \)
\( {\text{Pressure}}\;{\text{derivative,}}\,{\text{dimensionless}} \)
- \( q_{\text{D}} \)
\( {\text{Sandface}}\;{\text{flow}}\;{\text{rate,}}\,{\text{dimensionless}} \)
- \( q_{\text{o}} \)
\( {\text{oil}}\;{\text{flow}}\;{\text{rate,}}\,{\text{STB/day}} \)
- \( q_{\text{t}} \)
\( {\text{Total}}\;{\text{flow}}\;{\text{rate,}}\,{\text{bbl/day}} \)
- \( q_{\text{w}} \)
\( {\text{water}}\; {\text{flow}}\;{\text{rate,}}\,{\text{STB/day}} \)
- \( q_{\text{sc}} \)
\( {\text{Gas}}\;{\text{flow}}\;{\text{rate,}}\,{\text{MScf/day}} \)
- \( R_{\text{s}} \)
\( {\text{Solution}}\;{\text{gas}} - {\text{oil}}\;{\text{ratio}} \)
- \( R_{\text{s}}^{{\prime }} \)
\( {\text{Derivative}}\;{\text{of}}\;{\text{solution}}\;{\text{gas}} - {\text{oil}}\;{\text{ratio}} \)
- \( R_{\text{sb}} \)
\( {\text{Solution}}\;{\text{gas}} - {\text{oil}}\;{\text{ratio}}\;{\text{at}}\;{\text{bubble}}\;{\text{point}}\;{\text{pressure}} \)
- \( R_{\text{sw}} \)
\( {\text{Solution}}\;{\text{gas}} - {\text{water}}\;{\text{ratio}} \)
- \( R_{\text{sw}}^{{\prime }} \)
\( {\text{Derivative}}\;{\text{of}}\;{\text{solution}}\;{\text{gas}} - {\text{water}}\;{\text{ratio}} \)
- \( s \)
\( {\text{Laplace}}\;{\text{operator}} \)
- \( S_{\text{g}} \)
\( {\text{Gas}}\;{\text{saturation}} \)
- \( S_{\text{o}} \)
\( {\text{Oil}}\;{\text{saturation}} \)
- \( S_{\text{w}} \)
\( {\text{Water}}\;{\text{saturation}} \)
- \( T \)
\( {\text{Reservoir}}\;{\text{temperature}} \)
- \( t \)
\( {\text{Time,}}\,{\text{h}} \)
- \( t_{\text{D}} \)
\( {\text{Time,}}\,{\text{dimensionless}} \)
- \( \mu_{\text{g}} \)
\( {\text{Gas - phase}}\;{\text{viscosity,}}\,{\text{cp}} \)
- \( \mu_{\text{o}} \)
\( {\text{Oil - phase}}\;{\text{viscosity,}}\,{\text{cp}} \)
- \( \mu_{\text{w}} \)
\( {\text{Water - phase}}\;{\text{viscosity,}}\,{\text{cp}} \)
- \( w_{\text{f}} \)
\( {\text{Hydraulic}}\;{\text{fracture}} - {\text{half - length}},\,{\text{ft}} \)
- \( x_{\text{e}} \)
\( {\text{Reservoir}}\;{\text{boundary,}}\,{\text{ft}} \)
- \( x_{\text{f}} \)
\( {\text{Hydraulic}}\;{\text{fracture}}\;{\text{width,}}\,{\text{ft}} \)
- \( y_{\text{e}} \)
\( {\text{Reservoir}}\;{\text{boundary,}}\,{\text{ft}} \)
- \( \omega \)
\( {\text{Storativity}} \)
- \( \emptyset \)
\( {\text{Porosity}} \)
- \( \lambda \)
\( {\text{Interporosity}}\;{\text{flow}}\;{\text{coefficient}} \)
1 Introduction
The literature review of the two topics of interest in this paper is covered briefly. The first is the pressure behavior, decline rate, and productivity index of hydraulically fractured reservoirs, while the second focuses on the attempts of assembling the impact of multiphase flow with reservoir performance models. To avoid the excessive length of the manuscript, the literature review of the first part will not be discussed in details, while the second will be the focus of the literature review.
In the last couple decades, hydraulic fracturing stimulation technique has boosted up developing economically unconventional resources in spite of the great challenges. Improving the ultralow permeability of these resources and creating high-conductivity flow paths in the porous media are the key points in the process. As a matter of fact, starting from the first succeed of this technique and later on until the moment, a lot of researches in different topics and disciplines have been conducted and presented in the petroleum industry literature. Within the scope of this paper, topics such as pressure transient analysis (PTA), rate transit analysis (RTA), and productivity index of hydraulically fractured reservoirs are very well covered. At early 1960s, the physical meaning of dimensionless fracture conductivity was explained by Prats and Levine (1963) as the ratio of the ability of hydraulic fractures to transmit reservoir fluid to the wellbore, while 1970s have witnessed serious attempts to formulate pressure behavior and decline rate of fractured formations conducted by Gringarten and Ramey (1973), Gringarten et al. (1974), Cinco-Ley (1974), Holditch and Morse (1976), Raghavan et al. (1978), Cinco-Ley et al. (1975), Cinco et al. (1978), and Agarwal et al. (1979). These attempts have continued during the 1980s and 1990s by Cinco-Ley and Samaniego (1981), Bennett et al. (1986), Camacho et al. (1987), Ozkan (1988), Guppy et al. (1988), Ozkan and Raghavan (1991a, b), Soliman et al. (1990), Kuchuk (1990), Larsen and Hegre (1994), Raghavan et al. (1997), El-Banbi (1998), and Wan and Aziz (1999). The later 20 years have witnessed a great attention given to rate transient analysis (RTA) as well as pressure transient analysis (PTA). Because of the ultralow permeability in unconventional reservoirs, pressure pulse may need a very long time to reach the boundary and thereby pseudo-steady-state or boundary-dominated flow regime might not be observed. Therefore, RTA has been used confidentially as an excellent tool for unconventional reservoir characterization more than PTA. This has come out with a lot of models that describe decline rate with production time such as those proposed by Hagoort (2003), Levitan (2005), Ilk et al. (2006), Ibrahim and Wattenbarger (2006), Camacho et al. (2008), Bello (2008), Izadi and Yildiz (2009), and Cipolla (2009). Very recently, the two topics (RTA and PTA) have been supported by more research papers presented by Ozkan et al. (2011), Brown et al. (2011), Duong (2011), Chen and Jones (2012), Nobakht et al. (2012), Torcuk et al. (2013), Fuentes-Cruz et al. (2014), Luo et al. (2014), Shahamat et al. (2015), Fuentes-Cruz and Valko (2015), Behmanesh et al. (2015, 2018), and Kanfar and Clarkson (2018).
In the above-mentioned studies, single-phase flow is assumed as the dominant flow pattern in the porous media. This assumption might have led to misleading results for reservoirs that undergo multiphase flow conditions. In conventional reservoirs, single-phase flow is the common flow pattern in oil reservoirs with pressure greater than bubble point pressure. Single-phase flow is also the dominant flow pattern for natural dry gas reservoir and to some extent wet gas reservoirs. While in oil reservoirs where the pressure could drawdown the bubble point pressure, volatile oil reservoirs, and retrograde-condensate or near critical gas-condensate reservoirs, multiphase flow is the norm. Unconventional reservoirs may not have an exemption from the above-mentioned classification even though most shale layers are considered gas-producing plays. However, North Dakota Bakken formation and Texas Eagle Ford formation in the USA are two examples of ultralow permeability porous media where multiphase flow (black oil +gas) is the overwhelming flow pattern (Uzun et al. 2016).
No doubt, dealing with single-phase flow is much easier than with multiphase flow in terms of physical properties of reservoir fluids and petrophysical properties of porous media. Reservoir modeling for fluid flow and pressure distribution in drainage areas that undergo single-phase (oil or gas) flow can be accomplished either analytically or numerically with a consideration given to the absolute permeability of the porous media and total reservoir compressibility only. This would not be the case for multiphase flow where the considerations should be focused on relative permeability and saturation of each phase as well as the continuously changing reservoir fluid properties such as density, viscosity, formation volume factor, gas solubility, and compressibility. The problem could be more complicated considering that associated linear behavior of most reservoir rock and fluid properties in single-phase flow may not be applicable for multiphase flow. Nonlinear scheme (Tabatabaie and Pooladi-Darvish 2017) demonstrates most of the pressure- and time-dependent reservoir fluid properties as well as the changes in relative permeabilities with saturation that in turn could change with time and reservoir pressure.
Not until recently, multiphase flow has been given the attention in petroleum industry. The mathematical treatment proposed by Muskat and Meres (1936) for fluid flow in hydrocarbon reservoirs was considered by Perrine (1956) for multiphase flow and 3 years later Martin (1959) introduced a simplified equation for multiphase flow in gas drive reservoirs wherein relative permeability of reservoir fluid phases and their physical properties were considered. This simplified equation was obtained by the assumption of neglecting the pressure–saturation gradient as their vector products are small compared with the magnitude of pressure vector or saturation vector. Martin and James (1963) applied the proposed model by Martin (1959) for pressure transient analysis of two-phase radial flow considering constant pressure and no flow at the outer boundary. Because of the difficulties that governed utilizing multiphase flow in well test analysis, very limited attempts were conducted after the one proposed by Martin and James (1963). Chu et al. (1986) reconsidered two-phase flow problem in pressure transient applications. In that study, the primary concern was the saturation gradient and the relative permeability of each phase of reservoir fluid. The conceptual approach presented by Martin (1959) was adopted by Raghavan (1976, 1989) for the applications of well test analysis for a well producing from solution gas driven at constant surface production rate. At the same time, Fraim and Wattenbarger (1988) used the basic model presented by Muskat (1937) for the applications of decline curve analysis considering multiphase flow in porous media. They concentrated on the effect of saturation gradient on rate–time profile as reservoir pressure moves down the bubble point pressure.
Ayan and Lee (1988) stated that even though the approaches presented by Perrine (1956) and Martin (1959) are simple, they could yield misleading results under some circumstances as a consequence of the inherently assumptions in those two approaches including the non-uniform distribution of saturation and compressibility in the porous media. For this reason, Boe et al. (1989) believed that multiphase flow effect can be adapted to the liquid model solutions if total mobility and compressibility are used and pseudo-pressure function is developed. However, creating pseudo-pressure function needs very well-defined relationship between oil saturation and pressure that in turn needs the assumptions followed by Perrine (1956) and Martin (1963). For more easiness and less assumptions, Al-Khalifa et al. (1989) suggested using pressure square approach for multiphase flow instead of pseudo-pressure approach. They stated that the rate normalization using pressure square approach may yield reasonable estimates for the individual phase permeability and thereby accurate total mobility. More than 10 years later, Kamal and Pan (2010, 2011) demonstrated the applicability of pressure transient analysis for reservoir characterization under multiphase flow conditions. Recently, Li et al. (2017) proposed new method for construction gas–water two-phase steady-state flow productivity of fractured horizontal wells depleting tight gas reservoirs.
Unfortunately, despite the great attention given to multiphase flow in porous media, the above-mentioned attempts have not reached to robust techniques that rigorously included the impact of multiphase flow on reservoir performance. This could be explained by the difficulties that governed the proposed models for predicting the variances of physical and petrophysical properties such as saturation, viscosity, density, formation volume factor, and relative permeability with time. Therefore, most of the proposed models for the IPR assumed single-phase flow either oil or gas. However, from time to time, two-phase flow or multiphase flow IPRs have been suggested by several authors. Gallice and Wiggins (2004) investigated the IPRs for predicting pressure/production behavior of reservoirs dominated by two-phase flow. Ten years before, Wiggins (1994) introduced a study for three-phase flow generalized IPR, while Camacho and Raghavan (1989) used numerical model for examining the influence of reservoir pressure on the IPR for reservoirs producing under solution gas derive.
In this paper, the IPR of unconventional reservoirs considering multiphase flow conditions in the porous media is introduced. Early production time transient state flow IPR and late production time pseudo-state IPR are considered for constant Sandface flow rate and constant wellbore pressure. Several deterministic models for reservoir total mobility and compressibility are generated from PVT data and relative permeability curves. The analytical models of the flow regimes, developed in bounded hydraulically fractured unconventional reservoirs, are included the impact of multiphase flow condition and used to generate transient and pseudo-steady-state IPRs.
2 Multiphase Reservoir Total Compressibility \( (c_{\text{t}} )_{\text{mp}} \) and Mobility \( \left( {\frac{k}{\mu }} \right)_{\text{mp}} \)
A set of PVT data (Boe et al. 1989)
P, psi | B_{o} bbl/STB | B_{Go} bbl/SCF | R_{S} SCF/STB | μ_{o}, cp | μ_{g}, cp |
---|---|---|---|---|---|
5705 | 1.806 | 0.000596 | 1499.00 | 0.298 | 0.0298 |
5633 | 1.791 | 0.000600 | 1470.38 | 0.3 | 0.0295 |
5204 | 1.702 | 0.000625 | 1305.56 | 0.317 | 0.0281 |
4703 | 1.605 | 0.000661 | 1127.45 | 0.348 | 0.0263 |
4202 | 1.516 | 0.000709 | 963.48 | 0.391 | 0.0246 |
3700 | 1.434 | 0.000775 | 812.57 | 0.446 | 0.0228 |
3200 | 1.36 | 0.000870 | 673.78 | 0.515 | 0.021 |
2770 | 1.302 | 0.000991 | 563.79 | 0.587 | 0.0195 |
2340 | 1.249 | 0.001172 | 461.53 | 0.671 | 0.0181 |
1911 | 1.202 | 0.001455 | 366.52 | 0.768 | 0.0166 |
1482 | 1.159 | 0.001929 | 278.24 | 0.881 | 0.0152 |
1052 | 1.121 | 0.002826 | 196.10 | 1.011 | 0.0138 |
623 | 1.088 | 0.004998 | 119.14 | 1.164 | 0.0125 |
193 | 1.058 | 0.016881 | 44.29 | 1.35 | 0.0113 |
Reservoir physical and petrophysical properties (Boe et al. 1989)
\( {\text{Porosity}},\emptyset \) | 30% |
\( {\text{Permeability,}}\,{\text{k}} \) | 10 md |
\( {\text{Intial}}\;{\text{reservoir}}\;{\text{pressure}},P_{\text{i}} \) | 5705 psi |
\( {\text{Bubble}}\;{\text{point}}\;{\text{pressure}},P_{\text{b}} \) | 5705 psi |
\( {\text{Initial}}\;{\text{water}}\;{\text{saturation}},S_{\text{wi}} \) | 30% |
\( {\text{Initial}}\;{\text{gas}}\;{\text{saturation}},S_{\text{gi}} \) | 0.0 |
\( {\text{Wellbore}}\;{\text{radius}},r_{\text{w}} \) | 0.33 ft |
\( {\text{Reservoir}}\;{\text{radius}},r_{\text{e}} \) | 656 ft |
\( {\text{Formation}}\;{\text{thickness}},h \) | 15.6ft |
\( {\text{Gas}}\;{\text{specific}}\;{\text{gravity}},\gamma_{\text{g}} \) | 0.75 |
\( {\text{Reservoir}}\;{\text{temperature}},T \) | 200 °F |
\( {\text{Crude}}\;{\text{oil}}\;{\text{API}} \) | 35 |
Calculated formation water properties
P, psi | R_{sw} SCF/STB | μ_{w}, cp | B_{w} bbl/STB |
---|---|---|---|
5705 | 10.132 | 1.3304 | 1.0162 |
5633 | 10.051 | 1.3249 | 1.0170 |
5204 | 9.554 | 1.2932 | 1.0210 |
4703 | 8.929 | 1.2576 | 1.0250 |
4202 | 8.257 | 1.2236 | 1.0282 |
3700 | 7.538 | 1.1910 | 1.0308 |
3200 | 6.775 | 1.1602 | 1.0329 |
2770 | 6.081 | 1.1349 | 1.0343 |
2340 | 5.353 | 1.1107 | 1.0355 |
1911 | 4.592 | 1.0877 | 1.0365 |
1482 | 3.797 | 1.0659 | 1.0373 |
1052 | 2.966 | 1.0452 | 1.0379 |
623 | 2.103 | 1.0257 | 1.0384 |
193 | 1.203 | 1.0073 | 1.0388 |
Reservoir total mobility results
P, psi | k _{ro} | k _{rg} | k _{rwo} | k_{ro}/_{o} | k_{rg}/_{g} | k_{rw}/_{w} | (k/μ)_{mp} |
---|---|---|---|---|---|---|---|
5705 | 0.805 | 0 | 0 | 2.701 | 0.000 | 0.000 | 2.701 |
5633 | 0.802 | 0.6383 | 0.02 | 2.673 | 21.637 | 0.015 | 24.326 |
5204 | 0.751 | 0.6253 | 0.05 | 2.369 | 22.253 | 0.039 | 24.660 |
4703 | 0.71 | 0.606 | 0.08 | 2.040 | 23.042 | 0.064 | 25.146 |
4202 | 0.615 | 0.5885 | 0.1 | 1.573 | 23.923 | 0.082 | 25.577 |
3700 | 0.554 | 0.5681 | 0.11 | 1.242 | 24.917 | 0.092 | 26.251 |
3200 | 0.505 | 0.547 | 0.13 | 0.981 | 26.048 | 0.112 | 27.140 |
2770 | 0.462 | 0.5453 | 0.14 | 0.787 | 27.964 | 0.123 | 28.875 |
2340 | 0.421 | 0.5437 | 0.15 | 0.627 | 30.039 | 0.135 | 30.801 |
1911 | 0.382 | 0.5242 | 0.16 | 0.497 | 31.578 | 0.147 | 32.223 |
1482 | 0.351 | 0.5094 | 0.17 | 0.398 | 33.513 | 0.159 | 34.071 |
1052 | 0.331 | 0.508 | 0.18 | 0.327 | 36.812 | 0.172 | 37.311 |
623 | 0.302 | 0.5052 | 0.19 | 0.259 | 40.416 | 0.185 | 40.861 |
193 | 0.252 | 0.4856 | 0.2 | 0.187 | 42.973 | 0.199 | 43.359 |
- 1.
At the bubble point pressure, the three compressibility functions given by Eqs. (3)–(5) are calculated as well as reservoir total mobility.
- 2.
Using initial saturations at bubble point pressure, reservoir fluid total compressibility is calculated using Eq. (2) and multiphase reservoir total compressibility is calculated using Eq. (1).
- 3.
Calculate oil and water saturation derivatives given by Eqs. (6) and (7) for the second pressure interval.
- 4.
Calculate oil and water saturation for the new pressure interval. Gas saturation is calculated by \( \left( {S_{\text{g}} = 1 - S_{\text{w}} - S_{\text{o}} } \right) \).
- 5.
Calculate the relative permeability of the three phases of reservoir fluid using Stone’s models and reservoir total mobility using Eq. (8).
- 6.
Calculate the three compressibility functions given by Eqs. (3)–(5) and repeat steps for all pressure intervals.
Multiphase reservoir fluid total compressibility and reservoir total compressibility results
P, psi | f(C_{o}) | f(C_{w}) | f(C_{g}) | k _{ro} | k _{rg} | k _{rw} | dS_{o}/dP | dS_{w}/dP |
---|---|---|---|---|---|---|---|---|
5705 | 2.84217E−05 | 0.000115124 | 0.000151997 | 0.805 | 0 | 0 | 0.00011804 | − 3.42818E−05 |
5633 | 2.92528E−05 | 0.000112842 | 0.000154937 | 0.802 | 0.6383 | 0.02 | 6.8879E−05 | − 3.38365E−05 |
5204 | 3.47861E−05 | 0.000100175 | 0.000174279 | 0.751 | 0.6253 | 0.05 | 6.5541E−05 | − 3.13714E−05 |
4703 | 4.25923E−05 | 8.70976E−05 | 0.000201683 | 0.71 | 0.606 | 0.08 | 6.1618E−05 | − 2.84847E−05 |
4202 | 5.24708E−05 | 7.54926E−05 | 0.000235598 | 0.615 | 0.5885 | 0.1 | 5.723E−05 | − 2.55859E−05 |
3700 | 6.55027E−05 | 6.50462E−05 | 0.000277965 | 0.554 | 0.5681 | 0.11 | 5.3049E−05 | − 2.26747E−05 |
3200 | 8.34266E−05 | 5.56038E−05 | 0.000330732 | 0.505 | 0.547 | 0.13 | 4.8924E−05 | − 1.97131E−05 |
2770 | 0.000104926 | 4.81375E−05 | 0.000387535 | 0.462 | 0.5453 | 0.14 | 4.5284E−05 | − 1.71586E−05 |
2340 | 0.000135407 | 4.12212E−05 | 0.000459398 | 0.421 | 0.5437 | 0.15 | 4.1688E−05 | − 1.45672E−05 |
1911 | 0.000180454 | 3.48757E−05 | 0.000554911 | 0.382 | 0.5242 | 0.16 | 3.8175E−05 | − 1.19111E−05 |
1482 | 0.000251826 | 2.91979E−05 | 0.000695811 | 0.351 | 0.5094 | 0.17 | 3.4774E−05 | − 9.15998E−06 |
1052 | 0.000378684 | 2.45917E−05 | 0.000942774 | 0.331 | 0.508 | 0.18 | 3.1512E−05 | − 6.24259E−06 |
623 | 0.000669479 | 2.28134E−05 | 0.001519608 | 0.302 | 0.5052 | 0.19 | 2.8671E−05 | − 2.77317E−06 |
193 | 0.002199146 | 4.13836E−05 | 0.004688456 | 0.252 | 0.4856 | 0.2 | 2.9237E−05 | 5.23169E−06 |
S _{o} | S _{g} | S _{w} | k_{ro}/μ_{o} | k_{rg}/μ_{g} | k_{rw}/μ_{w} | c _{F} | (c_{t})_{mp} |
---|---|---|---|---|---|---|---|
0.7 | 0 | 0.3 | 2.701342282 | 0 | 0 | 5.44324E−05 | 6.44324E−05 |
0.69150133 | 0.006030382 | 0.302468288 | 2.673333333 | 21.63728814 | 0.015094982 | 5.52937E−05 | 6.52937E−05 |
0.661952103 | 0.021063747 | 0.31698415 | 2.369085174 | 22.25266904 | 0.038663299 | 5.84516E−05 | 6.84516E−05 |
0.629116026 | 0.038182736 | 0.332701238 | 2.040229885 | 23.0418251 | 0.063612677 | 6.34737E−05 | 7.34737E−05 |
0.598245605 | 0.054782346 | 0.346972049 | 1.572890026 | 23.92276423 | 0.081728382 | 7.04909E−05 | 8.04909E−05 |
0.569516207 | 0.070667602 | 0.35981619 | 1.242152466 | 24.91666667 | 0.092358169 | 8.03526E−05 | 9.03526E−05 |
0.542991711 | 0.085854733 | 0.371153555 | 0.980582524 | 26.04761905 | 0.112054332 | 9.43324E−05 | 0.000104332 |
0.521954508 | 0.098415291 | 0.379630201 | 0.787052811 | 27.96410256 | 0.123364244 | 0.00011118 | 0.00012118 |
0.502482397 | 0.110509214 | 0.387008388 | 0.627421759 | 30.03867403 | 0.135050138 | 0.00013476 | 0.00014476 |
0.484598046 | 0.122144237 | 0.393257717 | 0.497395833 | 31.57831325 | 0.147092982 | 0.000168942 | 0.000178942 |
0.468221156 | 0.133411276 | 0.398367568 | 0.398410897 | 33.51315789 | 0.159483736 | 0.000222371 | 0.000232371 |
0.453268406 | 0.144425233 | 0.402306361 | 0.327398615 | 36.8115942 | 0.172211233 | 0.000317699 | 0.000327699 |
0.439749641 | 0.155265928 | 0.404984431 | 0.259450172 | 40.416 | 0.185237653 | 0.000539586 | 0.000549586 |
0.427420959 | 0.166402147 | 0.406176894 | 0.186666667 | 42.97345133 | 0.198552032 | 0.001736939 | 0.001746939 |
The constants \( \left( A \right) \) and \( \left( B \right) \) in Eqs. (14) and (15) are determined from PVT data of the reservoir of interest. For example, the two constants in Eq. (15) are \( \left( {A = 0.3099} \right) \) and \( \left( {B = - 0.896} \right) \) calculated by the PVT data taken from Boe et al. (1989).
PVT data (Fraim and Wattenbarger 1988)
P, psi | B_{o} bbl/STB | R_{s} SCF/STB | μ_{o}, cp | B_{g} bbl/STB | μ_{g}, cp |
---|---|---|---|---|---|
0 | 1.09 | 0.0 | 1.673 | 1.35 | 0.011 |
200 | 1.12 | 40.0 | 0.954 | 0.0938 | 0.0115 |
400 | 1.151 | 101.0 | 0.873 | 0.0463 | 0.0124 |
700 | 1.193 | 176.0 | 0.733 | 0.0261 | 0.0135 |
1000 | 1.229 | 245.0 | 0.662 | 0.018 | 0.0146 |
1300 | 1.265 | 315.0 | 0.598 | 0.0137 | 0.0156 |
1600 | 1.302 | 386.0 | 0.565 | 0.011 | 0.0166 |
1900 | 1.341 | 462.0 | 0.519 | 0.0091 | 0.0178 |
2200 | 1.328 | 543.0 | 0.488 | 0.0078 | 0.019 |
2400 | 1.41 | 598.0 | 0.479 | 0.0072 | 0.0198 |
2600 | 1.442 | 658.0 | 0.468 | 0.0067 | 0.0208 |
2755 | 1.467 | 705.0 | 0.460 | 0.0064 | 0.0212 |
3500 | 1.57 | 920.0 | 0.425 | ||
4000 | 1.64 | 1060.0 | 0.400 | ||
4500 | 0.005 | 0.0276 | |||
5000 | 1.73 | 1275.0 | 0.360 | ||
6000 | 1.87 | 1550.0 | 0.325 | ||
7000 | 2 | 1850.0 | 0.280 | 0.004 | 0.0374 |
Reservoir information (Fraim and Wattenbarger 1988)
Oil density, \( \rho_{\text{o}} \) | \( 46.24\,{\text{Ib/ft}}^{3} \) |
Formation water density, \( \rho_{\text{w}} \) | \( 62.23\,{\text{Ib/ft}}^{3} \) |
Formation water compressibility, \( c_{\text{tw}} \) | \( 1*10^{ - 5} \,{\text{psi}}^{ - 1} \) |
Formation water viscosity, \( \mu_{\text{w}} \) | \( 0.31\,{\text{cp}} \) |
Initial water saturation | \( 35\% \) |
Effect rock compressibility, \( c_{\text{f}} \) | \( 7*10^{ - 6} \,{\text{psi}}^{ - 1} \) |
Initial reservoir pressure | \( 3600\,{\text{psi}} \) |
Bubble point pressure, P_{b} | \( 2755\,{\text{psi}} \) |
Formation porosity, \( \emptyset \) | \( 12\% \) |
Formation absolute permeability,\( k \) | \( 15\,{\text{md}} \) |
Formation thickness, \( h \) | 170 ft |
Wellbore radius, r_{w} | 0.025 ft |
Reservoir radius, r_{e} | 600 ft |
In the next sections, reservoir total mobility and compressibility, calculated by Eqs. (13) and (15), respectively, will be used in predicting transient and pseudo-steady-state IPR for unconventional reservoirs.
3 Transient Inflow Performance Relationship
Unconventional reservoirs are characterized by dominating transient state flow for a very long production time. The reason for that refers to a very slow transferring rate of pressure pulse in the porous media due to ultralow permeability. Accordingly, the conclusions that most of the production comes from transient state flow and pseudo-steady-state flow may not be reached are not unrealistic. Unlike pseudo-steady-state flow, transient state flow is known by varied productivity index. Because of that and considering multiphase flow conditions, conventional IPR models (Vogel 1968; Standing 1971; Fetkovich 1973; Wiggins 1994; Golan and Whitson 1995) may give misleading results. Therefore, new methodology is proposed in this study for pressure–flow rate relationships of unconventional reservoirs undergoing multiphase flow. The new methodology states that the IPRs of this type of reservoirs are not constant. They are changed with production time and the dominant flow regime in the porous media. Accordingly, different IPRs should be constructed for different flow regimes and production time.
More details for these two models, Eqs. (16) and (17), are given in “Appendix.”
The IPRs of the three flow regimes during transient state flow and pseudo-steady-state IPR are explained as follows.
3.1 Hydraulic Fracture Linear Flow Regime
3.2 Bilinear Flow Regime
3.3 Trilinear Flow Regime
3.4 Pseudo-steady-State (Boundary-Dominated) Flow Regime
Even though the two models in Eqs. (38) and (39) include multiphase stimulated reservoir total mobility \( \left( {\frac{k}{\mu }} \right)_{\text{mp}} \), productivity indices are considered constant as the rate of change in the mobility with pressure at late production time is very minor.
4 Application
Bakken and Eagle Ford formations (Uzun et al. 2014)
Black oil-1 Well Bakken formation | Volatile oil Well Eagle Ford formation | |
---|---|---|
Initial reservoir pressure, P_{i} | 7802 psi | 8428 psi |
Bubble point pressure, P_{b} | 2130 psi | 4350 psi |
Bottom hole temperature, T | \( 240\,^{{^\circ }} {\text{F}} \) | \( 269\,^{{^\circ }} {\text{F}} \) |
Solution gas–oil ratio, R_{s} | \( 850\,{\text{SCF/STB}} \) | \( 1112\,{\text{SCF/STB}} \) |
Oil formation volume factor, B_{o} | \( 1.61\,{\text{RBBL/STB}} \) | \( 1.56\,{\text{RBBL/STB}} \) |
Water formation volume, B_{w} | \( 1.04\,{\text{RBBL/STB}} \) | \( 1.04\,{\text{RBBL/STB}} \) |
Gas formation volume factor, B_{g} | \( 0.9\,{\text{RBBL/MScf}} \) | \( 0.55\,{\text{RBBL/MScf}} \) |
Oil viscosity, \( \mu_{\text{o}} \) | \( 0.39\,{\text{cp}} \) | \( 0.29\,{\text{cp}} \) |
Formation water viscosity, \( \mu_{\text{w}} \) | 1.0 cp | 1.0 cp |
Gas viscosity, \( \mu_{\text{g}} \) | 0.01 cp | 0.034 cp |
Spacing between stages | 587 ft | 297 ft |
No. of stages | 15 | 16 |
Wellbore length, \( L \) | 8800 ft | 5860 ft |
Formation thickness, h | 49.5 ft | 120 ft |
Well spacing | 1320 ft | 700 ft |
Porosity | 0.055 | 0.055 |
Estimated hydraulic fracture half-length, \( x_{\text{f}} \) | \( 396\,{\text{ft}} \) | \( 146\,{\text{ft}} \) |
Total compressibility, \( c_{\text{t}} \) | \( 1*10^{ - 5} \,{\text{psi}}^{ - 1} \) | \( 1.38*10^{ - 5} \,{\text{psi}}^{ - 1} \) |
Matrix permeability,\( k_{{\rm m}} \) | \( 6.27*10^{ - 4} \,{\text{md}} \) | \( 0.5*10^{ - 6} \,{\text{md}} \) |
Natural fracture permeability, \( k_{\text{f}} \) | \( 5.9*10^{ - 3} \,{\text{md}} \) | |
Hydraulic fracture permeability, \( k_{\text{hf}} \) | 600 md | 600 md |
Hydraulic fracture width, \( w_{\text{f}} \) | 0.25 in | 0.25 in |
Dimensionless parameters
Black oil-1 well Bakken formation | Volatile oil well Eagle Ford formation | |
---|---|---|
\( x_{\text{eD}} \) | \( 1.667 \) | \( 2.4 \) |
\( y_{\text{eD}} \) | \( 0.741 \) | \( 1.017 \) |
\( F_{\text{CD}} \) | \( 5.35 \) | \( 14.51 \) |
\( \omega \) | \( 1000 \) | \( 1000 \) |
\( \lambda \) | \( 100 \) | \( 100 \) |
\( R_{\text{CD}} \) | \( 12.7 \) | \( 1.16 * 10^{4} \) |
\( w_{\text{D}} \) | \( 5.26 *10^{ - 5} \) | \( 1.427 *10^{ - 4} \) |
\( \eta_{\text{FD}} \) | \( 1.017 *10^{ - 5} \) | \( 1.017 *10^{5} \) |
\( \eta_{\text{mD}} \) | \( 0.1063 \) | \( 8.474 *10^{ - 5} \) |
- 1.
For each pressure step, the following parameters are calculated:
- 2.
Calculate total flow rate \( \left( {q_{t} } \right) \) using Eq. (22) and total formation volume factor \( \left( {B_{\text{t}} } \right) \) using Eq. (23).
- 3.
Calculate multiphase reservoir total compressibility \( \left( {c_{\text{t}} } \right)_{\text{mp}} \) using Eq. (45).
- 4.
Calculate multiphase reservoir total mobility using Eq. (44).
- 5.
Calculate wellbore pressure drop \( \left( {\Delta P_{\text{wf}} } \right) \) by:
- 6.
Real production time is calculated from dimensionless production time by:
- 1.
The productivity of Eagle Ford formation for the same pressure drop is better than of Bakken formation.
- 2.
It takes longer time in Eagle Ford formation to reach pseudo-steady-state flow compared with Bakken formation because Eagle Ford formation has ultralow matrix permeability \( \left( {k_{\text{m}} = 0.5*10^{ - 6} \,{\text{md}}} \right) \) less than the one in Bakken formation \( \left( {k_{\text{m}} = 6.27*10^{ - 4} \,{\text{md}}} \right) \). Similar thing is observed for all flow regimes.
- 3.
Trilinear flow regime dominates fluid flow in Eagle Ford formation longer than in Bakken formation as the fracture half-length is shorter than the fracture half-length of Bakken formation.
5 Conclusions
- 1.
Multiphase flow may have significant impact on pressure heavier, decline rate pattern, and productivity index as well as inflow performance relationship of reservoirs regardless the inner boundary condition of the wellbore whether it is constant Sandface flow rate or constant wellbore pressure.
- 2.
Multiphase reservoir total mobility and compressibility significantly change with reservoir pressure. Reservoir compressibility demonstrates more changes than the mobility especially at low reservoir pressure; however, it shows more steady-state changes at high reservoir pressure compared with the mobility.
- 3.
Multiphase reservoir total mobility and compressibility show similar behavior for different reservoirs and different reservoir fluids. Therefore, very slight differences in the mathematical models of these two parameters are seen for different PVT data sets.
- 4.
Multiphase flow may not have significant impact on the inflow performance relationship at late production time when pseudo-steady-sate or boundary-dominated flow regime is the dominant flow pattern.
- 5.
There is no significant difference between the constant Sandface flow rate and constant wellbore pressure conditions; however, the inflow performance relationship of constant Sandface flow rate is better than of constant wellbore pressure.
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