Advertisement

Transport in Porous Media

, Volume 126, Issue 3, pp 743–777 | Cite as

Transient and Pseudo-Steady-State Inflow Performance Relationships for Multiphase Flow in Fractured Unconventional Reservoirs

  • Salam Al-RbeawiEmail author
Article
  • 64 Downloads

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 relationship 

List 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 \( \left( {\frac{k}{\mu }} \right)_{\text{mp}} \)

When the pressure in oil reservoirs declines less than bubble point pressure, free gas phase is developed as well as liquid phase. Therefore, two-phase flow (oil and gas) or even multiphase flow (oil, gas, and water) in case of water production is expected to occur in porous media. Therefore, the assumption for developing multiphase reservoir total compressibility and mobility models is \( \left( {P < P_{\text{b}} } \right) \). For multiphase flow conditions, reservoir total compressibility depends on reservoir fluids’ saturation that in turn depends on reservoir pressure. The mathematical models for this compressibility can be written as:
$$ \left( {c_{\text{t}} } \right)_{\text{mp}} = c_{\text{f}} + S_{\text{o}} c_{\text{o}} + S_{\text{g}} c_{\text{g}} + S_{\text{w}} c_{\text{w}} . $$
(1)
It is well known that formation compressibility (cf) may not significantly change with pressure, while reservoir fluid compressibilities, oil compressibility (co), gas compressibility (cg), and water compressibility (cw), are pressure-variant parameters. Several models were introduced in the literature for the changes in the three-phase reservoir fluid compressibilities. The following model was presented by Martin (1959), Raghvan (1989), and Tabatabaie and Pooladi-Darvish (2017) for reservoir fluid total compressibility:
$$ c_{\text{F}} = S_{\text{o}} f\left( {C_{\text{o}} } \right) + S_{\text{w}} f\left( {C_{\text{w}} } \right) + S_{\text{g}} f\left( {C_{\text{g}} } \right), $$
(2)
where
$$ f\left( {C_{\text{o}} } \right) = \frac{{B_{\text{g}} R_{\text{s}}^{{\prime }} }}{{B_{\text{o}} }} - \frac{{B_{\text{o}}^{{\prime }} }}{{B_{\text{o}} }}, $$
(3)
$$ f\left( {C_{\text{w}} } \right) = \frac{{B_{\text{g}} R_{\text{sw}}^{{\prime }} }}{{B_{\text{w}} }} - \frac{{B_{\text{w}}^{{\prime }} }}{{B_{\text{w}} }}, $$
(4)
$$ f\left( {C_{\text{g}} } \right) = - \frac{{B_{\text{g}}^{{\prime }} }}{{B_{\text{g}} }}. $$
(5)
It can be clearly understood that the functions in Eqs. (3)–(5) are pressure-dependent functions since all parameters included in these functions such as oil, water, and gas formation volume factors as well as the gas solubility in oil and water are functions of pressure. Therefore, reservoir fluid total compressibility (cF), given by Eq. (2), is expected to be pressure-dependent parameter. To develop deterministic models for these parameters, the three situations (oil \( S_{\text{o}} \), gas Sg, and water Sw) in Eq. (2) should be determined first. For this purpose, it is required to calculate oil and water saturation derivatives with pressure. In this study, the following two models (Martin 1959) are used to calculate oil and water saturation derivatives with pressure:
$$ \frac{{{\text{d}}S_{0} }}{{{\text{d}}P}} = \frac{{S_{\text{o}} B_{\text{o}}^{{\prime }} }}{{B_{\text{o}} }} + \frac{{k_{\text{ro}} /\mu_{0} }}{{\left( {\frac{k}{\mu }} \right)_{\text{mp}} }}c_{\text{F}} , $$
(6)
$$ \frac{{{\text{d}}S_{\text{w}} }}{{{\text{d}}P}} = \frac{{S_{\text{w}} B_{\text{w}}^{'} }}{{B_{\text{w}} }} + \frac{{k_{\text{rw}} /\mu_{\text{w}} }}{{\left( {\frac{k}{\mu }} \right)_{\text{mp}} }}c_{\text{F}} . $$
(7)
To solve Eqs. (6) and (7), reservoir total mobility \( \left( {\frac{k}{\mu }} \right)_{\text{mp}} \) should be calculated as well as oil, gas, and water saturation for each pressure step. Therefore, relative permeabilities of the three phases have to be estimated. From oil–water system using water saturation, relative permeability of water \( \left( {k_{\text{rw}} } \right) \) can be estimated while relative permeability of gas phase \( \left( {k_{\text{rg}} } \right) \) is predicted from gas–water system knowing gas saturation. These two relative permeabilities are used to calculate the relative permeability of oil phase \( \left( {k_{\text{ro}} } \right) \) by several models proposed in the literature such as Stone I and II models (Stone 1970, 1973) or it could be determined graphically as shown in Fig. 1 using the two saturations of oil and water calculated by Stones’ models. Mathematically, multiphase reservoir total mobility is calculated by:
Fig. 1

Oil relative permeability behavior with water and gas saturations

$$ \left( {\frac{k}{\mu }} \right)_{\text{mp}} = \frac{{k_{\text{ro}} }}{{\mu_{\text{o}} }} + \frac{{k_{\text{rg}} }}{{\mu_{\text{g}} }} + \frac{{k_{\text{rw}} }}{{\mu_{\text{w}} }}. $$
(8)
For better understanding the methodology proposed in this study for calculating reservoir total mobility and compressibility, a set of PVT data given in Table 1, taken from Boe et al. (1989), is used. Reservoir and fluid properties are given in Table 2. Because neither water formation volume factor \( \left( {B_{\text{w}} } \right) \) nor solution gas–water ratio is given in Table 1, these two parameters are calculated using the following two models for reservoir temperature \( \left( {T = 200\,^{{^\circ }} {\text{F}}} \right) \) (Lee and Wattenbarger 1996):
$$ B_{\text{w}} = 1.039 - 4.1*10^{ - 6} P - 7.4*10^{ - 9} P^{2} , $$
(9)
$$ R_{\text{sw}} = 0.7883 + 2.1683*10^{ - 3} P - 0.93*10^{ - 7} P^{2} , $$
(10)
while formation water viscosity is calculated by:
$$ \mu_{\text{w}} = \mu_{1} \left[ {0.9993 + 4.0295*10^{ - 5} P + 3.1062*10^{ - 9} P^{2} } \right], $$
(11)
where \( \left( {\mu_{1} } \right) \) formation water viscosity at atmospheric pressure and reservoir temperature and calculated by:
$$ \mu_{1} = AT^{B} , $$
(12)
where \( \left( A \right) \) and \( \left( B \right) \) are two constants and determined by several models presented in the literature based on formation water solid content or reservoir temperature (Lee and Wattenbarger 1996).
Table 1

A set of PVT data (Boe et al. 1989)

P, psi

Bo bbl/STB

BGo bbl/SCF

RS SCF/STB

μo, cp

μg, 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

Table 2

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

The calculated properties of formation water are given in Table 3.
Table 3

Calculated formation water properties

P, psi

Rsw SCF/STB

μw, cp

Bw 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

The calculated relative permeabilities of the three phases of reservoir fluids are used to calculate multiphase reservoir total mobility, given by Eq. (8). The results are given in Table 4 and plotted in Fig. 2. Multi-regression analysis is used to generate a deterministic model for multiphase reservoir total mobility with reservoir pressure. This model is:
$$ \left( {\frac{k}{\mu }} \right)_{\text{mp}} = AP^{2} + BP + C, $$
(13)
where the constants \( \left( {A = 8*10^{ - 7} } \right) \), \( \left( {B = - 0.0079} \right) \), and \( \left( {C = 44.9} \right) \) are determined by the PVT data of the reservoir of interest.
Table 4

Reservoir total mobility results

P, psi

k ro

k rg

k rwo

kro/o

krg/g

krw/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

Fig. 2

Multiphase reservoir total mobility behavior with pressure

To calculate reservoir total compressibility \( \left( {c_{\text{t}} } \right)_{\text{mp}} \), reservoir fluid total compressibility \( \left( {c_{F} } \right) \) should be calculated first. The following procedures illustrate the methodology used for calculating reservoir total compressibility.
  1. 1.

    At the bubble point pressure, the three compressibility functions given by Eqs. (3)–(5) are calculated as well as reservoir total mobility.

     
  2. 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. 3.

    Calculate oil and water saturation derivatives given by Eqs. (6) and (7) for the second pressure interval.

     
  4. 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. 5.

    Calculate the relative permeability of the three phases of reservoir fluid using Stone’s models and reservoir total mobility using Eq. (8).

     
  6. 6.

    Calculate the three compressibility functions given by Eqs. (3)–(5) and repeat steps for all pressure intervals.

     
The results of multiphase reservoir fluid total compressibility \( \left( {c_{\text{F}} } \right) \) and multiphase reservoir total compressibility \( \left( {c_{\text{t}} } \right)_{\text{mp}} \) are given in Table 5. Figure 3 shows the behavior of multiphase reservoir total compressibility with pressure. Multi-regression analysis is used to generate a deterministic model for multiphase reservoir fluid total compressibility:
$$ c_{\text{F}} = AP^{B} . $$
(14)
Table 5

Multiphase reservoir fluid total compressibility and reservoir total compressibility results

P, psi

f(Co)

f(Cw)

f(Cg)

k ro

k rg

k rw

dSo/dP

dSw/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

kro/μo

krg/μg

krw/μw

c F

(ct)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

Fig. 3

Multiphase reservoir total compressibility behavior with pressure

According to Eq. (1), multiphase reservoir total compressibility is calculated by the summation of reservoir fluid total compressibility \( \left( {c_{\text{F}} } \right) \) and formation compressibility. It has been found that multiphase reservoir total compressibility has the same power model for reservoir fluid total compressibility regardless of the value of formation compressibility. However, the two constants \( \left( A \right) \) and \( \left( B \right) \) may change very slightly. Therefore, the deterministic model of multiphase reservoir total compressibility is:
$$ \left( {c_{\text{t}} } \right)_{\text{mp}} = AP^{B} . $$
(15)

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).

To support the approach presented above for generating deterministic models for multiphase reservoir total mobility and compressibility, another set of PVT data taken from Fraim and Wattenbarger (1988) is given in Table 6. The available information of the reservoir of interest is given in Table 7. The behavior of multiphase reservoir total mobility with pressure is shown in Fig. 4, while Fig. 5 demonstrates multiphase reservoir total compressibility behavior with pressure. The two deterministic models of these two parameters are similar to those given in Eqs. (13) and (15); however, there are very slight differences in the constants of these two models obtained by analyzing PVT data given by Boe et al. (1989) and PVT data given by Fraim and Wattenbarger (1988). The differences of the constants in these two models come from the differences in the two reservoir fluid properties and the differences in reservoir conditions. Therefore, for more accuracy to the proposed approach used in this manuscript, more PVT data sets can be used for developing different multiphase reservoir total compressibility and mobility models. Accordingly, the average values of the parameters included in these two models can be calculated and used for generating single and unique set of the two models of compressibility and mobility.
Table 6

PVT data (Fraim and Wattenbarger 1988)

P, psi

Bo bbl/STB

Rs SCF/STB

μo, cp

Bg bbl/STB

μg, 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

Table 7

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, Pb

\( 2755\,{\text{psi}} \)

Formation porosity, \( \emptyset \)

\( 12\% \)

Formation absolute permeability,\( k \)

\( 15\,{\text{md}} \)

Formation thickness, \( h \)

170 ft

Wellbore radius, rw

0.025 ft

Reservoir radius, re

600 ft

Fig. 4

Multiphase reservoir fluid total compressibility behavior with pressure

Fig. 5

Multiphase reservoir total compressibility behavior with pressure

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.

It is very well documented in the literature that the two dominant flow regimes in unconventional reservoirs during transient state flow are hydraulic fracture linear flow followed by formation linear flow regime represented by bilinear flow and trilinear flow regime. While pseudo-steady-state flow regime develops at late production time when pressure pulse reaches the boundary of reservoirs depleted by constant Sandface flow rate. For reservoirs depleted by constant wellbore pressure, the observed flow regime at late production time is called boundary-dominated flow regime. These flow regimes can be characterized either from pressure records with time (PTA) as shown in Fig. 6 or from decline rate behavior with time (RTA) as shown in Fig. 7.
Fig. 6

Pressure behavior with time for fractured reservoirs

Fig. 7

Rate behavior with time for fractured reservoirs

Figure 6 is prepared using the following mathematical model, in dimensionless form, for wellbore pressure drop assuming constant Sandface flow rate (Brown et al. 2011; Ozkan et al. 2011):
$$ \overline{{P_{\text{wD}} }} = \frac{\pi }{{sF_{\text{CD}} \sqrt {A_{\text{F}} } \tanh \left( {\sqrt {A_{\text{F}} } } \right)}}, $$
(16)
while Fig. 7 is prepared using the following mathematical model for Sandface flow rate assuming constant wellbore pressure (van Everdingen and Hurst 1949):
$$ \overline{{q_{\text{D}} }} = \frac{1}{{s^{2} \overline{{P_{\text{wD}} }} }}. $$
(17)

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

This flow regime represents linear fluid flow inside hydraulic fractures toward horizontal wellbore. It is characterized by a slope of \( \left( {1/2} \right) \) on pressure derivative curve. The mathematical model, in dimensionless form, of wellbore pressure drop assuming constant Sandface flow rate is given by:
$$ P_{\text{wD}} = \frac{{2\sqrt {\pi \omega t_{\text{D}} } }}{{F_{\text{CD}} }} + s, $$
(18)
while dimensionless flow rate of constant wellbore pressure is:
$$ q_{\text{D}} = \frac{{F_{\text{CD}} }}{{\pi \sqrt {\pi \omega t_{\text{D}} } }}. $$
(19)
In field units, Eqs. (18) and (19) can be written as:
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = \frac{1}{{\left[ {\frac{8.126}{{x_{\text{f}} hF_{\text{CD}} }}\sqrt {\frac{\omega }{{k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} \emptyset \left( {c_{\text{t}} } \right)_{\text{mp}} }}t} + \frac{141.2}{{k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} h}}s} \right]}}, $$
(20)
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = 0.07838h\frac{{F_{\text{CD}} x_{\text{f}} \sqrt {k_{i} \left( {\frac{k}{\mu }} \right)_{\text{mp}} } }}{{\sqrt {\frac{\omega }{{\left( {c_{\text{t}} } \right)_{\text{mp}} \emptyset }}t} }}, $$
(21)
where \( \left( {k_{\text{i}} } \right) \) is the induced matrix permeability or the permeability of stimulated porous media between hydraulic fractures. It could be more than the original reservoir permeability because of the induced fractures caused by fracturing process, while \( \left( {\frac{k}{\mu }} \right)_{\text{mp}} \) is the mobility of stimulated porous media.
$$ q_{\text{t}} = q_{\text{o}} B_{\text{o}} + 1000B_{\text{g}} \left\{ {q_{\text{sc}} - \left( {q_{\text{o}} R_{\text{s}} + q_{\text{w}} R_{\text{sw}} } \right)} \right\} + q_{\text{w}} B_{\text{w}} , $$
(22)
$$ B_{\text{t}} = B_{\text{o}} + B_{\text{g}} \left( {R_{\text{sb}} - R_{\text{s}} } \right). $$
(23)
Log–log plot of production time versus \( \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} \) gives the IPR of hydraulic fracture linear flow for the two cases: constant Sandface flow rate and constant wellbore pressure as shown in Fig. 8.
Fig. 8

IPR of hydraulic fracture linear flow

3.2 Bilinear Flow Regime

This flow regime represents simultaneous linear fluid flow from the matrix to the hydraulic fractures and from hydraulic fractures to the horizontal wellbore. It is characterized by a slope of \( \left( {1/4} \right) \) on pressure derivative curve. The mathematical models, in dimensionless form, of wellbore pressure drop assuming constant Sandface flow rate are given by:
$$ P_{\text{wD}} = \frac{\pi }{{\varGamma \left( {5/4} \right)\sqrt {2F_{\text{CD}} } }}\sqrt[4]{{\frac{{t_{\text{D}} }}{\omega }}} + s\quad {\text{For}}\,{\text{hydraulic}}\,{\text{fractures}}\,{\text{with}}\,{\text{fracture}}\,{\text{conductivity}}\;\left( {F_{\text{CD}} \le 100} \right), $$
(24)
$$ P_{\text{wD}} = \frac{2\pi }{{\varGamma \left( {5/4} \right)\sqrt {F_{\text{CD}} } }}\sqrt[4]{{\frac{{t_{\text{D}} }}{\omega }}} + s\quad {\text{For}}\,{\text{hydraulic}}\,{\text{fractures}}\,{\text{with}}\,{\text{fracture}}\,{\text{conductivity}}\;\left( {F_{\text{CD}} > 100} \right), $$
(25)
while dimensionless flow rates of constant wellbore pressure are:
$$ q_{\text{D}} = \frac{{\sqrt {F_{\text{CD/2}} } }}{\pi }\frac{1}{{\sqrt[4]{{\frac{{t_{\text{D}} }}{\omega }}}}}\quad {\text{For}}\,{\text{hydraulic}}\,{\text{fractures}}\,{\text{with}}\,{\text{fracture}}\,{\text{conductivity}}\;\left( {F_{\text{CD}} \le 100} \right), $$
(26)
$$ q_{\text{D}} = \frac{{\sqrt {F_{\text{CD/2}} } }}{2\pi }\frac{1}{{\sqrt[4]{{\frac{{t_{\text{D}} }}{\omega }}}}}\quad {\text{For}}\,{\text{hydraulic}}\,{\text{fractures}}\,{\text{with}}\,{\text{fracture}}\,{\text{conductivity}}\;\left( {F_{\text{CD}} > 100} \right). $$
(27)
In field units, Eqs. (24)–(27) can be written, respectively, as:
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = \frac{1}{{\left[ {\frac{44}{{h\sqrt {x_{\text{f}} F_{\text{CD}} } }}\sqrt[4]{{\frac{1}{{\omega \emptyset k_{\text{i}}^{3} \left( {\frac{k}{\mu }} \right)_{\text{mp}}^{3} \left( {c_{\text{t}} } \right)_{\text{mp}} }}t}} + \frac{141.2}{{k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} h}}s} \right]}}, $$
(28)
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = \frac{1}{{\left[ {\frac{62.33}{{h\sqrt {x_{\text{f}} F_{\text{CD}} } }}\sqrt[4]{{\frac{1}{{\omega \emptyset k_{\text{i}}^{3} \left( {\frac{k}{\mu }} \right)_{\text{mp}}^{3} \left( {c_{\text{t}} } \right)_{\text{mp}} }}t}} + \frac{141.2}{{k_{\text{i}}^{3} \left( {\frac{k}{\mu }} \right)_{\text{mp}} h}}s} \right]}}, $$
(29)
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = 0.0125h\frac{{\sqrt {F_{\text{CD}} x_{\text{f}} } \sqrt[4]{{k_{\text{i}}^{3} \left( {\frac{k}{\mu }} \right)_{\text{mp}}^{3} }}}}{{\sqrt[4]{{\frac{1}{{\omega \left( {c_{\text{t}} } \right)_{\text{mp}} \emptyset }}t}}}}, $$
(30)
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = 0.00626h\frac{{\sqrt {F_{\text{CD}} x_{\text{f}} } \sqrt[4]{{k_{\text{i}}^{3} \left( {\frac{k}{\mu }} \right)_{\text{mp}}^{3} }}}}{{\sqrt[4]{{\frac{1}{{\omega \left( {c_{\text{t}} } \right)_{\text{mp}} \emptyset }}t}}}}. $$
(31)
The IPRs of bilinear flow for constant Sandface flow rate and constant wellbore pressure are shown in Figs. 9 and 10 for \( \left[ {F_{\text{CD}} \le 100.0} \right] \) and \( \left[ {F_{\text{CD}} > 100.0} \right] \), respectively.
Fig. 9

IPR of bilinear flow, \( F_{\text{CD}} \le 100.0 \)

Fig. 10

IPR of bilinear flow, \( F_{\text{CD}} > 100.0 \)

3.3 Trilinear Flow Regime

Unconventional reservoirs may consist of stimulated reservoir volume (SRV) where hydraulic fractures propagate in the porous media and unstimulated reservoir volume (USRV) where no hydraulic fractures. Trilinear flow regime is observed in this type of reservoirs wherein petrophysical properties are significantly different in these two volumes. It represents three simultaneous linear flow regimes. The first is the flow from unstimulated to stimulated reservoir volume. The second is the flow from stimulated reservoir volume to the hydraulic fractures, while the third is the linear flow inside these fractures. It is characterized by a slop of \( \left( {1/8} \right) \) on pressure derivative curve. It is always seen after bilinear flow regime and before pseudo-steady-state flow or boundary-dominated flow regime as shown in Fig. 11. The mathematical model of pressure drop caused by this flow regime assuming constant Sandface flow rate is:
$$ P_{\text{wD}} = \frac{2\pi }{{\varGamma \left( {9/8} \right)F_{\text{CD}} }}\sqrt[8]{{\frac{{t_{\text{D}} }}{\omega }}} + s, $$
(32)
while dimensionless flow rate for constant wellbore pressure is:
Fig. 11

Trilinear flow regimes

$$ q_{\text{D}} = \frac{{F_{\text{CD}} }}{\pi }\frac{1}{{\sqrt[8]{{\frac{{t_{\text{D}} }}{\omega }}}}}. $$
(33)
In field units, Eqs. (32) and (33) can be written, respectively, as:
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = \frac{1}{{\left[ {\frac{336}{{k_{\text{i}} \left( {\frac{k}{\pi }} \right)_{\text{mp}} hF_{\text{CD}} }}\sqrt[8]{{\frac{{k_{\text{i}} \left( {\frac{k}{\pi }} \right)_{\text{mp}} }}{{\omega \emptyset x_{\text{f}}^{2} \left( {c_{\text{t}} } \right)_{\text{mp}} }}t}} + \frac{141.2}{{k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} h}}s} \right]}}, $$
(34)
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = 0.0265h\frac{{F_{\text{CD}} k_{\text{i}} \left( {\frac{k}{u}} \right)_{\text{mp}} }}{{\sqrt[8]{{\frac{t}{{\omega x_{\text{f}}^{2} \left( {c_{\text{t}} } \right)_{\text{mp}} \emptyset }}}}}}. $$
(35)
The IPRs of trilinear flow of constant Sandface flow rate and constant wellbore pressure are shown in Fig. 12.
Fig. 12

IPR of trilinear flow regime

3.4 Pseudo-steady-State (Boundary-Dominated) Flow Regime

Unlike hydraulic fracture linear flow, bilinear flow, and trilinear flow regimes, pseudo-steady-state flow regime (constant Sandface flow rate) or boundary-dominated flow regime (constant wellbore pressure) is characterized by constant productivity index as shown in Fig. 13. These two flow regimes represent the impact of reservoir boundary on pressure behavior. For constant Sandface flow rate, stabilized productivity index means that the pressure drop is constant with time and pseudo-steady state has been reached, while boundary-dominated flow could exhibit asymptotically constant productivity index even though production rate and reservoir pressure change with time (Aulisa et al. 2009). The productivity index during pseudo-steady-state flow regimes is given by:
$$ J_{\text{Dq}} = \frac{1}{{0.5\ln \left( {\frac{{4x_{\text{eD}} y_{\text{eD}} }}{{1.781C_{\text{AFq}} }}} \right) + s}}, $$
(36)
and during boundary-dominated flow regime, the index is:
Fig. 13

Productivity index behavior

$$ J_{\text{DP}} = \frac{1}{{0.5\ln \left( {\frac{{4x_{\text{eD}} y_{\text{eD}} }}{{1.781C_{\text{AFP}} }}} \right) + s}}. $$
(37)
\( \left( {C_{\text{AFq}} } \right) \) and \( \left( {C_{\text{AFP}} } \right) \) in Eqs. (36) and (37) are the shape factors of hydraulically fractured reservoirs for constant Sandface flow rate and constant wellbore pressure, respectively. Figure 14 shows these two shape factors for hydraulically fractures reservoirs.
Fig. 14

Shape factor of hydraulically fractured reservoirs

In field units, Eqs. (36) and (37) can be written as:
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = \frac{{k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} }}{{141.2\left[ {0.5\ln \left( {\frac{{4x_{\text{e}} y_{\text{e}} }}{{1.781C_{\text{AFq}} x_{\text{f}}^{2} }}} \right)} \right]}}, $$
(38)
$$ \left( {\frac{{q_{\text{t}} B_{\text{t}} }}{{\Delta P_{\text{wf}} }}} \right)_{\text{mp}} = \frac{{k\left( {\frac{k}{\mu }} \right)_{\text{mp}} }}{{141.2\left[ {0.5\ln \left( {\frac{{4x_{\text{e}} y_{\text{e}} }}{{1.781C_{\text{AFP}} x_{\text{f}}^{2} }}} \right)} \right]}}. $$
(39)

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

The proposed approach in this study is applied for two PVT data sets of two different reservoirs. The first is given in Table 1 (Boe et al. 1989), and the second is given in Table 6 (Fraim and Wattenbarger 1988). Analyzing these PVT data and following the methodology presented in section 2, multiphase flow reservoir total mobility and compressibility given by Eqs. (13) and (15) are determined, respectively, for the first reservoir:
$$ \left( {\frac{k}{\mu }} \right)_{\text{mp}} = 8*10^{ - 7} P^{2} - 0.0079P + 44.9, $$
(40)
$$ \left( {c_{\text{t}} } \right)_{\text{mp}} = 0.3099P^{ - 0.896} , $$
(41)
while for the second reservoir:
$$ \left( {\frac{k}{\mu }} \right)_{\text{mp}} = 1.1*10^{ - 6} P^{2} - 0.0102P + 46.75, $$
(42)
$$ \left( {c_{\text{t}} } \right)_{\text{mp}} = 0.3164P^{ - 0.964} . $$
(43)
The above-mentioned four models indicate that multiphase reservoir total mobility and compressibility may not be significantly affected by reservoir type and reservoir fluid type. Even though two PVT data sets are used only in this study, the conclusion that the behaviors of multiphase flow reservoir mobility and compressibility with reservoir pressure are similar regardless of reservoir type might be true. This conclusion could be supported by using different PVT data sets for different reservoirs, generate the two models of multiphase reservoir mobility and compressibility, and compare them with the models given by Eqs. (40)–(43). Figure 15 depicts the behaviors of the two models of multiphase flow reservoir total mobility and compressibility with pressure. It can be seen that the two models demonstrate similar behavior for the two PVT data sets. Therefore, it is quite reasonable using the average values of the two models obtained from the two PVT data sets for calculating multiphase total mobility and compressibility of other reservoirs. These two models can be written as:
Fig. 15

Comparison of multiphase flow reservoir total mobility and compressibility for two different reservoirs

$$ \left( {\frac{k}{\mu }} \right)_{\text{mp}} = 9.7*10^{ - 7} P^{2} - 0.009P + 46.0, $$
(44)
$$ \left( {c_{\text{t}} } \right)_{\text{mp}} = 0.313P^{ - 0.93} . $$
(45)
The proposed models of multiphase flow reservoir total mobility and compressibility given by Eqs. (44) and (45) can be used in predicting transient and pseudo-steady-state IPRs of different reservoirs. To examine these two models, two hydraulically fractured unconventional reservoirs are considered in this study. The first is the Bakken formation located in Williston Basin in North Dakota, USA, and the second is the Eagle Ford formation in Texas, USA. The available information for these two formations is given in Table 8 (Uzun et al. 2014). Dimensionless parameters of these two reservoirs are calculated and given in Table 9.
Table 8

Bakken and Eagle Ford formations (Uzun et al. 2014)

 

Black oil-1

Well Bakken formation

Volatile oil

Well Eagle Ford formation

Initial reservoir pressure, Pi

7802 psi

8428 psi

Bubble point pressure, Pb

2130 psi

4350 psi

Bottom hole temperature, T

\( 240\,^{{^\circ }} {\text{F}} \)

\( 269\,^{{^\circ }} {\text{F}} \)

Solution gas–oil ratio, Rs

\( 850\,{\text{SCF/STB}} \)

\( 1112\,{\text{SCF/STB}} \)

Oil formation volume factor, Bo

\( 1.61\,{\text{RBBL/STB}} \)

\( 1.56\,{\text{RBBL/STB}} \)

Water formation volume, Bw

\( 1.04\,{\text{RBBL/STB}} \)

\( 1.04\,{\text{RBBL/STB}} \)

Gas formation volume factor, Bg

\( 0.9\,{\text{RBBL/MScf}} \)

\( 0.55\,{\text{RBBL/MScf}} \)

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

Table 9

Dimensionless parameters

 

Black oil-1 well

Bakken formation

Volatile oil well

Eagle Ford formation

\( x_{\text{eD}} \)

\( 1.667 \)

\( 2.4 \)

\( y_{\text{eD}} \)

\( 0.741 \)

\( 1.017 \)

\( F_{\text{CD}} \)

\( 5.35 \)

\( 14.51 \)

\( \omega \)

\( 1000 \)

\( 1000 \)

\( \lambda \)

\( 100 \)

\( 100 \)

\( R_{\text{CD}} \)

\( 12.7 \)

\( 1.16 * 10^{4} \)

\( w_{\text{D}} \)

\( 5.26 *10^{ - 5} \)

\( 1.427 *10^{ - 4} \)

\( \eta_{\text{FD}} \)

\( 1.017 *10^{ - 5} \)

\( 1.017 *10^{5} \)

\( \eta_{\text{mD}} \)

\( 0.1063 \)

\( 8.474 *10^{ - 5} \)

Dimensionless pressure and pressure derivative of Bakken and Eagle Ford formations are calculated using dimensionless parameters given in Table 9 and plotted as shown in Fig. 16, while dimensionless flow rate behavior is calculated and plotted as shown in Fig. 17. Transient and pseudo-steady-state productivity index of the two reservoirs in dimensionless form is calculated for the two wellbore conditions, constant Sandface flow rate and constant wellbore pressure, and plotted in Fig. 18. Four flow regimes are observed for the two reservoirs: Hydraulic fracture linear flow, bilinear flow, trilinear flow, and pseudo-steady-state flow regime.
Fig. 16

Pressure and pressure derivative behavior of Bakken and Eagle Ford formations

Fig. 17

Flow rate behavior

Fig. 18

Transient and pseudo-steady-state productivity index behaviors

To validate the proposed models of multiphase flow properties, bottom hole flowing pressure of the two wells, black oil well-1 (Bakken formation) and volatile oil well (Eagle Ford formation), is calculated as follows:
  1. 1.

    For each pressure step, the following parameters are calculated:

     
$$ B_{\text{o}} \quad B_{\text{g }} \quad B_{\text{w}} \quad R_{\text{s}} \quad R_{\text{sw}} . $$
It is important to emphasize that oil formation volume factor \( \left( {B_{\text{o}} } \right) \) and solution gas oil–oil ratio \( \left( {R_{\text{s}} } \right) \) are determined by the following two models, respectively:
$$ B_{\text{o}} = aP^{2} + bP + c, $$
(46)
$$ R_{\text{s}} = aP^{2} + bP + c, $$
(47)
where \( \left( a \right), \left( b \right), \) and \( \left( c \right) \) are determined from PVT data. For PVT data given in Table 1, these two relationships are plotted in Fig. 19, while water formation volume factor \( \left( {B_{\text{w}} } \right) \) and solution gas–water ratio \( \left( {R_{\text{sw}} } \right) \) are calculated using the mathematical models presented in the literature, Eqs. (9) and (10), and plotted in Fig. 20. Similarly, gas formation volume factor is calculated and plotted in Fig. 21.
Fig. 19

Oil formation volume factor and solution gas–oil ratio

Fig. 20

Water formation volume factor and solution gas–water ratio

Fig. 21

Gas formation volume factor

  1. 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).

     
  2. 3.

    Calculate multiphase reservoir total compressibility \( \left( {c_{\text{t}} } \right)_{\text{mp}} \) using Eq. (45).

     
  3. 4.

    Calculate multiphase reservoir total mobility using Eq. (44).

     
  4. 5.

    Calculate wellbore pressure drop \( \left( {\Delta P_{\text{wf}} } \right) \) by:

     
$$ \Delta P_{\text{wf}} = \frac{{141.2\left( {q_{\text{t}} B_{\text{t}} } \right)P_{\text{wD}} }}{{k_{\text{m}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} }}, $$
(48)
where \( \left( {P_{wD} } \right) \) is dimensionless wellbore pressure drop calculated by Eq. (16) for different dimensionless production times.
  1. 6.

    Real production time is calculated from dimensionless production time by:

     
$$ t = \frac{{\emptyset \left( {c_{\text{t}} } \right)_{\text{mp}} x_{\text{f}}^{2} t_{\text{D}} }}{{0.0002637k_{\text{m}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} }}. $$
(49)
The calculated bottom hole flowing pressure is plotted and compared with the pressure records of the two reservoirs. Excellent matching is obtained as shown in Fig. 22.
Fig. 22

Comparison for bottom hole flowing pressure; a Bakken formation and b Eagle Ford formation

The IPRs of Bakken and Eagle Ford reservoirs assuming constant Sandface flow rate for different flow regimes are plotted in Figs. 23 and 24, respectively. The starting time of each flow regime is determined using Eq. (49). The following points are inferred from these two plots:
Fig. 23

The IPR for different flow regimes—Bakken formation

Fig. 24

The IPR for different flow regimes—Eagle Ford formation

  1. 1.

    The productivity of Eagle Ford formation for the same pressure drop is better than of Bakken formation.

     
  2. 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. 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. 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. 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. 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. 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. 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.

     

References

  1. Agarwal, R.G., Carter, R.D., Pollock, C.B.: Evaluation and performance prediction of low-permeability gas wells stimulated by massive hydraulic fracturing. JPT 31(03), 362–372 (1979).  https://doi.org/10.2118/6838-pa CrossRefGoogle Scholar
  2. Al-Khalifa, A.J., Horne, R.N., Aziz, K.: Multiphase well test analysis: pressure and pressure-squared methods. Paper presented at the SPE California regional meeting held in Bakersfield, CA, USA, April 5–7 (1989).  https://doi.org/10.2118/18803-ms
  3. Aulisa, E., Ibragimov, A., Walton, J.: A new method for evaluating the productivity index of nonlinear flows. SPE J. 14(04), 693–706 (2009).  https://doi.org/10.2118/108984-pa CrossRefGoogle Scholar
  4. Ayan, C., Lee, W.J.: Multiphase pressure buildup analysis: field examples. Paper presented at the SPE California regional meeting held in Long Beach, California, USA, March 23–25 (1988).  https://doi.org/10.2118/17412-ms
  5. Bello, R.O.: Rate transit analysis in shale gas reservoirs with transient linear behavior. PhD, Texas A&M University, Collage Station, TX, USA (2008)Google Scholar
  6. Behmanesh, H., Mattar, L., Thompson, J.M., Anderson, D.M., Nakaska, D.W., Clarkson, C.R.: Treatment of rate-transient analysis during boundary-dominated flow. SPEJ (2018).  https://doi.org/10.2118/189967-pa Google Scholar
  7. Behmanesh, H., Clarkson, C.R., Tabatabaie, S.H., Heidari Sureshjani, M.: Impact of distance-of-investigation calculations on rate-transient analysis of unconventional gas and light-oil reservoirs: new formulations for linear flow. JCPT 54(06), 509–519 (2015).  https://doi.org/10.2118/178928-pa CrossRefGoogle Scholar
  8. Bennett, C.O., Reynolds, A.C., Raghavan, R., Elbel, J.L.: Performance of finite-conductivity, vertically fractured wells in single-layer reservoirs. SPE Form. Eval. 1(04), 399–412 (1986).  https://doi.org/10.2118/11029-pa CrossRefGoogle Scholar
  9. Boe, A., Skjaeveland, S.M., Whitson, C.H.: Two-phase pressure test analysis. SPE Form. Eval. 04(04), 601–610 (1989).  https://doi.org/10.2118/10224-pa CrossRefGoogle Scholar
  10. Brown, M., Ozkan, E., Raghavan, R., Kazemi, H.: practical solutions for pressure-transient responses of fractured horizontal wells in unconventional shale reservoirs. SPE Reserv. Eval. Eng. 14(6), 663–676 (2011).  https://doi.org/10.2118/125043-pa CrossRefGoogle Scholar
  11. Camacho, R.G., Raghavan, R.: Inflow performance relationships for solution-gas-drive reservoirs. JPT 41(05), 541–550 (1989).  https://doi.org/10.2118/16204-pa CrossRefGoogle Scholar
  12. Camacho, R.G., Raghavan, R., Reynolds, A.C.: Response of wells producing layered reservoirs: unequal fracture length. SPE Form. Eval. 2(01), 9–28 (1987).  https://doi.org/10.2118/12844-pa CrossRefGoogle Scholar
  13. Camacho Velazquez, R., Fuentes-Cruz, G., Vasquez-Cruz, M.A.: Decline-curve analysis of fractured reservoirs with fractal geometry. SPE Reserv. Eval. Eng. 11(03), 606–619 (2008).  https://doi.org/10.2118/104009-pa CrossRefGoogle Scholar
  14. Cinco-Ley, H.: Unsteady-state pressure distribution created by a slanted well or a well with an inclined fracture. PhD dissertation, Stanford University, California, USA (1974)Google Scholar
  15. Cinco-Ley, H., Samaniego, F.: Transient pressure analysis for fractured wells. JPT 33(09), 1749–1766 (1981).  https://doi.org/10.2118/7490-pa CrossRefGoogle Scholar
  16. Cinco-Ley, H., Ramey, H.J., Miller, F.G.: Unsteady-state pressure distribution created by a well with an inclined fracture. Paper presented at the 50th annual fall meeting of the society of petroleum engineers of AIME held in Dallas, TX, USA (1975).  https://doi.org/10.2118/5591-ms
  17. Cinco, L., Samaniego, V., Dominguez, A.: Transient pressure behavior for a well with a finite-conductivity vertical fracture. SPEJ 18(04), 253–264 (1978).  https://doi.org/10.2118/6014-pa CrossRefGoogle Scholar
  18. Cipolla, C.L.: Modeling production and evaluating fracture performance in unconventional gas reservoirs. JPT 61(09), 84–90 (2009).  https://doi.org/10.2118/118536-jpt CrossRefGoogle Scholar
  19. Chen, A., Jones, J.R.: Use of pressure/rate deconvolution to estimate connected reservoir-drainage volume in naturally fractured unconventional-gas reservoirs from canadian rockies foothills. SPE Reserv. Eval. Eng. 15(03), 290–299 (2012).  https://doi.org/10.2118/143016-pa CrossRefGoogle Scholar
  20. Chu, W.-C., Reynolds, A.C., Raghavan, R.: Pressure transient analysis of two-phase flow problems. SPE Form. Eval. 01(02), 151–164 (1986).  https://doi.org/10.2118/10223-pa CrossRefGoogle Scholar
  21. Duong, A.N.: Rate-decline analysis for fracture-dominated shale reservoirs. SPE Reserv. Eval. Eng. 14(03), 377–387 (2011).  https://doi.org/10.2118/137748-pa CrossRefGoogle Scholar
  22. El-Banbi, A.H.: Analysis of tight gas wells. PhD, Texas A&M University, Collage Station, TX, USA (1998)Google Scholar
  23. Fetkovich, M.J.: The isochronal testing of oil wells. Paper presented at the 48th Annual fall meeting of the SPE of AIME in Las Vegas, Nevada, USA, 30 Oct–3 Nov (1973).  https://doi.org/10.2118/4529-ms
  24. Fraim, M.L., Wattenbarger, R.A.: Decline curve analysis for multiphase flow. Paper presented at the SPE 63rd annual technical conference & exhibition held in Houston, TX, USA, October 2–5 (1988).  https://doi.org/10.2118/18274-ms
  25. Fuentes-Cruz, G., Valko, P.P.: Revisiting the dual-porosity/dual-permeability modeling of unconventional reservoirs: the induced-interporosity flow field. SPE J. 20(01), 125–141 (2015).  https://doi.org/10.2118/173895-pa CrossRefGoogle Scholar
  26. Fuentes-Cruz, G., Gildin, E., Valko, P.P.: Analyzing production data from hydraulically fractured wells: the concept of induced permeability field. SPE Reserv. Eval. Eng. 17(02), 220–232 (2014).  https://doi.org/10.2118/163843-pa CrossRefGoogle Scholar
  27. Gallice, F., Wiggins, M.L.: A comparison of two-phase inflow performance relationships. SPE Prod. Facil. 19(02), 100–104 (2004).  https://doi.org/10.2118/88445-pa CrossRefGoogle Scholar
  28. Gringarten, A.C., Ramey, H.J.: The use of source and green’s function in solving unsteady-flow problems in reservoirs. SPEJ 13(05), 286–296 (1973).  https://doi.org/10.2118/3818-pa CrossRefGoogle Scholar
  29. Gringarten, A.C., Ramey, H.J., Raghavan, R.: Unsteady-state pressure distributions created by a well with a single infinite-conductivity vertical fracture. SPEJ 14(04), 347–360 (1974).  https://doi.org/10.2118/4051-pa CrossRefGoogle Scholar
  30. Golan, M., Whitson, C.H.: Well Performance, 2nd edn. Prentice-Hall Inc (1995). Printed in Norway by Tapir (1996)Google Scholar
  31. Guppy, K.H., Kumar, S., Kagawan, V.D.: Pressure-transient analysis for fractured wells producing at constant pressure. SPE Form. Eval. 3(01), 169–178 (1988).  https://doi.org/10.2118/13629-pa CrossRefGoogle Scholar
  32. Hagoort, J.: Automatic decline-curve analysis of wells in gas reservoirs. SPE Reserv. Eval. Eng. 6(06), 433–440 (2003).  https://doi.org/10.2118/77187-pa CrossRefGoogle Scholar
  33. Holditch, S.A., Morse, R.A.: The effects of non-Darcy flow on the behavior of hydraulically fractured gas wells (includes associated paper 6417). JPT 28(10), 1169–1179 (1976).  https://doi.org/10.2118/5586-pa CrossRefGoogle Scholar
  34. Kamal, M.M., Pan, Y.: Use of transient data to calculate absolute permeability and average fluid saturations. SPE Reserv. Eval. Eng. 13(02), 306–312 (2010).  https://doi.org/10.2118/113903-pa CrossRefGoogle Scholar
  35. Kamal, M.M., Pan, Y.: Pressure transient testing under multiphase flow conditions. Paper presented at the SPE Middle East Oil and gas show and conference held in Manama, Bahrain, Sep. 25–28 (2011).  https://doi.org/10.2118/113903-pa
  36. Kanfar, M.S., Clarkson, C.R.: Rate dependence of bilinear flow in unconventional gas reservoirs. SPE Reserv. Eval. Eng. 5, 7 (2018).  https://doi.org/10.2118/186092-pa Google Scholar
  37. Kuchuk, F.J.: Applications of convolution and deconvolution to transient well tests. SPE Form. Eval. 5(04), 375–384 (1990).  https://doi.org/10.2118/16394-pa CrossRefGoogle Scholar
  38. Ibrahim, M., Wattenbarger, R.A.: Rate dependence of transient linear flow in tight gas wells. JCPT 45(10), 18–20 (2006).  https://doi.org/10.2118/06-10-tn2 CrossRefGoogle Scholar
  39. Ilk, D., Valko, P.P., Blasingame, T.A.: Deconvolution of variable-rate reservoir performance data using B-splines. SPE Reserv. Eval. Eng. 9(05), 582–595 (2006).  https://doi.org/10.2118/95571-pa CrossRefGoogle Scholar
  40. Izadi, M., Yildiz, T.: Transient flow in discretely fractured porous media. SPEJ 14(02), 362–373 (2009).  https://doi.org/10.2118/108190-pa CrossRefGoogle Scholar
  41. Larsen, L., Hegre, T.M.: Pressure transient analysis of multifractured horizontal wells. Paper (SPE-28389) presented at the SPE Annual technical conference & exhibition held in New Orleans, Louisiana, USA, 25–28 September (1994). https://doi.org/10.2118/28389-ms
  42. Lee, J., Wattenbarger, R.A.: Gas Reservoir Engineering. SPE Textbook Series, vol. 5. Society of Petroleum Engineers, Houston (1996)Google Scholar
  43. Levitan, M.M.: Practical application of pressure-rate deconvolution to analysis of real well tests. SPE Reserv. Eval. Eng. 8(02), 113–121 (2005).  https://doi.org/10.2118/84290-pa CrossRefGoogle Scholar
  44. Li, X., Liang, J., Xu, W., Li, X., Tan, X.: The new method on gas-water two phase steady state productivity of fractured horizontal well in tight gas reservoir. Geo Energy Res. 1(02), 105–111 (2017).  https://doi.org/10.26804/ager.2017.02.06 Google Scholar
  45. Luo, H., Mahiya, G., Pannett, S., Benham, P.H.: The use of rate-transient-analysis modeling to quantify uncertainties in commingled tight gas production-forecasting and decline-analysis parameters in the Alberta deep basin. SPE Reserv. Eval. Eng. 17(02), 209–219 (2014).  https://doi.org/10.2118/147529-pa CrossRefGoogle Scholar
  46. Martin, J.C.: Simplified equations of flow in gas drive reservoirs and the theoretical foundation of multiphase pressure buildup analyses. SPE Gen. 261(01), 321–323 (1959)Google Scholar
  47. Martin, J.C., James, D.M.: Analysis of pressure transients in two-phase radial flow. SPE J. 3(02), 116–126 (1963).  https://doi.org/10.2118/425-pa Google Scholar
  48. Muskat, M.: The flow of homogenous fluids through porous media. McGraw Hill Book Co., Inc., New York (1937)Google Scholar
  49. Muskat, M., Meres, M.W.: The flow of heterogeneous fluids in porous media. J. Appl. Phys. 7(09), 346–363 (1936).  https://doi.org/10.1063/1.1745403 Google Scholar
  50. Nobakht, M., Clarkson, C.R., Kaviani, D.: New and improved methods for performing rate-transient analysis of shale gas reservoirs. SPE Reserv. Eval. Eng. 15(03), 335–350 (2012).  https://doi.org/10.2118/147869-pa CrossRefGoogle Scholar
  51. Ozkan, E.: Performance of horizontal wells. PhD dissertation, The University of Tulsa, OK, USA (1988)Google Scholar
  52. Ozkan, E., Raghavan, R.: New solutions for well-test-analysis problems: part 1 analytical considerations. SPE Form. Eval. 6(03), 359–368 (1991a).  https://doi.org/10.2118/18616-pa CrossRefGoogle Scholar
  53. Ozkan, E., Raghavan, R.: New solutions for well-test-analysis problems: part 2 computational considerations and applications. SPE Form. Eval. 6(03), 369–378 (1991b).  https://doi.org/10.2118/18616-pa CrossRefGoogle Scholar
  54. Ozkan, E., Brown, M.L., Raghavan, R., Kazemi, H.: Comparison of fractured-horizontal-well performance in tight sand and shale reservoirs. SPE Reserv. Eng. Eval. 14(02), 248–256 (2011).  https://doi.org/10.2118/121290-pa CrossRefGoogle Scholar
  55. Perrine, R. L.: Analysis of pressure-buildup curves. Drilling and production practice, API-56-482, New York, USA (1956)Google Scholar
  56. Prats, M., Levine, J.S.: Effect of vertical fractures on reservoir behavior- results on oil and gas flow. JPT 15(10), 1119–1126 (1963).  https://doi.org/10.2118/593-pa CrossRefGoogle Scholar
  57. Raghavan, R.: Well test analysis: wells producing by solution gas drive. SPEJ 16(04), 196–208 (1976).  https://doi.org/10.2118/5588-pa CrossRefGoogle Scholar
  58. Raghavan, R.: Well-test analysis for multiphase flow. SPE Form. Eval. 4(04), 585–594 (1989).  https://doi.org/10.2118/14098-pa CrossRefGoogle Scholar
  59. Raghavan, R., Uraiet, A., Thomas, G.W.: Vertical fracture height: effect on transient flow behavior. SPEJ 18(04), 265–277 (1978).  https://doi.org/10.2118/6016-pa CrossRefGoogle Scholar
  60. Raghavan, R.S., Chen, C.-C., Agarwal, B.: An analysis of horizontal wells intercepted by multiple fractures. SPEJ 2(03), 235–245 (1997).  https://doi.org/10.2118/27652-pa CrossRefGoogle Scholar
  61. Shahamat, M.S., Mattar, L., Aguilera, R.: Analysis of decline curves on the basis of beta-derivative. SPE Reserv. Eval. Eng. 18(02), 214–227 (2015).  https://doi.org/10.2118/169570-pa CrossRefGoogle Scholar
  62. Soliman, M.Y., Hunt, J.L., El Rabaa, A.M.: Fracturing aspects of horizontal wells. JPT 42(08), 966–973 (1990).  https://doi.org/10.2118/18542-pa CrossRefGoogle Scholar
  63. Standing, M.B.: Concerning the calculation of inflow performance of wells producing from solution gas drive reservoirs. JPT 23(09), 1141–1142 (1971).  https://doi.org/10.2118/3332-pa CrossRefGoogle Scholar
  64. Stone, H.L.: Probability model for estimating three-phase relative permeability. JPT 22(2), 214–218 (1970).  https://doi.org/10.2118/2116-pa CrossRefGoogle Scholar
  65. Stone, H.L.: Estimation of three-phase relative permeability and residual oil data. JCPT 12(4), 53–61 (1973).  https://doi.org/10.2118/73-04-06 CrossRefGoogle Scholar
  66. Tabatabaie, S.H., Pooladi-Darvish, M.: Multiphase linear flow in tight oil reservoirs. SPE Reserv. Eval. Eng. 20(01), 184–196 (2017).  https://doi.org/10.2118/180932-pa CrossRefGoogle Scholar
  67. Torcuk, M.A., Kurtoglu, B., Alharthy, N., Kazemi, H.: Analytical solutions for multiple matrix in fractured reservoirs: application to conventional and unconventional reservoirs. SPEJ 18(05), 969–981 (2013).  https://doi.org/10.2118/164528-pa CrossRefGoogle Scholar
  68. Uzun, I., Kurtoglu, B., Kazemi, H.: Multiphase rate-transient analysis in unconventional reservoirs: theory and application. Paper presented at the SPE/CSUR unconventional resources conference held in Calgary, Alberta, Canada, September 30–October 2 (2014). https://doi.org/10.2118/171657-pa
  69. Uzun, I., Kurtoglu, B., Kazemi, H.: Multiphase rate-transient analysis in unconventional reservoirs: theory and application. SPE Reserv. Eval. Eng. 19(04), 553–566 (2016).  https://doi.org/10.2118/171657-pa CrossRefGoogle Scholar
  70. Van Everdingen, A.F., Hurst, W.: The application of the Laplace transformation to flow problems in reservoirs. Pet. Trans. AIME 186, 305–324 (1949).  https://doi.org/10.2118/949305-g Google Scholar
  71. Vogel, J.V.: Inflow performance relationships for solution-gas drive wells. JPT 20(01), 83–92 (1968).  https://doi.org/10.2118/1476-pa CrossRefGoogle Scholar
  72. Wan, J., Aziz, K.: Multiple hydraulic fractures in horizontal wells. Paper (SPE-54627) presented at the SPE Western regional meeting held in Anchorage, Alaska, USA, 26–27 May (1999).  https://doi.org/10.2118/54627-ms
  73. Wiggins, M.L.: Generalized inflow performance relationships for three-phase flow. SPE Reserv. Eng. 9(03), 181–182 (1994).  https://doi.org/10.2118/25458-pa CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.METU-Northern Cyprus CampusTurkey

Personalised recommendations