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Compositional simulation of three-phase flow in mixed-wet shale oil reservoir
Fuel 260 (2020) 116361
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Full Length Article
Compositional simulation of three-phase flow in mixed-wet shale oil reservoir
T
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Jingwei Huang , Tianying Jin, Zhi Chai, Maria Barrufet, John Killough Department of Petroleum Engineering, Texas A&M University, College Station, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Three-phase flow Mixed wettability Relative permeability Digital rock analysis Shale oil reservoir
Three-phase flow is often involved during the production of shale oil reservoir. Relative permeability, which is often used to characterize multiphase flow, is found a strong function of wettability. Meanwhile, the mixed wettability in shale reservoir has been well characterized and may have impacts on fluid flow. In this work, to describe the mix-wet condition in shale, matrix is divided into organic and inorganic matrix which are hydrophobic and hydrophilic respectively. Relative permeabilities under different condition of wettability are applied in each kind of matrix. To construct permeability for three-phase flow, water-oil and oil-gas systems are evaluated separately with the assumption that water/gas permeabilities are only functions of their own saturations. Specifically, water-oil relative permeability is obtained by simulating multiphase flow directly on a digital rock sample using the lattice Boltzmann method. Oil-gas relative permeability is calculated from capillary pressure obtained from confined vapor liquid equilibria (VLE) coupled with Young-Laplace equation. Compositional simulation shows that the rich hydrocarbon in organic matrix is difficult to be produced due to the relatively lower permeability and poorer oil mobility under oil-wet condition. Since organic matter is a dispersed phase inside inorganic matter, fluids in organic matrix must flow through inorganic matrix before reaching the fractures. The dispersed nature of organic matter makes the relative permeability in inorganic matrix a dominant factor that controls the overall production in shale oil reservoir. The effect of relative permeability in organic matrix however has limited effect.
1. Introduction
used to characterize multiphase flow in porous media. In a two-phase system, it is straightforward to measure the relative permeabilities since only two possible saturation paths exist [10]. In a three-phase system, relative permeabilities can be functions of any saturation and direct measurement is difficult [10]. Measurements of relative permeability can be classified into steady state and unsteady state methods [11]. The number of three-phase flow experiments is limited, and even contradictory results were reported [12–14]. Some experimental data demonstrate that the permeabilities of strongly wetting phase or nonwetting phase primarily depend on their own saturations and are weak functions of saturations of other phases [15–17]. The intermediatewetting phase is more influenced by other phases and may show hysteresis [15]. In other cases, however, the dependency of intermediatewetting phase’s permeability on saturations of other phases is not apparent [15]. Theoretical and empirical models have been proposed to fit the experimental data such as the models of Corey et al. [18], Stone [19,20], Naar and Wygal [21], Land et al. [22] and each model has its assumptions and application range. However, it has been reported that these existing models are hard to match experimental results [23,24]. In
Shale oil reservoirs have been a major contributor to US oil production in recent years due to the unconventional resource revolution [1]. It is reported that the technically recoverable shale oil resource can reach 345 billion barrels around the world [2]. The production of tight oil is predicted to exceed conventional oil by the year of 2040 [3]. However, difficulties have been encountered to model and predict productions from shale reservoirs due to the unconventional characteristics such as the ultralow porosity/permeability, complex facture networks, effects of nanopores etc. [4,5]. To accurately capture the complexity of flow and transport phenomenon in shale reservoir, certain physics must be taken into consideration including gas slippage, adsorption/desorption, diffusion, wettability, geo-mechanical effect, alteration of phase behavior in nanopores etc. [6-9]. The subsurface flow in shale oil reservoir often involves an oil phase, a gas phase and water. Modeling the interactions among the three phases are critical for numerical simulation and production forecast of shale oil reservoir. Relative permeability has been widely
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Corresponding author. E-mail address: [email protected] (J. Huang).
https://doi.org/10.1016/j.fuel.2019.116361 Received 10 August 2019; Received in revised form 1 October 2019; Accepted 4 October 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
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pronounced than wetting phase [34]. With the aforementioned discussions, we notice that there are limited number of work studying multicomponent three-phase flow in shale oil reservoir, especially under the condition of mixed wettability. In this work, the different minerals in shale reservoir are distinguished and the mixed wettability is studied explicitly. Specifically, the shale matrix is first divided into organic matrix and inorganic matrix which are hydrophobic and hydrophilic respectively. The organic matrix has a smaller pore size and lower permeability than the inorganic matrix. Fractures in shale reservoir are modeled using embedded discrete fracture model (EDFM). To include the effects of wettability, relative permeabilities under different contact angles are calculated independently for both organic and inorganic matrix. To obtain threephase relative permeability, it is assumed that oil and gas do not dissolve in water and mass transfer is allowed between oil and gas by phase equilibria calculation. Relative permeabilities of water and gas are functions of their own saturations and relative permeabilities of oil are influenced by other saturations. The water-oil system is modeled by direct numerical simulation using a multiphase lattice Boltzmann method. The oil-gas system is evaluated by confined vapor liquid equilibria coupled with Young-Laplace equation and relative permeabilities are obtained from the calculated capillary pressure curve. The rest of paper are organized as follows: we first review the multi-porosity model and EDFM approach. Then the governing equations for threephase flow, phase behavior and relative permeability calculation are briefly introduced. With calculated three-phase relative permeabilities, compositional simulation is conducted in a fractured shale oil reservoir. Production performances with different total organic carbon (TOC) in shale matrix are compared. Finally, effects of water-oil contact angle and oil-gas contact angle are investigated.
Fig. 1. FIB/SEM image of a shale sample, reproduced from [32].
reservoir simulation, relative permeabilities are obtained by history matching the production data [25,26]. For unconventional reservoir, it is more difficult to measure relative permeability by experiment due to the ultralow permeability of shale rocks [27]. The measurements are time-consuming, expensive and results are hard to interpret [27]. In recent years, pore-scale direct numerical simulation has been proven an efficient tool to calculate relative permeability in porous media [28,29]. With the advance of imaging techniques and high-performance computing, it is possible to directly simulate multiphase flow and transport on digital rock samples reconstructed from scanned images [28,30,31]. Recent studies show that the shale rock contains different minerals that can be classified into inorganic matter and organic matter [5,32]. The inorganic matter includes quartz, feldspar, dolomite, clays and the organic matter is primarily pyrobitumen [32]. The petrophysical properties of organic and inorganic matter are obviously distinct and should be treated independently [5]. From the FIB/SEM image in Fig. 1, it is seen that micropores primarily exist within the inorganic matter, while nanopores are widely distributed in organic matter. Thus, the organic matter has much smaller pore sizes and lower permeability than the inorganic matter. Meanwhile, it has been pointed out that the organic matter is a dispersed phase inside the inorganic matter and connectivity between organic matter is poor [33]. Therefore, it is essential to divide the shale matrix into organic matter and inorganic matter [33]. Besides, dual-wettability condition has been investigated in shale rock, that is the organic matter is strongly hydrophobic while the inorganic matter is hydrophilic. It is known that wettability is a strong factor that affects relative permeabilities. Extensive work has been conducted to study relative permeability in porous media under the condition of uniform wettability [15,28,29]. It has been found that the non-wetting phase tends to path through larger pores while the wetting phase occupies the small pores. Therefore, a larger relative permeability is often observed for non-wetting phase [32]. However, due to the mixed-wet condition in shale rock, the relative permeabilities obtained under uniform-wet condition may not be suitable for modeling shale reservoir. Theoretical analysis and experimental measurements have been applied to study relative permeability in mixed-wet porous media [34–36]. Capillary pressure-saturation-relative permeability relations are measured in mixed-wet porous media and models are proposed to interpret the results [34]. It is reported that oil-water system is more influenced by mixed wettability and gas-water system is not sensitive to wettability due to the small contact angle [34]. Pore-scale numerical simulation has also been used to model multiphase flow under mixed-wet condition [34,37,38]. The simulation results show that the effect of mixed wettability on non-wetting phase is more
2. Methodologies 2.1. Reservoir model built by multi-porosity and EDFM approach To model the fractured shale reservoir considering mixed wettability, the multi-porosity model and EDFM approach are applied. As discussed in the introduction part, it is essential to divide the shale matrix into organic matrix and inorganic matrix. The multi-porosity model has been proven an efficient method to subdivide the shale matrix in the reservoir scale [5,39,40]. Following the work in [33], we simplify the organic matter as a sphere embedded in the inorganic matter. In reservoir simulation, transmissibility is always used to model the mass transfer between grid cells. There exist three kinds of interblock transmissibilities in shale matrix after subdivision. The transmissibility within organic matrix is set as zero due to the dispersed nature of organic matter. The transmissibility within inorganic matrix is given by [6]
Ti, j = Aij
k i / d i × k j / dj k i / d i + k j / dj
(1)
where Aij is the contact area, ki and kj are the permeabilities of grid blocks, di and dj are the nodal distance from block center to interface. The connection between organic and inorganic matrix is a kind of nonneighbor connection and transmissibility can be calculated by [6]
Ti, j = σ
ki Vi × kj Vj ki Vi + kj Vj
(2)
where Vi and Vj are the bulk volume of grid blocks, σ is the shape factor. The shape factor between organic matrix and inorganic matrix can be obtained using [33]
σ=
3 (3Vo/4π )2/3
where Vo is the volume of organic matrix. 2
(3)
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Fig. 2. Fractured shale oil reservoir model; bold black line represents the horizontal well and red lines represent hydraulic fractures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Similarly, the mass balance equation of water is
Apart from shale matrix, description of the complex fracture systems is also critical for shale reservoir modeling. In this work, the enhanced embedded discrete fracture model (EDFM) is applied to model fractures explicitly [41,42]. Hydraulic/natural fractures with arbitrary strikes and angles can be efficiently described by EDFM. When implementing EDFM, three more kinds of non-neighbor connections need to be determined including the ones between fractures and matrix grids, between two neighboring fracture grids in the same fracture and between two intersecting fracture grids from different fractures [41]. The transmissibility for the three types of connections can be generally written as [41]
Tnnc
A k = nnc nnc dnnc
Vj Δt
Δ(ϕSw ρ~w ) −
s
λrα =
krα μα
In compositional simulation, vapor–liquid equilibrium (VLE) is used to calculate properties of oil and gas phases. The mass balance of components can be described by the Rachford-Rice equation [43] Nc
Rr =
In reservoir simulation, Darcy’s law is applied to model the transfer of species. The mass balance of a hydrocarbon component i in a cell j can be expressed as [6]
s
well
z (Ki − 1) =0 v (Ki − 1)
∑ 1 +i f
(8)
where z i is total molar fraction of component i, fv is the volume fraction of vapor phase, Nc is number of components and Ki is equilibrium ratio of component. The splitting of components between oil and gas phases is carried out by solving the thermodynamic equilibrium equations for fugacity. At equilibrium,
∑ Ts (λro ρ~o xi ΔΦo + λrg ρ~g yi ΔΦg ) −
∑ (ρ~o xi qop + ρ~g yi qgp) = 0
(7)
2.3. Phase equilibrium calculation
i=1
Δ(ϕ (So ρ~o x i + Sg ρ~g yi )) −
(6)
where μα is the viscosity of phase α , krα is the relative permeability of phase α . The relative permeability is involved in the multiphase Darcy flux term which is essential for the modeling of multiphase flow. The detailed calculations of three-phase relative permeability will be discussed in the following section.
(4)
2.2. Multi-component and multi-phase flow simulation
Vj
well
where Sw is water saturation, λrw is the relative mobility of water, ρw is molar density of water, Φw is the potential of water, qwp is the source/ sink strength of water. In Eqs. (5) and (6), the relative mobility λrα (α = o, g , w ) is defined as
where Annc is the contact area and dnnc is distance of connection. The calculations of contact area and distance are based on how the fractures are interesting with matrix grids and other fractures, which can be derived analytically. The detailed calculation of transmissibility when using EDFM can be found in our previous work [41,42]. Besides, it has been found that most of the fractures exist in the inorganic matter [33]. Thus, it is reasonable to assume that the EDFM factures are only in contact with inorganic matrix [33]. Then the mass transfer in shale oil reservoir will follow this sequence: organic matrix → inorganic matrix → fractures → wells.
Δt
∑ Ts λrw ρ~w ΔΦw − ∑ ρ~w qwp = 0
fil (T , P l, x i ) = fil (T , P v , yi )
(5)
where Vj is the bulk volume of cell j, ϕ is porosity, So and Sg are oil and gas saturation respectively, ρo and ρg are the molar density of oil and gas respectively, x i and yi are molar fraction of component i in oil and gas phase respectively, Φo and Φg are the potential of oil and gas respectively, λro and λrg are the relative mobility of oil and gas respectively, qop and qgp are the source/sink strength of oil and gas phase respectively, Ts is the transmissibility in connection via grid surface s. The transmissibility Ts is determined by using the method introduced in Section 2.1, based on the type of connections. In Eq. (5), the first term represents the accumulation of component i in unit cell volume, the second term is Darcy flux term, and the third term is a source/sink term.
(9)
where fil and f iv are fugacity of component i at liquid and vapor phase, T is temperature, x i and yi are the mole fraction of component i in liquid and vapor phase respectively, P l and P v are the pressure of liquid and vapor phase respectively. The phase behavior and fugacity of vapor and liquid can be calculated by using Peng-Robinson equation of state
Z 3 − (1 − B ) Z 2 + (A − 3B2 − 2B ) Z − (AB − B2 − B3) = 0
(10)
where Z is compressibility, A and B are the parameters as a function of pressure, temperature, critical properties of pure components. The detailed calculations of fugacity can be found in Refs. [6,44]. 3
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collision operator that generates interfacial tension [47]
Table 1 Basic parameters for shale oil reservoir.
(Ωi )2 =
Reservoir conditions Initial Pressure (Psia) Initial Temperature (oF) Producer BHP (Psia) Porosity (fraction) Grid number Grid size (ft)
3500 160 500 0.08 45 × 21 × 1 50 × 50 × 50
(c . ∇ρ N )2 A |∇ρ N | ⎡ωi i N 2 − Bi⎤ ⎢ ⎥ |∇ρ | 2 ⎣ ⎦
where ωi is the weighting coefficient, Bi are parameters following the work of [46], A is a parameter that controls the value of interfacial tension. ρ N is the phase field function defined as ρ (x , t ) − ρ (x , t )
ρ N (x , t ) = ρR (x , t ) + ρB (x , t ) . (Ωik )3 is the recoloring operator that forces R B the phase separation [48]
Organic Matrix Pore diameter (nm) Permeability (nd) Water saturation (fraction)
15 10 0.25
Inorganic matrix Pore diameter (nm) Permeability (nd) Water saturation (fraction) Hydraulic Fracture 20 300
2.4. Three-phase relative permeability calculation Relative permeabilities are a function of saturation, fluid and rock properties. Following the widely used method of Stone [19,20], it is assumed that the water and gas relative permeabilities are only functions of their own saturations. In this way, water-oil and oil-gas system can be evaluated separately. Note that it is assumed oil/gas do not dissolve in water. For the water-oil system, the two phases are treated as nearly immiscible. Direct numerical simulation has been proven an efficient tool for simulating immiscible two-phase displacement. Among the techniques, lattice Boltzmann (LB) method has been widely used to calculate relative permeability in porous media. In this work, the color-gradient model is applied which can model immiscible two-phase flow accurately and handle a wide range of viscosity contrast and interfacial tension [28]. In LB method, the fluid dynamics and equilibrium are described by the distribution of pseudoparticles. The evolution equation can be written as [28]
fik (x + ci Δt , t + Δt ) = fik (x , t ) + Ωik (x , t )
(Ωi
=
ij [f j
−
(ΩiB )3 (fiB ) =
ρB ρ ρ f − β R 2 B cos (φi ) fieq (ρ , 0) ρ i ρ
(16)
∑ χi (xi /Vml − yi /Vmv)
(18)
i=1
Vml
Vmv
and are the molar vowhere χi is the Parachor of component i, lume of liquid and vapor phase respectively, and the exponent parameter E is given as [50]
E = 3.583 + 0.16(1/ Vml − 1/ Vmv)
(11)
(19)
For an isothermal system, capillary pressure changes with reservoir pressure and there is a certain oil saturation at every pressure. Thus, capillary pressure can also be described as a function of oil saturation. With the capillary pressure curve, relative permeability of oil and gas can be obtained using [51]
(12)
f jeq ]
(17)
Nc
σ 1/ E =
So∗ 1 dSo∗ Pc2 ∗ 2 0 (So ) 1 1 dSo∗ 0 Pc2
∫
is the single-phase collision operator that simulates viscous where relaxation of stress [46]
−(M−1SM )
(15)
where r is pore radius and θ is the oil-gas contact angle. Interfacial tension σ can be calculated by Ref. [49]
(Ωik )1
)1
ρR ρ ρ f + β R 2 B cos (φi ) fieq (ρ , 0) ρ i ρ
Pc = 2σcosθ / r
where fik (x , t ) is the distribution function of pseudoparticles at position x and time t, ci is lattice velocity, Δt is the time step, Ωik is the collision operator, k = R or B denotes red or blue fluid. For the color gradient LB model, Ωik includes three parts [45]:
Ωik = (Ωik )3 [(Ωik )1 + (Ωik )2]
(ΩiR )3 (fiR ) =
where fi is total distribution function, ρ is total density, ρR and ρB are the density of red fluid and blue fluid respectively, φi is the angle between the phase field function gradient ∇ρ N and ci , β is a free parameter controls interface thickness. To implement wettability in color-gradient model, it is assumed that the rock surface is a mixture of two fluids with a certain value of ρ N . For instance, if the contact angle of red fluid is θ , ρ N on rock surface will be set ascosθ . To construct water-oil relative permeability, the porous media is first fully saturated with oil and water saturation is increased step by step. Simulation in each step is conducted until a quasi-stationary state is reached. For the oil-gas system in shale reservoir, the relative permeability largely depends on the phase behavior. When limited data provided, predictive modeling from first principle models is a proper choice to obtain relative permeability [44] and a commonly used one is to infer from capillary pressure correlation. In a confined pore space, capillary pressure can be calculated by Young-Laplace equation
100 100 0.35
Fracture number Fracture conductivity (mD.ft)
(14)
kro =
∫
(13)
(20)
1
krg = (1 − So∗)2
where M is the transformation matrix, S is the diagonal collision matrix, f jeq is equilibrium distribution function. (Ωik )2 is the two-phase
1 dSo∗ Pc2 1 1 dSo∗ 0 Pc2
∫So∗ ∫
(21)
Table 2 Compositional parameters of fluid model [6]. Component
Mole fraction
Critical Pressure (psia)
Critical temperature (°F)
Acentric factor
Molecular weight
Parachor
CO2 N2 + C1 C2–C3 C4–C6 C7+
0.001924 0.503155 0.139236 0.104671 0.251013
1069.87 661.78 661.59 492.99 277.38
87.89 −120.18 148.59 371.24 811.47
0.22450 0.00899 0.12527 0.23813 0.76355
44.01 16.26 35.83 69.32 198.88
78 76.18 130.62 227.18 785.52
4
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3. Simulation setup To study three-phase flow in shale oil reservoir, a reservoir model shown in Fig. 2 is built with parameters presented in Table 1. The shale matrix is divided into organic matrix and inorganic matrix with different petrophysical properties. The organic matrix has a smaller pore size, lower permeability and lower initial water saturation due to its hydrophobic condition. By subdividing the shale matrix, TOC can be modeled explicitly. To quantify the content of organic matter, a volumetric TOC is defined by Ref. [33]
TOC =
Fig. 4. Oil-water relative permeabilities from lattice Boltzmann simulations under different conditions of wettability [52].
where
So − So, r 1 − So, r
(23)
where Vo is volume of organic matrix and V is the total volume of matrix. TOC is set as 0.4 at the beginning and effect of TOC will be evaluated in the following section. Besides, the complex hydraulic fractures shown in Fig. 2 are modeled using EDFM. The fluid model used in this study is a volatile oil from north America [6]. The compositional parameters of the fluid are provided in Table 2. The three-phase relative permeability in shale matrix is obtained by using a two-step method. In the first step, the oil-water relative permeability is obtained by digital rock analysis. Fig. 3 presents the digital rock sample of a shale rock. The oil-water relative permeability is then calculated by directly simulating two-phase flow on the digital rock. Different contact angles are used in simulation to investigate effect of wettability. Fig. 4 shows the calculated relative permeabilities under conditions of oil-wet (water-oil contact angle 135°) and water-wet (water-oil contact angle 54°). Secondly, the oil-gas relative permeability is obtained by using first principle models. Provided with the compositional parameters in Table 2, interfacial tension between oil and gas can be calculated using Eq. (18) as well as the capillary pressures at different saturations (Fig. 5a). With capillary pressure curves, oil-gas relative permeabilities are obtained by calculating the integral in Eqs. (20) and (21). The gas phase is treated as the nonwetting phase in both organic and inorganic matrix and the relative permeabilities in Fig. 5b is under the condition when the oil-gas contact angle is 60°. With wateroil and oil-gas relative permeabilities, the three-phase relative permeabilities in matrix are constructed using the method of Stone II [20]. Besides, water-oil and oil-gas relative permeabilities in fractures are assumed to be a linear function for simplicity. Provided with the reservoir model, fluid model and calculated relative permeabilities, compositional simulation of this shale oil reservoir then can be conducted. Production is constrained by a constant bottomhole pressure of 500 Psi and no flow boundary condition is applied in the outer boundary.
Fig. 3. Digital rock sample for the calculation of water-oil relative permeability; gray part is solid grain, black part is pore, image reproduced from [52].
So∗ =
Vo V
(22)
In this work, the capillary pressure curve is fitted using a rational polynomial to calculate the integral in Eqs. (20) and (21).
Fig. 5. Calculated oil-gas capillary pressures (a) and relative permeabilities (b) when oil-gas contact angle is 60°. 5
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Fig. 6. Pressure distribution (a), oil saturation (b), water saturation (c) and gas saturation (d) in mixed-wet shale reservoir after 10 years of production.
4. Results and discussion
the oil mobility is lower and water mobility is higher in organic matrix. Thus, the rich hydrocarbon in organic matrix is more difficult to be produced. The overall production performance of this mixed-wet shale oil reservoir is shown in Fig. 7. To analyze the effect of organic matter, three different TOC in shale matrix are compared. From Fig. 7, it is seen that a higher reservoir pressure can be maintained when TOC is high due to the relative lower permeability of organic matrix. The oil and water production from shale reservoir with high TOC are lower at early stage of production, but higher at later period. Gas production in high-TOC case, on the other hand, is lower at the beginning but close to those in low-TOC cases in the end. From the comparisons, we find that effects of TOC can be classified into two stages: (1) at early stage, the low permeability of organic matrix leads to lower production rates but (2) at later stage, the rich hydrocarbon in organic matrix starts to take effect and increases the overall production.
In this section, the three-phase flow in mixed-wet shale oil reservoir is modeled using the proposed approach. Phase saturations in organic and inorganic matrix are obtained and production rates of water, oil and gas are particularly analyzed. By changing the volume fraction of organic matrix, effects of TOC are evaluated explicitly. Simulations results under mixed-wet condition are then compared to those where a uniform wettability is applied. Finally, the effects of contact angles on production are investigated including different water-oil contact angles and oil-gas contact angles, in both organic and inorganic matrix.
4.1. Production of shale oil reservoir under mixed-wet condition In this part, the mixed-wet condition is considered explicitly in shale matrix. Specifically, the organic matrix is set as oil-wet while the inorganic matrix is water-wet. The water-oil relative permeabilities in organic and inorganic matrix are using the ones presented in Fig. 4. Besides, the oil-gas relative permeabilities in Fig. 5b are applied. Fig. 6 presents the pressure distributions, water saturations, oil saturations and gas saturations in both organic and inorganic matrix, after ten years of production. It is seen that most of the pressure-decline and production of fluids happens in the stimulated area. Though having similar pressure distributions, the inorganic matrix and organic matrix have very different saturations of fluids. A higher oil/gas saturation and lower water saturation are observed in organic matrix. Since the organic matrix is not directly connected to fractures, fluids must go through the inorganic matrix before reaching fractures. Besides, because the organic matrix is oil-wet while inorganic matrix is water-wet,
4.2. Comparison with uniform-wet condition To further illustrate the effects of mixed wettability, another two cases with uniform wettability are conducted for comparison. In the water-wet case, all matrix is set as water-wet and the water-oil relative permeability with contact angle of 54° is applied in both organic and inorganic matrix. In the oil-wet case, matrix is oil wet and the water-oil relative permeability with contact angle of 135° is applied. Fig. 8 compares the reservoir pressure, oil, gas and water production under different conditions of wettability. Compared to mixed-wet case, the oil production is much lower, gas production is higher and water production is much higher when the shale matrix is uniformly oil wet. The 6
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Fig. 7. Effect of TOC on the production performance of shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
significant difference comes from the fact that the inorganic matrix is water wet in mixed-wet case but is oil wet in oil-wet case. Based on the relative permeabilities in Fig. 4, the oil mobility is largely suppressed, and water mobility is much increased when inorganic matrix is oil wet. Thus, the lower oil production and higher water production have been observed. When the shale matrix is uniformly water wet, both oil and water production is lower and gas production is slightly higher compared to mixed-wet case. It is seen that the difference between mixedwet case and water-wet case is not very significant. The organic matrix becomes water wet in water-wet case which means that water mobility should be lower and oil mobility should be higher in organic matrix. However, because organic matrix is embedded in inorganic matrix, all the fluids in organic matrix must go through inorganic matrix first before being produced through fractures. Therefore, the relative permeability in inorganic matrix is the dominant factor that controls the production rates of fluids. Since the wettability of inorganic matrix is the same in mixed-wet case and water-wet case, the production performances are thus closer.
water permeability is lower and oil permeability is higher in inorganic matrix, when the contact angle is decreased to 30°. On the other hand, the water permeability is increased, and oil permeability is decreased in organic matrix, when contact angle is set to 160°. The altered relative permeabilities due to contact angles in turn affect the production performance of the shale reservoir. Fig. 10 compares the pressure decline and production rates under different conditions of contact angles. No obvious differences are observed in terms of pressure decline and gas production. By comparing case (1) and case (2), the change of contact angle in organic matrix slightly affects oil and water productions. Results from case (1) and case (3) indicate that the contact angle in inorganic matrix has a significant effect on oil and water production: a stronger water-wet inorganic matrix leads to a higher oil production and lower water production. This finding further confirms the conclusion made in Section 4.2 that the relative permeability (wettability) in inorganic matrix controls the overall water and oil productions.
4.3. Effect of water-oil contact angle (WOCA)
Oil-gas relative permeability is also affected by oil-gas contact angle (OGCA). With the same fluid compositions and pore size, contact angle controls oil-gas capillary pressure, from which oil-gas relative permeability is extracted. Fig. 11 shows the oil-gas relative permeabilities with different OGCA in both organic and inorganic matrix. In inorganic matrix where pore size is relatively larger, the oil permeability is more sensitive to contact angle. Oil permeability is increased when contact angle is decreased to 30°. Gas permeability is lower at high oil saturation and higher at lower oil saturation when contact angle is turned to 30°. On the other hand, gas permeability is more sensitive to contact angle in organic matrix where pore size is smaller. Gas permeability is
4.4. Effect of oil–gas contact angle (OGCA)
It has been found that the relative permeability depends on contact angles [28,29]. Thus, it is essential to evaluate the three-phase flow in shale reservoir using different contact angles. In this part, the effects of water–oil contact angle (WOCA) are investigated. Three cases are conducted: (1) WOCA in organic matrix is 135° and in inorganic matrix is 54°, (2) WOCA in organic matrix is 160° and in inorganic matrix is 54°, (3) WOCA in organic matrix is 135° and in inorganic matrix is 30°. Fig. 9 presents the water-oil relative permeabilities under different contact angles, in both organic and inorganic matrix. It is seen that the 7
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Fig. 8. Comparison of production performance between mixed-wet shale reservoir and uniform-wet shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
Fig. 9. Effect of contact angle on water-oil relative permeability in inorganic matrix (a) and organic matrix (b).
oil-gas contact angle in inorganic matrix has much larger effects than that in organic matrix.
increased, and oil permeability is slightly decreased when contact angle becomes 30°. The production performance with different OGCA are compared in Fig. 12. The water production rate when contact angle is 30° is slightly higher than that when contact angle is 60°. Oil production rate is significantly increased when contact angle is 30° due to the increased oil permeability in the inorganic matrix. Though gas permeability is higher in organic matrix, the gas production rate has negligible changes from different contact angles. Again, it is found that the
5. Conclusions In this work, compositional simulation of three-phase flow is conducted considering mixed wettability in shale oil reservoir. By using the multi-porosity model, mixed-wettability is modeled explicitly by 8
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Fig. 10. Effect of water-oil contact angle on the production performance of shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
Fig. 11. Effect of oil-gas contact angle on the relative permeabilities in inorganic matrix (a) and organic matrix (b).
permeabilities are applied to organic and inorganic matrix according to the condition of wettability. From the simulation results, we obtain the following findings and conclusions:
dividing the shale matrix into organic and inorganic matrix, which are oil-wet and water-wet respectively. To construct relative permeability for three-phase flow, the Stone II method is applied that assumes the permeabilities of water and gas are only functions of their own saturations. Water-oil relative permeability is obtained by performing immiscible two-phase flow directly on digital rock sample using a multiphase lattice Boltzmann method. Oil-gas relative permeability is extracted from oil-gas capillary pressure curve, which is calculated from fluid compositions and phase behavior. Different three-phase relative
(1) The rich hydrocarbon in organic matrix is hard to be produced due to its low permeability, low oil mobility and poor connection to hydraulic fractures. (2) A higher TOC in shale matrix leads to lower production rates at 9
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Fig. 12. Effect of oil-gas contact angle on the production performance of shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
References
early stage due to the low permeability of organic matter, but oil/ gas production is increased at later stage when the rich hydrocarbon in organic matrix starts to take effect. (3) Comparisons with uniform-wet cases indicate that productions from mixed-wet and water-wet reservoir are close. On the other hand, oil production is much lower and water production is higher from oilwet reservoir. (4) Different water-oil contact angles and oil-gas contact angles are evaluated, in both organic and inorganic matrix. It is found that the contact angles in inorganic matrix have more significant effects on production. (5) Due to the dispersed nature of organic matter, the fluids in organic matrix must go through inorganic matrix first before reaching fractures. Therefore, the production rates largely depend on the mass transfer between inorganic matrix and fractures. The relative permeability in inorganic matrix, which is affected by wettability, is thus a dominant factor that controls the overall production in shale reservoir.
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11
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Full Length Article
Compositional simulation of three-phase flow in mixed-wet shale oil reservoir
T
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Jingwei Huang , Tianying Jin, Zhi Chai, Maria Barrufet, John Killough Department of Petroleum Engineering, Texas A&M University, College Station, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Three-phase flow Mixed wettability Relative permeability Digital rock analysis Shale oil reservoir
Three-phase flow is often involved during the production of shale oil reservoir. Relative permeability, which is often used to characterize multiphase flow, is found a strong function of wettability. Meanwhile, the mixed wettability in shale reservoir has been well characterized and may have impacts on fluid flow. In this work, to describe the mix-wet condition in shale, matrix is divided into organic and inorganic matrix which are hydrophobic and hydrophilic respectively. Relative permeabilities under different condition of wettability are applied in each kind of matrix. To construct permeability for three-phase flow, water-oil and oil-gas systems are evaluated separately with the assumption that water/gas permeabilities are only functions of their own saturations. Specifically, water-oil relative permeability is obtained by simulating multiphase flow directly on a digital rock sample using the lattice Boltzmann method. Oil-gas relative permeability is calculated from capillary pressure obtained from confined vapor liquid equilibria (VLE) coupled with Young-Laplace equation. Compositional simulation shows that the rich hydrocarbon in organic matrix is difficult to be produced due to the relatively lower permeability and poorer oil mobility under oil-wet condition. Since organic matter is a dispersed phase inside inorganic matter, fluids in organic matrix must flow through inorganic matrix before reaching the fractures. The dispersed nature of organic matter makes the relative permeability in inorganic matrix a dominant factor that controls the overall production in shale oil reservoir. The effect of relative permeability in organic matrix however has limited effect.
1. Introduction
used to characterize multiphase flow in porous media. In a two-phase system, it is straightforward to measure the relative permeabilities since only two possible saturation paths exist [10]. In a three-phase system, relative permeabilities can be functions of any saturation and direct measurement is difficult [10]. Measurements of relative permeability can be classified into steady state and unsteady state methods [11]. The number of three-phase flow experiments is limited, and even contradictory results were reported [12–14]. Some experimental data demonstrate that the permeabilities of strongly wetting phase or nonwetting phase primarily depend on their own saturations and are weak functions of saturations of other phases [15–17]. The intermediatewetting phase is more influenced by other phases and may show hysteresis [15]. In other cases, however, the dependency of intermediatewetting phase’s permeability on saturations of other phases is not apparent [15]. Theoretical and empirical models have been proposed to fit the experimental data such as the models of Corey et al. [18], Stone [19,20], Naar and Wygal [21], Land et al. [22] and each model has its assumptions and application range. However, it has been reported that these existing models are hard to match experimental results [23,24]. In
Shale oil reservoirs have been a major contributor to US oil production in recent years due to the unconventional resource revolution [1]. It is reported that the technically recoverable shale oil resource can reach 345 billion barrels around the world [2]. The production of tight oil is predicted to exceed conventional oil by the year of 2040 [3]. However, difficulties have been encountered to model and predict productions from shale reservoirs due to the unconventional characteristics such as the ultralow porosity/permeability, complex facture networks, effects of nanopores etc. [4,5]. To accurately capture the complexity of flow and transport phenomenon in shale reservoir, certain physics must be taken into consideration including gas slippage, adsorption/desorption, diffusion, wettability, geo-mechanical effect, alteration of phase behavior in nanopores etc. [6-9]. The subsurface flow in shale oil reservoir often involves an oil phase, a gas phase and water. Modeling the interactions among the three phases are critical for numerical simulation and production forecast of shale oil reservoir. Relative permeability has been widely
⁎
Corresponding author. E-mail address: [email protected] (J. Huang).
https://doi.org/10.1016/j.fuel.2019.116361 Received 10 August 2019; Received in revised form 1 October 2019; Accepted 4 October 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
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pronounced than wetting phase [34]. With the aforementioned discussions, we notice that there are limited number of work studying multicomponent three-phase flow in shale oil reservoir, especially under the condition of mixed wettability. In this work, the different minerals in shale reservoir are distinguished and the mixed wettability is studied explicitly. Specifically, the shale matrix is first divided into organic matrix and inorganic matrix which are hydrophobic and hydrophilic respectively. The organic matrix has a smaller pore size and lower permeability than the inorganic matrix. Fractures in shale reservoir are modeled using embedded discrete fracture model (EDFM). To include the effects of wettability, relative permeabilities under different contact angles are calculated independently for both organic and inorganic matrix. To obtain threephase relative permeability, it is assumed that oil and gas do not dissolve in water and mass transfer is allowed between oil and gas by phase equilibria calculation. Relative permeabilities of water and gas are functions of their own saturations and relative permeabilities of oil are influenced by other saturations. The water-oil system is modeled by direct numerical simulation using a multiphase lattice Boltzmann method. The oil-gas system is evaluated by confined vapor liquid equilibria coupled with Young-Laplace equation and relative permeabilities are obtained from the calculated capillary pressure curve. The rest of paper are organized as follows: we first review the multi-porosity model and EDFM approach. Then the governing equations for threephase flow, phase behavior and relative permeability calculation are briefly introduced. With calculated three-phase relative permeabilities, compositional simulation is conducted in a fractured shale oil reservoir. Production performances with different total organic carbon (TOC) in shale matrix are compared. Finally, effects of water-oil contact angle and oil-gas contact angle are investigated.
Fig. 1. FIB/SEM image of a shale sample, reproduced from [32].
reservoir simulation, relative permeabilities are obtained by history matching the production data [25,26]. For unconventional reservoir, it is more difficult to measure relative permeability by experiment due to the ultralow permeability of shale rocks [27]. The measurements are time-consuming, expensive and results are hard to interpret [27]. In recent years, pore-scale direct numerical simulation has been proven an efficient tool to calculate relative permeability in porous media [28,29]. With the advance of imaging techniques and high-performance computing, it is possible to directly simulate multiphase flow and transport on digital rock samples reconstructed from scanned images [28,30,31]. Recent studies show that the shale rock contains different minerals that can be classified into inorganic matter and organic matter [5,32]. The inorganic matter includes quartz, feldspar, dolomite, clays and the organic matter is primarily pyrobitumen [32]. The petrophysical properties of organic and inorganic matter are obviously distinct and should be treated independently [5]. From the FIB/SEM image in Fig. 1, it is seen that micropores primarily exist within the inorganic matter, while nanopores are widely distributed in organic matter. Thus, the organic matter has much smaller pore sizes and lower permeability than the inorganic matter. Meanwhile, it has been pointed out that the organic matter is a dispersed phase inside the inorganic matter and connectivity between organic matter is poor [33]. Therefore, it is essential to divide the shale matrix into organic matter and inorganic matter [33]. Besides, dual-wettability condition has been investigated in shale rock, that is the organic matter is strongly hydrophobic while the inorganic matter is hydrophilic. It is known that wettability is a strong factor that affects relative permeabilities. Extensive work has been conducted to study relative permeability in porous media under the condition of uniform wettability [15,28,29]. It has been found that the non-wetting phase tends to path through larger pores while the wetting phase occupies the small pores. Therefore, a larger relative permeability is often observed for non-wetting phase [32]. However, due to the mixed-wet condition in shale rock, the relative permeabilities obtained under uniform-wet condition may not be suitable for modeling shale reservoir. Theoretical analysis and experimental measurements have been applied to study relative permeability in mixed-wet porous media [34–36]. Capillary pressure-saturation-relative permeability relations are measured in mixed-wet porous media and models are proposed to interpret the results [34]. It is reported that oil-water system is more influenced by mixed wettability and gas-water system is not sensitive to wettability due to the small contact angle [34]. Pore-scale numerical simulation has also been used to model multiphase flow under mixed-wet condition [34,37,38]. The simulation results show that the effect of mixed wettability on non-wetting phase is more
2. Methodologies 2.1. Reservoir model built by multi-porosity and EDFM approach To model the fractured shale reservoir considering mixed wettability, the multi-porosity model and EDFM approach are applied. As discussed in the introduction part, it is essential to divide the shale matrix into organic matrix and inorganic matrix. The multi-porosity model has been proven an efficient method to subdivide the shale matrix in the reservoir scale [5,39,40]. Following the work in [33], we simplify the organic matter as a sphere embedded in the inorganic matter. In reservoir simulation, transmissibility is always used to model the mass transfer between grid cells. There exist three kinds of interblock transmissibilities in shale matrix after subdivision. The transmissibility within organic matrix is set as zero due to the dispersed nature of organic matter. The transmissibility within inorganic matrix is given by [6]
Ti, j = Aij
k i / d i × k j / dj k i / d i + k j / dj
(1)
where Aij is the contact area, ki and kj are the permeabilities of grid blocks, di and dj are the nodal distance from block center to interface. The connection between organic and inorganic matrix is a kind of nonneighbor connection and transmissibility can be calculated by [6]
Ti, j = σ
ki Vi × kj Vj ki Vi + kj Vj
(2)
where Vi and Vj are the bulk volume of grid blocks, σ is the shape factor. The shape factor between organic matrix and inorganic matrix can be obtained using [33]
σ=
3 (3Vo/4π )2/3
where Vo is the volume of organic matrix. 2
(3)
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Fig. 2. Fractured shale oil reservoir model; bold black line represents the horizontal well and red lines represent hydraulic fractures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Similarly, the mass balance equation of water is
Apart from shale matrix, description of the complex fracture systems is also critical for shale reservoir modeling. In this work, the enhanced embedded discrete fracture model (EDFM) is applied to model fractures explicitly [41,42]. Hydraulic/natural fractures with arbitrary strikes and angles can be efficiently described by EDFM. When implementing EDFM, three more kinds of non-neighbor connections need to be determined including the ones between fractures and matrix grids, between two neighboring fracture grids in the same fracture and between two intersecting fracture grids from different fractures [41]. The transmissibility for the three types of connections can be generally written as [41]
Tnnc
A k = nnc nnc dnnc
Vj Δt
Δ(ϕSw ρ~w ) −
s
λrα =
krα μα
In compositional simulation, vapor–liquid equilibrium (VLE) is used to calculate properties of oil and gas phases. The mass balance of components can be described by the Rachford-Rice equation [43] Nc
Rr =
In reservoir simulation, Darcy’s law is applied to model the transfer of species. The mass balance of a hydrocarbon component i in a cell j can be expressed as [6]
s
well
z (Ki − 1) =0 v (Ki − 1)
∑ 1 +i f
(8)
where z i is total molar fraction of component i, fv is the volume fraction of vapor phase, Nc is number of components and Ki is equilibrium ratio of component. The splitting of components between oil and gas phases is carried out by solving the thermodynamic equilibrium equations for fugacity. At equilibrium,
∑ Ts (λro ρ~o xi ΔΦo + λrg ρ~g yi ΔΦg ) −
∑ (ρ~o xi qop + ρ~g yi qgp) = 0
(7)
2.3. Phase equilibrium calculation
i=1
Δ(ϕ (So ρ~o x i + Sg ρ~g yi )) −
(6)
where μα is the viscosity of phase α , krα is the relative permeability of phase α . The relative permeability is involved in the multiphase Darcy flux term which is essential for the modeling of multiphase flow. The detailed calculations of three-phase relative permeability will be discussed in the following section.
(4)
2.2. Multi-component and multi-phase flow simulation
Vj
well
where Sw is water saturation, λrw is the relative mobility of water, ρw is molar density of water, Φw is the potential of water, qwp is the source/ sink strength of water. In Eqs. (5) and (6), the relative mobility λrα (α = o, g , w ) is defined as
where Annc is the contact area and dnnc is distance of connection. The calculations of contact area and distance are based on how the fractures are interesting with matrix grids and other fractures, which can be derived analytically. The detailed calculation of transmissibility when using EDFM can be found in our previous work [41,42]. Besides, it has been found that most of the fractures exist in the inorganic matter [33]. Thus, it is reasonable to assume that the EDFM factures are only in contact with inorganic matrix [33]. Then the mass transfer in shale oil reservoir will follow this sequence: organic matrix → inorganic matrix → fractures → wells.
Δt
∑ Ts λrw ρ~w ΔΦw − ∑ ρ~w qwp = 0
fil (T , P l, x i ) = fil (T , P v , yi )
(5)
where Vj is the bulk volume of cell j, ϕ is porosity, So and Sg are oil and gas saturation respectively, ρo and ρg are the molar density of oil and gas respectively, x i and yi are molar fraction of component i in oil and gas phase respectively, Φo and Φg are the potential of oil and gas respectively, λro and λrg are the relative mobility of oil and gas respectively, qop and qgp are the source/sink strength of oil and gas phase respectively, Ts is the transmissibility in connection via grid surface s. The transmissibility Ts is determined by using the method introduced in Section 2.1, based on the type of connections. In Eq. (5), the first term represents the accumulation of component i in unit cell volume, the second term is Darcy flux term, and the third term is a source/sink term.
(9)
where fil and f iv are fugacity of component i at liquid and vapor phase, T is temperature, x i and yi are the mole fraction of component i in liquid and vapor phase respectively, P l and P v are the pressure of liquid and vapor phase respectively. The phase behavior and fugacity of vapor and liquid can be calculated by using Peng-Robinson equation of state
Z 3 − (1 − B ) Z 2 + (A − 3B2 − 2B ) Z − (AB − B2 − B3) = 0
(10)
where Z is compressibility, A and B are the parameters as a function of pressure, temperature, critical properties of pure components. The detailed calculations of fugacity can be found in Refs. [6,44]. 3
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collision operator that generates interfacial tension [47]
Table 1 Basic parameters for shale oil reservoir.
(Ωi )2 =
Reservoir conditions Initial Pressure (Psia) Initial Temperature (oF) Producer BHP (Psia) Porosity (fraction) Grid number Grid size (ft)
3500 160 500 0.08 45 × 21 × 1 50 × 50 × 50
(c . ∇ρ N )2 A |∇ρ N | ⎡ωi i N 2 − Bi⎤ ⎢ ⎥ |∇ρ | 2 ⎣ ⎦
where ωi is the weighting coefficient, Bi are parameters following the work of [46], A is a parameter that controls the value of interfacial tension. ρ N is the phase field function defined as ρ (x , t ) − ρ (x , t )
ρ N (x , t ) = ρR (x , t ) + ρB (x , t ) . (Ωik )3 is the recoloring operator that forces R B the phase separation [48]
Organic Matrix Pore diameter (nm) Permeability (nd) Water saturation (fraction)
15 10 0.25
Inorganic matrix Pore diameter (nm) Permeability (nd) Water saturation (fraction) Hydraulic Fracture 20 300
2.4. Three-phase relative permeability calculation Relative permeabilities are a function of saturation, fluid and rock properties. Following the widely used method of Stone [19,20], it is assumed that the water and gas relative permeabilities are only functions of their own saturations. In this way, water-oil and oil-gas system can be evaluated separately. Note that it is assumed oil/gas do not dissolve in water. For the water-oil system, the two phases are treated as nearly immiscible. Direct numerical simulation has been proven an efficient tool for simulating immiscible two-phase displacement. Among the techniques, lattice Boltzmann (LB) method has been widely used to calculate relative permeability in porous media. In this work, the color-gradient model is applied which can model immiscible two-phase flow accurately and handle a wide range of viscosity contrast and interfacial tension [28]. In LB method, the fluid dynamics and equilibrium are described by the distribution of pseudoparticles. The evolution equation can be written as [28]
fik (x + ci Δt , t + Δt ) = fik (x , t ) + Ωik (x , t )
(Ωi
=
ij [f j
−
(ΩiB )3 (fiB ) =
ρB ρ ρ f − β R 2 B cos (φi ) fieq (ρ , 0) ρ i ρ
(16)
∑ χi (xi /Vml − yi /Vmv)
(18)
i=1
Vml
Vmv
and are the molar vowhere χi is the Parachor of component i, lume of liquid and vapor phase respectively, and the exponent parameter E is given as [50]
E = 3.583 + 0.16(1/ Vml − 1/ Vmv)
(11)
(19)
For an isothermal system, capillary pressure changes with reservoir pressure and there is a certain oil saturation at every pressure. Thus, capillary pressure can also be described as a function of oil saturation. With the capillary pressure curve, relative permeability of oil and gas can be obtained using [51]
(12)
f jeq ]
(17)
Nc
σ 1/ E =
So∗ 1 dSo∗ Pc2 ∗ 2 0 (So ) 1 1 dSo∗ 0 Pc2
∫
is the single-phase collision operator that simulates viscous where relaxation of stress [46]
−(M−1SM )
(15)
where r is pore radius and θ is the oil-gas contact angle. Interfacial tension σ can be calculated by Ref. [49]
(Ωik )1
)1
ρR ρ ρ f + β R 2 B cos (φi ) fieq (ρ , 0) ρ i ρ
Pc = 2σcosθ / r
where fik (x , t ) is the distribution function of pseudoparticles at position x and time t, ci is lattice velocity, Δt is the time step, Ωik is the collision operator, k = R or B denotes red or blue fluid. For the color gradient LB model, Ωik includes three parts [45]:
Ωik = (Ωik )3 [(Ωik )1 + (Ωik )2]
(ΩiR )3 (fiR ) =
where fi is total distribution function, ρ is total density, ρR and ρB are the density of red fluid and blue fluid respectively, φi is the angle between the phase field function gradient ∇ρ N and ci , β is a free parameter controls interface thickness. To implement wettability in color-gradient model, it is assumed that the rock surface is a mixture of two fluids with a certain value of ρ N . For instance, if the contact angle of red fluid is θ , ρ N on rock surface will be set ascosθ . To construct water-oil relative permeability, the porous media is first fully saturated with oil and water saturation is increased step by step. Simulation in each step is conducted until a quasi-stationary state is reached. For the oil-gas system in shale reservoir, the relative permeability largely depends on the phase behavior. When limited data provided, predictive modeling from first principle models is a proper choice to obtain relative permeability [44] and a commonly used one is to infer from capillary pressure correlation. In a confined pore space, capillary pressure can be calculated by Young-Laplace equation
100 100 0.35
Fracture number Fracture conductivity (mD.ft)
(14)
kro =
∫
(13)
(20)
1
krg = (1 − So∗)2
where M is the transformation matrix, S is the diagonal collision matrix, f jeq is equilibrium distribution function. (Ωik )2 is the two-phase
1 dSo∗ Pc2 1 1 dSo∗ 0 Pc2
∫So∗ ∫
(21)
Table 2 Compositional parameters of fluid model [6]. Component
Mole fraction
Critical Pressure (psia)
Critical temperature (°F)
Acentric factor
Molecular weight
Parachor
CO2 N2 + C1 C2–C3 C4–C6 C7+
0.001924 0.503155 0.139236 0.104671 0.251013
1069.87 661.78 661.59 492.99 277.38
87.89 −120.18 148.59 371.24 811.47
0.22450 0.00899 0.12527 0.23813 0.76355
44.01 16.26 35.83 69.32 198.88
78 76.18 130.62 227.18 785.52
4
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3. Simulation setup To study three-phase flow in shale oil reservoir, a reservoir model shown in Fig. 2 is built with parameters presented in Table 1. The shale matrix is divided into organic matrix and inorganic matrix with different petrophysical properties. The organic matrix has a smaller pore size, lower permeability and lower initial water saturation due to its hydrophobic condition. By subdividing the shale matrix, TOC can be modeled explicitly. To quantify the content of organic matter, a volumetric TOC is defined by Ref. [33]
TOC =
Fig. 4. Oil-water relative permeabilities from lattice Boltzmann simulations under different conditions of wettability [52].
where
So − So, r 1 − So, r
(23)
where Vo is volume of organic matrix and V is the total volume of matrix. TOC is set as 0.4 at the beginning and effect of TOC will be evaluated in the following section. Besides, the complex hydraulic fractures shown in Fig. 2 are modeled using EDFM. The fluid model used in this study is a volatile oil from north America [6]. The compositional parameters of the fluid are provided in Table 2. The three-phase relative permeability in shale matrix is obtained by using a two-step method. In the first step, the oil-water relative permeability is obtained by digital rock analysis. Fig. 3 presents the digital rock sample of a shale rock. The oil-water relative permeability is then calculated by directly simulating two-phase flow on the digital rock. Different contact angles are used in simulation to investigate effect of wettability. Fig. 4 shows the calculated relative permeabilities under conditions of oil-wet (water-oil contact angle 135°) and water-wet (water-oil contact angle 54°). Secondly, the oil-gas relative permeability is obtained by using first principle models. Provided with the compositional parameters in Table 2, interfacial tension between oil and gas can be calculated using Eq. (18) as well as the capillary pressures at different saturations (Fig. 5a). With capillary pressure curves, oil-gas relative permeabilities are obtained by calculating the integral in Eqs. (20) and (21). The gas phase is treated as the nonwetting phase in both organic and inorganic matrix and the relative permeabilities in Fig. 5b is under the condition when the oil-gas contact angle is 60°. With wateroil and oil-gas relative permeabilities, the three-phase relative permeabilities in matrix are constructed using the method of Stone II [20]. Besides, water-oil and oil-gas relative permeabilities in fractures are assumed to be a linear function for simplicity. Provided with the reservoir model, fluid model and calculated relative permeabilities, compositional simulation of this shale oil reservoir then can be conducted. Production is constrained by a constant bottomhole pressure of 500 Psi and no flow boundary condition is applied in the outer boundary.
Fig. 3. Digital rock sample for the calculation of water-oil relative permeability; gray part is solid grain, black part is pore, image reproduced from [52].
So∗ =
Vo V
(22)
In this work, the capillary pressure curve is fitted using a rational polynomial to calculate the integral in Eqs. (20) and (21).
Fig. 5. Calculated oil-gas capillary pressures (a) and relative permeabilities (b) when oil-gas contact angle is 60°. 5
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Fig. 6. Pressure distribution (a), oil saturation (b), water saturation (c) and gas saturation (d) in mixed-wet shale reservoir after 10 years of production.
4. Results and discussion
the oil mobility is lower and water mobility is higher in organic matrix. Thus, the rich hydrocarbon in organic matrix is more difficult to be produced. The overall production performance of this mixed-wet shale oil reservoir is shown in Fig. 7. To analyze the effect of organic matter, three different TOC in shale matrix are compared. From Fig. 7, it is seen that a higher reservoir pressure can be maintained when TOC is high due to the relative lower permeability of organic matrix. The oil and water production from shale reservoir with high TOC are lower at early stage of production, but higher at later period. Gas production in high-TOC case, on the other hand, is lower at the beginning but close to those in low-TOC cases in the end. From the comparisons, we find that effects of TOC can be classified into two stages: (1) at early stage, the low permeability of organic matrix leads to lower production rates but (2) at later stage, the rich hydrocarbon in organic matrix starts to take effect and increases the overall production.
In this section, the three-phase flow in mixed-wet shale oil reservoir is modeled using the proposed approach. Phase saturations in organic and inorganic matrix are obtained and production rates of water, oil and gas are particularly analyzed. By changing the volume fraction of organic matrix, effects of TOC are evaluated explicitly. Simulations results under mixed-wet condition are then compared to those where a uniform wettability is applied. Finally, the effects of contact angles on production are investigated including different water-oil contact angles and oil-gas contact angles, in both organic and inorganic matrix.
4.1. Production of shale oil reservoir under mixed-wet condition In this part, the mixed-wet condition is considered explicitly in shale matrix. Specifically, the organic matrix is set as oil-wet while the inorganic matrix is water-wet. The water-oil relative permeabilities in organic and inorganic matrix are using the ones presented in Fig. 4. Besides, the oil-gas relative permeabilities in Fig. 5b are applied. Fig. 6 presents the pressure distributions, water saturations, oil saturations and gas saturations in both organic and inorganic matrix, after ten years of production. It is seen that most of the pressure-decline and production of fluids happens in the stimulated area. Though having similar pressure distributions, the inorganic matrix and organic matrix have very different saturations of fluids. A higher oil/gas saturation and lower water saturation are observed in organic matrix. Since the organic matrix is not directly connected to fractures, fluids must go through the inorganic matrix before reaching fractures. Besides, because the organic matrix is oil-wet while inorganic matrix is water-wet,
4.2. Comparison with uniform-wet condition To further illustrate the effects of mixed wettability, another two cases with uniform wettability are conducted for comparison. In the water-wet case, all matrix is set as water-wet and the water-oil relative permeability with contact angle of 54° is applied in both organic and inorganic matrix. In the oil-wet case, matrix is oil wet and the water-oil relative permeability with contact angle of 135° is applied. Fig. 8 compares the reservoir pressure, oil, gas and water production under different conditions of wettability. Compared to mixed-wet case, the oil production is much lower, gas production is higher and water production is much higher when the shale matrix is uniformly oil wet. The 6
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Fig. 7. Effect of TOC on the production performance of shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
significant difference comes from the fact that the inorganic matrix is water wet in mixed-wet case but is oil wet in oil-wet case. Based on the relative permeabilities in Fig. 4, the oil mobility is largely suppressed, and water mobility is much increased when inorganic matrix is oil wet. Thus, the lower oil production and higher water production have been observed. When the shale matrix is uniformly water wet, both oil and water production is lower and gas production is slightly higher compared to mixed-wet case. It is seen that the difference between mixedwet case and water-wet case is not very significant. The organic matrix becomes water wet in water-wet case which means that water mobility should be lower and oil mobility should be higher in organic matrix. However, because organic matrix is embedded in inorganic matrix, all the fluids in organic matrix must go through inorganic matrix first before being produced through fractures. Therefore, the relative permeability in inorganic matrix is the dominant factor that controls the production rates of fluids. Since the wettability of inorganic matrix is the same in mixed-wet case and water-wet case, the production performances are thus closer.
water permeability is lower and oil permeability is higher in inorganic matrix, when the contact angle is decreased to 30°. On the other hand, the water permeability is increased, and oil permeability is decreased in organic matrix, when contact angle is set to 160°. The altered relative permeabilities due to contact angles in turn affect the production performance of the shale reservoir. Fig. 10 compares the pressure decline and production rates under different conditions of contact angles. No obvious differences are observed in terms of pressure decline and gas production. By comparing case (1) and case (2), the change of contact angle in organic matrix slightly affects oil and water productions. Results from case (1) and case (3) indicate that the contact angle in inorganic matrix has a significant effect on oil and water production: a stronger water-wet inorganic matrix leads to a higher oil production and lower water production. This finding further confirms the conclusion made in Section 4.2 that the relative permeability (wettability) in inorganic matrix controls the overall water and oil productions.
4.3. Effect of water-oil contact angle (WOCA)
Oil-gas relative permeability is also affected by oil-gas contact angle (OGCA). With the same fluid compositions and pore size, contact angle controls oil-gas capillary pressure, from which oil-gas relative permeability is extracted. Fig. 11 shows the oil-gas relative permeabilities with different OGCA in both organic and inorganic matrix. In inorganic matrix where pore size is relatively larger, the oil permeability is more sensitive to contact angle. Oil permeability is increased when contact angle is decreased to 30°. Gas permeability is lower at high oil saturation and higher at lower oil saturation when contact angle is turned to 30°. On the other hand, gas permeability is more sensitive to contact angle in organic matrix where pore size is smaller. Gas permeability is
4.4. Effect of oil–gas contact angle (OGCA)
It has been found that the relative permeability depends on contact angles [28,29]. Thus, it is essential to evaluate the three-phase flow in shale reservoir using different contact angles. In this part, the effects of water–oil contact angle (WOCA) are investigated. Three cases are conducted: (1) WOCA in organic matrix is 135° and in inorganic matrix is 54°, (2) WOCA in organic matrix is 160° and in inorganic matrix is 54°, (3) WOCA in organic matrix is 135° and in inorganic matrix is 30°. Fig. 9 presents the water-oil relative permeabilities under different contact angles, in both organic and inorganic matrix. It is seen that the 7
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Fig. 8. Comparison of production performance between mixed-wet shale reservoir and uniform-wet shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
Fig. 9. Effect of contact angle on water-oil relative permeability in inorganic matrix (a) and organic matrix (b).
oil-gas contact angle in inorganic matrix has much larger effects than that in organic matrix.
increased, and oil permeability is slightly decreased when contact angle becomes 30°. The production performance with different OGCA are compared in Fig. 12. The water production rate when contact angle is 30° is slightly higher than that when contact angle is 60°. Oil production rate is significantly increased when contact angle is 30° due to the increased oil permeability in the inorganic matrix. Though gas permeability is higher in organic matrix, the gas production rate has negligible changes from different contact angles. Again, it is found that the
5. Conclusions In this work, compositional simulation of three-phase flow is conducted considering mixed wettability in shale oil reservoir. By using the multi-porosity model, mixed-wettability is modeled explicitly by 8
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Fig. 10. Effect of water-oil contact angle on the production performance of shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
Fig. 11. Effect of oil-gas contact angle on the relative permeabilities in inorganic matrix (a) and organic matrix (b).
permeabilities are applied to organic and inorganic matrix according to the condition of wettability. From the simulation results, we obtain the following findings and conclusions:
dividing the shale matrix into organic and inorganic matrix, which are oil-wet and water-wet respectively. To construct relative permeability for three-phase flow, the Stone II method is applied that assumes the permeabilities of water and gas are only functions of their own saturations. Water-oil relative permeability is obtained by performing immiscible two-phase flow directly on digital rock sample using a multiphase lattice Boltzmann method. Oil-gas relative permeability is extracted from oil-gas capillary pressure curve, which is calculated from fluid compositions and phase behavior. Different three-phase relative
(1) The rich hydrocarbon in organic matrix is hard to be produced due to its low permeability, low oil mobility and poor connection to hydraulic fractures. (2) A higher TOC in shale matrix leads to lower production rates at 9
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Fig. 12. Effect of oil-gas contact angle on the production performance of shale reservoir; (a) pressure, (b) oil production, (c) gas production (d) water production.
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