Fractional-Flow Theory for Non-Newtonian Surfactant-Alternating-Gas Foam Processes
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Abstract
Foam can improve sweep efficiency in gas-injection-enhanced oil recovery. Surfactant-alternating-gas (SAG) is a favored method of foam injection. Laboratory data indicate that foam can be non-Newtonian at low water fractional flow f_{w}, and therefore during gas injection in a SAG process. We investigate the implications of this finding for mobility control and injectivity, by extending fractional-flow theory to gas injection in a non-Newtonian SAG process in radial flow. We make most of the standard assumptions of fractional-flow theory (incompressible phases, one-dimensional displacement through a homogeneous reservoir, instantaneous attainment of local equilibrium), excluding Newtonian mobilities. For this initial study, we ignore the effect of changing or non-uniform oil saturation on foam. Non-Newtonian behavior at low f_{w} implies that the limiting water saturation for foam stability varies as superficial velocity decreases with radial distance from the well. We discretize the domain radially and perform Buckley–Leverett analysis on each narrow increment in radius. Solution characteristics move outward with fixed f_{w}. We base the foam model parameters and non-Newtonian behavior on laboratory data in the absence of oil. We compare results to mobility and injectivity determined by conventional simulation, where grid resolution is usually limited. For shear-thinning foam, mobility control improves as the foam front propagates from the well, but injectivity declines somewhat with time. This change in mobility ratio is not that at steady state at fixed water fractional flow in the laboratory, however, because the shock front in a non-Newtonian SAG process does not propagate at fixed fractional flow (though individual characteristics do). Moreover, the shock front is not governed by the conventional condition of tangency to the fractional-flow curve, though it continually approaches this condition. Injectivity benefits from the increased mobility of shear-thinning foam near the well. The foam front, which maintains a constant dimensionless velocity for Newtonian foam, decelerates somewhat with time for shear-thinning foam. For shear-thickening foam, mobility control deteriorates as the foam front advances, though injectivity improves somewhat with time. Overall, however, injectivity suffers from reduced foam mobility at high superficial velocity near the well. The foam front accelerates somewhat with time. Conventional simulators cannot adequately represent these effects, or estimate injectivity accurately, in the absence of extraordinarily fine grid resolution near the injection well.
Keywords
Surfactant-alternating-gas Foam Non-Newtonian fluids Fractional-flow theory Enhanced oil recoveryList of Symbols
- C
Constant (Eq. 4)
- epdry
Parameter in the STARS foam model controlling the abruptness of foam collapse at the limiting liquid saturation, unitless (Eq. A6)
- F_{2}
Dry-out function in the STARS foam model, unitless. (Eq. A5)
- f_{g}
Foam quality (gas fractional flow), unitless
- f_{w}
Water fractional flow, unitless
- FM
Mobility reduction factor, unitless
- fmdry
Limiting liquid saturation in the STARS foam model, unitless. (Eq. A6)
- fmmob
Reference gas-mobility-reduction factor in the STARS foam model, unitless. (Eq. A5)
- h
Thickness of the reservoir, m
- k
Permeability, unitless
- k_{rg}
Gas relative permeability in the absence of foam, unitless
- \(k_{\text{rg}}^{0}\)
End-point gas relative permeability, unitless. (Eq. A2)
- \(k_{\text{rg}}^{f}\)
Gas relative permeability in the presence of foam, unitless. (Eq. A4)
- k_{rw}
Liquid relative permeability, unitless
- \(k_{\text{rw}}^{0}\)
End-point liquid relative permeability, unitless. (Eq. A1)
- λ_{w}
Liquid mobility, (Pa s)^{−1}
- λ_{rt}
Total relative mobility, (Pa s)^{−1}
- n
Power-law exponent, unitless
- n_{g}
Gas Corey–Brooks exponent, unitless. (Eq. A2)
- n_{w}
Liquid Corey–Brooks exponent, unitless. (Eq. A2)
- μ_{g}
Gas viscosity, Pa s
- μ_{w}
Water viscosity, Pa s
- P_{D}
Dimensionless pressure rise at well, unitless
- P_{i−1}
Pressure at the inner boundary of the ith grid block, Pa. (Eq. B2)
- P_{i}
Pressure at the outer boundary of the ith grid block, Pa. (Eq. B2)
- \(\nabla P\)
Pressure gradient, Pa/m
- ∆P_{f}
Pressure difference across the foam bank, Pa
- ∆P_{w}
Pressure difference across the liquid bank, Pa
- Q
Total volumetric injection rate, m^{3}/s
- r
Radius, m
- r_{w}
Wellbore radius, m
- r_{e}
Outer radius, m
- r_{i}
Radius at the inner boundary of the ith grid block, unitless
- r_{i−1}
Radius at the outer boundary of the ith grid block, unitless
- S
Normalized liquid saturation, unitless
- S_{gr}
Residual gas saturation, unitless
- S_{w}
Liquid saturation, unitless
- S_{wi}
Liquid saturation in the ith grid block, unitless
- S_{wi−1}
Liquid saturation of the grid-block upstream of the ith grid block, unitless
- S_{wr}
Residual liquid saturation, unitless
- S_{w}*
Limiting liquid saturation, unitless
- t
Time, s
- t_{D}
Dimensionless time
- ∆t
Time increment, s
- U_{t}
Total superficial velocity, m/s
- U_{w}
Liquid superficial velocity, m/s
- U_{g}
Liquid superficial velocity, m/s
- x
Position, m
- x_{D}
Dimensionless position, unitless
- ∆x
Position increment, m
1 Introduction
1.1 Fractional-Flow Theory
- 1.
One-dimensional (radial or linear) flow.
- 2.
Two mobile and incompressible phases.
- 3.
Instantaneous equilibrium adsorption of surfactant on the rock. In this study, for simplicity, we assume adsorption was satisfied during the injection of the preceding liquid slug.
- 4.
No dispersive processes, including fingering, capillary diffusion or dispersion.
- 5.
Instantaneous attainment of local steady-state mobilities, which depend on local saturations. In this study, mobilities depend on total superficial velocity as well.
- 6.
No chemical or biological reactions.
- 7.
Fixed total volumetric injection rate Q.
The superficial velocity of an incompressible fluid injected at a fixed rate Q into a cylindrical reservoir decreases continuously from the wellbore radius to the outer radius. This implies that non-Newtonian fluids experience not only a different superficial velocity but different rheology, as a function of radial position. Mathematically, this means that f_{w} is a function of S_{w} and x_{D}. The fractional-flow analysis with this additional constraint results in characteristics that in general do not have a fixed dimensionless velocity (Rossen et al. 2011; Wu et al. 1993). Liquid fractional flow is fixed for each characteristic as it travels through the porous medium, although liquid saturation is not.
1.2 Foam in Porous Media
Foam increases sweep efficiency during gas injection in enhanced oil recovery applications (Holm 1968; Schramm 1994; Patzek 1996; Skauge et al. 2002; Blaker et al. 2002; Zhu et al. 2013; Lake et al. 2014). It is also used in aquifer remediation and in acid diversion in well-stimulation treatments (Thompson and Gdanski 1993; Behenna 1995; Hirasaki et al. 1997a, b; Talabani et al. 2001; Cheng et al. 2002; Maire et al. 2015; Portois et al. 2018). Foam flow in porous media exhibits two flow regimes: a low-quality (large f_{w}) regime and a high-quality (small f_{w}) regime (Osterloh and Jante 1992; Alvarez et al. 2001). The low-quality regime is characterized by a gas-mobility reduction and a pressure gradient independent of liquid superficial velocity, whereas the high-quality regime is characterized by a limiting capillary pressure P_{c}* and a pressure gradient independent of gas superficial velocity.
Foam can be non-Newtonian in both regimes. There are several reasons for this behavior. Foam flowing through smooth tubes shows shear-thinning behavior (Hirasaki and Lawson 1985). Flow in periodically constricted tubes is even more shear thinning (Xu and Rossen 2003). Gas trapping further reduces gas mobility (Falls et al. 1989), so if gas trapping declines as pressure gradient increases (Tang and Kovscek 2006); this provides a separate mechanism of shear-thinning behavior. Rheology in the low-quality regime is consistently found to be shear thinning with respect to total superficial velocity at fixed f_{w}, but foam can be either shear thickening or shear thinning in the high-quality regime. Rheology in the high-quality regime is controlled by the limiting capillary pressure P_{c}* (Khatib et al. 1988). Jimenez and Radke (1989) present a model that predicts that P_{c}* decreases with increasing gas velocity, which would imply that pressure gradient would decrease, rather than hold constant, as gas velocity increases with at liquid velocity.
“Implicit Texture” models, here referred to as “IT’ models for simplicity, are regularly used in combination with fractional-flow theory to describe foam displacements (Rossen et al. 2011; Al Ayesh et al. 2017). Unlike “Population Balance” models (Patzek 1985; Falls et al. 1988; Fergui et al. 1998; Kovscek et al. 1994), they assume local equilibrium in the dynamics of bubble creation and destruction and represent the effects of foam on gas mobility through a mobility-reduction factor (Cheng et al. 2000). “Population Balance” models can also be constrained to conditions of local equilibrium between foam-generation and foam-destruction processes (Kam et al. 2007; Kovscek et al. 2010). In the present study, we use an IT model because it requires fewer parameters and avoids some of the numerical challenges that are present in “Population Balance” models (Ashoori et al. 2012).
Most IT models allow for non-Newtonian behavior in the low-quality regime (Computer Modeling Group 2015; Cheng et al. 2000) but not in the high-quality regime. The limiting capillary pressure, which controls behavior in the high-quality regime (Khatib et al. 1988), corresponds to a limiting water saturation S_{w}*. Since foam does not alter the liquid relative permeability function (Eftekhari and Farajzadeh 2017), a stronger foam corresponds to lower limiting liquid saturation, S_{w}* (Zhou and Rossen 1995). Thus, non-Newtonian behavior in the high-quality regime requires that S_{w}* be a function of water superficial velocity. If it depended on gas superficial velocity, the pressure-gradient contours in Fig. 1 would not be vertical. In this study, however, for simplicity, we assume that S_{w}* is a function of total superficial velocity.
1.3 Fractional-Flow Solution for Gas Injection in a Newtonian SAG
The methodology described above has been shown to be more accurate than numerical simulation as long as the assumptions of fractional-flow theory apply (Rossen 2013). In particular, the abrupt transition imposed by the limiting capillary pressure is difficult to model correctly using finite-difference methods without using an extremely refined grid near the wellbore (Leeftink et al. 2015). Boeije and Rossen (2015) use the theory to derive an analytical formula to estimate the injectivity of the first gas slug in a SAG process. The formula predicts that, soon after injection begins, the pressure gradient across the foam bank is nearly constant as it advances.
2 Fractional-Flow Solution for Gas Injection in Non-Newtonian SAG Processes
There are a number of studies of non-Newtonian behavior in 1D linear flow, in which a single fractional-flow curve describes the process: e.g., Wu et al. (1992) and Yi (2004), and Subramanian et al. (1997, 1999) describe flow from one porous medium to another, described by a different fractional-flow function. Bedrikovetsky (1993) describes the solution for a 1D drainage of a dome-shaped reservoir with a gas cap. In this case, the fractional-flow function varies with vertical position. Jamshidnezhad et al. (2008) solve for steady-state gravity segregation in 2D radial flow for non-Newtonian fluids, where the fractional-flow function varies with radial distance from the well. Wu et al. (1991) solve for 1D linear displacement of a Newtonian fluid by a non-Newtonian fluid. El-Khatib (2006) studied numerically the immiscible displacement of non-Newtonian fluids in communicating stratified reservoirs.
A previous work on non-Newtonian foam displacements (Rossen et al., 2011) was limited to foam injection or gas injection in a SAG process behind the shock front and included only shear-thinning behavior. The SAG analysis showed the effect of changing gas saturation and non-Newtonian behavior in the near-well region after the shock passes out of this region. The results showed that shear-thinning behavior affects mobility near the well; this implies that the injectivity is better than that predicted by a Newtonian model. As in the Newtonian case, the finite-difference simulator requires an extremely refined grid (especially near the wellbore) to match the analytical solution.
In this study, we extend the previous work to include both shear-thinning and shear-thickening behavior, as observed in the laboratory (Fig. 1). Equally important, we provide a methodology to solve issues that arise, e.g., when new characteristics emerge from the shock or when a characteristic and the shock collide. Finally, we show the consequences of non-Newtonian behavior for overall injectivity and mobility control at the leading edge of the foam bank.
We consider a homogenous cylindrical reservoir that is initially fully saturated with surfactant solution. Starting at time zero, gas is injected at a fixed volumetric rate Q into the reservoir. In this situation, the gas-injection process is governed by the high-quality regime, where foam strength depends on the limiting water saturation, S_{w}*, named fmdry in the foam model described in “Appendix A.” Under these assumptions (including incompressible fluids), total superficial velocity u_{t} is a function of radial distance from the well.
In a non-Newtonian scenario, a characteristic no longer carries its water-saturation value as it crosses between increments, but it does carry its water fractional-flow value instead (Wu et al. 1993; Rossen et al., 2011). The individual characteristics within the spreading wave do not collide with each other, because, at any value of x_{D}, velocity df_{w}/dS_{w} decreases monotonically with S_{w}. Thus, individual characteristics spread further apart as they move downstream. Interactions with the shock are possible, however. We use the approach of Lake et al. (2003) to resolve the complications raised by the collision between a characteristic and a shock or by an accelerating shock that sheds additional characteristics.
We carry out the calculations as follows. We discretize x_{D} into 1000 increments, spaced so that total superficial velocity decreases by 0.7% between consecutive increments of increasing radial position. Thus, increments are smaller near the wellbore, where total superficial velocity changes rapidly. For shear-thinning foam, we calculate velocities for 300 characteristics in the first increment. In each new increment in x_{D} moving outward, we calculate the intersection point between the shock and the characteristic immediately behind it. If the intersection is within the increment, we recalculate the new shock velocity and eliminate the characteristic from that point forward. We then check whether the next characteristic would then intersect the new shock trajectory within the increment; if so, we update shock velocity and eliminate the next characteristic, and so forth.
For shear-thickening foam, we calculate velocities for 300 characteristics starting in the outermost increment, corresponding to r_{e}. We carry out calculations moving inward; in essence working from Fig. 9a–c at each new increment moving inward. At each new increment, we calculate the shock velocity using the tangency condition. Any characteristics with larger velocities are eliminated. (Thus, in our calculations, characteristics are eliminated as we move inward, rather than created in moving outward.) In the example shown below, there are 169 characteristics left at r_{w}, so resolution is good throughout the domain of interest.
For both shear-thinning and shear-thickening cases, results were substantially unchanged whether we started with 200 or 300 characteristics, or whether they were initially spaced equally in S_{w} or in f_{w}. Also, for both rheologies, results were unchanged whether we started with 1000 or 1100 increments in x_{D}.
In the end, we have a table of dimensionless times at which each characteristic passes the outer boundary of each increment in x_{D}, along with the values of f_{w} and S_{w} for that characteristic. From S_{w}, the total mobility corresponding to that characteristic in that increment can be determined.
2.1 Application
We apply the methodology described above to a homogeneous cylindrical reservoir with wellbore and open-outer-boundary radii of 0.1 m and 100 m, respectively. The superficial velocity varies by a factor of 1000 from the outer radius to the wellbore radius. Experimental data on the non-Newtonian behavior of foam in the high-quality regime extend over ranges very much smaller than this (cf. Fig. 1); thus, our results illustrate the implications if these trends continue over a much-wider range of velocities.
The STARS foam model is able to reproduce an abrupt, though not complete, foam collapse at a water-saturation value fmdry (previously referred as S_{w}*) (Computer Modeling Group. 2015; Cheng et al. 2000). However, this version of the dry-out function can underestimate the injectivity observed in the field during gas injection in a SAG (Rossen et al. 2017). Therefore, in this study, we use the Namdar-Zanganeh modification of this model, which assumes complete foam collapse at residual water saturation S_{wr} (Namdar-Zanganeh et al. 2011; Rossen et al. 2016). See “Appendix A” for a description of the foam model used in this study.
Summary of the input parameters used in this study (Kapetas et al. 2017)
Viscosities | |
µ_{w} = 0.001 Pa s | µ_{g} = 0.00002 Pa s |
Corey relative permeability parameters | |
S_{wr} = 0.25 | S_{gr} = 0.2 |
k_{rw}^{0} = 0.39 | k_{rg}^{0} = 0.59 |
n_{w} = 2.86 | n_{g} = 0.7 |
Foam parameters | |
fmdry = 0.271 | |
fmmob = 47,700 | |
epdry = 400 |
We apply power-law exponents similar to those reported by Alvarez et al. (2001) (n = 1.34) and by Osterloh and Jante (1992) (n = 0.33) to the entire range of velocities. (Alvarez et al. present estimates for several sets of data; most are less shear thickening than that used for illustration in Fig. 1.) As noted, these trends were determined experimentally over a much-narrower range of velocities than assumed here. Our results illustrate the implications if those trends continue over the entire range of velocities around an injection well.
2.2 Results
2.2.1 Shear-thinning foam (n = 0.33)
These results illustrate two general trends for gas injection in SAG with shear-thinning foam: mobility control at the foam front improves as foam advances from the well, but injectivity declines. The mobility at the shock is consistently less than that estimated from the tangency condition for the f_{w}(S_{w}) function starting at a radius, r = 0.13 m. Mobility at the shock decreases to about a factor of 0.04 of its original value, instead of (1000)^{−0.67} ≈ 0.01 suggested by the power law.
2.2.2 Shear-Thickening Foam (n = 1.34)
We also modeled a shear-thickening foam with n = 1.67, value reported by Alvarez et al. (2001). As seen in Fig. 5, for n = 1.34, fmdry approaches S_{wr} near the wellbore. For n = 1.67, and assuming a strong foam at r_{e}, fmdry approaches so close to S_{wr} near the wellbore that the adjustment of Namdar-Zanganeh, which requires foam collapse at S_{wr}, gives shear-thinning behavior very near the well. Therefore, we do not show that case. Details are in Ter Haar (2018).
2.2.3 Finite-Difference Simulation
In this section, we present the evolution of total dimensionless pressure for non-Newtonian SAG processes during gas injection, calculated using a finite-difference simulator. A description of the discretization scheme used in the simulator is included in Appendix B; see also Bos (2017). As in the fractional-flow calculations, the input parameters correspond to the petrophysical and foam parameters listed in Table 1.
3 Conclusions
In this paper, we present a method of solution for initial gas injection in a non-Newtonian SAG process that includes the interactions between the shock and the characteristics. The methodology can be applied to both shear-thinning and shear-thickening behavior.
For a shear-thinning foam, we find that mobility control improves as the foam front propagates from the well, but injectivity declines somewhat with time. However, the injectivity is still more favorable than for a Newtonian foam with the same mobility at the outer radius. In a case of a foam with marginal mobility control, there could be problems with viscous fingering as foam initially advances from the near-well region. For a shear-thinning foam, the shock does not necessarily satisfy the conventional tangency condition that applies to Newtonian foam, though it does continually approach it. In addition, the mobility at the front needs not fit the power-law behavior seen at fixed gas fraction in the laboratory.
For a shear-thickening foam, mobility control deteriorates as the foam front advances, though injectivity improves somewhat with dimensionless time. However, injectivity is less favorable than for a Newtonian foam with the same mobility far from the well. In case of marginal mobility control, the foam could have problems with viscous fingering far from the injection well.
Overall, injectivity is a complex result of changing saturations and varying superficial velocities very near the well. Conventional simulators cannot adequately represent these effects, or estimate injectivity accurately, in the absence of exceptional grid resolution near the injection well.
Notes
Funding
This work was funded by a grant from Instituto Mexicano del Petróleo, Consejo Nacional de Ciencia y Tecnología (Grant No. 508403), and the Joint Industry Project on Foam for EOR at Delft University of Technology, with support from ConocoPhillips, Equión Energía, Neptune Energy, PEMEX and Shell.
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