Consistent upwinding for sequential fully implicit multiscale compositional simulation

Abstract

There is strong interest to design sequential fully implicit (SFI) methods for compositional flow simulations with convergence properties that are comparable to fully implicit (FI) methods. SFI methods decompose the fully coupled system into a pressure equation and a transport system of the components. During the pressure update, the compositions are frozen, and during the transport calculations, both the pressure and total velocity are kept constant. The two systems are solved sequentially, and the solution, which is a fully implicit one, is obtained by controlling the splitting errors due to the decoupling. Having an SFI scheme that enjoys a convergence rate similar to FI makes it possible to design specialized numerical methods optimized for the different parabolic and the hyperbolic operators, as well as the use of high-order spatial and temporal discretization schemes. Here, we use the multiscale restriction-smoothed basis (MsRSB) method for the parabolic operator. We also show that phase-potential upwinding is incompatible with the total velocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong gravity or capillary pressure, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. In this work, we employ phase upwinding (PU) as well as implicit hybrid upwinding (IHU) with a SFI method. IHU determines the upwinding direction differently for the viscous, buoyancy, and capillary pressure terms in the phase velocity expressions. The use of IHU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong buoyancy or capillarity. We demonstrate the robustness of the IHU-based SFI algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.

Appendices

Appendix A: Description of the models

The 1D model used in this study has 220 cells of dimensions of dx = 20 ft, dy = 10 ft, dz = 2 ft. The porosity is ϕ = 0.2 and the permeability K = 100 mD. The 2D model has 100×100 cells of dimensions of dx = dy = dz = 3 ft, porosity is ϕ = 0.2 and the permeability K = 50 mD in every direction. The rock compressibility for the 1D and 2D cases is 1.78e − 5 1/psi. The relative permeabilities are quadratic for the gas the oil and the water phases. The shape of the capillary pressure between the gas and the oil phases is provided by Table 11 and the shape of the capillary pressure between the oil and the water phases is provided by Table 12. For the 1D cases, the maximum values of the capillary pressure between the gas and the oil and between the oil and the water phases are taken to the very large values of 100.0 psi. For the 2D cases, unless specified, the maximum value of the capillary pressure between the gas and the oil phases is taken at 2.0 psi and between the oil and the water phases is taken at 25.0 psi. The fluids is taken from the SPE 5 comparative solution project [18]. The test cases have initial conditions with P = 4000.0 psi, T = 160 ^∘F and S_o = 1.0 except for the three-phase 2D test cases that have the initial pressure at saturated condition of P = 2302.0 psi. Table 13 provides the initial composition for the gas and the oil phases. The connection factors for all the cell-well connections are fixed at 0.3 Rbbl.cP/day-psi. For the first 1D test case, 4.0 STB/day of water is injected and the producer is controled at 4000 psi. For the second 1D test case, 3.0 STB/day of component C1 is injected and the producer is controled at 4000 psi. For the third 1D test case, 3.0 STB/day of water is injected and the producer is controled at 4000 psi. For the 2D test cases, the top-left corner is kept at the initial pressure.

Table 11 Shape of the capillary pressure between the gas and the oil phases

Table 12 Shape of the capillary pressure between the oil and the water phases

Table 13 Initial gas and oil compositions for SPE 5 fluid

Appendix B: Partial molar volumes for black-oil formulation

In this appendix, we give more details on the computation of the partial molar volumes for the specific case of the black-oil formulation [1, 30]. The black-oil model contains as many components as phases and assumes that, when mass exchange occurs between the phases, the phase compositions are direct functions of the gas phase pressure P; that is, for each phase p and each component c

$$ x_{p,c}= x_{p,c}^{sat}(P). $$

(57)

As a result, when mass exchange occurs between the phases, the phase mole densities ρ_p are also only functions of the pressure. We can express the number of moles of each component c

$$ N_{c} = \sum\limits_{p} x_{p,c} N_{p} $$

(58)

in function of the number of moles for each phase pN_p. In the black-oil assumption, as there are as many components as phases, we can reverse the relationship:

$$ N_{p} = \sum\limits_{c} {\gamma}_{p,c} N_{c} $$

(59)

with the coefficients γ_p,c only functions of pressure. The partial molar volume of component c is then written as:

$$ V_{T_{c}}=\frac{\partial V_{T}}{\partial N_{c}} = \sum\limits_{p} \frac{\partial \left( \frac{N_{p}}{\rho_{p}} \right)}{\partial N_{c}} = \sum\limits_{p} \frac{\partial N_{p}}{\partial N_{c}} \frac{1}{\rho_{p}} = \sum\limits_{p} {\gamma}_{p,c} \frac{1}{\rho_{p}}. $$

(60)

The phase mole densities have been removed of the partial derivatives as they are only functions of pressure. Using Eqs. 59 and 60 gives

$$ \begin{array}{@{}rcl@{}} \sum\limits_{c} z_{c} V_{T_{c}} &=& \frac{1}{{\sum}_{c} N_{c}} \sum\limits_{c} N_{c} V_{T_{c}} \end{array} $$

(61)

$$ \begin{array}{@{}rcl@{}} &=& \frac{1}{{\sum}_{c} N_{c}} \sum\limits_{c} N_{c} \left( \sum\limits_{p} {\gamma}_{p,c} \frac{1}{\rho_{p}}\right) \end{array} $$

(62)

$$ \begin{array}{@{}rcl@{}} &=& \frac{1}{{\sum}_{c} N_{c}} \sum\limits_{p} \left( \sum\limits_{c} {\gamma}_{p,c} N_{c} \right) \frac{1}{\rho_{p}} \end{array} $$

(63)

$$ \begin{array}{@{}rcl@{}} &=& \frac{1}{{\sum}_{c} N_{c}} \sum\limits_{p} \frac{N_{p}}{\rho_{p}} \end{array} $$

(64)

$$ \begin{array}{@{}rcl@{}} &=& \sum\limits_{p} \frac{\frac{N_{p}}{{\sum}_{c} N_{c}}}{\rho_{p}} = \sum\limits_{p} \frac{\beta_{p}}{\rho_{p}}. \end{array} $$

(65)

with β_p the mole fraction of the phase p. By definition, the quantity $\frac {\beta _{p}}{\rho _{p}}$ is the volume of phase p divided by the total number of moles. It is then equivalent to

$$ \frac{\beta_{p}}{\rho_{p}} = \frac{S_{p}}{{\sum}_{q} \rho_{q} S_{q}}, $$

(66)

giving the relationship

$$ \sum\limits_{c} z_{c} V_{T_{c}} = \sum\limits_{p} \frac{\beta_{p}}{\rho_{p}} = \sum\limits_{p} \frac{S_{p}}{{\sum}_{q} \rho_{q} S_{q}} = \frac{1}{{\sum}_{q} \rho_{q} S_{q}}. $$

(67)

For a system with two hydrocarbon phases, gas and oil, and two hydrocarbon components, light component l and heavy component h, the partial molar volumes are written as:

$$ \begin{array}{@{}rcl@{}} V_{T_{l}} &=& \left( \frac{1-x_{o,l}^{sat}}{\rho_{g}} - \frac{x_{g,h}^{sat}}{\rho_{o}} \right) \gamma \end{array} $$

(68)

$$ \begin{array}{@{}rcl@{}} V_{T_{h}} &=& \left( -\frac{x_{o,l}^{sat}}{\rho_{g}} + \frac{1-x_{g,h}^{sat}}{\rho_{o}} \right) \gamma \end{array} $$

(69)

with γ the inverse of the determinant of the matrix composed by the mole fractions of the components in phases

$$ \gamma = \frac{1}{(1-x_{g,h}^{sat}) (1-x_{o,l}^{sat}) - x_{g,h}^{sat} x_{o,l}^{sat}}. $$

(70)

Usually, we express this system of two hydrocarbon components with four functions of pressure: R_s, R_v, B_g, B_o [1, 30]. The phase compositions are then written as

$$ \begin{array}{@{}rcl@{}} x_{o,l}^{sat} &=& \frac{R_{s}}{\alpha_{b}+R_{s}} \end{array} $$

(71)

$$ \begin{array}{@{}rcl@{}} \text{and } x_{g,h}^{sat} &=& \frac{R_{v}}{\frac{1}{\alpha_{b}}+R_{v}}, \end{array} $$

(72)

the phase mole densities as

$$ \begin{array}{@{}rcl@{}} \rho_{g} &=& \frac{1}{{M^{w}_{g}}} \frac{\overline{\rho_{l}^{sc}} + \overline{\rho_{h}^{sc}} R_{v}}{B_{g}} \end{array} $$

(73)

$$ \begin{array}{@{}rcl@{}} \text{and } \rho_{o} &=& \frac{1}{{M^{w}_{o}}} \frac{\overline{\rho_{l}^{sc}}R_{s} + \overline{\rho_{h}^{sc}}}{B_{o}}, \end{array} $$

(74)

with the molar weights of phase gas and oil as

$$ \begin{array}{@{}rcl@{}} {M^{w}_{g}} &=& (1-x_{g,h}^{sat})\cdot {M^{w}_{l}} + x_{g,h}^{sat}\cdot {M^{w}_{h}} \end{array} $$

(75)

$$ \begin{array}{@{}rcl@{}} \text{and } {M^{w}_{o}} &=& x_{o,l}^{sat}\cdot {M^{w}_{l}} + (1-x_{o,l}^{sat})\cdot {M^{w}_{h}}, \end{array} $$

(76)

in function of the coefficient

$$ \alpha_{b} = \frac{{M^{w}_{l}}}{{M^{w}_{h}}} \frac{\overline{\rho_{h}^{sc}}}{\overline{\rho_{l}^{sc}}} $$

(77)

with ${M^{w}_{l}}$ and ${M^{w}_{l}}$ the molar weights of components l and h, respectively, and $\overline {\rho _{l}^{sc}}$ and $\overline {\rho _{h}^{sc}}$ the mass densities at surface conditions of components l and h, respectively.

Appendix C: SI metric conversion factors

1	psi	=	100000/14.5037	Pa
1	ft	=	1/3.2808	m
1	lb	=	1/2.20462262	kg
1	mD	=	9.869e-16	m2
1	cP	=	0.001	Pa.s
1	day	=	86400	s