# American Institute of Mathematical Sciences

September  2019, 14(3): 489-536. doi: 10.3934/nhm.2019020

## Compressible and viscous two-phase flow in porous media based on mixture theory formulation

 1 Department of Energy and Petroleum Engineering, University of Stavanger, Stavanger, NO-4068, Norway 2 School of Mathematics, South China University of Technology, Guangzhou, 510641, China

* Corresponding author: Steinar Evje

Received  June 2018 Revised  March 2019 Published  May 2019

Fund Project: Wen was supported by the National Natural Science Foundation of China (Grant No. 11671150, 11722104) and by GDUPS (2016).

The purpose of this work is to carry out investigations of a generalized two-phase model for porous media flow. The momentum balance equations account for fluid-rock resistance forces as well as fluid-fluid drag force effects, in addition, to internal viscosity through a Brinkmann type viscous term. We carry out detailed investigations of a one-dimensional version of the general model. Various a priori estimates are derived that give rise to an existence result. More precisely, we rely on the energy method and use compressibility in combination with the structure of the viscous term to obtain $H^1$-estimates as well upper and lower uniform bounds of mass variables. These a priori estimates imply existence of solutions in a suitable functional space for a global time $T>0$. We also derive discrete schemes both for the incompressible and compressible case to explore the role of the viscosity term (Brinkmann type) as well as the incompressible versus the compressible case. We demonstrate similarities and differences between a formulation that is based, respectively, on interstitial velocity and Darcy velocity in the viscous term. The investigations may suggest that interstitial velocity seems more natural to use in the formulation of momentum balance than Darcy velocity.

Citation: Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020
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##### References:
(left figure) and initial water saturation (3.84) profile (right), both similar to that used in [9]">Figure 1.  Water fractional flow function $\hat{f}_w(s_w)$ as given by (4.127) for the incompressible model obtained by using the parameters specified in Table 1 (left figure) and initial water saturation (3.84) profile (right), both similar to that used in [9]
] to solve the model (1.13). From these computations we see that the solution is sensitive to whether the interstitial velocity $u_i$ or the Darcy velocity $U_i$ appear in the viscous term. In particular, the use of Darcy velocity seems to generate considerably more oscillatory behavior behind the "water bank" formed at the front">Figure 2.  Upper row: Results produced by the discrete scheme described in Appendix D (incompressible model). Three kinds of curves are plotted including the case without viscous effect, i.e., $\varepsilon_w = \varepsilon_o = 0$, the one based on using Darcy velocity $U_i$ $(i = w, o)$ in the viscous term, and the one with interstitial velocity $u_i$ $(i = w, o)$ in the viscous term. The left figure shows results with $\varepsilon = \varepsilon_w = \varepsilon_o = 10^7$ whereas the right figure shows results with $\varepsilon = \varepsilon_w = \varepsilon_o = 10^6$. Lower row: The results of two corresponding cases with $\varepsilon_w = \varepsilon_o = 10^7$ and $\varepsilon_w = \varepsilon_o = 10^6$ after a dimensionless time, 0.65, produced by the numerical scheme described in [9] to solve the model (1.13). From these computations we see that the solution is sensitive to whether the interstitial velocity $u_i$ or the Darcy velocity $U_i$ appear in the viscous term. In particular, the use of Darcy velocity seems to generate considerably more oscillatory behavior behind the "water bank" formed at the front
Simulation results with smaller viscous parameters after 10 days of water flooding. Three kinds of curves are compared: zero viscous effect, Darcy velocity $U_i$ in viscous term and interstitial velocity $u_i$ in viscous term. It shows that the viscous constant water level gradually vanishes when $\varepsilon$ is as low as $10^3$ and $10^2$
with interstitial velocity in viscous term. Four curves are compared: the one with large values of $\varepsilon_w$ and $\varepsilon_o$, $10^6$; the second one with large $\varepsilon_o$, $10^6$ and small $\varepsilon_w$, $10^4$; the third one with large $\varepsilon_w$, $10^6$ and small $\varepsilon_o$, $10^4$ and the last one with small values of $\varepsilon_w$ and $\varepsilon_o$, $10^4$. It shows that the displaced oil influences significantly on the constant level of water which displaces oil">Figure 4.  The results after 10 days with initial data are shown in Fig. 1 with interstitial velocity in viscous term. Four curves are compared: the one with large values of $\varepsilon_w$ and $\varepsilon_o$, $10^6$; the second one with large $\varepsilon_o$, $10^6$ and small $\varepsilon_w$, $10^4$; the third one with large $\varepsilon_w$, $10^6$ and small $\varepsilon_o$, $10^4$ and the last one with small values of $\varepsilon_w$ and $\varepsilon_o$, $10^4$. It shows that the displaced oil influences significantly on the constant level of water which displaces oil
]">Figure 5.  Initial water saturation profile from Coclite et al. [9]
] based on Darcy velocity in viscous term. Right: Numerical scheme (after 8 days) which uses interstitial velocity in viscous term with different viscous values $\varepsilon = 0, 10^3, 10^4, 10^5$ and $\varepsilon_w = \varepsilon_o = \varepsilon$">Figure 6.  Left: The results from Coclite et al. [9] based on Darcy velocity in viscous term. Right: Numerical scheme (after 8 days) which uses interstitial velocity in viscous term with different viscous values $\varepsilon = 0, 10^3, 10^4, 10^5$ and $\varepsilon_w = \varepsilon_o = \varepsilon$
Comparison between the compressible model and the incompressible model for water-oil flow with $\varepsilon_w = \varepsilon_o = \varepsilon = 10^7, 10^6$. After the same period of 10 days, water flow in the compressible model is delayed compared with water profiles in the incompressible model, for both situations with interstitial velocity and Darcy velocity in viscous terms
The water pressure evolution in the compressible model for the case with Darcy velocity in viscous term (left figure) and the case with interstitial velocity in viscous term (right figure). Water pressure increases with time in the water displacing part of the reservoir layer which leads to a compression effect where the magnitude of the viscous terms increase and thereby slows down the displacement of the water front
Left: Comparison of saturation profiles for water injection and gas injection, respectively, after the same time period (10 days) in the compressible model using interstitial velocity in viscous term ($\varepsilon_w = \varepsilon_o = \varepsilon = 10^7$). Right: The gas saturation profile shown at different times
Input parameters of reservoir and fluid properties used for for the below simulations. Note that $P_{wL}$ is the boundary pressure at left for the incompressible model whereas for the compressible model it represents the initial pressure distribution
 Parameter Dimensional Value Parameter Dimensional Value $\, L$ $100$ $\text{m}$ $\, I_w$ $1.5$ $\, \phi$ $1$ $\, I_o$ $1.5$ $\, A$ $1$ $\text{m}^2$ $\, I$ $0$ (Pa$\cdot$s)$^{-1}$ $\, \tilde{\rho}_{w0}$ $1$ $\text{g}/\text{cm}^3$ $\, \alpha$ $0$ $\, \tilde{\rho}_{o0}$ $1$ $\text{g}/\text{cm}^3$ $\, \beta$ $0$ $\, C_{w}$ $10^6$ $\text{m}^2/\text{s}^2$ $\, \varepsilon_w$ $10^7, 10^6, 10^5, 10^4, 10^3, 10^2$ $\text{cP}$ $\, C_{o}$ $5\cdot10^5$ $\text{m}^2/\text{s}^2$ $\, \varepsilon_o$ $10^7, 10^6, 10^5, 10^4, 10^3, 10^2$ $\text{cP}$ $\, \mu_w$ $1$ $\text{cP}$ $\, K$ $1000$ $\text{mD}$ $\, \mu_o$ $1$ $\text{cP}$ $\, k_{rw}^{max}=1/I_w$ $0.667$ $\, Q$ $8.004$ $\text{m}^3/\text{day}$ $\, k_{ro}^{max}=1/I_o$ $0.667$ $\, P_{wL}$ $10^6$ $\text{Pa}$ $\, T$ $10$ $\text{days}$ $\, N_x$ $2001$ $\, \triangle t$ $8640$ $\text{s}$
 Parameter Dimensional Value Parameter Dimensional Value $\, L$ $100$ $\text{m}$ $\, I_w$ $1.5$ $\, \phi$ $1$ $\, I_o$ $1.5$ $\, A$ $1$ $\text{m}^2$ $\, I$ $0$ (Pa$\cdot$s)$^{-1}$ $\, \tilde{\rho}_{w0}$ $1$ $\text{g}/\text{cm}^3$ $\, \alpha$ $0$ $\, \tilde{\rho}_{o0}$ $1$ $\text{g}/\text{cm}^3$ $\, \beta$ $0$ $\, C_{w}$ $10^6$ $\text{m}^2/\text{s}^2$ $\, \varepsilon_w$ $10^7, 10^6, 10^5, 10^4, 10^3, 10^2$ $\text{cP}$ $\, C_{o}$ $5\cdot10^5$ $\text{m}^2/\text{s}^2$ $\, \varepsilon_o$ $10^7, 10^6, 10^5, 10^4, 10^3, 10^2$ $\text{cP}$ $\, \mu_w$ $1$ $\text{cP}$ $\, K$ $1000$ $\text{mD}$ $\, \mu_o$ $1$ $\text{cP}$ $\, k_{rw}^{max}=1/I_w$ $0.667$ $\, Q$ $8.004$ $\text{m}^3/\text{day}$ $\, k_{ro}^{max}=1/I_o$ $0.667$ $\, P_{wL}$ $10^6$ $\text{Pa}$ $\, T$ $10$ $\text{days}$ $\, N_x$ $2001$ $\, \triangle t$ $8640$ $\text{s}$
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