# Buckley–Leverett Theory for Two-Phase Immiscible Fluids Flow Model with Explicit Phase-Coupling Terms

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Buckley–Leverett Theory

#### 3.1. Buckley–Leverett Equations

- Comparison: classical Darcy’s system vs coupled system with $k{r}_{o,w}=k{r}_{w,o}$ (see (19))
- Effect of non-symmetric coupling: coupled system with $k{r}_{o,w}=\u03f5k{r}_{w,o}$ (we refer to $\u03f5$ as tolerance). For more examples, see (see (20))

- (a)
- First set of data
**Parameters****Value****Unit**Length of formation, L 1000.0 [m] Cross-area of reservoir, A ${10}^{4}$ [m${}^{2}$] Absolute permeability, k 2.96 $\times {10}^{-13}$ [m${}^{2}$] Oil phase viscosity, ${\mu}_{o}$ 5 $\times {10}^{-3}$ [Pa.s] Water phase viscosity, ${\mu}_{w}$ ${10}^{-3}$ [Pa.s] Oil density, ${\rho}_{o}$ 0.8 $\times {10}^{3}$ [kg/m${}^{3}$] Water density, ${\rho}_{w}$ ${10}^{3}$ [kg/m${}^{3}$] Initial water injection rate, ${q}_{t}$ 400 [m${}^{3}$/s] Porosity, $\varphi $ 0.30 [-] Residual oil saturation, ${S}_{or}$ 0.25 [-] Connate water saturation, ${S}_{wc}$ 0.20 [-] Maximal relative permeability for oil, ${k}_{ro}^{max}$ 0.8 [-] Maximal relative permeability for water, ${k}_{rw}^{max}$ 0.5 [-] Power index of water relative permeability ${n}_{w}$ 2.00 [-] Power index of oil relative permeability, ${n}_{o}$ 2.00 [-] Power index of first coupling term $k{r}_{o,w}$, ${n}_{o,w}$ 2.00 [-] Power index of second coupling term $k{r}_{w,o}$, , ${n}_{w,o}$ 2.00 [-] - (b)
- Second set of data
**Parameters****Value****Unit**Length of formation, L 3280 [ft] Cross-area of reservoir, A $10,764\times {10}^{4}$ [ft${}^{2}$] Absolute permeability, k 300 [mD] Oil phase viscosity, ${\mu}_{o}$ 5 [cP] Water phase viscosity, ${\mu}_{w}$ 1 [cP] Oil density, ${\rho}_{o}$ 22.653 [kg/ft${}^{3}$] Water density, ${\rho}_{w}$ 28.316 [kg/ft${}^{3}$] Initial water injection rate, ${q}_{t}$ 2515.924 [bbl/s] Porosity, $\varphi $ 0.30 [-] Residual oil saturation, ${S}_{or}$ 0.25 [-] Connate water saturation, ${S}_{wc}$ 0.20 [-] Maximal relative permeability for oil, ${k}_{ro}^{max}$ 0.8 [-] Maximal relative permeability for water, ${k}_{rw}^{max}$ 0.5 [-] Power index of water relative permeability ${n}_{w}$ 2.00 [-] Power index of oil relative permeability, ${n}_{o}$ 2.00 [-] Power index of first coupling term $k{r}_{o,w}$, ${n}_{o,w}$ 2.00 [-] Power index of second coupling term $k{r}_{w,o}$, , ${n}_{w,o}$ 2.00 [-]

- The case of $k{r}_{o,w}=k{r}_{w,o}$, we choose$$\begin{array}{c}k{r}_{o,w}=k{r}_{w,o}={\left({S}_{w}-{S}_{wc}\right)}^{2}{\left(1-{S}_{w}-{S}_{or}\right)}^{2}\hfill \end{array}$$
- The case of $k{r}_{o,w}=\epsilon k{r}_{w,o}$ we choose:$$\begin{array}{c}k{r}_{o,w}={\left({S}_{w}-{S}_{wc}\right)}^{2}{\left(1-{S}_{w}-{S}_{or}\right)}^{2},\phantom{\rule{2.84544pt}{0ex}}k{r}_{w,o}={\displaystyle \frac{7}{5}}{\left({S}_{w}-{S}_{wc}\right)}^{2}{\left(1-{S}_{w}-{S}_{or}\right)}^{2}\phantom{\rule{5.69046pt}{0ex}}\hfill \\ k{r}_{o,w}={\left({S}_{w}-{S}_{wc}\right)}^{2}{\left(1-{S}_{w}-{S}_{or}\right)}^{2},\phantom{\rule{5.69046pt}{0ex}}k{r}_{w,o}=2{\left({S}_{w}-{S}_{wc}\right)}^{2}{\left(1-{S}_{w}-{S}_{or}\right)}^{2}\hfill \end{array}$$

#### 3.2. Numerical Results

Classical Darcy | Tolerance
$\mathbf{\epsilon}=\mathbf{0}.\mathbf{8}$ | Tolerance
$\mathbf{\epsilon}=\mathbf{1}.\mathbf{4}$ | Tolerance
$\mathbf{\epsilon}=\mathbf{2}$ |

${t}_{L}=2.7219\times {10}^{3}$ days | ${t}_{L}=2.5843\times {10}^{3}$ days | ${t}_{L}=2.5397\times {10}^{3}$ days | ${t}_{L}=2.5024\times {10}^{3}$ days |

Classical Darcy | Coupled Darcy with
${\mathit{kr}}_{\mathbf{w},\mathbf{o}}=\mathit{kr}\mathbf{o},\mathbf{w}$ |

${t}_{L}=2.7219\times {10}^{3}$ days | ${t}_{L}=2.7071\times {10}^{3}$ days |

## 4. Discussion

#### 4.1. On the Fractional Flow

#### 4.2. Solution of Coupled System by a Decoupling Approach

**Lemma**

**1.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Van Genuchten capillary pressure and its derivative with respect to water saturation versus water saturation.

**Figure 3.**Typical fractional flow curve with respect to water saturation (

**left**), and its derivative with respect to water saturation (

**right**) versus water saturation.

**Figure 4.**(

**Left**) Relative permeability $k{r}_{o}$ and $k{r}_{w}$ in terms of water saturation. (

**Right**) Coupling term permeability $k{r}_{w,o}$ versus water saturation.

**Figure 6.**Water Saturation profile ${S}_{w}$ in terms of the distance from the inlet (Time = 1500 days).

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**MDPI and ACS Style**

Guérillot, D.; Kadiri, M.; Trabelsi, S.
Buckley–Leverett Theory for Two-Phase Immiscible Fluids Flow Model with Explicit Phase-Coupling Terms. *Water* **2020**, *12*, 3041.
https://doi.org/10.3390/w12113041

**AMA Style**

Guérillot D, Kadiri M, Trabelsi S.
Buckley–Leverett Theory for Two-Phase Immiscible Fluids Flow Model with Explicit Phase-Coupling Terms. *Water*. 2020; 12(11):3041.
https://doi.org/10.3390/w12113041

**Chicago/Turabian Style**

Guérillot, Dominique, Mostafa Kadiri, and Saber Trabelsi.
2020. "Buckley–Leverett Theory for Two-Phase Immiscible Fluids Flow Model with Explicit Phase-Coupling Terms" *Water* 12, no. 11: 3041.
https://doi.org/10.3390/w12113041