# A Novel Technique to Determine Concentration-Dependent Solvent Dispersion in Vapex

^{*}

## Abstract

**:**

## 1. Introduction

- The amount and flow rate of solvent required to mobilize the heavy oil.
- The extent of heavy oil and bitumen reserves that would undergo viscosity reduction.
- The time required to mobilize the heavy oil and bitumen for drainage under gravity.
- The production rate of live oil.

## 2. Experimental Setup

^{3}collection tube. The tube is connected to a viscosity measurement unit to measure the online live oil viscosity.

^{3}capacity. The flash tank is wrapped with an electrical heating tape. The volume of propane separated from the live oil inside the flash tank is measured by a gas measurement unit. It is composed of two cylinders of respective capacities 2,600 cm

^{3}and 2,900 cm

^{3}. A needle thermocouple and two resistance temperature detectors respectively measure the temperature of the packed medium, propane gas, and the flash separation tank. A data acquisition system records the system properties online.

#### 2.1. Experimental Procedure

^{3}density at 22 °C) was heated to 60 °C. Glass beads of known permeability were gradually added to the heated heavy oil ensuring proper mixing without trapping air bubbles. The saturated mixture of the heavy oil and glass beads was packed into a cylindrical wire mesh of 25 cm height, and 6 cm diameter. During this operation, the mesh lay inside an ice bath to prevent the bitumen from oozing out. The cylindrical packed medium, i.e., the physical model of Vapex, of heavy oil saturated with glass beads was weighed, and left at room temperature for one day to reach thermal equilibrium prior to an experiment.

^{3}of live oil was collected, the oil was drained through a capillary tube into the flash tank. The propane liberated from the live oil in the flash tank was directed to the gas-measurement unit filled initially with water. The displaced volume of water determined the propane volume. The propane-free oil residual in the flash tank was weighed. The amount of live oil produced with time was recorded.

Parameter | Value |
---|---|

permeability (K), Darcy | 204 |

porosity (φ) | 0.38 |

temperature,°C | 21 |

pressure, MPa | 0.689 |

## 3. Theoretical Development

#### 3.1. Mass Transfer Model

- Vapex is carried out at constant temperature and pressure.
- Solvent dispersion is along the radial direction only.
- The velocity of the live oil along the vertical direction is governed by Darcy law in a porous medium.
- The porous medium has uniform porosity and permeability.
- There are no chemical reactions.
- Any volume change results and corresponds to drainage of the live oil.
- The heavy oil is non-volatile.

_{f}in the radial direction.

_{t}is relative permeability of the medium, K is its permeability, ρ is the density of live oil, g is gravity, and μ is the live oil viscosity.

_{0}ω

^{−2}with a high value of 0.982 for the r

^{2}-coefficient of determination.

_{0}. The packing surface has the solvent gas concentration equal to its interface saturation concentration under prevailing temperature and pressure. Thus, the initial conditions at t = 0 are as follows:

_{int}

#### 3.2. The Mathematical Objective

_{e}(t) is the experimental cumulative mass of the live oil produced at any time t, and m

_{c}(t) is the model cumulative predicted mass of the live oil produced at any time t. The calculated mass m

_{c}(t) is given by

_{2}= −v(t,r,0) = −α[ω(t,r,0)]

^{2}

#### 3.3. Determination of Necessary Conditions

#### 3.4. Adjoint Equations

## 4. Computational Algorithm

- Initialize dispersion function.
- Simultaneously integrate the continuity equation [Equation (12)] and Equation (14), subject to the initial and boundary conditions, to obtain the values of ω(t,r,z) and Z(t,r) for each node.
- Calculate the objective function given by Equation (9).
- Simultaneously integrate Equation (43) and Equation (45) backward, subject to the final boundary conditions, using stored values of ω and Z to get the values of λ(t,r,z) and γ(t,r) for each grid point.
- Improve D(ω) using the gradient correction given by Equation (41).
- Go to Step 2 until the improvement in J is negligible.

#### 4.1. Implementation

_{D}was scaled to the magnitude of current dispersion values as follows:

^{2}is a small adjustable parameter, J

_{DS}(k) is the scaled gradient correction at a specified gas mass fraction of the dispersion function, and n is the number of specified gas mass fractions of the dispersion function. Using the scaled gradient correction, the iterative improvement in the value of D(ω) was given by

_{i+1}= D

_{i}− β

_{i}J

_{DS,i}

_{i}is the optimal step length along the search direction in the ith iteration. The dispersion was considered to be discrete function, D(ω), at specified gas mass fractions between zero and the maximum concentration of the solvent in bitumen. D(ω) was initialized to a uniform value as high as possible without causing m

_{c}(t) to intersect m

_{e}(t). Equations (12), (14), (43), and (45) were finite-differenced along r and z directions. The resulting set of ordinary differential equations written for corresponding grid points are given in Appendix A. The details of the variational derivative of J with respect to D are provided in Appendix B.

^{−6}in the time domain, the equations were numerically integrated using semi-implicit Bader-Deuflhard algorithm, and adaptive step size control [13]. Analytical Jacobian of Equations (12), (14), (43), and (45) was employed in the calculations. To fix the number of grid points along the r and z directions, N

_{t}and N

_{z}, the equations were integrated with increasing the number of grid points until the changes in the solution became negligible. The gradient correction in D(ω) was applied to the dispersion using Broyden-Fletcher-Goldfarb-Shanno algorithm [14,15].

_{c}(t) and m

_{e}(t); ω(t) and Z(t); and the variational derivative J

_{D}at each specified solvent mass fraction.

## 5. Results and Discussion

Parameter | Value |
---|---|

φ | 0.38 |

K_{t} | 1 |

K, m^{2} | 2.013 × 10^{−10} |

R, m | 0.03 |

Z_{0}, m | 0.25 |

μ_{0}, kg/m·s | 1.158 × 10^{−3} |

ρ, kg/m³ | 830 |

_{int}in the range of 0.70–0.9 and the initial uniform dispersion function D(ω) in the range of 10

^{−7}− 2.5 × 10

^{−5}m

^{2}/s. In order to obtain the optimal value for propane mass fraction at the interface to solve equations, the minimum resultant objective functions were plotted against ω

_{int}as shown in Figure 3. The optimal propane interface mass fraction ω

_{int}was found to be 0.76. The above value of ω

_{int}was used to determine the dispersion of propane as a function of its concentration in heavy oil. The results are presented in Figure 4, Figure 5 and Figure 6.

_{int}= 0.76, the application of the algorithm resulted in an iterative reduction of the objective function accompanied by a corresponding improvement in D(ω). The objective function decreased monotonically to the minimum as shown in Figure 4. The final optimal function D(ω) was obtained in 29 iterations after which no further improvement was observed. The initial and final D(ω) are plotted Figure 5. It shows that the final, optimally determined D(ω) rises to a maximum value, and then drops toward the end. The maximum value of propane dispersion is 4.048 × 10

^{−5}m

^{2}/s at the propane mass fraction of 0.336. This result is for the propane–heavy oil system at 21–22 °C and 0.689 MPa. Figure 6 compares the experimental live oil production to the calculated one with the optimally determined propane dispersion. It is observed that the experimental and calculated live oil productions agree very well. The calculated production follows experimental production very closely during the operation time of about 60 minutes.

**Figure 5.**Dispersion coefficient of propane in heavy oil at 21 °C and 0.689 MPa (diamond: the final, optimal dispersion coefficient, square: initial guess).

^{−5}m

^{2}/s; about twice the average value of dispersion). Hence, in addition to enabling more accurate reservoir simulations, the concentration-dependent dispersion function provide insights into optimizing Vapex operations as well.

**Figure 6.**Experimental and calculated mass of live oil produced with time (diamond: experimental mass, line: calculated mass).

## 6. Conclusions

## Acknowledgments

## Nomenclature

A | area, m ^{2} |

D | dispersion coefficient of solvent in medium, m²/s |

g | gravity, m/s² |

I | objective functional defined by Equation (9) |

J | augmented objective functional defined by Equation (11) |

J_{f} | dispersive flux of solvent in the medium along the radial direction, kg/m ^{2}·s |

K | permeability of the medium, m ^{2} |

K_{r} | relative permeability of the medium |

m_{c} | calculated mass of the produced live oil, kg |

m_{e} | experimental mass of the produced live oil, kg |

r | distance along the radial direction, m |

R | radius of cylindrical medium, m |

S | surface area, m ^{2} |

t | time, s |

V | volume of a finite element in the medium, m ^{3} |

v | Darcy velocity, m/s |

z | distance along the vertical direction, m |

Z | bitumen height in the medium at a given r and t, m |

Z_{0} | initial height, cm |

## Greek Symbols

φ | porosity of the medium |

γ | adjoint variable defined by Equation (43) |

λ | adjoint variable defined by Equation (45) |

μ | viscosity of the live oil, kg/m.s |

μ_{0} | viscosity coefficient of the live oil, kg/m·s |

ρ | density of the live oil, kg/m³ |

ω | mass fraction of solvent in bitumen |

ω_{int} | mass fraction of solvent at the solvent–heavy oil interface |

## Appendix A

#### A.1. The Mathematical Model

_{r}− 1) and j < (N

_{z}− 1)

**a.**- for i = 0 and 0 < j < (N
_{z}− 1) **b.**- for i = 0 and j = 0
**c.**- for i = 0 and j = (N
_{z}− 1)

**a.**- for i = (N
_{r}− 1) and j = 0 **b.**- for i = (N
_{r}− 1) and 0 < j < (N_{z}− 1) **c.**- for i = (N
_{r}− 1) and j = (N_{z}− 1)

_{r}− 1) and j = 0

_{r}− 1) and j = (N

_{z}− 1)

_{r}− 1) is given by

#### A.2. The Adjoint Equations

_{r}− 1) and j < (N

_{z}− 1)

_{z}− 1)

**a.**- for i = (N
_{r}− 1) and j = 0 **b.**- for i = (N
_{r}− 1) and 0 < j < (N_{z}− 1) **c.**- for i = (N
_{r}− 1) and j = (N_{z}− 1)

_{r}− 1) and j = 0

_{r}− 1) and j = (N

_{z}− 1)

## Appendix B

_{r}− 1) and j < (N

_{z}− 1)

_{z}− 1)

**a.**- for i = (N
_{r}− 1) and j = 0 **b.**- for i = (N
_{r}− 1) and 0 < j < (N_{z}− 1) **c.**- for i = (N
_{r}− 1) and j = (N_{z}− 1)

_{r}− 1) and j = 0

_{r}− 1) and j = (N

_{z}− 1)

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**MDPI and ACS Style**

Abukhalifeh, H.; Lohi, A.; Upreti, S.R.
A Novel Technique to Determine Concentration-Dependent Solvent Dispersion in Vapex. *Energies* **2009**, *2*, 851-872.
https://doi.org/10.3390/en20400851

**AMA Style**

Abukhalifeh H, Lohi A, Upreti SR.
A Novel Technique to Determine Concentration-Dependent Solvent Dispersion in Vapex. *Energies*. 2009; 2(4):851-872.
https://doi.org/10.3390/en20400851

**Chicago/Turabian Style**

Abukhalifeh, Hadil, Ali Lohi, and Simant Ranjan Upreti.
2009. "A Novel Technique to Determine Concentration-Dependent Solvent Dispersion in Vapex" *Energies* 2, no. 4: 851-872.
https://doi.org/10.3390/en20400851