Development of a fast EoS based compositional model for three-phase core flooding
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Abstract
Compositional simulators are of great importance to model core flow experiments and enhanced oil recovery processes. In the present study, a semi-implicit, three dimensional EoS-based formulation is developed for modeling and simulation of core-flooding process. The model treats three-phase flow in cylindrical coordinates with no need to use an iterative numerical method. Overall composition, pressure and water saturation are calculated implicitly for each grid block. Using this approach, a linear set of equations with constant coefficients would be solved at each time step. It is assumed that there is no mass transfer between hydrocarbon phases and water phase. In addition, a new technique has been suggested to calculate transmissibility coefficients in each direction, which is a major problem in compositional modeling. Each transmissibility coefficient is divided into two general parts which are treated differently. The first part is constant and the second part is time-dependent. This model yields a robust and fast calculation procedure, which is validated using Eclipse simulation results. Simulation of several miscible flooding processes were performed using Eclipse core-flood models in order to compare the results with the developed in-house model. Results are matched reasonably while the computing time is considerably decreased using the in-house model. It is shown that using second Stone’s relative permeability model and Advanced Peng–Robinson Equation of State, best results are achieved with an average absolute relative deviation of 7.7% from Eclipse results for ultimate recovery factor. In addition, sensitivity analysis has been performed on several input data including time-step size and block number. Results show that in a reasonable range of these parameters, convergence problems are not encountered. We have developed an in-house simulator based on our model, which can be a basis for other compositional simulation purposes. In addition to decreasing computing time, convergence problems encountered in most implicit techniques would be avoided as well.
Keywords
Compositional model Coreflooding Gas injection PC-SAFT EoS Semi-implicit Transmissibility coefficientList of symbols
- A_{r}
Cross-sectional area normal to r direction
- A_{z}
Cross-sectional area normal to z direction
- A_{θ}
Cross-sectional area normal to θ direction
- H
Height
- k_{rg}
Relative permeability to gas phase
- k_{ro}
Relative permeability to oil phase
- k_{rog}
Relative permeability to oil phase in two-phase oil–gas system
- k_{row}
Relative permeability to oil phase in two-phase oil–water system
- k_{rw}
Relative permeability to water phase
- K_{r}
Permeability in the direction of the r-axis
- K_{z}
Permeability in the direction of the z-axis
- K_{θ}
Permeability in the direction of the θ-axis
- n
Old time level
- n_{c}
Number of components
- N_{b}
Number of gridblocks
- P
Pressure
- P_{cgo}
Gas–oil capillary pressure
- P_{cwo}
Water–oil capillary pressure
- S_{g}
Saturation of gas phase
- S_{o}
Saturation of oil phase
- S_{w}
Saturation of water phase
- t
Time
- T
Transmissibility
- V
Bulk volume, control volume or gridblock bulk volume
- x_{i}
Mole fraction of component i in oil phase
- y_{i}
Mole fraction of component i in gas phase
- z_{i}
Overall composition of component i
- Z
Elevation
- γ_{g}
Gravity of the gas phase
- γ_{o}
Gravity of the oil phase
- γ_{w}
Gravity of the water phase
- \(\bar{\delta }\)
Difference operator in the time domain
- Δ
Central difference operator in the spatial domain
- Δt
Time step
- θ
Angle in the θ direction in cylindrical coordinate system
- μ_{g}
Gas-phase viscosity
- μ_{o}
Oil-phase viscosity
- μ_{w}
Water-phase viscosity
- ρ_{g}
Gas-phase density
- ρ_{o}
Oil-phase density
- ρ_{w}
Water-phase density
- ϕ
Porosity
1 Introduction
“Black-oil” modeling allows an assumption that reservoir gas and oil have different but fixed composition, with the solubility of gas in oil being dependent on pressure alone. In “compositional” models, oil and gas equilibrium compositions vary considerably with spatial position and time [1].
Van Quy et al. [2] developed a compositional model describing a one-dimensional, two-phase (gas/oil) system, neglecting capillary and gravity forces. A three-component correlation was used which guaranteed consistency of phase compositions and properties at the critical point. Corteville et al. [3] presented additional comparisons between experimental and linear calculated results using the same model.
Metcalfe et al. [4] and Fussel et al. [5] published researches in order to simulate the multi-contact miscibility process using a cell-to-cell flash calculation model. Iterative methods were described by Fussell and Yanosik [6] for phase equilibria calculations using Redlich–Kwong equation of state. Fussel and Fussel [7] developed a formulation for a multidimensional compositional model. They used their model for an immiscible gas injection case, as an example.
Coats [1] described an equation-of-state implicit compositional model formulation for three-dimensional, three-phase flow under viscous, gravity and capillary forces. In his formulation, 2N_{c} + 4 equations should be solved simultaneously, using an iterative procedure which makes the calculations time-consuming.
Although fully-implicit approach is an accurate method, it has some drawbacks including numerical dispersion and lower computational speed. IMPES model was presented in 1981 by Nghiem et al. [8] which solves a set of equations using an iterative-sequential method. Implicit formulation for pressure equation which results from combination of mass balance equation for water phase and molar balance for hydrocarbon system, would be solved using a direct method or Newton-like iterative procedure. After calculating pressures in each iteration step, water saturation and overall molar hydrocarbon composition would be calculated explicitly. Flash calculations are performed to find molar composition, density and mole fraction of each phase. Then saturation of each hydrocarbon phase could be calculated. The main drawback of this formulation is limitation on time step length which is related to explicit solving for composition and transmissibility parameter. Young and Stephenson [9] suggested a more efficient procedure on the basis of Newton–Raphson method to solve IMPES equations.
Branco and Rodrigues [10] developed a semi-implicit model for compositional simulation where mole fractions of components in mass balance equations are calculated explicitly one step earlier than each iteration. Equations would be divided into two sets. One of them including three equations with three unknowns like black-oil equations and the other one including N_{c}-2 equations for mole fraction of components in one phase. The level of implicitness of this method is something between IMPES and fully implicit methods.
In 2011, Wei et al. applied APR EoS in Coats method and presented a new algorithm for multiphase flash calculations. They used Michelsen stability analysis in their model [11].
In 2013, Zaydullin et al. presented a new framework for the general compositional problem associated with multicomponent multiphase flow in porous media. Adaptive construction and interpolation using supporting tie lines were used in their model to obtain the phase state and the phase compositions. The computation of the phase behavior in the course of simulations becomes an iteration-free, table look-up procedure [12].
In 2015, Fleming developed a method to locally lump the fluid for phase-behavior calculations. In his method, the regions that are able to maintain sufficient compositional accuracy with fewer components can use less-expensive EoS calculations. In addition, different lumpings can be used at different times in the life of the reservoir, as compositional effects become more or less important in different regions in the reservoir [13].
In 2018, Khorsandi et al. developed a fully compositional simulation model using an equation of state (EoS) for relative permeabilities to eliminate the unphysical discontinuities in flux functions caused by phase labeling. The hysteresis effects were also considered in their three-phase relative permeability model. They used their mode for simulation of multi-cycle WAG injection [14].
The purpose of this research is to develop a three-phase, three-dimensional, compositional model in cylindrical coordinate system to simulate core-flooding process. An In-house simulator has been developed based on the proposed model, which is used to obtain results, validate and analyze the model.
2 Description of the model
Three-phase including oil, gas and water
Three-dimensional, cylindrical, in vertical or horizontal directions
Compositional
Describing phase behavior using equations of state
The proposed model is based on finite difference method of discretization, originally suggested by Coats [1], associated with minor modifications. Since flash calculations using an equation of state is performed at each time step for each grid block, the computation is naturally time-consuming. The main focus of this research is on the method of discretization of the flow equations, and on proper selection of the parameters that are going to be solved implicitly/explicitly. In most previous researches, the discretization method of the flow equations results in a set of equations with multiple unknown parameters. These equations are then combined into a single matrix equation. If the final multiplier matrix of the unknown parameters contains unknowns itself, the equation is to be solved using iterative numerical methods. However, it has been tried in the proposed method to reach a multiplier matrix of constant values, at each time step. It should be noted that the multiplier matrix of constant values changes at each time step and is not constant throughout the whole simulation.
Capillary pressure is neglected
No mass transfer occurs between water and hydrocarbon phases
Variables that are calculated implicitly are: p, S_{w}, z_{1}, z_{2}, …, \(z_{{n_{c} }}\).
- Transmissibility coefficients are assumed to be the product of two parameters, T and W, multiplied by each other. T does not change as time proceeds, while W is dependent on time. They are defined as the following:$$T_{{r,i \pm \frac{1}{2},j,k}} = \beta_{c} K_{r} \frac{{A_{r} }}{\Delta r}|_{{i \pm \frac{1}{2}}}$$(7)$$T_{{\theta ,i,j \pm \frac{1}{2},k}} = \beta_{c} \frac{{K_{\theta } A_{\theta } }}{{r_{ijk} \Delta \theta }}|_{{j \pm \frac{1}{2}}}$$(8)$$T_{{z,i,j,k \pm \frac{1}{2}}} = \beta_{c} \frac{{K_{z} A_{z} }}{\Delta z}|_{{k \pm \frac{1}{2}}}$$(9)$$W_{o,a, i,j,k} = \left( {\rho_{o} x_{a} \frac{{k_{ro} }}{{\mu_{o} }}} \right)_{i,j,k}$$(10)$$W_{g,a, i,j,k} = \left( {\rho_{g} y_{a} \frac{{k_{rg} }}{{\mu_{g} }}} \right)_{i,j,k}$$(11)$$W_{w,i,j,k} = \left( {\rho_{w} \frac{{k_{rw} }}{{\mu_{w} }}} \right)_{i,j,k}$$(12)
The subscripts i, j, and k refer to radial, angular and height coordinates, respectively and n refers to time step number. Subscript a denotes component a in the terms W_{g,a,i,j,k} and W_{o,a,i,j,k}.
After determining unknowns in each gridblock, flash calculation is performed using an equation of state. We have applied APR and PC-SAFT equations of state in the proposed model. Three viscosity models have been applied to the developed simulator; LBC (Lohrenz-Bray-Clark) [15], modified LBC [16] and EF Correlation (Expanded Fluid) by Satyro and Yarranton [17] considering a modification in mixing rules, suggested by Motahhari [18].
Four relative permeability models have been applied; the first and the second Stone’s models [19, 20], Naar, Henderson and Wygal [21, 22] and Corey’s relative permeability models [23].
Eight user-defined multipliers have been introduced to find the optimum way of treating inter-blocks transmissibility parameters; g_{1}, g_{2}, g_{3}, g_{4}, l_{1}, l_{2}, l_{3}, l_{4}. Four adjacent blocks are selected in the flow direction such that, g_{i} and l_{i} would be multiplied by W_{g} and W_{o} of the respective blocks, assuming fluid flow between block numbers 2 and 3. These multipliers have been optimized and reported in results section.
The in-house simulator has been developed in Visual C# with a user-friendly interface. It is specially developed for core-flooding process, though it can be generalized to other processes occurring in reservoir scale.
3 Model validation
Hypothetical core–fluid composition
Component | Composition (mol%) |
---|---|
C_{1} | 10 |
C_{5} | 10 |
C_{10} | 10 |
C_{15} | 10 |
C_{33} | 60 |
Input data of the validation model
Property | Quantity | Unit |
---|---|---|
Core length | 34.56 | cm |
Core diameter | 2.376 | cm |
Porosity | 33.57 | % |
Absolute permeability | 45.23 | md |
Water saturation | 29.44 | % |
Irreducible water saturation | 29.44 | % |
Injection rate | 0.10 | cc/min |
Models used in validation
Viscosity model | LBC (Lohrenz–Bray–Clark) |
EoS | APR (Advanced Peng–Robinson) |
Relative permeability model | Stone 2 |
The second Stone’s relative permeability model needs four sets of data including k_{rw}, k_{rg}, k_{rog}, and k_{row}. Three scenarios have been assumed as input data sets of the second Stone’s relative permeability model. In these scenarios, k_{rw} has been assumed to be constant, since the main goal of this research is to model gas injection, which is not considerably affected by water phase movement.
Gridblocks numbers in different axis of the horizontal and vertical model
Simulator | Flow direction | Coordinate direction | Blocks number |
---|---|---|---|
Eclipse | Vertical (cylindrical) | r | 5 |
θ | 5 | ||
z | 10 | ||
Horizontal (Cartesian, eight-points) | x | 5 | |
y | 5 | ||
z | 20 | ||
In-house simulator | Vertical (cylindrical) | r | 4 |
θ | 6 | ||
z | 10 | ||
Horizontal (cylindrical) | r | 4 | |
θ | 6 | ||
z | 20 |
The simulation time for horizontal cases is 900 s and for vertical cases is 2000s.
3.1 Scenario #1
3.2 Scenario #2
3.3 Scenario #3
4 Results
5 Semi-miscible displacement scenario
During semi-miscible displacement scenario, both miscible and immiscible displacements happen. Therefore, this case could be considered as a general case. We have designed the validation problem in horizontal direction in such a way that it falls into a semi-miscible region to observe both miscible and immiscible cases at the same time.
Gridding parameters for semi-miscible displacement scenario
Horizontal (cylindrical) | r | 4 |
θ | 6 | |
z | 6 |
According to Fig. 15, it can be seen that upon the initiation of the gas injection process, the fluid pressure begins to increase. After a while, the increase in methane fraction in overall composition of the fluid results in formation of gas in this grid block. After about 70 min, gas saturation reaches its maximum (1-S_{w}).
Meanwhile, Fig. 16 shows that after formation of the gas phase in the lowest block, gravity segregation phenomenon moves the gas to the higher blocks. Consequently, gas saturation in the lowest block decreases to zero again.
6 Sensitivity analysis
In this section, the effect of different models and a few parameters are investigated on the results obtained from the developed in-house simulator.
6.1 Relative permeability models effect
6.2 Viscosity models effect
6.3 Grid blocks number effect
6.4 Time step effect
6.5 Core diameter effect
6.6 Core length effect
6.7 Pressure effect
6.8 EoS effect
In addition to APR EoS, PC-SAFT EoS was also implemented to the developed in-house simulator. In the developed simulator, APR EoS was chosen as the primary EoS, because it is faster and comparable to Eclipse results.
Composition of core fluid used to investigate EoS effect
Component | Composition (mol%) |
---|---|
Methane | 10 |
n-Decane | 10 |
n-Tetradecane | 10 |
n-Hexadecane | 10 |
n-Eicosane | 60 |
7 Conclusion
Optimized tuning parameters for vertical models
g_{1} | g_{2} | g_{3} | g_{4} |
0 | 0.005 | 0 | 0.995 |
l_{1} | l_{2} | l_{3} | l_{4} |
0 | 0.005 | 0 | 0.995 |
Optimized tuning parameters for horizontal models
g_{1} | g_{2} | g_{3} | g_{4} |
0 | 0.5 | 0.5 | 0 |
l_{1} | l_{2} | l_{3} | l_{4} |
0 | 0.5 | 0.5 | 0 |
Average absolute deviation for calculation of ultimate recovery was 7.68% and 0% for the horizontal and vertical scenarios, respectively. The reason for zero AAD of the vertical case is that the assumed core models are homogenous in absolute permeability and porosity. As a result, theoretically the injected gas would be distributed in the core homogenously and all the initial oil in place would be recovered. It must be noted that in reality, inhomogeneity in porosity and permeability can result in fingering phenomena, which decreases ultimate oil recovery.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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