Abstract
As the main way to reproduce flow response in subsurface reservoirs, the reservoir simulation could drastically assist in reducing the uncertainties in the geological characterization and in optimizing the field development strategies. However, the challenges in providing efficient and accurate solutions for complex field cases constrain further utilization of this technology. In this work, we develop a new reservoir simulation framework based on advanced spatial discretization and linearization schemes, the mimetic finite difference (MFD) and operator-based linearization (OBL), for fully implicit temporal discretization. The MFD has gained some popularity lately since it was developed to solve for unstructured grids and full tensor properties while mimicking the fundamental properties of the system (i.e. conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus). On the other hand, in the OBL the mass-based formulations are written in an operator form where the parametric space of the unknowns is treated in a piece-wise manner for the linearization process. Moreover, the values of these operators are usually precomputed into a nodal tabulation and with the implementation of multi-linear interpolation, the values of these operators and their derivatives during a simulation run can be determined in an efficient way for the Jacobian assembly at any time-step. This saves computational time during complex phase behavior computations. By first coupling these two schemes within a parallel framework, we can solve large and complex reservoir simulation problems in an efficient manner. Finally, we present a benchmark case that compares the numerical solutions to a Buckley-Leverett analytical solution to assure their accuracy and convergence. Moreover, we test three challenging field cases to demonstrate the performance of the advanced parallel framework for complex reservoir simulation.
Similar content being viewed by others
References
Abushaikha, A.S., Voskov, D.V., Tchelepi, H.A.: Fully implicit mixed-hybrid finite-element discretization forgeneral purpose subsurface reservoir simulation. J. Comput. Phys. 346, 514–538 (2017)
Hjeij, D., Abushaikha, A. S.: Comparing advanced discretization methods for complex hydrocarbon reservoirs. In: SPE Reservoir characterisation and simulation conference and exhibitions. Society of Petroleum Engineers (2019)
Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal, curvilinear grids for multi-phase flow. In: Proceedings of the fourth European conference on the mathematics of oil recovery, Røros, Norway (1994)
Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127, 2–14 (1996)
Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: A class of discretization methods for structured and unstructured grids in anisotropic, inhomogeneous media. In: Proceedings of 5th European conference on the mathematics of oil recovery, Leoben, Austria (1996)
Aavatsmark, I., Barkve, T., Mannseth, T.: Control-volume discretization methods for 3D quadrilateral grids in inhomogeneous, anisotropic reservoirs. SPE J. 3, 146–154 (1998)
Edwards, M. G., Rogers, C. F.: A flux continuous scheme for the full tensor pressure equation. In: Proceedings of the fourth European conference on the mathematics of oil recovery, Røros (1994)
Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259–290 (1998)
Li, L., Khait, M., Voskov, D. V., Abushaikha, A. S.: Parallel framework for complex reservoir simulation with advanced discretization and linearization schemes. In: the SPE Europec featured at 82nd EAGE Conference and Exhibition, Amsterdam, Netherlands. SPE-200615-MS (2020)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
Chavent, G., Roberts, J.: A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in water flow problems. Adv. Water Resour. 14, 329–348 (1991)
Younes, A., Fontaine, V.: Efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media. Int. J. Numer. Methods Eng. 76(3), 314–336 (2008)
Tikhonov, A.N., Samarskii, A.A.: Homogeneous difference schemes. Comput. Math. Math. Phys. 1(1), 5–67 (1962)
Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163–1227 (2014)
Alpak, F.O.: A mimetic finite volume discretization method for reservoir simulation. SPE J. 15(2), 436–453 (2010)
Abushaikha, A.S., Terekhov, K.M.: A fully implicit mimetic finite difference scheme for general purpose subsurfacereservoir simulation with full tensor permeability. J. Comput. Phys. 406, 109194 (2020)
Zhang, N., Abushaikha, A.: An implementation of mimetic finite difference method for fractured reservoirs using a fully implicit approach and discrete fracture models. J. Comput. Phys. 446, 110665 (2021)
Zhang, N., Abushaikha, A. S.: Fully implicit reservoir simulation using mimetic finite difference method in fractured carbonate reservoirs. In: SPE reservoir characterisation and simulation conference and exhibition, 17–19 September, Abu Dhabi, UAE. SPE-196711-MS (2019)
Li, L., Voskov, D.V., Yao, J., et al.: Multiphase transient analysis for monitoring of CO2 flooding. J. Pet. Sci. Eng. 160, 537–554 (2018)
Liu, L., Yao, J., Sun, H., et al.: Compositional modeling of shale condensate gas flow with multiple transportmechanisms. J. Pet. Sci. Eng. 172, 1186–1201 (2018)
Sun, H., Yao, J., Gao, S., Fan, D.Y., Wang, C.C., Sun, Z.X.: Numerical study of CO2 enhanced natural gas recovery and sequestration in shale gas reservoirs. Int. J. Greenh. Gas Control. 19, 406–419 (2013)
Yao, J., Sun, H., Fan, D., Wang, C.C., Sun, Z.X.: Numerical simulation of gas transport mechanisms in tight shale gas reservoirs. Pet. Sci. 10(4), 528–537 (2013)
Liu, P., Li, J., Sun, S., Yao, J., Zhang, K.: Numerical investigation of carbonate acidizing with gelled acid using a coupled thermal-hydrologic-chemical model. Int. J. Therm. Sci. 160, 106700 (2021)
Namdar Zanganeh, M.: Simulation and optimization of foam EOR processes. Master dissertation, TU Delft, Netherlands (July 2011)
Voskov, D.V.: Operator-based linearization approach for modeling of multiphase multi-component flow in porous media. J. Comput. Phys. 337, 275–288 (2017)
Khait, M., Voskov, D.V.: Operator-based linearization for general purpose reservoir simulation. J. Pet. Sci. Eng. 157, 990–998 (2017)
Khait, M., Voskov, D.V.: Operator-based linearization for efficient modeling of geothermal processes. Geothermics. 74, 7–18 (2018)
Khait, M., Voskov, D. V.: Adaptive parameterization for solving of thermal/ compositional nonlinear flow and transport with buoyancy. SPE J. 23 (02) (2018)
Wang, Y., Voskov, D.V., Khait, M., Bruhn, D.: An efficient numerical simulator for geothermal simulation: a benchmark study. Appl. Energy. 264, 114693 (2020)
Al-Jundi, A., Li, L., Abushaikhaa, A. S.: Investigation of the accuracy and efficiency of the operator-based linearization through an advanced reservoir simulation framework. In: 17th European conference on the mathematics of oil recovery, Edinburgh, United Kingdom (2020)
Lyu, X., Voskov, D. V., Rossen, W.: Simulation of foam-assisted CO2 storage in saline aquifers. In: 17th European conference on the mathematics of oil recovery, Edinburgh, United Kingdom (2020)
Li, L., Abushaikha, A. S.: Development of an advancing parallel framework for reservoir simulation. In: Third EAGE WIPIC workshop: Reservoir Management in Carbonates, Doha, Qatar (2019)
Li, L., Abushaikha, A.: An advanced parallel framework for reservoir simulation with mimetic finite difference discretization and operator-based linearization. In ECMOR XVII-17th European conference on the mathematics of oil recovery (2020)
Li, B., Chen, Z., Huan, G.: Comparison of solution schemes for black oil reservoir simulations with unstructured grids. Comput. Methods Appl. Mech. Eng. 193, 319–355 (2004)
Killough, J. E.: Ninth SPE comparative solution project: A Reexamination of Black-Oil Simulation. In: SPE Reservoir simulation symposium, San Antonio, Texas. SPE-29110-MS (1995)
Jansen, J.D., Fonseca, R.M., Kahrobaei, S., Siraj, M.M., van Essen, G.M., van den Hof, P.M.J.: The egg model: a geological ensemble for reservoir simulation. Geosci. Data J. 1, 192–195 (2014)
Nardean, S., Ferronato, M., Abushaikha, A.: A novel block non-symmetric preconditioner for mixed-hybrid finite-element-based Darcy flow simulations. J. Comput. Phys. 442, 110513 (2021)
Acknowledgments
This publication was supported by the National Priorities Research Program grant NPRP11S-1210-170079 from Qatar National Research Fund.
Nomenclature
B Formation volume factor
cr Rock compressibility factor
K Permeability tensor
krα Relative permeability of phase α
Nf Number of interfaces of element E
ni Outward normal on interface i
p Pressure
pref Reference pressure for the porosity ϕ0
Qα,i Flux of phase α on interface i
q Component rate per unit volume
R Gas solubility
S Saturation
t Time
u Velocity
zc Mass fraction of component c
ρ Phase density
ϕ Reservoir porosity
μ Viscosity
g Gas phase
o Oil phase
w Water phase
st Standard condition
n Previous time step
n + 1 Current time step
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, L., Abushaikha, A. A fully-implicit parallel framework for complex reservoir simulation with mimetic finite difference discretization and operator-based linearization. Comput Geosci 26, 915–931 (2022). https://doi.org/10.1007/s10596-021-10096-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-021-10096-5