Efficient integration of stiff kinetics with phase change detection for reactive reservoir processes

We propose the use of implicit one-step Explicit Singly Diagonal Implicit Runge–Kutta (ESDIRK) methods for integration of the stiff kinetics in reactive, compositional and thermal processes that are solved using operator-splitting type approaches. To facilitate the algorithmic development we construct a virtual kinetic cell model. The model serves both as a tool for the development and testing of tailored solvers as well as a testbed for studying the interactions between chemical kinetics and phase behavior. As case study, two chemical kinetics models with 6 and 14 components, respectively, are implemented for in situ combustion, a thermal oil recovery process. Through benchmark studies using the 14 component reaction model the new ESDIRK solvers are shown to improve computational speed when compared to the widely used multi-step BDF methods DASSL and LSODE. Phase changes are known to cause convergence problems for the integration method. We propose an algorithm for detection and location of phase changes based on discrete event system theory. Experiments show that the algorithm improves the robustness of the integration process near phase boundaries by lowering the number convergence and error test failures by more than 50% compared to direct integration without the new algorithm.

[1]  John C. Friedly,et al.  Solute transport with multiple equilibrium‐controlled or kinetically controlled chemical reactions , 1992 .

[2]  M. R. Islam,et al.  State-of-the-Art of In-Situ Combustion Modeling and Operations , 1989 .

[3]  T. H. Gillham,et al.  Keys to Increasing Production Via Air Injection in Gulf Coast Light Oil Reservoirs , 1997 .

[4]  Paul I. Barton,et al.  State event location in differential-algebraic models , 1996, TOMC.

[5]  Ian T. Cameron,et al.  A four-stage index 2 diagonally implicit Runge-Kutta method , 2002 .

[6]  N. Mahinpey,et al.  A Simple Kinetic Model for Coke Combustion During an In-Situ Combustion (ISC) Process , 2005 .

[7]  C. Runge.  Ueber die numerische Auflösung von Differentialgleichungen , 1895 .

[8]  Adrian Sandu,et al.  Benchmarking Stiff ODE Solvers for Atmospheric Chemistry Problems I: Implicit versus Explicit , 1996 .

[9]  M. Gerritsen,et al.  Multiscale Process Coupling by Adaptive Fractional-Stepping; An In-Situ Combustion Model , 2006 .

[10]  Robert Gordon Moore,et al.  A Guide to High Pressure Air Injection (HPAI) Based Oil Recovery , 2002 .

[11]  Hans Bock,et al.  Numerical methods for optimum experimental design in DAE systems , 2000 .

[12]  Kjell Gustafsson,et al.  Control of Error and Convergence in ODE Solvers , 1992 .

[13]  H. H. Rosenbrock,et al.  Some general implicit processes for the numerical solution of differential equations , 1963, Comput. J..

[14]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[15]  K. H. Coats.  In-Situ Combustion Model , 1980 .

[16]  Rutherford Aris,et al.  Elementary Chemical Reactor Analysis , 1969 .

[17]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[18]  Norman P. Freitag,et al.  Low-temperature oxidation of oils in terms of SARA fractions: Why simple reaction models don't work , 2003 .

[19]  M. Durandeau,et al.  Laboratory Studies for Light-Oil Air Injection Projects: Potential Application in Handil Field , 2000 .

[20]  Laurent Hascoët,et al.  TAPENADE 2.1 user's guide , 2004 .

[21]  Adrian Sandu,et al.  Benchmarking stiff ode solvers for atmospheric chemistry problems II: Rosenbrock solvers , 1997 .

[22]  W. Kutta.  Beitrag zur Naherungsweisen Integration Totaler Differentialgleichungen , 1901 .

[23]  Wayne H. Enright,et al.  Interpolants for Runge-Kutta formulas , 1986, TOMS.

[24]  N. Freitag,et al.  A SARA-Based Model for Simulating the Pyrolysis Reactions That Occur in High- Temperature EOR Processes “Waxy ” Mixtures ™ , 2002 .

[25]  Peter Knabner,et al.  A new numerical reduction scheme for fully coupled multicomponent transport‐reaction problems in porous media , 2005 .

[26]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[27]  Michael L. Michelsen,et al.  Increasing the Computational Speed of Flash Calculations With Applications for Compositional, Transient Simulations , 2006 .

[28]  Henry J. Ramey,et al.  Reaction Kinetics of In-Situ Combustion: Part 2--Modeling , 1984 .

[29]  Ernst Hairer,et al.  Examples of Stiff Equations , 1996 .

[30]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[31]  Dafydd Gibbon,et al.  1 User’s guide , 1998 .

[32]  J. Friedly.  Extent of reaction in open systems with multiple heterogeneous reactions , 1991 .

[33]  R. Alexander,et al.  Design and implementation of DIRK integrators for stiff systems , 2003 .

[34]  J. W. Grabowski,et al.  A Fully Implicit General Purpose Finite-Difference Thermal Model For In Situ Combustion And Steam , 1979 .

[35]  W. H. Chen,et al.  A Numerical Simulation Model for Thermal Recovery Processes , 1979 .

[36]  Robert Gordon Moore,et al.  Low-Temperature Oxidation Kinetic Parameters for In-Situ Combustion Numerical Simulation , 1987 .

[37]  Hester Bijl,et al.  Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow , 2002 .

[38]  A. Kværnø,et al.  Norges Teknisk-naturvitenskapelige Universitet Singly Diagonally Implicit Runge-kutta Methods with an Explicit First Stage Singly Diagonally Implicit Runge-kutta Methods with an Explicit First Stage , 2022 .