Sequential Fully Implicit Formulation for Compositional Simulation using Natural Variables

The Sequential Fully Implicit (SFI) method was proposed (Jenny et al., JCP 2006), in the context of a Multiscale Finite Volume (MSFV) formulation, to simulate coupled immiscible multiphase fluid flow in porous media. Later, Lee et al. (Comp. Geosci. 2008) extended the SFI formulation to the black-oil model, whereby the gas component is allowed to dissolve in the oil phase. Most recently, the SFI approach was extended to fully compositional isothermal displacements by Moncorgé et al., (JCP 2017). SFI schemes solve the fully coupled system in two steps: (1) Construct and solve the pressure equation (flow problem). (2) Solve the coupled species transport equations for the phase saturations and phase compositions. In SFI, each outer iteration involves this two-step sequence. Experience indicates that complex interphase mass transfer behaviors often lead to large numbers of SFI outer iterations compared with the Fully Implicit (FI) method. Here, we demonstrate that the convergence difficulties are directly related to the treatment of the coupling between the flow and transport problems, and we propose a new SFI variant based on a nonlinear overall-volume balance equation. The first step consists of forming and solving a nonlinear pressure equation, which is a weighted sum of all the component mass conservation equations. A Newton-based scheme is used to iterate out all the pressure dependent nonlinearities in both the accumulation and flux terms of the overall-volume balance equation. The resulting pressure field is used to compute the Darcy phase velocities and the total-velocity. The second step of the new SFI scheme entails introducing the overall-mass density as a degree-of-freedom, and solving the full set of component conservation equations cast in the natural-variables form (i.e., saturations and phase compositions). During the second step, the pressure and the total-velocity fields are fixed. The SFI scheme with a nonlinear pressure extends the SFI approach of Jenny et al. (JCP 2006) to multi-component compositional processes with interphase mass transfer. The proposed compositional SFI approach employs an overall balance for the pressure equation; however, unlike existing volume-balance Sequential Implicit (SI) schemes (Acs et al. and Doster et al., CRC 2014), which use overall compositions, this SFI formulation is well suited for the natural variables (saturations and phase compositions). We analyze the 'splitting errors' associated with the compositional SFI scheme, and we show how to control these errors in order to converge to the same solution as the Fully Implicit (FI) method. We then demonstrate that the compositional SFI has convergence properties that are very comparable to those of the FI approach. This robust sequential-implicit solution scheme allows for designing numerical methods and linear solvers that are optimized for the sub-problems of flow and transport. The SFI scheme with a nonlinear pressure formulation is well suited for multiscale formulations, and it promises to replace the widely used FI approach for compositional reservoir simulation

Pressure-stabilized fixed-stress iterative solutions of compositional poromechanics

We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence

An Improved Multi-Stage Preconditioner on GPUs for Compositional Reservoir Simulation

The compositional model is often used to describe multicomponent multiphase porous media flows in the petroleum industry. The fully implicit method with strong stability and weak constraints on time-step sizes is commonly used in the mainstream commercial reservoir simulators. In this paper, we develop an efficient multi-stage preconditioner for the fully implicit compositional flow simulation. The method employs an adaptive setup phase to improve the parallel efficiency on GPUs. Furthermore, a multi-color Gauss-Seidel algorithm based on the adjacency matrix is applied in the algebraic multigrid methods for the pressure part. Numerical results demonstrate that the proposed algorithm achieves good parallel speedup while yields the same convergence behavior as the corresponding sequential version.Comment: 24 pages, 4 figures, and 8 tables. arXiv admin note: text overlap with arXiv:2201.0197

Nonlinear Acceleration of Sequential Fully Implicit (SFI) Method for Coupled Flow and Transport in Porous Media

The sequential fully implicit (SFI) method was introduced along with the development of the multiscale finite volume (MSFV) framework, and has received considerable attention in recent years. Each time step for SFI consists of an outer loop to solve the coupled system, in which there is one inner Newton loop to implicitly solve the pressure equation and another loop to implicitly solve the transport equations. Limited research has been conducted that deals with the outer coupling level to investigate the convergence performance. In this paper we extend the basic SFI method with several nonlinear acceleration techniques for improving the outer-loop convergence. Specifically, we consider numerical relaxation, quasi-Newton (QN) and Anderson acceleration (AA) methods. The acceleration techniques are adapted and studied for the first time within the context of SFI for coupled flow and transport in porous media. We reveal that the iterative form of SFI is equivalent to a nonlinear block Gauss-Seidel (BGS) process. The effectiveness of the acceleration techniques is demonstrated using several challenging examples. The results show that the basic SFI method is quite inefficient, suffering from slow convergence or even convergence failure. In order to better understand the behaviors of SFI, we carry out detailed analysis on the coupling mechanisms between the sub-problems. Compared with the basic SFI method, superior convergence performance is achieved by the acceleration techniques, which can resolve the convergence difficulties associated with various types of coupling effects. We show across a wide range of flow conditions that the acceleration techniques can stabilize the iterative process, and largely reduce the outer iteration count

Modelling In-situ Upgrading of Heavy Oil Using Operator Splitting Methods

Heavy oil and oil sands are important hydrocarbon resources that account for over 10 trillion barrels (Meyer et al., 2007), nearly three times the conventional oil in place in the world. There are huge, wellknown resources of heavy oil, extra-heavy oil, and bitumen in Canada, Venezuela, Russia, the USA and many other countries. The oil sands of Alberta alone contain over two trillion barrels of oil. In Canada, approximately 20% of oil production is from heavy oil and oil sand resources

An implicit local time-stepping method based on cell reordering for multiphase flow in porous media

We discuss how to introduce local time-step refinements in a sequential implicit method for multiphase flow in porous media. Our approach relies heavily on causality-based optimal ordering, which implies that cells can be ordered according to total fluxes after the pressure field has been computed, leaving the transport problem as a sequence of ordinary differential equations, which can be solved cell-by-cell or block-by-block. The method is suitable for arbitrary local time steps and grids, is mass-conservative, and reduces to the standard implicit upwind finite-volume method in the case of equal time steps in adjacent cells. The method is validated by a series of numerical simulations. We discuss various strategies for selecting local time steps and demonstrate the efficiency of the method and several of these strategies by through a series of numerical examples.publishedVersio

Modelling In-situ Upgrading (ISU) of heavy oil using dimensionless analysis and operator splitting method

The In-Situ Upgrading (ISU) of bitumen and oil shale is a very challenging process to model numerically because a large number of components need to be modelled using a system of equations that are both highly non-linear and strongly coupled. In addition to the transport of heat by conduction and convection, and the change of properties with varying pressure and temperature, these processes involve transport of mass by convection, evaporation, condensation and pyrolysis chemical reactions. The behaviours of these systems are difficult to predict as relatively small changes in the material composition can significantly change the thermophysical properties. Accurate prediction is further complicated by the fact that many of the inputs needed to describe these processes are uncertain, e.g. the reaction constants and the temperature dependence of the material properties. The large number of components and chemical reactions involves a non-linear system that is often too large for full field simulation using the Fully Implicit Method (FIM). Operator splitting (OS) methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as splitting error. Obviously the best splitting method for modelling a given process is the one that minimises the splitting error whilst improving computational performance over that obtained from using a fully implicit approach. Although operator splitting has been widely used for the modelling of reactive-transport problems, it has not yet been applied to models that involve the coupling of mass transport, heat transfer and chemical reactions. One reason is that it is not clear which operator splitting technique to use. Numerous such techniques are described in the literature and each leads to a different splitting error, which depends significantly on the relative importance of the mechanisms involved in the system. While this error has been extensively analysed for linear operators for a wide range of methods, the results observed cannot be extended to general non-linear systems. It is therefore not clear which of these techniques is most appropriate for the modelling of ISU. Analysis using dimensionless numbers can provide a useful insight into the relative importance of different parameters and processes. Scaling reduces the number of parameters in the problem statement and quantifies the relative importance of the various dimensional parameters such as permeability, thermal conduction and reaction constants. Combined with Design of Experiments (DOE), which allows quantification of the impact of the parameters with a minimal number of numerical experiments, dimensionless analysis enables experimental programmes to be focused on acquiring the relevant data with the appropriate accuracy by ranking the different parameters controlling the process. It can also help us design a better splitting method by identifying the couplings that need to be conserved and the ones that can be relaxed. This work has three main objectives: (1) to quantify the main interactions between the heat conduction, the heat and mass convection and the chemical reactions, (2) to identify the primary parameters for the efficiency of the process and (3) to design a numerical method that reduces the CPU time of the simulations with limited loss in accuracy. We first consider a simplified model of the ISU process in which a solid reactant decomposes into non-reactive gas. This model allows us to draw a parallel between the in-situ conversion of kerogen and the thermal decomposition of polymer composite when used as heat-shield. The model is later extended to include a liquid phase and several reactions. We demonstrate that a ISU model with nf fluid components, ns solid components and k chemical reactions depends on 9+k*(3+nf+ns-2)+8nf+2ns dimensionless numbers. The sensitivity analysis shows that (1) the heat conduction is the primary operator controlling the time scale of the process and (2) the chemical reactions control the efficiency of the process through the extended Damköhler numbers, which quantify the ratio of chemical rate to heat conduction rate at the heater temperature for each reaction in the model. In the absence of heat loss and gravity effects, we show that the ISU process is most efficient at a heater temperature for which the minimum of the extended Damköhler numbers of all reactions included in the model was between 10 and 20. For the numerical method, the standard Iterative Split Operator (ISO) does not perform well due to many convergence failures, whereas the standard Sequential Split Operator (SSO) and the Strang-Marchuk Split Operator (SMSO) give large discretization errors. We develop a new method, called SSO-CKA, which has smaller discretization error. This method simply applies SSO with three decoupled operators: the heat conduction (operator $C$), the chemical reactions (operator $K$) and the heat and mass convection (operator $A$), applied in this order. When we apply SSO-CKA with the second-order trapezoidal rule (TR) for solving the chemical reaction operator, we obtain a method which generally gives smaller discretization errors than FIM. We design an algorithm, called SSO-CKA-TR-AIM, which is faster and generally more accurate than FIM for simulations with a kinetic model including a large number of components that could be regrouped into a small number of chemical classes for the advection and heat conduction operator. SSO-CKA works best for ISU models with small reaction enthalpies and no other reaction than pyrolysis reactions, but can give a large discretization method for ISU models with non-equilibrium reactions.Open Acces