General Nonlinear Solution Strategies for Multiphase Multicomponent EoS Based Simulation

Abstract

We present a novel reservoir flow simulation framework built on top of an Automatic Differentiation with Expression Templates Library (ADETL). Given discrete algebraic expressions of the governing conservation equations, associated constraint relations, and equations of state, the ADETL library generates efficient computer code for the corresponding Jacobian matrix. The framework allows for complete flexibility in the specification of the independent variables, which leads to unified simulator for different formulations and solution strategies. We study the behavior of several widely used nonlinear formulations and solution strategies for general-purpose compositional simulation. The numerical behaviors using different variable sets, including the so called natural-and mass-type variables, are investigated. We also report on tie-line based parameterization strategies for compositional modeling. The focus is on the fully implicit method (FIM), where the full Jacobian (primary and secondary algebraic equations in terms of both the primary and secondary variables) is constructed. Schur-complement reduction of the full Jacobian is used to assemble the (reduced) Jacobian for the primary equations as a function of the primary unknowns only. The numerical challenges associated with the nonlinear hydrodynamic and thermodynamic interactions associated with compositional displacement processes are the focus of this investigation. These include phase appearance and disappearance, crossing into and out of the critical region, wide pressure variations in space and time. The behavior of the Newton iterations as a function of timestep size is analyzed for different variable sets, alignment of equations (including linear and nonlinear constraints), and nonlinear updating strategies. Comparisons are performed for several challenging multi-phase multi-component problems of practical interest.

Faster Simulation with Optimized Automatic Differentiation and Compiled Linear Solvers

“…The first implementation of AD we are aware of appeared in an early version of the commercial INTERSECT simulator [5]. Its use was not continued, and use of AD for reservoir simulation has instead primarily been pioneered by Stanford's AD-GPRS simulator [22,23,27], which builds on ADETL, a library for forward-mode AD implemented in terms of expression templates in C++ [25,26]. Seeing the success of AD-GPRS, we decided to implement the same type of methods in MRST [2,12,17], using ideas from Neidinger [18] to develop a variable-first forward-AD library suited for vectorized operations in MATLAB.…”

Multi-Stage Preconditioners for Thermal-Compositional-Reactive Flow in Porous Media

“…We use the Automatic-Differentiation General Purpose Research Simulator (AD-GPRS [42]) to solve the system of equations using a finite volume, fully implicit discretization and the natural variables formulation. More details about the reservoir simulation implementation can be found in Coats et al [7], Cao [16], Aziz and Settari [43], Voskov et al [44]. We omit details here as they are not the central focus of this work.…”

MRST-AD – an Open-Source Framework for Rapid Prototyping and Evaluation of Reservoir Simulation Problems

“…This technique was introduced in an early version of the commercial Intersect simulator (DeBaun et al 2005), but has mainly been pioneered through a reimplementation of the GPRS research simulator (Cao 2002). The new simulator, called AD-GPRS is primarily based on fully implicit formulations (Voskov et al 2009;Zhou et al 2011;Voskov and Tchelepi 2012) in which independent variables and residual equations are AD structures implemented using ADETL, a library for forward-mode AD realized by expression templates in C++ (Younis and Aziz 2007;Younis 2009). This way, the Jacobi matrices needed in the nonlinear Newton-type iterations can be constructed from the derivatives that are implicitly computed from when evaluating the residual equations.…”