A Semi-Implicit Formulation for Compositional Reservoir Simulation

Abstract

A new semi-implicit formulation to solve the set of nonlinear difference equations that describe multiphase compositional flow in a reservoir is presented in this paper. Newton's method is applied on the nonlinear difference equations obtained after treating compositions in the flow terms explicitly, one iteration behind. The linearized system of equations generated from this treatment possess a peculiar matrix structure, that allows the reduction of the system, via matrix operations, to three equations in three unknowns per gridblock regardless of the number of components. The level of implicitness of the formulation presented in this paper is intermediate to IMPECS and Fully-Implicit methods and has a higher level of implicitness than a semi-implicit method previously proposed in the literature. It provides an attractive approach to solve problems where the rate of change of pressures, saturations and compositions in the reservoir are important, such that IMPECS method becomes impractical and the Fully-Implicit method becomes too expensive. Applications to the simulation of a well-test in a gas-condensate well and the natural depletion and gas-cycling of a volatile-oil reservoir are presented. Introduction Compositional simulation is an important tool to optimize recovery of reservoirs subject to changes in the composition of the hydrocarbon phases along the productive life of the reservoir. Gas-condensate and volatile-oil reservoirs are typical examples in which compositional simulation is mandatory. The first compositional simulators of general application were developed by Kazemi et al. in 1978 and by Fussell and Fussell in 1979. Thermodynamic equilibrium for the hydrocarbon phases was handled by Kazemi et al. through experimentally obtained equilibrium ratios, while Fussell and Fussell introduced the use of an equation of state. The approach used by these authors for solving the compositional reservoir flow equations is an extension of IMPES method, applied in black-oil simulation, which could be referred to as IMPECS, implicit pressure explicit compositions and saturations, method. IMPECS methods are attractive due to the relatively small computer work requirements, that make numerical simulation inexpensive; the low level of implicitness used to solve the flow difference equations makes such methods impractical in some applications. In 1980, Coats presented an implicit formulation, based on Newton iteration, to simulate compositional reservoir problems with an equation of state. The high numerical stability of the implicit method, with enhanced efficiency and reliability to handle most compositional problems, demands larger computer work requirements, that increase with the number of pseudocomponents being considered. In 1981, Nghiem et al. presented a variation of Kazemi et al.'s method that used an equation of state. They suggested the use of two-point upstream weighting for transmissibilities in the flow terms. Watts in 1983, based on the ideas of Acs et al., presented another approach, the sequential-implicit method, to solve the compositional flow difference equations. An attempt was made of combining the advantages of IMPECS while retaining some of the stability characteristics of the implicit method. Quandalle and Savary presented in 1989 a method, implicit in pressure and saturations and explicit in compositions, for the compositional problem. For multiphase flow, oil-gas-water, the gridblock equations are reduced to three equations in three primary unknowns, po, Sg and Sw. In doing this, the subset of thermodynamic equations is arbitrarily decoupled and then used, after solving for the primary unknowns, in a sequential fashion to solve for the oil and gas phase compositions; this is done as in IMPECS method by performing flash calculations on a cell-by-cell basis.

Pore network model for retrograde gas flow in porous media

“…A simulação composicional torna-se necessária quando o processo de recuperação de hidrocarbonetosé sensível a mudanças na composição de suas fases ao longo da vida produtiva do reservatório. Essa técnica consiste em resolver as equações de fluxo para cada componente em cada fase do sistema (Branco et al, 1996). A figura 1.3 sugere a classe de simulador para cada região do envelope de fases em um diagrama (P, T ): os simuladores composicionais são os mais adequados para modelar o escoamento de hidrocarbonetos nas proximidades do ponto crítico; os térmicos, por incluírem Equação de Conservação da Energia, permitem modelar a recuperação deóleos pesados através de métodos térmicos; e os black-oil, devido a sua simplicidade e eficiência, permitem modelar todo o espectro restante.…”

“…Adicionalmente, seu algoritmo permitia resolver o problema implicitamente em apenas algumas células, resolvendo explicitamente as demais e ganhando eficiência computacional. Branco et al (1996) apresentaram uma formulação composicional semiimplícita: as composições dos termos de escoamento eram tratadas explicitamente, defasadas de uma iteração, e a próxima iteração do método de Newton era chamada para resolver o sistema para a pressão e as saturações. No entanto as composições dos termos fonte e de acumulação continuavam sendo tratadas implicitamente, conferindoà formulação nível de implicitude intermediário entre IMPEC (implícito nas pressões e explícito nas composições) e totalmente implícito.…”

Development of a fast EoS based compositional model for three-phase core flooding

“…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.…”

Comparison of Two Time-Stepping Methods for Compositional Reservoir Simulation

“…Other procedures for solving the pressure and saturation implicitly (IMPSAT) were proposed in [4] and [5]. However, the one presented here has similar advantages as the original IMPEC formulation in the UTCOMP simulator.…”