A unified multilevel framework of upscaling and domain decomposition

Presented at CMWR 2010 - XVIII International Conference on Computational Methods in Water Resources, June 21-24, 2010, Barcelona, SpainWe consider multiscale preconditioners for a class of mass-conservative domain-decomposition (MCDD) methods. For the application of reservoir simulation, we need to solve large linear systems, arising from finite-volume discretisations of elliptic PDEs with highly variable coefficients. We introduce an algebraic framework, based on probing, for constructing mass-conservative operators on a multiple of coarse scales. These operators may further be applied as coarse spaces for additive Schwarz preconditioners. By applying different local approximations to the Schur complement system based on a careful choice of probing vectors, we show how the MCDD preconditioners can be both efficient preconditioners for iterative methods or accurate upscaling techniques for the heterogeneous elliptic problem. Our results show that the probing technique yield better approximation properties compared with the reduced boundary condition commonly applied with multiscale methods.publishedVersio

A unified multilevel framework of upscaling and domain decomposition

We consider multiscale preconditioners for a class of mass-conservative domain-decomposition (MCDD) methods. For the application of reservoir simulation, we need to solve large linear systems, arising from finite-volume discretisations of elliptic PDEs with highly variable coefficients. We introduce an algebraic framework, based on probing, for constructing mass-conservative operators on a multiple of coarse scales. These operators may further be applied as coarse spaces for additive Schwarz preconditioners. By applying different local approximations to the Schur complement system based on a careful choice of probing vectors, we show how the MCDD preconditioners can be both efficient preconditioners for iterative methods or accurate upscaling techniques for the heterogeneous elliptic problem. Our results show that the probing technique yield better approximation properties compared with the reduced boundary condition commonly applied with multiscale methods

Sufficient criteria are necessary for monotone control volume methods

AbstractControl volume methods are prevailing for solving the potential equation arising in porous media flow. The continuous form of this equation is known to satisfy a maximum principle, and it is desirable that the numerical approximation shares this quality. Recently, sufficient criteria were derived guaranteeing a discrete maximum principle for a class of control volume methods. We show that most of these criteria are also necessary. An implication of our work is that no linear nine-point control volume method can be constructed for quadrilateral grids in 2D that is exact for linear solutions while remaining monotone for general problems

Upscaled modeling of CO2 injection and migration with coupled thermal processes

A practical modeling approach for CO2 storage over relatively large length and time scales is the vertical-equilibrium model, which solves partially integrated conservation equations for flow in two lateral dimensions. We couple heat transfer within the vertical equilibrium framework for fluid flow, focusing on thermal processes that most impact the CO2 plume. We investigate a simplified representation of heat exchange that also includes transport of heat within the plume. In addition, we explore available CO2 thermodynamic models for reliable prediction of density under different injection pressures and temperatures. The model concept is demonstrated on simplified systems.publishedVersio

PVTx Properties of a Two-phase CO2 Jet from Ruptured Pipeline

Span and Wagner equation of state (SW EOS) have been used to investigate changes in the thermodynamic properties of CO2 during a depressurization process from a pipeline into marine environment. The process is assumed to be isenthalpic, as only the thermodynamic change at the moment of depressurization is considered. The calculations show that the depth location of the pipeline influences greatly the density, temperature and volume changes, because of the difference in the surrounding pressures. In general the two-phase area is reached at depths shallower than 600 meters, which yields for the Norwegian Continental Shelf, as it is mainly shallower than 500 meters depth. There is a rapid decrease in density in the two-phase area causing a rapid expansion in the volume of CO2 from 4 MPa to 1 MPa. At the shallowest depth considered (100m) the volume fraction consist almost entirely of gas, and the density change give a significant increase in volume.publishedVersio

PVTx Properties of a Two-phase CO2 Jet from Ruptured Pipeline

Span and Wagner equation of state (SW EOS) have been used to investigate changes in the thermodynamic properties of CO2 during a depressurization process from a pipeline into marine environment. The process is assumed to be isenthalpic, as only the thermodynamic change at the moment of depressurization is considered. The calculations show that the depth location of the pipeline influences greatly the density, temperature and volume changes, because of the difference in the surrounding pressures. In general the two-phase area is reached at depths shallower than 600 meters, which yields for the Norwegian Continental Shelf, as it is mainly shallower than 500 meters depth. There is a rapid decrease in density in the two-phase area causing a rapid expansion in the volume of CO2 from 4 MPa to 1 MPa. At the shallowest depth considered (100m) the volume fraction consist almost entirely of gas, and the density change give a significant increase in volume

Dynamic PVT model for CO2-EOR black-oil simulations

A well-planned CO2EOR operation can help meet an ever-increasing need for energy and at the same time reduce the total CO2 footprint from the energy production. Good simulation studies are crucial for investment decisions where increased oil recovery is optimized and balanced with permanent CO2 storage. It is common to use a compositional simulator for CO2 injection to accurately calculate the PVT properties of the mixture of oil and CO2. Compositional simulations have significantly increased simulation time compared to blackoil simulations. Large simulation studies where many simulations are used either to represent uncertainty and/or optimize the results can thus be unpractical. On the other hand existing blackoil formulations often poorly represent the PVT properties of the oil-CO2 mixtures. We therefore present an extended blackoil formulation with dynamic blackoil properties that depends on the fraction of CO2 in the cell. These properties represents the density and viscosity of the hydrocarbon - CO2 mixture more accurately and thus give results closer to the compositional simulator. The dynamic blackoil functions are calculated from numerical slim-tube experiments based on one-dimensional equation-of-state (EOS) simulations. The same simulations also gives estimates of the minimum-miscibility pressure (MMP). We present examples based on data from the Fifth Comparative Solution Project: Evaluation of Miscible Flood Simulators, but uses CO2 as the injection gas. These examples shows that the new blackoil model gives results that are closer to compositional simulations compared to existing blackoil formulations. The model is implemented in the Flow simulator. The Flow simulator is developed as part of the open porous media (OPM) project and is an openly developed and free reservoir simulator that is capable of simulating industry relevant reservoir models with similar single and parallel performance as commercial simulators.publishedVersio

Dynamic PVT model for CO2-EOR black-oil simulations

A well-planned CO2EOR operation can help meet an ever-increasing need for energy and at the same time reduce the total CO2 footprint from the energy production. Good simulation studies are crucial for investment decisions where increased oil recovery is optimized and balanced with permanent CO2 storage. It is common to use a compositional simulator for CO2 injection to accurately calculate the PVT properties of the mixture of oil and CO2. Compositional simulations have significantly increased simulation time compared to blackoil simulations. Large simulation studies where many simulations are used either to represent uncertainty and/or optimize the results can thus be unpractical. On the other hand existing blackoil formulations often poorly represent the PVT properties of the oil-CO2 mixtures. We therefore present an extended blackoil formulation with dynamic blackoil properties that depends on the fraction of CO2 in the cell. These properties represents the density and viscosity of the hydrocarbon - CO2 mixture more accurately and thus give results closer to the compositional simulator. The dynamic blackoil functions are calculated from numerical slim-tube experiments based on one-dimensional equation-of-state (EOS) simulations. The same simulations also gives estimates of the minimum-miscibility pressure (MMP). We present examples based on data from the Fifth Comparative Solution Project: Evaluation of Miscible Flood Simulators, but uses CO2 as the injection gas. These examples shows that the new blackoil model gives results that are closer to compositional simulations compared to existing blackoil formulations. The model is implemented in the Flow simulator. The Flow simulator is developed as part of the open porous media (OPM) project and is an openly developed and free reservoir simulator that is capable of simulating industry relevant reservoir models with similar single and parallel performance as commercial simulators