## Abstract

This thesis represents a three year research project, resulting in the development of the full-scale three dimensional two-phase immiscible incompressible streamline simulator accounting for capillary effects.

Streamline simulation is a relatively new technique, with a potential to become one of the key tools in reservoir simulation. The first streamline simulator appeared around 10 years ago. The advantages of the streamline methods are their exceptional simulation speed and less dispersed numerical solutions [7, 17, 19, 67]. Tracing the streamlines with respect to Darcy flow velocity [31] makes it possible to account for the non-linearities associated with fluid mobilities and the capillary pressure. The streamlines allow to decouple complex 3D saturation equation into set of simple 1D solutions by means of the time-of-flight (TOF) concept [17, 32]. However, up to the current moment, they provide limited abilities, compared to industrial standard finite-difference simulators. The main drawback of the two-phase immiscible incompressible streamline simulator is lack of capillary effects. In heterogeneous reservoirs with alternated wettability these forces may be extremely important, and, in some cases, dominating. The developments of the streamline methods are presented in Chapter 1.

This thesis presents a methodology to introduce capillary effects into streamline simulation. The first chapter gives an introduction into the fluid flow in the porous media and into capillary effects. The second chapter presents mathematical formulation of the governing equations with capillary effects. Both the pressure and the saturation equations are modified to include capillary effects. Introduction of capillary effects into the pressure equation is necessary to correctly predict the phase pressures. The pressure values are used to compute the Darcy flow velocity. As a result the streamlines are traced with respect to viscous, gravity and capillary forces. The modification of the saturation equation is necessary to correctly predict the capillary cross-flow effects. Various aspects of the numerical solution of the governing equations are discussed. A Capillary-Viscous Potential (CVP) [14] is introduced as a modification to the pressure equation with capillary effects for a better handling of the heterogeneities of the porous media. The CVP method is shown to provide more stable solution, compared to the straightforward method (SFD) of accounting for capillary effects in the pressure equation. The saturation equation is solved using the operator splitting method. The viscous forces are accounted along the streamlines; afterwards the fluids are redistributed on the finite-difference grid with respect to the capillary and gravity forces.

The third chapter presents simulation results of a number of test cases and the discussion of the modifications. First, the CVP and the SFD methods of introduction of capillary effects into the pressure equation are compared. Afterwards the saturation equation modifications are discussed. The time step selection methods are evaluated. Finally the streamline simulator is applied to the reservoir-scale simulations.

The fourth chapter starts with illustrations of capillary effects in the heterogeneous and the alternated wet media. The investigation of the zone of application of the streamline simulator with capillary effects is presented. The CapSL is compared to the commercial finite-difference simulator Eclipse based on the laboratory experiments.

The last part of my thesis presents conclusions and briefly addresses possible research topics for the future.

Streamline simulation is a relatively new technique, with a potential to become one of the key tools in reservoir simulation. The first streamline simulator appeared around 10 years ago. The advantages of the streamline methods are their exceptional simulation speed and less dispersed numerical solutions [7, 17, 19, 67]. Tracing the streamlines with respect to Darcy flow velocity [31] makes it possible to account for the non-linearities associated with fluid mobilities and the capillary pressure. The streamlines allow to decouple complex 3D saturation equation into set of simple 1D solutions by means of the time-of-flight (TOF) concept [17, 32]. However, up to the current moment, they provide limited abilities, compared to industrial standard finite-difference simulators. The main drawback of the two-phase immiscible incompressible streamline simulator is lack of capillary effects. In heterogeneous reservoirs with alternated wettability these forces may be extremely important, and, in some cases, dominating. The developments of the streamline methods are presented in Chapter 1.

This thesis presents a methodology to introduce capillary effects into streamline simulation. The first chapter gives an introduction into the fluid flow in the porous media and into capillary effects. The second chapter presents mathematical formulation of the governing equations with capillary effects. Both the pressure and the saturation equations are modified to include capillary effects. Introduction of capillary effects into the pressure equation is necessary to correctly predict the phase pressures. The pressure values are used to compute the Darcy flow velocity. As a result the streamlines are traced with respect to viscous, gravity and capillary forces. The modification of the saturation equation is necessary to correctly predict the capillary cross-flow effects. Various aspects of the numerical solution of the governing equations are discussed. A Capillary-Viscous Potential (CVP) [14] is introduced as a modification to the pressure equation with capillary effects for a better handling of the heterogeneities of the porous media. The CVP method is shown to provide more stable solution, compared to the straightforward method (SFD) of accounting for capillary effects in the pressure equation. The saturation equation is solved using the operator splitting method. The viscous forces are accounted along the streamlines; afterwards the fluids are redistributed on the finite-difference grid with respect to the capillary and gravity forces.

The third chapter presents simulation results of a number of test cases and the discussion of the modifications. First, the CVP and the SFD methods of introduction of capillary effects into the pressure equation are compared. Afterwards the saturation equation modifications are discussed. The time step selection methods are evaluated. Finally the streamline simulator is applied to the reservoir-scale simulations.

The fourth chapter starts with illustrations of capillary effects in the heterogeneous and the alternated wet media. The investigation of the zone of application of the streamline simulator with capillary effects is presented. The CapSL is compared to the commercial finite-difference simulator Eclipse based on the laboratory experiments.

The last part of my thesis presents conclusions and briefly addresses possible research topics for the future.

Original language | English |
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Place of Publication | Kgs. Lyngby |
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Publisher | Technical University of Denmark |

Number of pages | 161 |

ISBN (Print) | 87-91435-08-0 |

Publication status | Published - 2004 |