By Jan Dirk Jansen
This textual content varieties a part of fabric taught in the course of a path in complex reservoir simulation at Delft collage of expertise during the last 10 years. The contents have additionally been provided at numerous brief classes for business and educational researchers attracted to historical past wisdom had to practice learn within the region of closed-loop reservoir administration, sometimes called clever fields, on the topic of e.g. model-based creation optimization, info assimilation (or background matching), version relief, or upscaling ideas. each one of those themes has connections to system-theoretical concepts.
The introductory a part of the direction, i.e. the structures description of circulation via porous media, types the subject of this short monograph. the most target is to offer the vintage reservoir simulation equations in a notation that enables using thoughts from the systems-and-control literature. even if the speculation is restricted to the rather easy scenario of horizontal two-phase (oil-water) move, it covers numerous regular facets of porous-media flow.
The first bankruptcy offers a short overview of the fundamental equations to symbolize single-phase and two-phase stream. It discusses the governing partial-differential equations, their actual interpretation, spatial discretization with finite variations, and the therapy of wells. It comprises recognized concept and is basically intended to shape a foundation for the subsequent bankruptcy the place the equations may be reformulated by way of systems-and-control notation.
The moment bankruptcy develops representations in state-space notation of the porous-media move equations. The systematic use of matrix partitioning to explain the differing kinds of inputs results in an outline when it comes to nonlinear ordinary-differential and algebraic equations with (state-dependent) procedure, enter, output and direct-throughput matrices. different subject matters comprise generalized state-space representations, linearization, removing of prescribed pressures, the tracing of movement strains, carry tables, computational points, and the derivation of an power stability for porous-media flow.
The 3rd bankruptcy first treats the analytical resolution of linear platforms of standard differential equations for single-phase circulation. subsequent it strikes directly to the numerical resolution of the two-phase circulate equations, protecting a number of features like implicit, specific or combined (IMPES) time discretizations and linked balance matters, Newton-Raphson new release, streamline simulation, computerized time-stepping, and different computational facets. The bankruptcy concludes with basic numerical examples to demonstrate those and different features akin to mobility results, well-constraint switching, time-stepping facts, and system-energy accounting.
The contents of this short could be of price to scholars and researchers attracted to the appliance of systems-and-control innovations to grease and gasoline reservoir simulation and different purposes of subsurface circulate simulation resembling CO2 garage, geothermal power, or groundwater remediation.
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Additional resources for A Systems Description of Flow Through Porous Media
Combining and reorganizing all terms results in V /Sw ðcw þ cr Þ op oSw þ/ ot ot ! ÀðTw ÞiÀ1;j piÀ1;j À ðTw Þi;jÀ1 pi;jÀ1 þ 2 i;j 2 h i À Á ðTw ÞiÀ1;j þðTw Þi;jÀ1 þðTw Þi;jþ1 þðTw Þiþ1;j pi;j À ðTw Þi;jþ1 pi;jþ1 À ðTw Þiþ1;j piþ1;j ¼ V q000 w i;j ; 2 2 2 2 2 2 ð1:110Þ where the transmissibilities are now defined as ðTw ÞiÀ1;j , 2 Dy h ðkkrw ÞiÀ1;j ; etc : 2 Dx lw ð1:111Þ A similar discretization can be obtained for Eq. 107): V /ð1 À Sw Þðco þ cr Þ op oSw À/ ot ot ! 112) can be written in matrix form as !
The unphysical solutions originate from neglecting the effect of capillary pressure which in reality produces a sharp increase in water saturation at a value of xD somewhere in the triple value region. This effect can be approximated in the form of a step-wise increase of saturation known as a shock in the theory of hyperbolic differential equations. 8 1 Water saturation Sw , - Fig. 8 Relative position along core xD , - 1 Fig. g. Lake (1989) for details) 10 We may either consider the water mass or the oil mass.
A continuous sequence of values of x over a certain time interval is often referred to as a trajectory in state space. In reservoir engineering applications it is sometimes preferred to start from Eqs. 2) rather than from Eq. 3), even if the functions ei are linear. We will refer to equations of the type of Eq. 4 In LTI form they can be written as ^ þ Bu ^ x_ ¼ Ax ^ : E 3 ð2:9Þ An alternative name for the input matrix is distribution matrix because it distributes the inputs u over the states x.
A Systems Description of Flow Through Porous Media by Jan Dirk Jansen