Download A Systems Description of Flow Through Porous Media by Jan Dirk Jansen PDF

By Jan Dirk Jansen

This textual content varieties a part of fabric taught in the course of a direction in complicated reservoir simulation at Delft college of expertise over the last 10 years. The contents have additionally been provided at a number of brief classes for business and educational researchers drawn to history wisdom had to practice examine within the quarter of closed-loop reservoir administration, sometimes called shrewdpermanent fields, on the topic of e.g. model-based creation optimization, info assimilation (or historical past matching), version aid, or upscaling options. every one of those issues has connections to system-theoretical concepts.
The introductory a part of the direction, i.e. the structures description of circulation via porous media, kinds the subject of this short monograph. the most target is to give the vintage reservoir simulation equations in a notation that allows using innovations from the systems-and-control literature. even supposing the speculation is restricted to the rather easy state of affairs of horizontal two-phase (oil-water) circulation, it covers a number of standard points of porous-media flow.
The first bankruptcy offers a short overview of the elemental equations to symbolize single-phase and two-phase circulation. It discusses the governing partial-differential equations, their actual interpretation, spatial discretization with finite variations, and the therapy of wells. It includes recognized idea and is basically intended to shape a foundation for the subsequent bankruptcy the place the equations should 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 different sorts of inputs results in an outline by way of nonlinear ordinary-differential and algebraic equations with (state-dependent) process, enter, output and direct-throughput matrices. different subject matters comprise generalized state-space representations, linearization, removal of prescribed pressures, the tracing of movement traces, elevate tables, computational elements, and the derivation of an power stability for porous-media flow.
The 3rd bankruptcy first treats the analytical answer of linear platforms of normal differential equations for single-phase stream. subsequent it strikes directly to the numerical answer of the two-phase circulate equations, overlaying a number of facets like implicit, specific or combined (IMPES) time discretizations and linked balance concerns, Newton-Raphson new release, streamline simulation, automated time-stepping, and different computational elements. The bankruptcy concludes with easy numerical examples to demonstrate those and different points corresponding to mobility results, well-constraint switching, time-stepping records, and system-energy accounting.
The contents of this short could be of worth to scholars and researchers attracted to the appliance of systems-and-control ideas to grease and fuel reservoir simulation and different functions of subsurface circulation simulation comparable to CO2 garage, geothermal strength, or groundwater remediation.

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11 Incompressible Flow In the special case that the fluid and rock compressibilities are so small that they may be neglected, it follows from Eqs. 118) that Vwp ¼ Vop ¼ 0. In that case we can rewrite Eq. 129) as ! ! ! Fw ðsÞ Tw ðsÞ 0 p 0 Vws p_ ¼ þ ð1:130Þ q ; s_ To ðsÞ 0 s Fo ð s Þ t 0 Vos which may also be expressed as: Vws s_ þ Tw ðsÞp ¼ Fw ðsÞqt ; ð1:131Þ Vos s_ þ To ðsÞp ¼ Fo ðsÞqt : ð1:132Þ Because Vws = -Vos we can add the two equations to obtain the pressure equation for incompressible flow Tt ðsÞp ¼ qt ; ð1:133Þ where we used the equality Fw þ Fo ¼ I, and where Tt ¼ Tw þ To is the total transmissibility matrix, which is still a function of saturation.

2:30Þ 46 2 System Models " ^ , ogðu; x; x_ Þ ; B ou ð2:31Þ " ^ , ogðu; x; x_ Þ ; E ox_ ð2:32Þ " ^ , ojðu; x; yÞ C ox ð2:33Þ " ^ , ojðu; x; yÞ ; D ou ð2:34Þ " ^ , ogðu; x; yÞ ; F oy ð2:35Þ where we have dropped the superscripts 0 for clarity. 2 Secant and Tangent Matrices In reservoir simulation one often encounters systems x_ ¼ fðu; xÞ that can be expressed in the form8 x_ ¼ AðxÞx þ BðxÞu : ð2:36Þ In that case we obtain the linearized Eq. 24) with Jacobians defined as 0 0 À Á À Á " u0 ; x0 , A x0 þ oAðx Þ x0 þ oBðx Þ u0 ; A ox ox À 0Á À 0Á " x ,B x : B ð2:37Þ ð2:38Þ If we linearize the state equations along all points of a given trajectory ðx0 ðtÞ; u0 ðtÞÞ in state-input space, the resulting model is referred to as the tangentlinear approximation of the nonlinear model, or simply the tangent-linear model.

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