A Systems Description of Flow Through Porous Media - download pdf or read online

By Jan Dirk Jansen

ISBN-10: 3319002597

ISBN-13: 9783319002590

ISBN-10: 3319002600

ISBN-13: 9783319002606

This textual content types a part of fabric taught in the course of a direction in complicated reservoir simulation at Delft collage of expertise over the last 10 years. The contents have additionally been offered at quite a few brief classes for business and educational researchers drawn to history wisdom had to practice learn within the sector of closed-loop reservoir administration, sometimes called clever fields, with regards to e.g. model-based construction optimization, information assimilation (or heritage matching), version aid, or upscaling recommendations. every one of those themes has connections to system-theoretical concepts.
The introductory a part of the path, i.e. the platforms description of stream via porous media, varieties the subject of this short monograph. the most aim is to provide the vintage reservoir simulation equations in a notation that allows using strategies from the systems-and-control literature. even if the speculation is proscribed to the quite uncomplicated scenario of horizontal two-phase (oil-water) movement, it covers a number of standard points of porous-media flow.
The first bankruptcy supplies a quick evaluation of the fundamental equations to symbolize single-phase and two-phase circulation. It discusses the governing partial-differential equations, their actual interpretation, spatial discretization with finite changes, and the remedy of wells. It comprises recognized concept and is basically intended to shape a foundation for the following bankruptcy the place the equations can be reformulated when it comes to systems-and-control notation.
The moment bankruptcy develops representations in state-space notation of the porous-media movement 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 issues comprise generalized state-space representations, linearization, removal of prescribed pressures, the tracing of movement traces, elevate tables, computational facets, and the derivation of an strength stability for porous-media flow.
The 3rd bankruptcy first treats the analytical answer of linear structures of standard differential equations for single-phase circulate. subsequent it strikes directly to the numerical answer of the two-phase move equations, overlaying a number of elements 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 facets similar to mobility results, well-constraint switching, time-stepping information, and system-energy accounting.
The contents of this short could be of worth to scholars and researchers drawn to the applying of systems-and-control ideas to grease and fuel reservoir simulation and different functions of subsurface circulation simulation equivalent to CO2 garage, geothermal strength, or groundwater remediation.

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Extra info for A Systems Description of Flow Through Porous Media

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Xn do not represent spatial coordinates. Also note that we have indicated that the source term u has m elements instead of n. This anticipates a situation where many of the source terms are equal to zero, such that m ( n, in which case it may be computationally advantageous to use a shorter vector u. 3) can now be written as x_ ðtÞ ¼ f ðt; uðtÞ; x_ ðtÞÞ ; ð2:5Þ where f is a nonlinear vector function of x, u and t, and where we have emphasized the dependence of x and u on t by writing x(t) and u(t).

28). However, the relative permeabilities krw need to be determined through upstream weighting to obtain the correct convective behavior; see Aziz and Settari (1979), p. 153. This implies that 13 To stay in line with the notation used in the single-phase flow case, we should have used /0, qo,0 and qw,0 to indicate the pressure-independence of these parameters, but we have dropped the subscripts 0 to simplify the notation. 4 Two-Phase Flow 29 & ðkrw Þiþ1;j , 2 ðkrw Þi;j if pi;j ! piþ1;j ; ðkrw Þiþ1;j if pi;j \piþ1;j ð1:109Þ The second term in Eq.

11 26 1 Porous-Media Flow l , lo ¼ lw ; ð1:96Þ we have k ; l ð1:97Þ fw ¼ Sw ; ð1:98Þ k, and such that we can rewrite Eqs. 73) as o op k ¼ 0; ox ox m oSw oSw þ/ ¼ 0; ox ot ð1:99Þ ð1:100Þ which are linear elliptic and hyperbolic (convection) equations with spatially varying coefficients, with m ¼ Àk op : ox ð1:101Þ If an additional diffusion term is introduced in Eq. 100) we obtain a linear convection–diffusion equation ÀD o2 Sw oSw oSw þ/ ¼ 0; þm 2 ox ox ot ð1:102Þ where D is the diffusion constant.

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