By Jan Dirk Jansen
This article types a part of fabric taught in the course of a direction in complex reservoir simulation at Delft college of expertise during the last 10 years. The contents have additionally been awarded at a number of brief classes for commercial and educational researchers attracted to history wisdom had to practice learn within the region of closed-loop reservoir administration, often referred to as shrewdpermanent fields, concerning e.g. model-based creation optimization, info assimilation (or background matching), version aid, or upscaling ideas. each one of those issues has connections to system-theoretical concepts.
The introductory a part of the path, i.e. the platforms description of movement via porous media, varieties the subject of this short monograph. the most goal is to offer the vintage reservoir simulation equations in a notation that allows using strategies from the systems-and-control literature. even supposing the speculation is proscribed to the fairly easy state of affairs of horizontal two-phase (oil-water) circulation, it covers numerous regular elements of porous-media flow.
The first bankruptcy offers a short evaluation 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 remedy of wells. It comprises famous idea and is essentially intended to shape a foundation for the following bankruptcy the place the equations could be reformulated by way of 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 differing kinds of inputs results in an outline when it comes to nonlinear ordinary-differential and algebraic equations with (state-dependent) method, enter, output and direct-throughput matrices. different themes comprise generalized state-space representations, linearization, removing of prescribed pressures, the tracing of movement strains, carry tables, computational facets, 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 circulate. subsequent it strikes directly to the numerical resolution of the two-phase move equations, protecting numerous points like implicit, specific or combined (IMPES) time discretizations and linked balance matters, Newton-Raphson new release, streamline simulation, computerized time-stepping, and different computational features. The bankruptcy concludes with easy numerical examples to demonstrate those and different facets resembling mobility results, well-constraint switching, time-stepping statistics, and system-energy accounting.
The contents of this short can be of price to scholars and researchers drawn to the appliance of systems-and-control ideas to grease and fuel reservoir simulation and different purposes of subsurface circulation simulation reminiscent of CO2 garage, geothermal power, or groundwater remediation.
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This article types a part of fabric taught in the course of a path in complicated reservoir simulation at Delft collage of know-how over the last 10 years. The contents have additionally been offered at quite a few brief classes for commercial and educational researchers attracted to history wisdom had to practice learn within the sector of closed-loop reservoir administration, sometimes called shrewdpermanent fields, concerning e.
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Extra info for A Systems Description of Flow Through Porous Media (SpringerBriefs in Earth Sciences)
N ; wn Þ; dt where the continuous dependent variable and the continuous source term w of Eq. 2 Normally the functions ^ei are linear in the derivatives dði Þ=dt, which makes it possible to transform the system of Eq. 2) such that the derivatives in the left-hand side terms are isolated, leading to: 8 d ð 1 Þ > > ¼ f1 ðt; 1 ; 2 ; . ; n ; w1 Þ; > > > dt > > > > d ð 2 Þ > < ¼ f2 ðt; 1 ; 2 ; . ; n ; w2 Þ; dt ð2:3Þ > .. > > > . > > > > > > : dðn Þ ¼ fn ðt; 1 ; 2 ; . ; n ; wn Þ; dt where the functions fi are different from the functions ^fi in Eq.
2118/ 10528-PA Russel TF, Wheeler MF (1983) Finite-element and finite-difference methods for continuous flows in porous media. In: Ewing RE (ed) The mathematics of reservoir simulation. SIAM, Philadelphia Welge HJ (1952) A simplified method for computing oil recovery by gas or water drive. Pet Trans AIME 195:91–98 Whitson CH, Brulé MR (2000) Phase behavior. SPE Monograph Series 20, SPE, Richardson Chapter 2 System Models Abstract This chapter develops representations in state-space notation of the porous-media flow equations derived in Chap.
E. it may not be possible to derive an explicit expression f for a given implicit representation g. However, usually the implicit _ typically using some form of time representation may be solved numerically for x, discretization and an iterative algorithm. 10) which is often preferred for analysis purposes. g. g could represent a complete reservoir simulator. Detailed examples of the state variable description of reservoir systems will be discussed below. 5 System equations expressed as gð. Þ ¼ 0 are sometimes referred to as equations in residual form.