By Jan Dirk Jansen
This textual content varieties a part of fabric taught in the course of a path in complicated reservoir simulation at Delft college of expertise during the last 10 years. The contents have additionally been awarded at a variety of brief classes for business and educational researchers attracted to history wisdom had to practice learn within the zone of closed-loop reservoir administration, often referred to as shrewdpermanent fields, on the topic of e.g. model-based creation optimization, information assimilation (or background matching), version aid, or upscaling options. every one of those subject matters has connections to system-theoretical concepts.
The introductory a part of the path, i.e. the structures description of circulation via porous media, types the subject of this short monograph. the most goal is to offer the vintage reservoir simulation equations in a notation that enables using ideas from the systems-and-control literature. even though the speculation is restricted to the rather basic state of affairs of horizontal two-phase (oil-water) move, it covers a number of standard elements of porous-media flow.
The first bankruptcy provides a short overview of the elemental equations to symbolize single-phase and two-phase move. It discusses the governing partial-differential equations, their actual interpretation, spatial discretization with finite variations, and the therapy of wells. It comprises famous thought and is essentially intended to shape a foundation for the subsequent bankruptcy the place the equations may be reformulated when it comes to systems-and-control notation.
The moment bankruptcy develops representations in state-space notation of the porous-media stream equations. The systematic use of matrix partitioning to explain the different sorts 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 issues comprise generalized state-space representations, linearization, removing of prescribed pressures, the tracing of circulation 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 structures of normal differential equations for single-phase move. subsequent it strikes directly to the numerical resolution of the two-phase stream equations, overlaying a variety 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 points. The bankruptcy concludes with basic numerical examples to demonstrate those and different elements comparable to mobility results, well-constraint switching, time-stepping facts, and system-energy accounting.
The contents of this short may be of worth to scholars and researchers attracted to the appliance of systems-and-control suggestions to grease and fuel reservoir simulation and different functions of subsurface stream simulation equivalent to CO2 garage, geothermal power, or groundwater remediation.
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Extra info for A Systems Description of Flow Through Porous Media
V is a diagonal matrix known as the accumulation matrix and T is a symmetric banded matrix, known as the transmissibility matrix. e. to wells, and are expressed in m3/s. Positive values imply injection and negative values imply production. Because usually only a few grid blocks are penetrated by wells, only a few elements of q have a non-zero value. In the case of a reservoir modeled with n grid blocks and produced with m wells, V and T would be n 9 n matrices, and p and q would be n 9 1 vectors, of which q would have m non-zero entries.
H o op o op op oSw kkro kkro À þ þ h /ð1 À Sw Þðco þ cr Þ À / À hq000 o lo ox ox oy oy ot ot ¼ 0: ð1:107Þ The first term in Eq. 106) can be rewritten as h o op h D Dp kkrw kkrw % lw ox ox lw Dx ÀDx Á À Á h ðkkrw Þiþ12;j piþ1;j À pi;j À ðkkrw ÞiÀ12;j pi;j À piÀ1;j ¼ ; ð1:108Þ lw ðDxÞ2 where the absolute permeabilities k are based on harmonic averages just as in the single-phase case; see Eq. 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.
2118/6893-PA Peaceman DW (1983) Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE J 23(3):531–543. 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.