By Jan Dirk Jansen
This textual content kinds a part of fabric taught in the course of a direction in complex reservoir simulation at Delft collage of expertise during the last 10 years. The contents have additionally been awarded at a variety of brief classes for commercial and educational researchers attracted to heritage wisdom had to practice study within the quarter of closed-loop reservoir administration, sometimes called clever fields, on the topic of e.g. model-based construction optimization, facts assimilation (or background matching), version relief, or upscaling recommendations. each one of those themes has connections to system-theoretical concepts.
The introductory a part of the direction, i.e. the platforms description of circulate via porous media, varieties the subject of this short monograph. the most goal is to give the vintage reservoir simulation equations in a notation that allows using techniques from the systems-and-control literature. even if the idea is restricted to the really uncomplicated scenario of horizontal two-phase (oil-water) circulate, it covers numerous general facets of porous-media flow.
The first bankruptcy supplies a quick evaluate 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 alterations, and the remedy of wells. It includes recognized concept and is basically intended to shape a foundation for the following bankruptcy the place the equations could be reformulated when it comes to systems-and-control notation.
The moment bankruptcy develops representations in state-space notation of the porous-media circulate 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 issues comprise generalized state-space representations, linearization, removal of prescribed pressures, the tracing of circulate strains, elevate tables, computational points, and the derivation of an strength stability for porous-media flow.
The 3rd bankruptcy first treats the analytical resolution of linear structures of normal differential equations for single-phase stream. subsequent it strikes directly to the numerical resolution of the two-phase circulation equations, overlaying quite a few elements like implicit, specific or combined (IMPES) time discretizations and linked balance concerns, Newton-Raphson generation, streamline simulation, computerized time-stepping, and different computational features. The bankruptcy concludes with basic numerical examples to demonstrate those and different elements comparable to mobility results, well-constraint switching, time-stepping records, and system-energy accounting.
The contents of this short will be of worth to scholars and researchers drawn to the applying of systems-and-control recommendations to grease and fuel reservoir simulation and different purposes of subsurface stream simulation comparable to CO2 garage, geothermal strength, or groundwater remediation.
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Additional info for A Systems Description of Flow Through Porous Media
85) resulting in vt t dfw ; ð1:86Þ xjSw ¼^Sw ¼ / dSw Sw ¼^Sw where the integration constant has been set equal to zero which implies that x ¼ 0 at t ¼ 0. If the core has length L and cross sectional area A, it is convenient to rescale Eq. 86) as dfw xD jSw ¼^Sw ¼ tD ; ð1:87Þ dSw Sw ¼^Sw which leads to the dimensionless Buckley-Leverett velocity dfw mD jSw ¼^Sw ¼ : dSw Sw ¼^Sw ð1:88Þ Here the dimensionless length and time are defined as x ; L ð1:89Þ Amt t qt t ¼ ; AL/ Vp ð1:90Þ xD , tD , where Vp is the pore volume of the core.
100)) can be interpreted as describing the flow of two incompressible miscible fluids with identical properties such as water with two different colors (sometimes referred to as a blue and red water situation). Alternatively, the equations can be interpreted to describe the flow of immiscible fluids, in which case D represents the effect of dispersion due to geological heterogeneities. 54) to relate pressure, temperature and densities of the reservoir fluids. g. for the oil we can write, instead of Eq.
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. 106) can be rewritten in a similar fashion. 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.
A Systems Description of Flow Through Porous Media by Jan Dirk Jansen