By Jan Dirk Jansen
This article kinds a part of fabric taught in the course of a path in complex reservoir simulation at Delft collage of know-how over the last 10 years. The contents have additionally been offered at a variety of brief classes for commercial and educational researchers drawn to history wisdom had to practice study within the region of closed-loop reservoir administration, often referred to as clever fields, regarding e.g. model-based construction optimization, info assimilation (or background matching), version relief, or upscaling suggestions. each one of those themes has connections to system-theoretical concepts.
The introductory a part of the path, i.e. the structures description of circulation via porous media, kinds the subject of this short monograph. the most aim is to provide the vintage reservoir simulation equations in a notation that enables using options from the systems-and-control literature. even if the idea is proscribed to the rather uncomplicated state of affairs of horizontal two-phase (oil-water) stream, it covers numerous average facets of porous-media flow.
The first bankruptcy supplies a short overview of the elemental equations to symbolize single-phase and two-phase movement. It discusses the governing partial-differential equations, their actual interpretation, spatial discretization with finite ameliorations, and the therapy of wells. It includes famous thought and is essentially intended to shape a foundation for the subsequent 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 stream equations. The systematic use of matrix partitioning to explain the different sorts of inputs ends up in an outline by way of nonlinear ordinary-differential and algebraic equations with (state-dependent) process, enter, output and direct-throughput matrices. different themes comprise generalized state-space representations, linearization, removal of prescribed pressures, the tracing of circulate 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 normal differential equations for single-phase circulate. subsequent it strikes directly to the numerical answer of the two-phase move equations, overlaying quite a few features like implicit, particular or combined (IMPES) time discretizations and linked balance concerns, Newton-Raphson new release, streamline simulation, computerized time-stepping, and different computational facets. The bankruptcy concludes with uncomplicated numerical examples to demonstrate those and different elements corresponding to mobility results, well-constraint switching, time-stepping information, and system-energy accounting.
The contents of this short may be of price to scholars and researchers attracted to the appliance of systems-and-control techniques to grease and gasoline reservoir simulation and different purposes of subsurface stream simulation resembling CO2 garage, geothermal power, or groundwater remediation.
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Additional resources for A Systems Description of Flow Through Porous Media (SpringerBriefs in Earth Sciences)
O op op oSw ko À þ / ð1 À Sw Þðco þ cr Þ À ¼ 0: ð1:61Þ ox ox ot ot Here we introduced the water and oil mobilities kw , kkrw ðSw Þ lw ð1:62Þ ko , kkro ðSw Þ : lo ð1:63Þ and Addition of Eqs. 61) results in a PDE with only the pressure as primary variable7 À o op op kt þ /ct ¼0; ox ox ot ð1:64Þ where the total mobility kt, and the total compressibility ct have been defined as 6 kt , kw þ ko ; ð1:65Þ ct , Sw cw þ ð1 À Sw Þco þ cr : ð1:66Þ Absence of source terms corresponds to considering the (1-D) flow between an injector and a producer, in which case the flow is driven through the boundary conditions.
References Aarnes JE, Gimse T, Lie KA (2007) An introduction to the numerics of flow in porous media using Matlab. In: Hasle G, Lie KA, Quack E (eds) Geometric modeling, numerical simulation, and optimization; applied mathematics at SINTEF. Springer, Berlin Aziz K, Settari A (1979) Petroleum reservoir simulation. Applied Science Publishers, London Bear J (1972) Dynamics of fluids in porous media. Elsevier, New York. Reprinted in 1988 by Dover, New York Buckley SE, Leverett MC (1942) Mechanisms of fluid displacement in sands.
In: Hasle G, Lie KA, Quack E (eds) Geometric modeling, numerical simulation, and optimization; applied mathematics at SINTEF. Springer, Berlin Aziz K, Settari A (1979) Petroleum reservoir simulation. Applied Science Publishers, London Bear J (1972) Dynamics of fluids in porous media. Elsevier, New York. Reprinted in 1988 by Dover, New York Buckley SE, Leverett MC (1942) Mechanisms of fluid displacement in sands. Pet Trans AIME 146:107–116 Chen Z, Huan G, Ma Y (2006) Computational methods for multiphase flows in porous media.