By Jan Dirk Jansen
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 awarded at a number of brief classes for commercial and educational researchers attracted to heritage wisdom had to practice learn within the zone of closed-loop reservoir administration, sometimes called clever fields, regarding e.g. model-based creation optimization, facts assimilation (or background matching), version relief, or upscaling thoughts. every one of those issues has connections to system-theoretical concepts.
The introductory a part of the direction, i.e. the platforms description of move via porous media, varieties the subject of this short monograph. the most target is to give the vintage reservoir simulation equations in a notation that enables using strategies from the systems-and-control literature. even supposing the speculation is restricted to the quite easy scenario of horizontal two-phase (oil-water) move, it covers numerous normal points of porous-media flow.
The first bankruptcy provides a quick evaluate of the elemental equations to symbolize single-phase and two-phase circulation. It discusses the governing partial-differential equations, their actual interpretation, spatial discretization with finite ameliorations, and the therapy of wells. It comprises famous idea and is essentially intended to shape a foundation for the subsequent bankruptcy the place the equations should be reformulated when it comes to systems-and-control notation.
The moment bankruptcy develops representations in state-space notation of the porous-media move 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) method, enter, output and direct-throughput matrices. different issues comprise generalized state-space representations, linearization, removal of prescribed pressures, the tracing of movement strains, raise tables, computational elements, 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 move. subsequent it strikes directly to the numerical resolution of the two-phase stream equations, masking a variety of 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 basic numerical examples to demonstrate those and different elements reminiscent of mobility results, well-constraint switching, time-stepping facts, and system-energy accounting.
The contents of this short can be of worth to scholars and researchers attracted to the appliance of systems-and-control thoughts to grease and fuel reservoir simulation and different functions of subsurface movement simulation reminiscent of CO2 garage, geothermal strength, or groundwater remediation.
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Additional resources for A Systems Description of Flow Through Porous Media
2), and in this text we will therefore make use of both formulations. 1 State Equations A more concise form of Eq. 3) can be obtained through the use of vector notation. We introduce the vectors x ¼ ½ x1 x2 . . xn T and u ¼ ½ u1 u2 . . um T to represent the discrete values of the dependent variables, instead of i and wi 2 Typically, most of the values i are equal to zero in a single equation. For example in the case of one-dimensional single-phase flow modeled with first-order finite differences, the only three non-zero values i in the ith equation in a system of n equations with 1 \ i \ n are given by: ^ei ði ; dði Þ=dtÞ ¼ ^fi ðiÀ1 ; i ; iþ1 ; wi Þ: 42 2 System Models which we used until now.
3 Relative Permeabilities The saturation-dependency of the relative permeabilities is usually determined from laboratory experiments where water is forced through a small core (a cylindrical piece of rock) initially saturated with oil. The values to be used in reservoir simulation are typically provided in the form of tables or simple mathematical expressions with parameters that have been fitted using the experimental results. Several of such expressions are known in the literature. Here we use the socalled Corey model given by 0 nw krw ¼ krw S ; ð1:74Þ 0 kro ¼ kro ð1 À SÞno ; ð1:75Þ where S is a scaled saturation defined as S, Sw À Swc ;0 1 À Sor À Swc S ð1:76Þ 1; 0 0 and kro are the end-point relative permeabilities, nw and no are the Corey krw exponents, Swc is the connate-water saturation and Sor is the residual-oil saturation.
73) is a first-order nonlinear hyperbolic equation, with a non-constant coefficient mt = mw ? mo which depends on the pressure according to Eqs. 69). In theory, the coupled Eqs. 73) are therefore both nonlinear. However, because the changes in saturations usually occur much slower 8 fw ðSw Þ is sometimes referred to as the flux function. 20 1 Porous-Media Flow than the pressure changes,9 the nonlinearity in the pressure Eq. 64) is weak, and the equation may often be considered as a linear one with slowly-varying coefficients.
A Systems Description of Flow Through Porous Media by Jan Dirk Jansen