... for all time; thus (9) lim r D → 1 [ p D ( r D ,   t D ) ] = 0 ;     t D > 0. The inner boundary condition was stated earlier. To obtain a solution in terms of the wellbore pressure, p wD , one could either rewrite the original PDE...

... petroleum engineer journal production control complex reservoir time period upstream oil & gas reservoir characterization flow in porous media boundary condition reservoir surveillance fluid dynamics samaniego-v We recently examined the response of vertically fractured wells in layered...

... front displacement flow in porous media combustion reservoir surveillance growth rate fluid dynamics porous medium linear analysis boundary condition equation disturbance enhanced recovery perturbation elliptical finger model aspect ratio mobility ratio production monitoring front...

... in porous media phase behavior fluid dynamics equation concentration logic programming oil displacement artificial intelligence chemical flooding methods surfactant concentration pvt measurement fleming boundary condition logic & formal reasoning relative permeability soc reservoir...

... pressure saturation equation water saturation equation saturation map upstream oil & gas heterogeneity effect vector field petroleum engineer journal boundary two-dimensional waterfiooding flow in porous media gravity reservoir simulation boundary condition enhanced recovery...

... upstream oil & gas necessary condition enhanced recovery boundary condition relation reservoir simulation constraint variation ox vector concentration weak minimum principle injection policy petroleum engineer journal In recent years, EOR techniques have received much attention...

... optimal injection policy costate equation dimensionless distance capillary number artificial intelligence boundary condition oil saturation viscosity dimensionless time surfactant concentration flow in porous media fluid dynamics chemical cost injection policy oil recovery equation...

... to surfactant flooding. In essence, this general theory is also based on the theory of characteristics, the application of which in chromatography is well advanced. In the special case of uniform initial and constant boundary conditions, the corresponding characteristic ground curves in the z,t plane...

... by ar variation in the vertical coordinate, and the fracture criterion, K i = K e , is satisfied at both major and minor axes. The net pressure at the tip, L f , of the long axis required by the boundary condition K i = K c does not seem crucial in determining fracture height or BHP (compare with one...

... multigrid method algorithm finest grid relaxation modeling & simulation boundary condition relaxation sweep test problem Simulation of large reservoirs or entire fields containing several thousand gridblocks entails solution of very large sets of linear equations. Because the amount...

... approach to the simulation of geothermal reservoirs. Use of this approach in a one-dimensional (ID) finite-difference simulator yields results that can be compared with a traditional numerical scheme. artificial intelligence catastrophe theory boundary condition derivative two-phase dome simulator...

... fluid inertia viscosity rod string artificial lift system plunger installation sucker rod polished-rod load coupling fluid column boundary condition valve upstream oil & gas frictional force traveling valve computer program pumping system friction factor dynagraph minimum polished...

... equation body surface time step approximation isaacson representation facet upstream oil & gas boundary condition equation wave force linear theory prediction free surface cylinder vector correspond computation variation closed-fonn solution procedure circular cylinder design wave...

... approximation velocity component fluid dynamics immiscible fluid flow boundary condition equation saturation numerical simulation simulation approximation boundary transport equation reservoir simulation miscible displacement difference equation concentration free surface pressure field...

... is the rock formation temperature. Figs. 1 and 2 compare simulation results for a test well using the insulated boundary condition and the wellbore water energy balance with several values of α . Fig. 1 shows wellbore shut-in temperatures at an arbitrary depth located above the injection zone...

... displacement dz 2 shear load drillstring design tangent tubing length upstream oil & gas helically buckled helix hammerlindl centerline dz dz exterior well tubing buckling behavior length change equilibrium equation boundary condition coefficient The buckling behavior of well...

... the sandface pressure and saturation boundary conditions can be calculated and how the well boundary itself can be replaced (approximately) by a source function. This paper and the following companion paper, SPE 9770, present a unified viewpoint of material, some of which may be already familiar to simulator...

... by solving Eq. 2 subject to appropriate boundary conditions. (4) z − η = 0. Figure 1 Cylindrical system. When a fluid displaces a more viscous fluid, the displacement front may become unstable, resulting in viscous fingering. This phenomenon raises both practical...

... show good agreement for cases where the wells are located at the block center regardless of the grid size or the position of the well block within the grid system. The paper also illustrates how pressure boundary conditions can be applied in cases where the wells are not located at the center of a grid...

... enhanced recovery Upstream Oil & Gas correspond fracture propagation hydraulic fracturing fracture face boundary condition Injection Rate waterflooding fluid loss replacement ratio waterflood-induced hydraulic fracture reservoir permeability Voidage Replacement Ratio propagation...