Features of Three-Component, Three-Phase Displacement in Porous Media

Abstract

Summary A theory of 1D dispersion-free displacement in porous media is used toreveal features of floods in which three phases flow simultaneously. Theanalysis shows how the sequence of compositions realized in given classes ofproblems can be recognized from a single diagram. It also explains whythree-phase routes of physical interest shock into and out of tie-triangles, why composition routes measured during relative permeability experiments oftenare identical through the three-phase region, and how an equilibrium tertiarydisplacement can evoke either an oil bank that runs ahead of the phasecontaining the active agent or one that breaks through with it. Introduction Three-phase flow arises in many commercial oil recovery processes, includingnatural or implemented waterfloods in oil reservoirs containing free gas;waterfloods in gas-condensate reservoirs; and such enhanced recovery methods assteam, immiscible gas, and surfactant injections. Although three-phase flowoccurs often in practice, there have been relatively few fundamentalinvestigations practice, there have been relatively few fundamentalinvestigations into understanding the roles it plays in oil recovery processes. The theory and methods delineated by Helfferich (called coherence theory) canmake known salient features of multicomponent, multiphase displacement inporous media. In coherence theory, which can be considered a generalization of Buckley-Leverett and fractional-flow theories, equations governing conservationof mass, momentum, and energy during idealized displacements in porous mediaare formulated and solved. As described in Ref. 6, the first step in solving agiven problem with coherence theory is to construct a composition path problemwith coherence theory is to construct a composition path grid, which depictssolutions (paths) to differential conservation equations before initial andboundary conditions are applied. After the composition path grid is determined, the next step is to find the composition route that will be followed underspecified initial and boundary conditions. A route shows the sequence ofcompositions (and their velocities) that arises when fluids of one compositionare injected into a system initially containing fluids of another composition. A few rules make determining routes in coherent solutions straightforward. Routes run "exclusively along paths" (Rule 1), and a route fromboundary to initial conditions "follows the paths in the sequence ofincreasing wave velocities" (Rule 2), Consequently, of all the pathsassociated with the composition corresponding to the boundary condition, theroute first takes the one having the lowest velocity. It then switches toprogressively faster paths until it ends at the initial condition progressivelyfaster paths until it ends at the initial condition on the path with thehighest velocity. If the velocities along a path in a route from the boundaryto the initial condition decrease, the resulting self-sharpening wave developsinto a shock (a discontinuous change in composition). To describe shocks, thedifferential conservation equations must be replaced by integral ones. Thedifferential coherence condition (Eq. 14 in Ref. 6) is superseded by anintegral condition, (1) must be the same for all j at the shock. To be an admissible solution, thewave velocities in a shock-containing route from boundary to initial conditionsmust be encountered in a nondecreasing sequence (Rule 3). Because the equationsof coherence theory are formulated without dispersive terms, however, manyadmissible shock-containing solutions can be constructed. From our experience, the physically correct route is the one that includes the fastest shock (Rule4). The reason is the same as that used to justify which from among possibleshocks are included in Buckley-Leverett solutions to two-phase flow problems(which can be shown graphically). Dispersion in solutions problems (which canbe shown graphically). Dispersion in solutions containing less stable shocksproduces compositions that form new, faster-moving shocks. The shockpropagating most rapidly is the only one that is stable during disturbancescaused by dispersion. Once the composition route is established, adistance/time diagram can be constructed. From a distance/time diagram, profiles, which are the distributions of phases or components in the porousmedium at a specified time, and histories, which are plots of the fractionalflow of components or phases at a given location as a function of time, can bedetermined easily; these outputs commonly are generated when numericallysolving the equations governing flow in porous media. In what follows, weillustrate features of composition routes in three-component, three-phasesystems in which relative permeabilities differ from straight-linefunctionalities. Rules permeabilities differ from straight-linefunctionalities. Rules for finding routes through two-phase zones are extendedso that they apply to routes through tie-triangles. Because three-phase routesof physical interest always contain shocks, we demonstrate how to constructplots that facilitate identifying routes. We subsequently use such plots toreveal the types of routes that arise in three-component, three-phasedisplacements. In doing so, we calculate specific routes, corroborate them withsolutions obtained with a finite-difference technique, and explain severalexperimentally observed phenomena. Last, we outline how the methodology andfindings reported here pertain to floods that encompass more than justthree-phase flow. Composition Routes Within Tie-Triangles Amplification of Rules for Determining Routes. In following Rules 1 and 2stated earlier, finding the route that a system takes under (constant) boundaryand (uniform) metal conditions can be as simple as tracing two curves throughthe composition path grid. If the slow path passing through the compositioncorresponding to the boundary condition (that is, the composition in the porousmedium that produces the fractional flow being injected) intersects the fastpath containing the initial condition, the route (1) begins at the pointcorresponding to the injected fluid, (2) follows the slow path through thispoint until this path intersects the fast path through the initial condition, and (3) tracks this fast path until it (4) ends at the initial condition. Theseprocedures (together with the application of Rules 3 and 4 to determine theproperties of any shocks that arise in the solution) suffice for determiningroutes in three-component, two-phase and constant-phase-velocity-ratiothree-phase displacements. As shown in Ref. 6, however, every slow path doesnot intersect every fast path in composition path grids for more generalthree-component, three-phase problems (nonlinear relative permeability curves). Thus, if the slow path through the boundary permeability curves). Thus, if theslow path through the boundary condition does not cross the fast path throughthe initial condition, the construction outlined above cannot be made. In suchcases, the only routes that do not involve switching from a faster path to aslower one incorporate one of the "special" paths path to a slower oneincorporate one of the "special" paths (described in Ref. 6 and shownin Fig. 6 therein) through the composition with the minimum mobility. SPERE P. 426