Two cubic equations of state (EOS) have been adopted to compute multicomponent two-phase compressibility, CO 2 /water and hydrocarbon/water phase behavior, and gas-and liquid-phase densities. The equations used in this paper are the Schmidt-Wenzel (SW) EOS and the Peng-Robinson (PR) EOS. While these cubic equations have the same form, the SW is reported to be more accurate for predicting hydrocarbon gas-and liquid-phase densities. Density predictions are compared with experimental data to confirm the superiority of the SW EOS.The use of EOS to predict equilibrium phase compositions of water/hydrocarbon and water/C0 2 systems is discussed. For the water/hydrocarbon systems, the aqueous-phase interaction coefficient between water and the dissolved component shows a strong temperature dependency, while in the gas phases, a constant value of interaction coefficient is adequate. In the case of the CO 2 /water systems, the interaction coefficients for both the aqueous and gas phases show temperature dependency.A scheme to compute the two-phase compressibility of multicomponent reservoir fluid systems is also introduced. Our results show the expected sharp change in the compressibility during phase change. Such computations are required in some reservoir simulators.

This paper provides an analytical exposition of the relationship between two fully compositional, isothermal, three-phase, numerical simulator types in the literature. The Newton-Raphson (NR) method is based on the standard NR solution of the mass-or mole-conservation equations and associated constraints. The volume-balance (VB) method is based on the principle that, in each simulator block, the PV equals the combined fluid volume of all the phases. These two types of models have been regarded as different approaches to the same problem. An analytical demonstration, however, proves rather surprisingly that, although the two model types were developed from fundamentally different viewpoints (i.e., mass conservation vs. volume conservation), they lead to the same matrix system of the finite-difference equations. This understanding yields many advantages, including (1) methods to improve efficiency in both model types, (2) fully implicit and iterative forms of the VB model, (3) the assignation of physical meaning to groupings of terms from the NR Jacobian matrix, and (4) an understanding of how to decouple the mass-or mole-conservation equations from the thermodynamic equilibrium constraints. Also, a unified approach is possible that offers the best features of both simulation model types.

Molecular diffusion is a proven oil recovery mechanism in fractured reservoirs. Neglecting diffusion during the simulation of gas injections can lead to underestimated oil recoveries. This is particularly important in fractured reservoirs with low permeabilities. This paper presents the implementation of a diffusion model in a reservoir simulator. The implementation is verified with discrete fracture and dual-porosity model cases. The adopted diffusion model is based on irreversible thermodynamics. It uses full matrices of diffusion coefficients and chemical potential gradients as the driving force. To evaluate how much diffusion is dominant during fluid flow in fractured reservoirs, a form of the Péclet number is proposed as the ratio of characteristic times of diffusion to gravity drainage for a given reservoir and fluid properties. The incremental oil recovery from gas injection simulation results confirms the flow regime predicted by the Péclet number. This paper also examines the performance of simulations that involve diffusion by using various solution schemes, including explicit, fully implicit, and partially implicit methods. Optimal performance is achieved with the partially implicit method in which diffusion coefficients are updated at each timestep, while driving forces are updated inside Newton iterations. The simulation results also show that constant diffusivities might not provide a good representative for diffusion coefficients during gasflooding. They can cause oil recovery overestimation or underestimation. The authors demonstrate a technique to forgo diffusion calculations in the regions with convection-dominated flow regimes to help reduce computational time of the simulations involving diffusion. The speedup obtained for gas injection cases with a wide range of Péclet numbers is also examined.

Couyt.gnl 1987 Soc,ely of Pwfoloum EngmeefsTh,s paper was p!cparcw Iof prescmlabon a! !hc? timth SPE Sympcmum on Reswvo,r S,mu$alml hcl~m San Anlomo. Texas. February 1-4. 1987 Th,s P3PIY was selected 10, prwcn!alron by an SPE Program Comm,ltee fol10wIn9 rewew Of mfo,mahon contained m an abSIfaCt submitted by !De aulhOr(S) Contents ot me paper as presente' have not been rewewed by the SOc,e!y of Petroleum Engineers and are Sublecl 10 UJrfeC!lOn by Ihe i3tilh0 r(S) Tho mater.,1 as presenleci does not necessaf!ly IefleC! any posmon of lhe SOcfety of Pelroleum Engineers. IIS Otficers 01 members Papers p,esen!ed al SPE meetings are subject to pvbhcat:on rewew by Ed,lo,:al Comm,llees of Ihe Society of Petroleum Engmeets Permmon to copy IS ,eslt,c led 10 an abS1,W1 01 cot more man 30+3 woms Hlusl, atrons may nol be co!3:ed The absrracl should contain cons~,cuoua acknowledgment Of where and by wnom me paper ,s presented Wie Publications Manager SPE. P O Box 833836. R,chatclson. TX 75083-3836 Telex. 730989 SPFDAL ABSTRACTThis paper examines the iela:ive strengths and weaknesses of solving a fully compositional isothermal three phase numerical simulotor by two fundamentally different approaches. One of [hc me[hods considered is based on the standard Newton-R:iphson Froccdure where the partial derivatives for all the pertinent equa[ions are written wi[h respect to a selected set of primary variitbles in a Jacobian mm-ix. This model can be solved fully implicitly as CXNS did, or implicidy for pressure and explicitly for saturation and composition (IMPEM) m Fussull and Fussell or Young and Stephenson did. The altr.mative method, first proposed by Acs et al., is based on a volume balimce where in each simulator block, the pore volumes arc equaled with the fluid volumes. AS Originally prcsposcd, [his method was also an IMPEM one, but Waus later extended it 10 include a sequential step implicit in saturation. As yet, no computational results have been published for this volume balimce method.The Young and Stephenson model is compared with the l\lPEM method of Acs et al. as well as the sequential method of Watts. The main area rsf comparison is the relative ease or difficulty with which each simulator handles v@ous compositional problems. The advantages and disadvantages of usink either method under differen: circumstances are discussed. A: WCII, the performance of these simulators in their corresponding black oil forms is compared.Both mode!s can be run using either the Peng Robinson, So~ve-Rcdlich-Kwong, or Schmidt-Wenzel equations of state. All the derivatives needed in both mwtels can b-e solved explicitly from analytical expressions that have been derived from these equa[ions of state.

Today, most unconventional reservoir simulations are performed within a single-porosity framework. Very fine, high permeability gridblocks are used to represent the discrete fractures. Geometric spacing of blocks away from the fractures is necessary to capture the pressure gradients and simultaneously keep the number of gridblocks manageable. Without further subgridding the effects of desorption, diffusion, and transport through macro- and micro- fractures must be lumped resulting in a possible loss of transient fidelity. Yet these processes readily lend themselves to analysis with additional porosity types where the transport sequence can be preserved and studied. In this paper, we use a full production simulator to explore the advantages and disadvantages of simulating unconventional reservoirs with one-, two-, three-, and four-porosity systems. Kerogen, macro-fractures, micro-fractures, and an inorganic matrix are represented directly in the four-porosity system.