K.H. Coats

The Use of Vertical Equilibrium in Two-Dimensional Simulation of Three-Dimensional Reservoir Performance

The use of the Vertical Equilibrium (VE) concept in simulating heterogeneous reservoirs is discussed. Where VE criteria are met, this technique allows 2-dimensional simulation of 3-dimensional problems with equivalent accuracy, and with attendant substantial savings in data preparation and machine time. The study presents the VE concept itself and a new dimensionless group as a possible criterion for the validity of VE, as applied to thick reservoirs or to reservoirs where the capillary transition zone is a small fraction of thickness. A description of the generation of the appropriate pseudo relative permeability and capillary pressure curves is presented. In addition to the dimensionless group criterion, an actual comparison of the results of an x-z cross section and a one-dimensional areal run with VE illustrates the validity of the VE concept. Numerical results of such a comparison along with the attendant machine-time requirements are presented. More than an order of magnitude difference in machine-time requirements was experienced. Finally, an actual field case example shows the utility of VE as applied to a reservoir containing one or multiple gas pools residing on a common aquifer.

Application of a Regression-Based EOS PVT Program to Laboratory Data

An equation-of-state (EOS) based PVT program was applied in matching laboratory PVT data for three published and nine additional reservoir fluid samples. The paper includes laboratory test data for the nine samples. The paper describes PVT program features, especially regression, which the authors find conducive to rapid determination of EOS parameter values necessary in matching data. With regression, both the Peng-Robinson and Zudkevitch-Joffe Redlich-Kwong EOS give comparable and generally good agreement with laboratory data. Without regression or significant adjustment of EOS parameters, neither EOS adequately predicts observed reservoir fluid PVT behavior. The EOS tuning approach described and illustrated here utilizes a small degree of C7+ fraction splitting. The agreement of these EOS results with data compares favorably with that obtained in previously published studies which utilized extensive, tuned C7+ splitting.

Implicit Compositional Simulation of Single-Porosity and Dual-Porosity Reservoirs

cubic equation-of-state 1 (EOS) forrepresentation of gas-oil phaee equilibriaand deneitiee. The generalized EOS This paper describeean iraplicit numerical model for repreaente the Redlich-Kw ong2 Soave-Redlich-Kw ong3 com positionalaimulation of single-porosityand dual.! Zudkevilch-JoffeRedlich-Kwong 95, and Peng-Robinsond poroeityoilor gaa condensatereservoirs.A 3-component E OS. A tabular,pressure-dependentK-valueoptionprovides equation-of-state com positionalapproach ia proposed as a an alternativeto EOS usage. EOS parametesaare obtained desirablealternativeto extended black oil modelling, using a regression-basedpVT program~. Different requiringlittlemore computing time than the letter. The parameter setsare ueed forrese~oir and surfaceseparation approachiaillustrated foran actualneas-critical volatileoil calculations. This eases the burden of determining reservoir. A aimple method for reducing implicit formulation time truncation error is described and parameters and increasesEOS accuracy at reservoirand surface conditions. Viscositiesare calculatedfrom the illustrated. A new bottomhole constraintfunction is Lohrenz et al correlationsand interfacieltension ia describedfor betterpreservationof productionwelltarget obtainedusingthe M acLeod-Sugdenm ethodg. ratee in com positionalmodels. A new matrix-fracture transferformulationincludingmatrix-fracturediffusionis The model simulates1-,2and 3-dimenaiortel flow in preeentedfor the dual-porosity description; itsaccuracy is Cartesianor cylindrical coordinates.Darey’sLaw modified examined in connection with severaltest problema where by relativepermeabilityand capillarypressurerepresenta correct reeulte are available from single-porosity the viacoue.,capillaryand gravity forcee. Effects of simulation. Results are discussedfor a 3D 600-block interracial tensionon capillarypressureare included. The aimulationof a highlyfracturednear-critical volatileoil model appliesto depletion,water injection,cycling(gas reeervoir. injection), and enrichedgas/solventinjectionoperationsin reeervoixtypes ranging from black oil to near-critical Introduction volatile oil and condensate to lean gas condensate. Applicationsincludeaimulationof laboratoryexpenm enta, This paper describeea fullyimplicitnumerical model cylindncel-coordinate single-wel.1etudiasand are~ crosefor compositionalaimulationof m ultidimenaion~ threasectionalor 3D field-scale studiee. phaae flow in single-porosityand naturally fractured reservoirs.A general descriptionof the model is given, Implicit formulafions generally have a tendency followedby a sectiongivingmore detailregardingcertain toward greatacnumericaldispersioneffectsthan theIX PES features. The model equationsare then preeented. The formulation. A dispersion-control feature is described major em phaaia here relates to the fracturedreservoir which reduces sensitivityof reaultato time step aiae in application. Therefore, the rem ainder of the paper some caees. describesa new matrix-fracturefluidtransferformulation and estimatesitsaccuracyin connectionwith a number of Production well rate is allocatedamong Layers by exam plaor testproblems. pressure and mobility,includingan implicitbottomhole constraintreatmentto preserveapecifiadtargetrate. The GeneralDescriptionof the Model well rate terms involved are implicitin all variables: com positions, saturationsand pressure.A new formulation The model iE fullycom poaitimmlwith a generalized for the impw bottomhole target rate constraintgives better preeenrationof specifiedrate for the case of R&erences and illustrations at end ~f paper. compositionalsimulation,

Simulation of Three-Dimensional, Two-Phase Flow In Oil and Gas Reservoirs

Two computer-oriented techniques for simulating the 3-dimensional flow behavior of 2 fluid phases in petroleum reservoirs were developed. Under the first technique the flow equations are solved to model 3-dimensional flow in a reservoir. The second technique was developed for modeling flow in 3-dimensional media that have sufficiently high permeability in the vertical direction so that vertical flow is not seriously restricted. Since this latter technique is a modified 2-dimensional areal analysis, suitably structured 3-dimensional reservoirs can be simulated at considerably lower computational expenses than is required using the 3-dimensional analysis. A quantitative criterion is provided for determining when vertical communication is good enough to permit use of the modified 2-dimensional areal analysis. The equations solved by both techniques treat both fluids as compressible, and, for gas-oil applications, provide for the evolution of dissolved gas. Accounted for are the effects of relative permeability, capillary pressure and gravity in addition to reservoir geometry and rock heterogeneity.

Compositional and black oil reservoir simulation

This paper describes a three-dimensional, three-phase reservoir simulation model for black oil and compositional applications. Both IMPES and fully implicit formulations are included. The model`s use of a relaxed volume balance concept effectively conserves both mass and volume and reduces Newton iterations. A new implicit well rate calculation method improves IMPES stability. It approximates wellbore crossflow effects with high efficiency and relative simplicity in both IMPES and fully implicit formulations. Multiphase flow in the tubing and near-well turbulent gas flow effects are treated implicitly. Initial saturations are calculated as a function of water-oil and gas-oil capillary pressures which are optimally dependent upon the Leverett J function or initial saturations may be entered as data arrays. A normalization of the relative permeability and capillary pressure curves is used to calculate these terms as a function of rock type and grid block residual saturations. Example problems are presented, including several of the SPE Comparative Solution problems and field simulations. 48 refs.

Three-Dimensional Simulation of Steamflooding

This paper describes a 3-dimensional model for numerical simulation of steam injection processes. The model describes 3-phase flow of water, oil, and steam and heat flow in the reservoir and overburden. The method of solution effects simultaneous solution of the mass and energy balances and eliminates the need for iterating on the mass transfer (condensation) term. Laboratory data are reported for steamfloods of 5,780 cp oil in a 1/4 five-spot sand pack exhibiting 3-dimensional flow effects. These experiments provide additional data for the purpose of checking accuracy and assumptions in numerical models. Comparisons of model results with several sets of experimental data indicate a need to account for effects of temperature on relative permeability. Calculated areal conformance of a steamflood in a confined 5-spot is strongly dependent upon the alignment of the x-y grid axes relative to the diagonal joining injection and production wells. It has not been determined which, if either, of the 2 grid types yields the correct areal conformance. (30 refs.)

Simulation of Gas Condensate Reservoir Performance

Generalized phase density and component fugacity equations are presented which represent several of the widely used cubic equations-of-state (EOS). These former equations are obtained by manipulation of Martins generalized equation-of-state and are applied in this study. A component pseudoization procedure preserves densities and viscosities of the pseudo components and original mixture as functions of pressure and temperature. This procedure is applied with material balance requirements in generation of 2-component, black oil properties for gas condensates. Agreement between resulting black oil and full conpositional simulations of gas condensate reservoir depletion is demonstrated for a very rich, near-critical condensate. Agreement also is shown between EOS compositional results and experimental laboratory expansion data for 2 rich condensates. The full compositional simulation necessary for below-dewpoint cycling is performed for the near-critical condensate using a wide range of component pseudoizations. 21 references.

IMPES Stability: The CFL Limit

This work considers cocurrent, three-dimensional, single-phase miscible and two-phase immiscible, hyperbolic flow in a general grid, structured or unstructured. A given grid block or control volume may have any number of neighbors. Heterogeneity, anisotropy, and viscous and gravity forces are included, while tensor considerations are neglected. The flow equations are discretized in space and time, with explicit composition and mobility used in the interblock flow terms (the Impes case). Published stability analyses for this flow in a less general framework indicate that the CFL number must be < 1 or < 2 for stability. A recent paper reported non-oscillatory stability of oneand two-dimensional Buckley-Leverett two-phase simulations for CFL < 2. A subsequent paper claimed to predict this CFL < 2 limit from a stability analysis. This work gives a different reason for that stability up to CFL < 2. This work shows that the eigenvalues of the stability matrix are equal to its diagonal entries, for any ordering scheme. The eigenvalues are in turn equal to 1-CFLi, which leads to a conclusion of an early paper that CFL < 1 is required for non-oscillatory stability. CFL values between 1 and 2 give oscillatory stability. In general, our Impes simulations require the non-oscillatory stability ensured by CFL < 1.

Prediction of polymer flood performance

This experimental and numerical study was performed to estimate the incremental oil recovery by pattern polymer flooding in a California viscous oil reservoir. Results indicate that adding 270 ppm Kelzan to the normal flood water will boost oil production by 42% (at 1 PV injected) and sharply reduce water handling costs. This corresponds to

Carbon Dioxide Well Stimulation: Part 1-A Parametric Study

8.35 incremental oil/