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Go**Summary**

For decades the effect of physical dispersion (in-situ mixing) in porous media has been of interest in reservoir engineering and groundwater hydrology. Dispersion can affect the development of multi-contact miscibility and bank breakdown in enriched gas drives and miscible solvent floods of any mobility ratio.

The magnitude or extent of dispersion is quantified by the rock property physical dispersivity (a) which is the order of 0.01 ft for consolidated rocks and several times smaller for sand-packs, from many laboratory measurements.

Numerical studies of the effect of dispersion on enriched gas drives and field tracer tests often use input values of dispersivity 100 to 1000 or more times larger than ~0.01 ft. These large input dispersivity values stem from large apparent dispersivities (a_{a}) determined by matching the one-dimensional convection-diffusion (1D CD) equation to production well effluent tracer concentration profiles observed in field tracer tests.

The large apparent dispersivities reflect conformance or other behavior not governed by the 1D CD equation and should not be used to justify large physical dispersivity as input to numerical studies. This paper shows that large apparent dispersivities observed in field tests can result with physical dispersivity no larger ~0.01 ft lab-measured value.

Heterogeneity alone (no physical dispersivity or molecular diffusion) causes no in-situ mixing and cannot explain observed large apparent echo dispersivities. Large apparent echo dispersivities for two reported field tracer tests are shown to result from the effect of drift alone with no dispersion.

The widely reported scale-dependence of apparent dispersivity is a simple and necessary consequence of mis-applying the 1D CD equation, with its single parameter of Peclet number *L*/a, to conformance it does not describe. Apparent dispersivity is scale-dependent but physical dispersivity is a rock property independent of scale and time.

apparent dispersivity, chemical tracer, concentration, conformance, dispersion, dispersivity, drift, enhanced recovery, equation, field, formation evaluation, heterogeneity, injection, input dispersivity, model, Muskat, permeability, physical dispersivity, profile, reservoir description and dynamics, solution, stratification, tracer test analysis, Upstream Oil & Gas

SPE Disciplines:

This paper describes methods for the simulation of first contact miscible, multi-contact miscible and vaporizing gas drive, which include logic to account for bypassed oil. Dispersion control is included for first contact miscible, FCM, cases.

First contact miscibility may be simulated in fully compositional mode or in pseudo two-component EOS mode. Pseudoization is performed internally in the model so that hydrocarbon fluid density and viscosity, as functions of pressure and composition, are the same as calculated in NC-component mode. Bypassed oil and dispersion control are based on an extension of Koval's method. This procedure allows the user to adjust the fractional flow of oil during upscaling or history matching to match fine grid or historical results. This feature can be used to simulate both WAG and tertiary recovery projects where solvent injection is preceded by a water injection period.

Example cases are included to illustrate the techniques presented in this paper. Results are also given on the efficiencies of the algorithms.

Oilfield Places: North America > United States > Alaska > North Slope > Prudhoe Bay Oil Field (0.99)

**Summary**

This work considers cocurrent, 3D, single-phase miscible and two-phase immiscible, hyperbolic flow in a general grid, structured or unstructured. A given gridblock 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 (implicit pressure, explicit saturation/ concentration) 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 stable 1- and 2D Buckley- Leverett two-phase simulations for CFL limits up to 2. A subsequent paper presented a stability analysis predicting a CFL limit of 2 for one of those simulations. This work gives a different reason for that stability up to a CFL limit of 2.

This work shows that the eigenvalues of the stability matrix are equal to its diagonal entries which, for any ordering scheme, are in turn equal to 1-CFLi. This leads to a conclusion of an early paper that CFL < 1 is required for nonoscillatory stability. This paper discusses cases in which larger CFL limits between 1 and 2 exhibit stability, the existence or absence of applicable theory in such cases, and the practical contribution of such larger CFL limits to increased model efficiency.

SPE Disciplines:

**Summary**

An IMPES stability criterion is derived for multidimensional three-phase flow for black-oil and compositional models. The grid may be structured or unstructured. Tensor considerations are neglected. The criterion can be used to set the time steps in an IMPES formulation or as a switching criterion in an adaptive implicit model.

The criterion extends previous work by accounting for three-phase flow, including capillary, gravity, and viscous forces, with all the possible cocurrent and countercurrent flow configurations in a general grid. The criterion derivation uses stability theory to the limits of its applicability, augmented by numerical experimentation, including extensive 1D tests and numerous field study datasets.

SPE Disciplines:

- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Multiphase flow (0.73)
- Reservoir Description and Dynamics > Formation Evaluation & Management (0.73)
- Reservoir Description and Dynamics > Reservoir Simulation (0.53)
- Reservoir Description and Dynamics > Fluid Characterization > Fluid modeling, equations of state (0.53)

Abstract

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 one- and 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-CFL_{i}, 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.

1. Introduction

The Impes formulation^{1-3} treats interblock flow rates implicitly in pressure, but explicitly in saturations and compositions. This explicit treatment gives rise to a conditional stability,

Equation (1)

*F _{i}* is some function of rates and/or reservoir and fluid properties associated with grid block

The Appendix gives a brief derivation of the well known explicit difference equation

Equation (2)

which describes one-dimensional (1D) two-phase flow. For gas-oil flow, *D _{i}* is (

Prior to 1950, mathematicians developed stability analyses for Eq. 2. Subsequent work used their methods and results to derive stability conditions for Impes.^{4-11} In 1968^{4} the following stability conditions were derived for 1D, 2D, or 3D flow:

Equation (3)

Equation (4)

Those conditions and the additional following result, when both *D _{i}* and

Equation (5)

Todd *et al *stated their Condition 5 applied if the total flow rates *q _{ x}*,

**Abstract **

An Impes stability criterion is derived for multidimensional three-phase flow for black oil and compositional models. The grid may be structured or unstructured. Tensor considerations are neglected. The criterion can be used to set the time steps in an Impes formulation or as a switching criterion in an adaptive implicit model.

The criterion extends previous work by accounting for three-phase flow, including capillary, gravity and viscous forces, with all the possible cocurrent and countercurrent flow configurations in a general grid. The criterion derivation uses stability theory, to the limits of its applicability, augmented by numerical experimentation, including extensive one-dimensional tests and numerous field study datasets.

**Introduction **

A reservoir simulation model consists of nonlinear *N N _{c}* difference equations which express conservation of mass of

Equation (1)

where *M _{I,n}* is the mass of component

The Impes formulation^{1,2} treats the interblock flow rates implicitly in pressure, but explicitly in saturations and compositions. This explicit treatment gives rise to a conditional stability for Impes,

Equation (2)

where Ä*t* is maximum stable timestep and *F _{i}* is some function of rates and reservoir and fluid properties. This paper derives the function

Two functions *F _{i}* are derived for use in the Condition 2. The first relates to effects of explicit treatment of the saturation-dependent terms (relative permeability and capillary pressure) in the interblock flow rates. The second relates to the explicit treatment of compositions in the interblock flow rates.

Derivations of the function *F _{i}* are lengthy and, at various points, tedious. This tends to obscure the simplicity and low cpu expense associated with the final results. Therefore, a Summary section gives the final results, followed by sections describing the derivations.

**Summary **

For the unstructured grid case, the subscript *i* denotes a grid block and the subscript *j* denotes one of its neigbors. Deriva-tions using a Cartesian grid use subscipts *i, j, k* as the grid block indices in the *x, y*, and *z* directions, respectively. In all equa-tions throughout the paper, each phase mobility and its derivatives are evaluated at the upstream block for the phase.

SPE Disciplines:

- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (0.87)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Multiphase flow (0.73)
- Reservoir Description and Dynamics > Fluid Characterization > Fluid modeling, equations of state (0.73)

**Summary **

The generalized IMPES method applies to simulation models involving any number of conservation equations. The IMPES pressure equation is a linear combination of the linearized conservation equations. This article shows the generality, simplicity, uniqueness, and derivational brevity of that equation. The associated IMPES reduction vector leads directly to the value of total compressibility in a multiphase gridblock. That compressibility in turn gives several error checks on black oil pressure/volume/temperature (PVT) data.

Three IMPES-type compositional models are compared and found to be very similar, with moderate differences in efficiencies and generality. An example problem serves two comparative purposes related to IMPES model efficiency.

**IMPES Method **

The acronym IMPES was used in 1968^{1} in a description of a numerical model for simulating black oil reservoir behavior. The IMPES method was generalized in 1980^{2} to apply to simulation models involving any number *n* of conservation equations, e.g., thermal, chemical flood, and compositional models. The basic principle of the method is elimination of differences in nonpressure variables from the model's set of *n* conservation equations to obtain a single pressure equation. This principle was attributed to Stone^{3} and Sheldon *et al.*^{4,5} used the same principle in deriving the total compressibility of multiphase black oil systems. Perhaps the first black oil IMPES model was published by Fagin and Stewart in 1966.^{6}

Later articles^{7-13} presented and compared derivations of the IMPES pressure equation for *n*-equation compositional models. The length of some of those presentations tends to obscure the fact that the IMPES pressure equation is unique, independent of the manner of derivation, choice and ordering of variables, and ordering of equations.

This article demonstrates the simplicity and uniqueness of the IMPES pressure equation, and the brevity of its derivation for the general *n*-equation model. A simple relationship is shown relating total compressibility of multiphase systems to the IMPES reduction vector.^{2} This compressibility yields error checks on black oil PVT data. Several IMPES compositional models are compared, with emphasis on their similarity. Their relative efficiencies are estimated. An example problem serves some comparative purposes.

**General Model Equations **

The numerical model consists of *n+N* equations written for each gridblock. The first *n* (primary) equations express conservation of *n* species, M {i}^{n+1} - M {i}^{n}=Q {i}\Delta t,\quad i=1,2,..,n,\eqno ({\rm 1}) where *n* denotes the timestep level and *Q _{i}* represents interblock flow and well terms. One of the

A set of *n+N* variables \{P {j}\} can always be found or chosen such that each *M _{i}* is a unique function of one or more of the

**Reduced Model Equations **

Writing Eq. 3 in matrix form gives an *n+N×n+N* matrix *G*, partitioned as shown in **Fig. 1**. The elements *Q _{i}* of the

**IMPES Pressure Equation ^{2} **

The IMPES pressure equation is obtained by multiplying the *i*th equation of Eq. 4 by *X _{i}* and summing the resulting

*This paper was prepared for presentation at the 1999 SPE Reservoir Simulation Symposium held in Houston, Texas, 14-17 February 1999.*

*This paper (SPE 50990) was revised for publication from paper SPE 29111, first presented at the 1995 SPE Symposium on Reservoir Simulation, 12-15 February. Original manuscript received for review 15 February 1995. Revised manuscript received 20 May 1998. Paper peer approved 3 June 1998.*

*Summary*

*This paper describes a three-dimensional (3D), three-phase reservoir simulation model for black oil and compositional applications. Both implicit pressure, explicit saturation/concentration (IMPES) and fully implicit formulations are included. The relaxed volume balance concept effectively conserves mass and volume and reduces Newton iterations in both formulations. 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 non-Darcy gas flow effects are treated implicitly.*

*Initial saturations are calculated as a function of water/oil and gas/oil capillary pressures, which are optionally dependent upon the Leverett J function. A normalization of the relative permeability and capillary pressure curves is used to calculate these terms as a function of rock type and gridblock residual saturations.*

*Example problems are presented, including several of the SPE comparative solution problems and field simulations.*

*P. 372*

- North America > Canada > Alberta > Carson Creek Oil Field (0.99)
- Europe > Norway > North Sea > Central North Sea > Ekofisk Oil Field (0.99)
- North America > United States > Arkansas > Smackover Oil Field (0.98)

*SPE Members*

*Abstract*

*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 nearwell turbulent gas flow effects are treated implicitly.*

*Initial saturations are calculated as a function of water-oil and gas-oil capillary pressures which are optionally 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.*

*Introduction*

*This paper describes a numerical model for simulating three- dimensional, three-phase flow in heterogeneous, single-porosity reservoirs. The model, which is referred to as Sensor, incorporates black oil and fully compositional capabilities formulated in both IMPES and fully implicit modes. The formulations include a relaxed volume Concept and a new method for implicit treatment of well rates with wellbore crossflow. Following model description, several example problems are presented. They include five SPE Comparative Solution Project problems, a turbulent gas flow problem, a crossflow problem, and three field studies.*

*General Description of the Model*

*The model simulates three-dimensional, three-phase flow in heterogeneous, single-porosity porous media. The usual viscous, gravity and capillary forces are represented by Darcy's law modified for relative permeability. The flow is isothermal although, as an option, a spatially variable, time invariant temperature distribution may be specified in the compositional case.*

*The conventional seven-point orthogonal Cartesian xyz grid and the cylindrical grid are used. Mapping or linear indexing is used to require storage and arithmetic only for active grid blocks.*

*The model includes both black oil and fully compositional capabilities. The black oil option includes the , stb/scf term as well as the normal solution gas term. It therefore applies to gas condensate and black oil problems. Interfacial tension, modifying gas-oil capillary pressure, is also entered versus pressure in the black oil PVT table.*

*P. 149*

- Europe > Norway > North Sea > Central North Sea > Ekofisk Oil Field (0.99)
- North America > United States > Texas > Permian Basin > Cowden Oil Field (0.98)
- North America > United States > Arkansas > Smackover Oil Field (0.98)
- Africa > Middle East > Libya > Phillips Oil Field (0.98)