Skip Nav Destination
Filter
Filter
Filter
Filter
Filter

Update search

Filter

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

### NARROW

Peer Reviewed

Format

Subjects

Journal

Date

Availability

1-4 of 4

Keywords: saturation change

Close
**Follow your search**

Access your saved searches in your account

Would you like to receive an alert when new items match your search?

*Close Modal*

Sort by

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*14 (03): 216–220.

Paper Number: SPE-4276-PA

Published: 01 June 1974

..., practical goal was to avoid running problems that would be annoying and mysterious to the field reservoir engineer. SPEJ P. 216 1 6 1974 1 6 1974 1974. Society of Petroleum Engineers Fluid Dynamics

**saturation****change**relative permeability Upstream Oil & Gas calculation time...
Abstract

Introduction The semi-implicit reservoir simulator has become a very important part of the total simulation package necessary for the practicing reservoir engineer. Any multiphase simulation of a single well tom, problem (e.g., a study of water-oil coning) is very problem (e.g., a study of water-oil coning) is very expensive unless such a simulator is available. With a "standard" reservoir simulator, this difficulty arises because the relative permeabilities and capillary pressures, which depend upon saturations, lag one time step behind the pressure calculation. For this reason such a simulator is said to be implicit in pressure and explicit in saturation (abbreviated as IMPES). When "new" saturations are obtained, they are formed from "old" relative permeabilities and capillary pressures. The mathematical form of the equations is pressures. The mathematical form of the equations is such that an uncontrolled oscillation in the saturation values develops if the time step is too large. Only by taking smaller time steps can this oscillation be suppressed in an IMPES simulator, and very small time steps are then necessary for simulating coning behavior. The same problem can also appear in any production well model (in an IMPES simulator) that production well model (in an IMPES simulator) that distributes fluid production proportionally to phase mobilities. These saturation oscillations can be eliminated by making the well model "implicit in saturation." To overcome this instability, Blair and Weinaug developed a fully implicit simulator. All coefficients were updated iteratively until convergence occurred. It was necessary for them to stabilize their solution technique by the use of Newtonian iteration. Then it was found that a time-step limitation occurred because of nonlinearities, since the Newtonian iteration would not converge without a good initial estimate. Coats and MacDonald proposed an effective solution to this problem. They suggested estimating the relative permeabilities and capillary pressures by an extrapolation; e.g., pressures by an extrapolation; e.g., (1) A mathematical investigation showed that since Sn + 1, the saturation at be new time step, is found simultaneously with the pressures as part of the solution, the mathematical time-step limitation inherent in the IMPES technique as a result of using "old" relative permeabilities would not occur. They also suggested that the equations be linearized by dropping products of (Sn+1 - Sn) and (pn+1 - pn). The equations are then more nearly linear. Hence, the difficulties in convergence of the solution technique are greatly reduced (Newtonian iteration is not needed). Nolen and Berry showed that linearization of the accumulation terms was not necessarily the best strategy (in problems that have solution gas), because material-balance errors would result. They felt that linearization of the flux terms made little difference. Many questions still remain unanswered by these papers. Nonlinearities remain in the equations, papers. Nonlinearities remain in the equations, particularly when a phase is near its immobile particularly when a phase is near its immobile saturation. Because of the use of upstream weighting of the relative permeabilities, another type of nonlinearity (potential reversal) can occur. Furthermore, the question of a practical procedure for selecting the time step must be settled. Finally, there are nonlinearities in the well model, which can cause slow convergence, or failure to converge, especially when dealing with a well completed in several layers or with a well that changes constraints. The problems just mentioned are all more severe if large time steps are used. Reducing time-step size is expensive, and in many cases difficult to automate. In this paper we present our experience in treating or circumventing these problems. We have felt that an important principle to follow is to eliminate time-step limitations due to mathematical instabilities. Thus, one should be able to run steady-state problems with essentially unlimited time-step size. For transient problems, it is expected that time truncation errors would normally govern the time-step size. One final, practical goal was to avoid running problems that would be annoying and mysterious to the field reservoir engineer. SPEJ P. 216

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*11 (04): 419–425.

Paper Number: SPE-3360-PA

Published: 01 December 1971

... hysteresis in the relationship of relative permeability to saturation has been recognized for many yews. Geden et al. and Osoba et al. called attention to the occurrence of hysteresis and the importance of the direction of

**saturation****change**on the relative permeability-saturation relations. It is generally...
Abstract

Two-phase imbibition relative permeability was measured in an attempt to validate a method of calculating imbibition relative permeability. The stationary-liquid-phase method was used to measure several hysteresis loops for alundum and Berea sandstone samples. The method of calculating imbibition relative permeability is described, and calculated relative permeability curves are compared with measured curves. The calculated relative Permeability is shown to be a reasonably good Permeability is shown to be a reasonably good approximation of measured values if an adjustment is made to some necessary data. Due to the compressibility of gas, which is used as the nonwetting phase, a correction to the measured trapped gas saturation is necessary to make it agree with the critical gas saturation of the imbibition relative permeability curve. Introduction The existence of hysteresis in the relationship of relative permeability to saturation has been recognized for many yews. Geden et al. and Osoba et al. called attention to the occurrence of hysteresis and the importance of the direction of saturation change on the relative permeability-saturation relations. It is generally believed that relative permeability is a function of saturation alone for a permeability is a function of saturation alone for a given direction of saturation change, but that there is a distinct difference in relative permeability curves for saturation changes in different directions. The reservoir engineer should be aware of this hysteresis, and he should select the relative permeability curve which is appropriate for the permeability curve which is appropriate for the recovery process of interest. The directions of saturation change have been designated "drainage" and "imbibition" in reference to changes in the wetting-phase saturation. In a two-phase system, an increase in the wetting-phase saturation is referred to as imbibition, while a decrease in wetting-phase saturation is called drainage. The solution-gas-drive recovery mechanism is controlled by relative permeability to oil and gas in which the saturation of oil, the wetting phase, is decreasing. In waterflooding a water-wet reservoir rock, the saturation of water, the wetting phase, is increasing. These two sets of relative permeability curves, gas-oil and oil-water, do not have the same relationship to the wetting-phase saturation. This difference is not due to the difference in fluid properties, but is a result of the difference in properties, but is a result of the difference in direction of saturation change. The flow properties of the drainage and imbibition systems differ because of the entrapment of the nonwetting phase during imbibition. As drainage occurs, the nonwetting phase occupies the most favorable flow channels. During imbibition, part of the nonwetting phase is bypassed by the increasing wetting phase, leaving a portion of the nonwetting phase in an immobile condition. This trapped part phase in an immobile condition. This trapped part of the nonwetting phase saturation does not contribute to the flow of that phase, and at a given saturation the relative permeability to the nonwetting phase is always less in the imbibition direction phase is always less in the imbibition direction than in the drainage direction. The concept that some of the nonwetting phase is mobile and some is immobile during a saturation change in the imbibition direction previously was used to develop equations for imbibition relative permeability. In this development, it was assumed permeability. In this development, it was assumed that the amount of entrapment at any saturation can be obtained from the relationship between initial nonwetting-phase saturations established in the drainage direction and residual saturations after complete imbibition. The equations for imbibition relative permeability were not verified by laboratory measurements. The purpose of this report is m give the results of a laboratory study of imbibition relative permeability and to present a comparison of calculated relative permeability with relative permeability from laboratory measurements. permeability from laboratory measurements. In two-phase systems, hysteresis is more prominent in the relative permeability to the nonwetting phase than in that to the wetting phase. The hysteresis in the wetting-phase relative permeability is believed to be very small, and thus difficult to distinguish tom normal experimental error. SPEJ P. 419

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*8 (02): 149–156.

Paper Number: SPE-1942-PA

Published: 01 June 1968

...Carlon S. Land Relative permeability functions are developed for both two- and three-phase systems with the

**saturation****changes**in the imbibition direction. An empirical relation between residual nonwetting-phase saturation after water imbibition and initial nonwetting-phase saturations is found...
Abstract

Relative permeability functions are developed for both two- and three-phase systems with the saturation changes in the imbibition direction. An empirical relation between residual nonwetting-phase saturation after water imbibition and initial nonwetting-phase saturations is found from published data. From this empirical relation, expressions are obtained for trapped and mobile nonwetting-phase saturations which are used in connection with established theory relating relative permeability to pore-size distribution. The resulting equations yield relative permeability as a function of saturation having characteristics believed to be representative of real systems. The relative permeability of water-wet rocks for both two- and three-phase systems, with the saturation change in the imbibition direction, may be obtained by this method after properly selecting two rock properties: the residual nonwetting-phase saturation after the complete imbibition cycle, and the capillary pressure curve. Introduction Relative permeability is a function of saturation history as well as of saturation. This fact was first pointed out for two-phase flow by Geffen et al. and by Osaba et al. Hysteresis in the relative permeability-saturation relation also has been reported for three-phase systems. Since saturations may change simultaneously in two directions in a three-phase system, four possible relationships arise between relative permeability and saturation for a water-wet system. The four saturation histories of this system were given by Snell as II, ID, DI and DD. I refer to the direction of saturation change (imbibition and drainage), with the first letter of the symbol indicating the direction of change of the water phase. As used in this paper, the second letter of the symbol refers to the direction of saturation change of the gas phase, i.e., D and I indicate an increase and decrease, respectively, in gas saturation. Only a few three-phase relative permeability curves have been published. Leverett and Lewis published three-phase curves for unconsolidated sand, and Snell reported results of several English authors for both drainage and imbibition three-phase relative permeability of unconsolidated sands. Three-phase relative permeability curves for a consolidated sand were published by Caudle et al. for increasing water and gas saturations (ID). Corey et al. reported drainage (DD) three-phase relative permeability for consolidated sands. Recently, Donaldson and Dean and Sarem calculated three- phase relative permeability curves from displacement data on consolidated sands, also for saturation changes in the drainage direction. The only published three - phase relative permeability curves for consolidated sands with saturation changes in the imbibition direction (II) are those of Naar and Wygal. These curves are based on at theoretical study of the model of Wyllie and Gardner as modified by Naar and Henderson. Interest in three-phase relative permeability has increased recently due to the introduction of new recovery methods and refinements in calculation procedures brought about by the use of large-scale digital computers. The scarcity of empirical relations for three-phase flow, and the experimental difficulty encountered in obtaining such data, have made the theoretical approach to this problem attractive. RELATIVE PERMEABILITY AS A FUNCTION OF PORE-SIZE DISTRIBUTION Purcell used pore sizes obtained from mercury-injection capillary pressure data to calculate the permeability of porous solids. Burdine extended the theory by developing a relative permeability-pore size distribution relation containing the correct tortuosity term. SPEJ P. 149ˆ

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*5 (01): 15–24.

Paper Number: SPE-1011-PA

Published: 01 March 1965

... equation of state

**saturation****change**capillary behavior flow in porous media equilibrium pressure deficiency Probe quantity fluid modeling Upstream Oil & Gas hydraulic contact imbibition model capillary behavior porous material pendular ring enhanced recovery Fluid Dynamics experiment...
Abstract

The experimental points which describe capillary pressure curves are determined at apparent equilibria which are observed after hydrodynamic flow has ceased. For most systems, the time required to obtain equalization of pressure throughout the discontinuous part of a phase is prohibitive. To permit experimental points to be described as equilibria, a model of capillary behavior is proposed where mass transfer is restricted to bulk fluid flow. Model capillary pressure curves follow if the path described by such points is independent of the rate at which the saturation was changed to attain a capillary pressure point. A modified suction potential technique is used to study cyclic relationships between capillary pressure and moisture content for a porous mass. The time taken to complete an experiment was greatly reduced by using small samples. Introduction Capillary retention of liquid by porous materials has been investigated in the fields of hydrology, soil science, oil reservoir engineering, chemical engineering, soil mechanics, textiles, paper making and building materials. In studies of the immiscible displacement of one fluid by another within a porous bed, drainage columns and suction potential techniques have been used to obtain relationships between pressure deficiency and saturation (Fig. 1). Except where there is no hysteresis of contact angle and the solid is of simple geometry, such as a tube of uniform cross section, there is hysteresis in the relationship between capillary pressure and saturation. The relationship which has received most attention is displacement of fluid from an initially saturated bed (Fig. 1, Curve Ro), the final condition being an irreducible minimum fluid saturation Swr. Imbibition (Fig. 1, Curve A), further desaturation (Fig. 1, Curve R), and intermediate scanning curves have been studied to a lesser but increasing extent. This paper first considers the nature of the experimental points tracing the capillary pressure curves with respect to the modes and rates of mass transfer which are operative during the course of measurement. There are clear indications that the experimental points which describe these curves are obtained at apparent equilibria which are observed when viscous fluid flow has ceased; and any further changes in the fluid distribution are the result of much slower mass transfer processes, such as diffusion. Unless stated otherwise, this discussion applies to a stable packing of equal, smooth, hydrophilic spheres supported by a suction plate with water as the wetting phase and air as the nonwetting phase. SPEJ P. 15ˆ