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

1971 ◽  
Vol 11 (01) ◽  
pp. 63-71 ◽  
Author(s):  
K.H. Coats ◽  
J.R. Dempsey ◽  
J.H. Henderson
Keyword(s):  
Capillary Pressure ◽  
Transition Zone ◽  
Relative Permeability ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensionless Group ◽  
Machine Time ◽  
Vertical Equilibrium ◽  
Time Requirements ◽  
Capillary Pressure Curves

Abstract This paper discusses the use of the Vertical Equilibrium (VE) concept in simulating heterogeneous reservoirs. Where VE criteria are met, this technique allows two-dimensional (2-D) simulation of three-dimensional (3-D) problems with equivalent accuracy, and with attendant substantial savings in data preparation and machine time. The paper 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 permeability and capillary pressure curves is presented. presented. In addition to the dimensionless group criterion, an actual comparison of the results of an x-z cross-section and a one-dimensional (1-D) 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. Introduction Numerical simulation of reservoir performance currently encompasses a wide variety of recovery processes, reservoir types and purposes or questions processes, reservoir types and purposes or questions to which answers are sought. A feature common to virtually all reservoir simulation studies, however, is the choice of simulation in one, two or three dimensions. Most frequently this choice is one between an areal (x-y) study and a 3-D study. While the areal study is considerably cheaper than a 3-D simulation, the validity or accuracy of the former is often questioned in light of its apparent inability to simulate flow and fluid saturation distributions in the vertical direction. Areal studies are frequently performed with little attention to or understanding of the extent to which the x-y calculations do or can be made to account for this vertical flow and fluid distribution. Previous papers describe a VE assumption or concept which leads to the definition of pseudo relative permeability and capillary pressure curves to be used in areal studies to simulate 3-D flow. A dimensionless group proposed as a criterion for the assumptions validity primarily treats the case of a reservoir where the capillary transition zone is an appreciable fraction of reservoir thickness. This paper neats the case of a reservoir where the capillary capillary transition zone is a small fraction of reservoir thickness (e.g., less than 10 percent). We propose to describe the VE concept as percent). We propose to describe the VE concept as applied to thick reservoirs or to reservoirs where capillary transition zone is a small fraction of thickness; to describe the generation of appropriate relative permeability and capillary pressure curves for such reservoirs to represent 3-D performance by 2-D areal calculations; to propose a new dimensionless group as a criterion for the VE assumptions' validity, obtained from an analysis of countercurrent gravity segregation; and finally, to present a cross-sectional vs 1-D (VE) comparison and a 2-D areal field case study. THE VERTICAL EQUILIBRIUM CONCEPT Most oil and gas reservoirs extend distances areally which are at least two orders of magnitude greater than reservoir thickness. Viewed in perspective, these reservoirs appear as "blankets" perspective, these reservoirs appear as "blankets" For a variety of reasons, some valid and some invalid, numerical simulations of such reservoirs are performed occasionally in three dimensions as opposed to only two areal (x-y) dimensions. SPEJ P. 63

Improved Characterization and Modeling of Capillary Transition Zones in Carbonate Reservoirs

2007 ◽  
Vol 10 (02) ◽  
pp. 191-204 ◽  
Author(s):  
Shehadeh K. Masalmeh ◽  
Issa M. Abu-Shiekah ◽  
Xudong Jing
Keyword(s):  
Capillary Pressure ◽  
Transition Zone ◽  
Relative Permeability ◽  
Dynamic Properties ◽  
Carbonate Reservoirs ◽  
Oil Saturation ◽  
Transition Zones ◽  
Heterogeneous Reservoirs ◽  
Oil Water ◽  
Capillary Pressure Curves

Summary An oil/water capillary transition zone often contains a sizable portion of a field's initial oil in place, especially for those carbonate reservoirs with low matrix permeability. The field-development plan and ultimate recovery may be influenced heavily by how much oil can be recovered from the transition zone. This in turn depends on a number of geological and petrophysical properties that influence the distribution of initial oil saturation (Sor) against depth, and on the rock and fluid interactions that control the residual oil saturation (Sor), capillary pressure, and relative permeability characteristics as a function of initial oil saturation. Because of the general lack of relevant experimental data and the insufficient physical understanding of the characteristics of the transition zone, modeling both the static and dynamic properties of carbonate fields with large transition zones remains an ongoing challenge. In this paper, we first review the transition-zone definition and the current limitations in modeling transition zones. We describe the methodology recently developed, based on extensive experimental measurements and numerical simulation, for modeling both static and dynamic properties in capillary transition zones. We then address how to calculate initial-oil-saturation distribution in the carbonate fields by reconciling log and core data and taking into account the effect of reservoir wettability and its impact on petrophysical interpretations. The effects of relative permeability and imbibition capillary pressure curves on oil recovery in heterogeneous reservoirs with large transition zones are assessed. It is shown that a proper description of relative permeability and capillary pressure curves including hysteresis, based on experimental special-core-analysis (SCAL) data, has a significant impact on the field-performance predictions, especially for heterogeneous reservoirs with transition zones. Introduction The reservoir interval from the oil/water contact (OWC) to a level at which water saturation reaches irreducible is referred to as the capillary transition zone. Fig. 1 illustrates a typical capillary transition zone in a homogeneous reservoir interval within which both the oil and water phases are mobile. The balance of capillary and buoyancy forces controls this so-called capillary transition zone during the primary-drainage process of oil migrating into an initially water-filled reservoir trap. Because the water-filled rock is originally water-wet, a certain threshold pressure must be reached before the capillary pressure in the largest pore can be overcome and the oil can start to enter the pore. Hence, the largest pore throat determines the minimum capillary rise above the free-water level (FWL). As shown schematically in Fig. 2, close to the OWC, the oil/water pressure differential (i.e., capillary pressure) is small; therefore, only the large pores can be filled with oil. As the distance above the OWC increases, an increasing proportion of smaller pores are entered by oil owing to the increasing capillary pressure with height above the FWL. The height of the transition zone and its saturation distribution is determined by the range and distribution of pore sizes within the rock, as well as the interfacial-force and density difference between the two immiscible fluids.

Scale Up of Pore-Network Models into Reservoir Scale: Optimization of Inflow Control Devices Placement

SPE Journal ◽  
2021 ◽  
pp. 1-18
Author(s):  
Farzad Bashtani ◽  
Mazda Irani ◽  
Apostolos Kantzas
Keyword(s):  
Pore Structure ◽  
Capillary Pressure ◽  
Relative Permeability ◽  
Scale Up ◽  
Pore Network ◽  
Flow In Porous Media ◽  
Well Completion ◽  
Control Devices ◽  
Relative Permeability Curves ◽  
Capillary Pressure Curves

Summary Improvements to more advanced tools, such as inflow control devices (ICDs), create a high drawdown regime close to wellbores. Gas liberation within the formation occurs when the drawdown pressure is reduced below the bubblepoint pressure, which in turn reduces oil mobility by reducing its relative permeability, and potentially reducing oil flow. The key input in any reservoir modeling to compare the competition between gas and liquid flow toward ICDs is the relative permeability of different phases. Pore-network modeling (PNM) has been used to compute the relative permeability curves of oil, gas, and water based on the pore structure of the formation. In this paper, we explain the variability of pore structure on its relative permeability, and for a similar formation and identical permeability, we explain how other factors, such as connectivity and throat radius distribution, can vary the characteristic curves. By using a boundary element method, we also incorporate the expected relative permeability and capillary pressure curves into the modeling. The results show that such variability in the pore network has a less than 10% impact on production gas rates, but its effect on oil production can be significant. Another important finding of such modeling is that providing the PNM-created relative permeabilities may provide totally different direction on setting the operational constraints. For example, in the case studied in this paper, PNM-created relative permeability curves suggest that a reduction of flowing bottomhole pressure (FBHP) increases the oil rate, but for the case modeled with a Corey correlation, changes in FBHP will not create any uplift. The results of such work show the importance of PNM in well completion design and probabilistic analysis of the performance, and can be extended based on different factors of the reservoir in future research. Although PNM has been widely used to study the multiphase flow in porous media in academia, the application of such modeling in reservoir and production engineering is quite narrow. In this study, we develop a framework that shows the general user the importance of PNM simulation and its implementation in day-to-day modeling. With this approach, the PNM can be used not just to provide relative permeability or capillary pressure curves on a core or pore- scale, but to preform simulations at the wellbore or reservoir scale as well to optimize the current completions.

The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions

1973 ◽  
Vol 13 (03) ◽  
pp. 175-185 ◽  
Author(s):  
Hugh H. Jacks ◽  
Owen J.E. Smith ◽  
C.C. Mattax
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Relative Permeability ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Relative Permeabilities ◽  
Large Reservoir ◽  
Model Simulations ◽  
Wide Range ◽  
Section Model

Abstract Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs. Introduction Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions". DESCRIPTION OF PSEUDO-RELATIVE PERMEABILITY FUNCTIONS PERMEABILITY FUNCTIONS Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities. SPEJ P. 175

A Pore-Network-Modeling Approach To Predict Petrophysical Properties of Diatomaceous Reservoir Rock

2007 ◽  
Vol 10 (06) ◽  
pp. 597-608 ◽  
Author(s):  
Liping Jia ◽  
Cynthia Marie Ross ◽  
Anthony Robert Kovscek
Keyword(s):  
Multiphase Flow ◽  
Capillary Pressure ◽  
Network Model ◽  
Relative Permeability ◽  
Oil Recovery ◽  
Pore Network ◽  
Reservoir Rock ◽  
Pore Network Model ◽  
Rock Types ◽  
Capillary Pressure Curves

Summary A 3D pore-network model of two-phase flow was developed to compute permeability, relative permeability, and capillary pressure curves from pore-type, -size, and -shape information measured by means of high-resolution image analysis of diatomaceous-reservoir-rock samples. The diatomite model is constructed using pore-type proportions obtained from image analysis of epoxy-impregnated polished samples and mercury-injection capillary pressure curves for diatomite cores. Multiple pore types are measured, and each pore type has a unique pore-size and throat-size distribution that is incorporated in the model. Network results present acceptable agreement when compared to experimental measurements of relative permeability. The pore-network model is applicable to both drainage and imbibition within diatomaceous reservoir rock. Correlation of network-model results to well log data is discussed, thereby interpolating limited experimental results across the entire reservoir column. Importantly, our method has potential to predict the petrophysical properties for reservoir rocks with either limited core material or those for which conventional experimental measurements are difficult, unsuitable, or expensive. Introduction Model generation for reservoir simulation requires accurate entering of physical properties such as porosity, permeability, initial water saturation, residual-oil saturation, capillary pressure functions, and relative permeability curves. These functions and parameters are necessary to estimate production rate and ultimate oil recovery, and thereby optimize reservoir development. Accurate measurement and representation of such information is, therefore, essential for reservoir modeling. Relative permeability and capillary pressure curves are the most important constitutive relations to represent multiphase flow. Often, it is difficult to sample experimentally the range of relevant multiphase-flow behavior of a reservoir. In addition to the availability of rock samples, measurements are frequently time consuming to conduct, and conventional techniques are not suitable for all rock types (Schembre and Kovscek 2003). It is impossible, therefore, to measure all the unique relative permeability functions of different reservoir-rock types and variations within a rock type. This lack of constitutive information limits the accuracy of reservoir simulators to predict oil recovery. Simply put, other available data must be queried for their relevance to multiphase flow and must be used to interpret the available relative permeability and capillary pressure information.

Investigation of Reduced Interfacial Tension by Surfactant Flooding in Mixed-Wet Reservoirs

2021 ◽  
Author(s):  
Usman Aslam
Keyword(s):  
Interfacial Tension ◽  
Capillary Pressure ◽  
Enhanced Oil Recovery ◽  
Relative Permeability ◽  
Oil Recovery ◽  
Simulation Studies ◽  
Surfactant Flooding ◽  
Gravity Segregation ◽  
Simulation Results ◽  
Capillary Pressure Curves

Abstract Surfactant flooding has long been considered a reliable solution for enhanced oil recovery, either by reducing oil-water interfacial tension (IFT) or through wettability alteration. This paper reveals the effect that reduced IFT has on capillary trapping in heterogeneous reservoirs. This effect is investigated through various numerical experiments on different simulation models where rock capillary pressure is assumed to scale with IFT. Capillary contrast on the scale of a few centimeters to a few tens of meters is reduced in the presence of surfactants. This reduction in IFT, under very specific circumstances, creates favorable conditions for increased or accelerated hydrocarbon production from mixed-wet reservoirs. The focus of this study is to ascertain the effectiveness of surfactant flooding in mixed-wet reservoirs. Simulation studies of different mechanisms which are believed to occur in mixed-wet reservoirs are presented. Simulation results indicate the promising effect of surfactant flooding on oil recovery, depending on the type of reservoir. Detailed fine-scale simulation studies are carried out with representative relative permeability and imbibition capillary pressure curves from mixed-wet cores. By designing and selecting a series of surfactants to lower the IFT to the range of 10-3dynes/cm, a recovery of 10 to 20% of the original oil-in-place is technically and economically feasible. The efficiency of surfactant flooding is investigated through sensitivity scenarios on formation rock/fluid parameters, including permeability, interfacial tension, rate flow, etc. Geological heterogeneity (layering and heterogeneous inclusions), imbibition capillary pressure curves, viscous/capillary balance (Nc), and gravitational forces were all found to have an impact on recovery by surfactant flooding. Numerical model dimensions, permeability, IFT, density contrast between oil and water, and injection flow rates were found to be the critical parameters influencing simulation results. Gravity segregation, typically ignored in earlier studies, was found to have a significant effect on reservoir performance. Two different numerical models, with and without impermeable shale streaks, were used to capture the gravity segregation effect. The results revealed that the reduction in interfacial tension helps gravity to segregate oil and water, ultimately resulting in improved oil recovery. Moreover, results from the numerical simulation studies revealed that either an inexpensive or a good quality surfactant at low concentration can be used to obtain the same enhanced oil recovery. The effect of change in oil relative permeability curvature, due to reduced interfacial tension, also revealed a reduction in the remaining oil saturation with an increase in the capillary number.

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