Representing Wells in Numerical Reservoir Simulation: Part 2 - Implementation

1981 ◽  
Vol 21 (03) ◽  
pp. 339-344 ◽  
Author(s):  
John E. Chappelear ◽  
Alexander S. Williamson
Keyword(s):  
Relative Permeability ◽  
Reservoir Simulation ◽  
Model Development ◽  
Preceding Paper ◽  
Fluid Saturation ◽  
Well Model ◽  
Grid Block ◽  
The Difference ◽  
Reservoir Simulator ◽  
Water Block

Abstract A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well, This model is called a well model. This paper presents a succinct but comprehensive description of the installation of a well model in a simulator, including problems which may be encountered and possible remedies. This and the preceding paper, SPE 7697, present possible remedies. This and the preceding paper, SPE 7697, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the inclusion of a well model and well boundary conditions in a reservoir simulator. The source representation and the wellbore flow model are the basic components of the well model. The usefulness of the working version finally installed in a reservoir simulator depends greatly on the numerical implementation. We accordingly discuss numerical aspects of the well model for black-oil, compositional, and thermal well models.We have omitted a discussion of the incorporation into well models of surface gathering facilities and what could be called "well group constraints" such as lease, platform or pipeline constraints. These subjects easily could be the topics of several other papers.A satisfactory well model is frequently a key to successful simulation. Many of the details of well model development have not appeared in the petroleum literature. It is our hope that this paper may provide a basis for further work and discussion of this paper may provide a basis for further work and discussion of this essential topic. Implementation We shall discuss the implementation of the following equation (developed in Part 1) for the flow of each phase per completion interval. (1) Here, p is the phase (either oil, water, or gas). We note here certain aspects of this well model.1. The rates are in standard units.2. The relative permeability is calculated using the grid-block (average) fluid saturation from a well (i.e., not necessarily the grid-block) relative permeability table. It is at this point that the saturation boundary condition is imposed.3. The oil pressure is used to calculate the potential for all phases. Thus, capillarity, is not treated (i.e., no capillary end effect or water block). Also, the difference in phase pressures within a grid block due to gravity segregation is ignored.4. Zk is the vertical distance from the center of the kth completion interval to the center of the (k + 1)th completion interval (positive downward).5. The viscosity, formation volume factor, solution GOR, and density are calculated at gridblock pressure. Only the grid block for the completion interval is used.6. The skin and well radius are the same for every completion interval for each well.7. The external radius re of the grid block is a function of the grid-block, geometry. JPT P. 339

Representing Wells in Numerical Reservoir Simulation: Part 1 - Theory

1981 ◽  
Vol 21 (03) ◽  
pp. 323-338 ◽  
Author(s):  
Alexander S. Williamson ◽  
John E. Chappelear
Keyword(s):  
Boundary Condition ◽  
Reservoir Simulation ◽  
Singular Solution ◽  
Numerical Solutions ◽  
Applied Mathematics ◽  
Internal Boundary ◽  
Local Flow ◽  
Well Model ◽  
Grid Block ◽  
Wellbore Flow

Abstract A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well. This model is called a well model. We give a description here of the theoretical background of a well model, including how the sandface pressure and saturation boundary conditions can be calculated and how the well boundary itself can be replaced (approximately) by a source function. This paper and the following companion paper, SPE 9770, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the theory of representation of wells and the well boundary condition in a reservoir simulator.It frequently has been noted that, except in the case of a central well in a problem involving cylindrical coordinates, it is impractical to represent a well with an internal boundary. The ratio of well radius to desired grid-block length can be of order 0.001 or less. In such cases, an alternative procedure has evolved in which the well is represented by a source. The relationship between the source strength, the wellbore flow, and the flow in the surrounding grid blocks composes an essential part of the well model. Even when the grid around a well is sufficiently fine to represent the well as an internal boundary, other features such as partial perforation, partial penetration, or skin may be important to the local flow but extend over a "small" interval in relation to the appropriate grid-block dimension. Here also, a suitable source representation is advantageous. We shall develop the source representation of a well for a variety of circumstances.The well boundary condition generally involves the sandface pressure and flow rate. However, these quantities also must be consistent with the requirements of wellbore flow - i.e., reservoir and wellbore flows are coupled, and a wellbore flow model is required. We describe a means of treating a wide variety of wellbore flows without creating a numerically cumbersome simulator. We hope that this paper may provide a basis for further work and discussion of this essential topic. Review of Literature The source representation of a well can be described as a local, approximate, steady, singular solution of the flow equations. The idea of separating a singularity of this type for special treatment is an old idea in applied mathematics. In series solutions to certain elliptic and parabolic equations, it was found that the convergence of the series could be improved considerably by first extracting the singular part. In these cases the singular solution extended through the entire domain. The analogous approach using numerical methods in place of the series solution is also well known. The use of singular solutions in a purely local role in numerical solutions was introduced before the general use of digital computers. Woods' use of a local logarithmic expression in a solution of Poisson's equation by relaxation methods corresponds closely to the source representation of a well recently proposed by Peaceman. SPEJ P. 323＾

Estimation of Depths of Fluid Contacts and Relative Permeability Curves by History Matching Using Iterative Ensemble-Kalman Smoothers

SPE Journal ◽  
2009 ◽  
Vol 15 (02) ◽  
pp. 509-525 ◽  
Author(s):  
Yudou Wang ◽  
Gaoming Li ◽  
Albert C. Reynolds
Keyword(s):  
Relative Permeability ◽  
History Matching ◽  
Ensemble Member ◽  
Production Data ◽  
Model Parameters ◽  
Future Performance ◽  
Grid Block ◽  
Relative Permeability Curves ◽  
Statistical Consistency ◽  
Reservoir Simulator

Summary With the ensemble Kalman filter (EnKF) or smoother (EnKS), it is easy to adjust a wide variety of model parameters by assimilation of dynamic data. We focus first on the case where realizations and estimates of the depths of the initial fluid contacts, as well as grid- block rock-property fields, are generated by matching production data with the EnKS. Then we add the parameters defining power law relative permeability curves to the set of parameters estimated by assimilating production data with EnKS. The efficiency of EnKF and EnKS arises because data are assimilated sequentially in time and so "history matching data" requires only one forward run of the reservoir simulator for each ensemble member. For EnKS and EnKF to yield reliable characterizations of the uncertainty in model parameters and future performance predictions, the updated reservoir-simulation variables (e.g., saturations and pressures) must be statistically consistent with the realizations of these variables that would be obtained by rerunning the simulator from time zero using the updated model parameters. This statistical consistency can be established only under assumptions of Gaussi- anity and linearity that do not normally hold. Here, we use iterative EnKS methods that are statistically consistent, and show that, for the problems considered here, iteration significantly improves the performance of EnKS.

scholarly journals Effect Study of Aperture Distribution on the Capillary Pressure-Saturation Relation for the Single Fracture

Geofluids ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Yuan Wang ◽  
Chenglong Wu ◽  
Yuntian Zhou
Keyword(s):  
Capillary Pressure ◽  
Effect Study ◽  
Single Fracture ◽  
Saturation Curve ◽  
Static Displacement ◽  
Statistical Parameters ◽  
Dual Connectivity ◽  
Fluid Saturation ◽  
The Difference ◽  
Aperture Distribution

A systematic numerical method was presented to investigate the effect of aperture distribution on the relation of capillary pressure versus fluid saturation (P-S relation) for a single fracture. The fracture was conceptualized as a two-dimensional lattice-grid model and its aperture field was described by a probability distribution. Based on the invasion percolation theory, a program was developed to simulate the quasi-static displacement. The simulation was verified validly by comparisons of the experimental results. The effects of the statistical parameters were further quantified. The results show that the largest local aperture on the fracture boundary determines the AEV. The larger mean decreases the variation coefficient, which causes the more uniform aperture field, smoother air invasion front, and steeper capillary pressure-saturation curve (CPSC). The larger standard deviation increases not only the range but also the contrast degree of the apertures, thus providing a nondeterministic rule in the P-S relation. The larger correlation length causes a more homogeneous aperture field and a dual connectivity of the fracture. The increase of the difference and contrast degree between the small and large apertures results in dual-aperture fields. The dual-aperture field and dual connectivity of the fracture both contribute to the bimodal characteristic of the CPSC.

Use of Horizontal Drift-Flux Models For Simulating Wellbore Flow in SAGD Operations

2021 ◽  
Author(s):  
Mohammad Heidari ◽  
Christopher Istchenko ◽  
William Bailey ◽  
Terry Stone
Keyword(s):  
Flux Flow ◽  
Pressure Drops ◽  
Steam Assisted Gravity Drainage ◽  
Drift Flux ◽  
Pressure Profiles ◽  
Well Model ◽  
Inclination Angles ◽  
Operation Pressure ◽  
Reservoir Simulator ◽  
High Degree

Abstract The paper examines new horizontal drift-flux correlations for their ability to accurately model phase flow rates and pressure drops in horizontal and undulating wells that are part of a Steam-Assisted Gravity Drainage (SAGD) field operation. Pressure profiles within each well correlate to the overall performance of the pair. SAGD is a low-pressure process that is sensitive to reservoir heterogeneity and other factors, hence accurate simulation of in situ wellbore pressures is critical for both mitigating uneven steam chamber evolution and optimizing wellbore design and operation. Recently published horizontal drift-flux correlations have been implemented in a commercial thermal reservoir simulator with a multi-segment well model. Valid for horizontally drilled wells with undulations, they complement previously reported drift-flux models developed for vertical and inclined wells down to approximately 5 degrees from horizontal. The formulation of these correlations has a high degree of nonlinearity. These models are tested in simulations of SAGD field operations. First, an overview of drift-flux models is discussed. This differentiates those based on vertical flow with gravity segregation to those that model horizontal flow with stratified and slug flow regimes. Second, the most recent and significant drift-flux correlation by Bailey et al. (2018, and hereafter referred to as Bailey-Tang-Stone) was robustly designed to be used in the well model of a reservoir simulator, can handle all inclination angles and was optimized to experimental data from the largest available databases to date. This and earlier drift-flux models are reviewed as to their strengths and weaknesses. Third, governing equations and implementation details are given of the Bailey-Tang-Stone model. Fourth, six case studies are presented that illustrate homogeneous and drift-flux flow model differences for various well scenarios.

scholarly journals 1 Assessing the predictive adequacy of simple and complex mathematical models

2019 ◽  
Vol 97 (Supplement_3) ◽  
pp. 24-24
Author(s):  
Luis O Tedeschi
Keyword(s):  
Model Development ◽  
Predictive Ability ◽  
Random Variable ◽  
Coefficient Of Determination ◽  
Skewed Data ◽  
Future Performance ◽  
Evaluation Phase ◽  
The Difference ◽  
Predicted Values ◽  
Analytical Approaches

Abstract The establishment of credibility for a mathematical model’s (MM) predictive ability is an essential component for improving the MM because it stimulates the evolutionary thinking (i.e., the next generation of the model) of mental conceptualizations, assumptions, and boundaries of the MM. Its predictive adequacy is commonly assessed through its ability to precisely or accurately predict observed (real) values. The precision component measures how closely the model predicted values are of each other or whether a defined pattern of predictions exists. The accuracy component, on the other hand, measures how closely the average of the model predicted values are to the actual (true) average. Many statistics exist to determine precision and accuracy of MM such as mean bias, resistant coefficient of determination, coefficient of determination, modeling efficiency, concordance correlation coefficient (CCC), the mean square error of prediction, Kleijnen’s statistic (regression of the difference between predicted and observed on their sum), and Altman and Bland’s limits of agreement statistics among many more. However, for complex models that use skewed data or repeated data in which the data is not independent (e.g., multiple measurements on the same subject), simple statistics may not suffice. For instance, four methods to compute CCC exist (moment, variance components, U-statistics, and generalized estimating equations—GEE), but only the last two methods are resilient to lightly skewed data. Another type of complexity arises when meta-analytical approaches are used at the model development phase or the model evaluation phase. In general, meta-analytical approaches remove errors (i.e., variation) associated with random variables that are believed to be known. Under these circumstances, MM tends to overperform (i.e., they have greater predictive adequacy) and their future performance may be deceitful when trying to forecast at scenarios in which the random variable(s) is(are) indeterminable or unquantifiable.

Use of Transient Data To Calculate Absolute Permeability and Average Fluid Saturations

2010 ◽  
Vol 13 (02) ◽  
pp. 306-312 ◽  
Author(s):  
Medhat M. Kamal ◽  
Yan Pan
Keyword(s):  
Relative Permeability ◽  
Reservoir Simulation ◽  
Surface Flow ◽  
Absolute Permeability ◽  
Phase Flow ◽  
Simulation Studies ◽  
Flow Rates ◽  
Reservoir Performance ◽  
Transient Data ◽  
Relative Permeability Curves

Summary A new well-testing-analysis method is presented. The method allows for calculating the absolute permeability of the formation in the area influenced by the test and the average saturations in this area. Traditional pressure-transient-analysis methods have been developed and are completely adequate for single-phase flow in the reservoir. The proposed method is not intended for these conditions. The method applies to two-phase flow in the reservoir (oil and water or oil and gas). Future expansion to three-phase flow is possible. Current analysis methods yield only the effective permeability for the dominant flowing phase and the "total mobility" of all phases. The new method uses the surface-flow rates and fluid properties of the flowing phases and the same relative permeability relations used in characterizing the reservoir and predicting its future performance. The method has been verified by comparing the results from analyzing several synthetic tests that were produced by a numerical simulator with the input values. Use of the method with field data is also described. The new method could be applied wherever values of absolute permeability or fluid saturations are used in predicting well and reservoir performance. Probably, the major impact would be in reservoir simulation studies in which the need to transform welltesting permeability to simulator input values is eliminated and additional parameters (fluids saturations) become available to help history match the reservoir performance. This work will also help in predicting well flow rates and in situations in which absolute permeability changes with time (e.g., from compaction). Results showed that the values of absolute permeability in water/oil cases could be reproduced within 3% of the correct values and within 5% of the correct values in gas/oil cases. Errors in calculating the fluid saturations were even lower. One of the main advantages of this method is that the relative permeability curves used in calculating the absolute permeability and average saturations, and later on in numerical reservoir simulation studies, are the same, ensuring a consistent process. The proposed method does not address the question of which set of relative permeability curves should be used. This question should be answered by the engineer performing the reservoir engineering/simulation study. The proposed method mainly is meant to provide consistent results for predicting the reservoir performance using whatever relative permeability relations that are being used in the reservoir simulation model. The method does not induce any additional errors in determining the average saturation or absolute permeability over what may result from using these specific relative permeability curves in the reservoir simulation study. The impact of this study will be to expand the use of information already contained in transient data and surface flow rates of all phases. The results will provide engineers with additional parameters to improve and speed up history matching and the prediction of well and reservoir performances in just about all studies.

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