Dual-permeability Model Research Articles

A two-phase flow simulation of a fractured reservoir using a new fissure element method

Modeling of a fissure network is conventionally done using a dual porosity–dual permeability model, but this type of model involves large-scale approximate description of fluid transfers between the matrix and fissure. For understanding of the simplifications involved in the dual porosity approach, a fine description of the fissure network is necessary. To obtain this fine scale description, a new methodology was developed for the modeling of flow in fractured reservoirs. This technique called “the fissure element methodology” is based on application of a specific grid to the fractured medium. This unstructured grid is made of two kinds of elements, linear elements for the fissure network and triangular elements for the matrix. An appropriate finite volume scheme has been developed to provide a good description of the flow at the fissure–matrix interface and a good representation of the geometry. A specific numerical treatment of fissure intersections is proposed. The method has been validated by comparison with results from a classical black-oil simulator run on a finely gridded Cartesian model. Comparisons have been done also with a dual porosity–dual permeability model. The new tool will permit to run reference simulations for the validation of new formulations of the matrix–fissure exchange in dual-porosity simulators.

天然フラクチャー型貯留層の有効浸透率予測

Appropriate modeling of naturally fractured reservoirs is one of the most important and challenging issues in reservoir characterization. In simulation, a dual porosity or dual permeability model is applied when fractures are well developed to form a fracture network. On the other hand, the single-continuum approach, where the fracture system is represented by effective permeability, is commonly used if fractures are discrete or disconnected. Focusing on the latter case, this paper proposes a semi-analytical technique to evaluate effective permeability for periodically or randomly fractured media including infinitely thin, infinite-conductivity fractures.The complex variable boundary element method is used to compute potential fields and streamlines in the two-dimensional space for discretely distributed fracture systems under the periodic boundary conditions. Effective permeability is evaluated first for discrete fracture systems of regular patterns to demonstrate the validity of the method and to examine the sensitivity of the effective permeability to the variations in the basic fracture parameters. With a constant total length of fractures, systems of varied fracture lengths show higher effective permeability than systems of uniform fracture length. 500 distributions of stochastic fractures are next generated to establish correlation between effective permeability and the fracture parameters, total length L, mean length m, and standard deviation of fracture length ?D. Sensitivity to the parameters shows that non-zero a increases effective permeability, that the incremental gain of effective permeability is proportional to L, and that the larger m, the larger effective permeability. The effective permeability tensors are also determined for oriented fractures. Analyses by non-parametric regression show that the diagonal elements, kxx and kyy, are highly affected by the angle between the oriented fractures and the pressure gradient, while the off-diagonal elements, kxy and kyx, are strongly affected by both the total length and the angle.

Thermal loading studies using the Yucca Mountain unsaturated zone model

A systematic modeling study of the effect of thermal loading on moisture, gas, and heat flow in the unsaturated zone of Yucca Mountain has been carried out. The study is based on a two-dimensional (2-D) north–south vertical cross-section, using both the effective continuum and the dual-permeability modeling approaches. A study was also conducted on a three-dimensional (3-D) model using the effective continuum approach. The 2-D model conducted in 1996 uses a average uniform infiltration rate of 4.4 mm/year and a thermal load of 83 kW/acre. The 3-D model uses an spatially varying infiltration rate and a revised average thermal load of 87.6 kW/acre. For the rock properties used in the Topopah Springs (TSw) hydrogeological unit, heat pipe conditions develop above and below the repository in 10–100 years in both models. The average temperature of the boiling zone is about 96°C. This boiling zone is confined to the TSw hydrogeological unit and lasts about 1000 years. At the top of the CHn (vitric/zeolitic interface), predicted maximum temperature is about 70–75°C after about 2000 years. The model predicts a temperature increase of approximately 30°C at the water table. The results show that thermal loading at the repository also results in significant changes in the moisture distributions at the repository horizon and the zone directly above and below it. A large increase in liquid and gas flux, several orders of magnitude above ambient conditions, is predicted near the repository. This study indicates the ECM and dual- k modeling approaches provide similar simulation results, in terms of temperature and moisture flow and distribution. The only difference is that at early times, the ECM model predicts more extensive boiling conditions. Localized dry-out is predicted in areas with low infiltration flux within the central part of the repository. However, because coarse grids were used, average saturation in the matrix blocks representing the repository indicates will remain high even when the regions near repository are completely dry. Analysis of the flux pattern at the top and bottom boundaries of the repository shows that the liquid always flows into the repository for most the thermal loading period (up to 10,000 years).

Influence of fracture truncation on dispersion: A dual permeability model

Simulations with a dual permeability model show that a few discrete fractures can have a major influence on plume geometry. A limited number of truncated fractures (1–4) within a permeable matrix can create nearly circular plumes, with about the same degree of spreading in the direction transverse to the average hydraulic gradient as in the longitudinal direction. By comparison, continuous fractures in the direction of flow tend to produce elongated plumes, similar to those typically seen in granular materials. Both types of plumes have been observed in tracer experiments in fractured porous media on the Oak Ridge Reservation. Understanding the influence of fracture geometry is important in planning field characterization and subsequent remediation in fractured porous media.

Single Phase Flow in Partially Fissured Media

Totally fissured media in which the individual cells are isolated by the fissure system are effectively described by double porosity models with microstructure. Such models contain the geometry of the individual cells in the medium and the flux across their interface with the fissure system which surrounds them. We extend these results to a dual-permeability model which accounts for the secondary flux arising from direct cell-to-cell diffusion within the solid matrix. Homogenization techniques are used to construct a new macroscopic model for the flow of a single phase compressible fluid through a partially fissured medium from an exact but highly singular microscopic model, and it is shown that this macroscopic model is mathematically well posed. Preliminary numerical experiments illustrate differences in the behaviour of solutions to the partially fissured from that of the totally fissured case.

Well-Test Analysis in Solution-Gas-Drive Systems for Fractured Reservoirs

Summary This study examines issues pertaining to two-phase (oil and gas) transient flow in fractured reservoirs. Dual-porosity and dual-permeability models are examined. Pressure, saturation, and flowing GOR behaviors are noted, and the influence of communication on these behaviors is documented. Methods to compute saturation as a function of pressure for the fracture and matrix systems are presented.

Eagle Lake Viking Miscible Displacement Laboratory Studies

Abstract A series of displacement tests was conducted on preserved reservoir core plugs to assist in estimating waterflood and miscible flood performance potential for the Eagle Lake Viking Reservoir in Saskatchewan. The reservoir rock is a heterogeneous sandstone, comprising thin interbeds of sand and shale with the sand lenses containing widely different amounts of smectite clays. To assess miscible recovery efficiencies in this complex sand, displacement tests were conducted with first-contact miscible propane and carbon dioxide solvents. Using standard waterflood relative permeabilities and a dual permeability model, all miscible displacements were matched by simulation with a Todd-Longstaff-type miscible mixing model. Introduction The Eagle Lake Viking Reservoir is located in southwest Saskatchewan as shown in Figure 1. It is a sandstone formation of Mississippian age. Discovered in 1957, one half of this 12.7 million m3 field was put on waterflood ten years later in 1967. Unfortunately, because of its low permeability, it has been possible to only inject 0.14 pore volumes of water in the subsequent 2O-year period. As a result, in the thirty years since discovery, only 15% of original oil- in-place has been produced. The cause of poor performance appears to be due to the abundant shales and days. Thin section analyses, however, show that bands of higher permeability are also found throughout the pay zone and often exist as thin to very thin sand stringers which are relatively free of days. In 1985, operations engineers decided that the half of the field which was still under primary depletion should be placed under some form of secondary recovery. Because the waterflood had been less than satisfactory, it was decided to examine miscible flooding as a potential alternative. Hence, a core flooding study using carbon dioxide as a flooding agent was designed with the following objectives;quantify carbon dioxide oil displacement efficiency;compare carbon dioxide with waterflood oil recovery; andcompare the potentially complex carbon dioxide miscible oil displacement process with a more conventional hydrocarbon solvent process using propane. To extend the initial objectives for reservoir design purposes, a fourth objective was to:obtain waterflood and miscible displacement parameters for use in numerical simulation of field applications, Reservoir Fluid and CO2 Miscibility Pressure The live oil used in this study was obtained by recombining separator gas and liquid to a bubble point pressure of 6509 kPa at 22 °C. Fluid densities were measured as a function of pressure at the reservoir temperatures of 22 °C and were matched with the Peng-Robinson(l) equation-of-state. The reference 2 characterization procedure was used and viscosity predictions were made with the Jossi, Stiel and Thodos correlation. Table 1 illustrates the accuracy or this prediction which involved proprietary correlations for heavy end parameters. This set of density and viscosity data was used in the simulation study. The oil composition is also shown in Table 1. To determine the first-contact miscibility pressure for CO2, oil and CO2 were blended in a visual cell at 22 °C. These results and the Peng-Robinson equation-of-state matches are presented in Table 2.

Simulation of Block-to-Block Processes in Naturally Fractured Reservoirs

Summary Simulation of fractured reservoirs with the dual-porosity/dual-permeability approach involves discretization of the solution domain into two collocated continua called the matrix and the fracture. The original idealized model assumes that the matrix acts essentially as a source or sink to the fracture, which is the primary conduit for fluid flow. In multiphase flow situations, this idealization was found to be inadequate. Enhancements are needed to represent the local matrix/fracture and matrix/matrix drainage and imbibition processes. Attempts to represent these processes include the gravity-segregated model, the subdomain model, the pseudofunction method, and the dual-permeability model. This work examines the mechanisms involved in gas/oil gravity drainage in terms of the block-to-block process. Current methods for treating this problem are reviewed to identify deficiencies. A new approach is proposed in which these mechanisms can be represented properly in the field-scale simulation of these reservoirs. The method involves the calculation of pseudo capillary potentials, which in an average sense (on a computational block basis) give the correct flow behaviors. These pseudos can be calculated a priori if a vertical equilibrium (VE) assumption can be made about the fluid distribution in the matrix blocks. When the VE assumption is not valid, the pseudos can be determined from fine-grid simulations.

An Evaluation Of The Improved Dual Porosity Model For The Simulation Of Gravity Effects In Naturally Fractured Reservoirs

Abstract The accurate representation of the interactions between matrix and fracture in the dual porosity model is crucial to the proper prediction of recovery from fractured petroleum reservoirs. In the standard dual porosity model, matrix-fracture transfer is represented by a single term in the mass balance equations, and the matrix-matrix flow is assumed to be negligible. This was found to be inadequate when gravity effects are important. Various methods were reported in the recent literature which account for the gravity drainage process. These include the gravity segregated model, the sub-domain method, and the dual permeability model. In this paper, we will examine the gas-oil gravity drainage process and evaluate the different approaches for solving this problem. Introduction The dual porosity approach has become the most widely accepted technique for the simulation of naturally fractured reservoirs. In this approach, the reservoir is represented by two collocated continua. They are referred to as fracture and matrix. The fracture continuum represents the interconnected network of fractures and/or vugs which constitute the primary conduits for fluid flow. The matrix continuum represents the pore volume in the rock which comprises the majority of the storage in the reservoir. Because the permeability in the matrix is substantially lower than that in the fracture, earlier models usually assume that the matrix does not contribute to the global mass transfer and act primarily as source and sink terms to the fracture. An early account of a full-field study of a model of this type can be found in Kazemi et al.(1). The fractured continuum is discretized into computational grid cells. The matrix continuum is assumed to be comprised of matrix blocks which are separated spatially by fractures. Strictly speaking, these matrix blocks can be highly irregular, and their sizes are not constant throughout the reservoir. To make the analysis manageable, it is typically assumed that the matrix blocks can be represented by regular geometries such as rectangular prisms(1,2) or cylinders(3). The dimensions of these matrix blocks can be variable throughout the reservoir and are primarily a function of the fracture spacing, orientation, and width. One computational grid cell may contain several matrix blocks. It is usually not feasible to insist on having one matrix block per computational cell. Because the matrix block is the primary unit for matrix-fracture exchange calculation, it is usually assumed that all the matrix blocks within one computational cell are geometrically and dynamically similar. Hence, the mass transfer between the matrix and the fracture in one computational grid is just the mass transfer between one matrix block and its surrounding fractures multiplied by the number of matrix blocks in the fracture cell. The adequacy of this assumption will be examined in later sections. The matrix-fracture transfer is usually represented by a single term with a geometric factor in the mass balance equations. The mass balance finite-difference equations have been described in several papers on the subject and will not be repeated here (see, for example, Reference 6).

Vertical-Interference Pressure Testing Across a Low-Permeability Zone With Unsteady Crossflow

Summary Existing analytical models for interpreting vertical-interference pressure tests across the low-permeability zone in a three-layered reservoir usually assume transient radial flow in the high-permeability layers and pseudosteady vertical crossflow in the intervening tight zone. The neglect of vertical flow in the high-permeability layers and of radial flow in the tight layer requires that the permeability contrast between these layers be large. When this assumption is satisfied, however, it is also very likely that the pressure diffusivity in the tight layer is much smaller than that in the high-permeability layers. Thus, to be consistent, the flow across the tight layer must be treated as an unsteady diffusive process driven by the transient pressure changes occurring in the high-permeability layers. In this paper, we extend the analysis of Ehlig-Economides and Ayoub by replacing their pseudosteady crossflow assumption with unsteady crossflow and interpret the crossflow response in terms of near- and far-boundary flow-response functions for the tight zone. The inclusion of unsteady crossflow requires one additional dimensionless parameter and results in a set of governing equations that are different from but similar to the equations of the dual-porosity, dual-permeability model with unsteady interporosity flow presented by Wijesinghe and Culham for naturally fractured reservoirs. As in the case of dual-porosity, dualpermeability reservoirs, the pressure response with unsteady crossflow is significantly different at early times from that for pseudosteady crossflow under practical conditions. Exact and asymptotic solutions and type curves for the wellbore pressures at the upper and lower perforations are presented for flow-rate-specified vertical interference tests. The solutions account for different formation properties, wellbore storage, and skin factors of the two high-permeability layers and the formation properties of the tight zone. The impact of unsteady and pseudosteady crossflow model assumptions on the wellbore pressures and on the spatial distributions of fluid pressures and crossflow rates is examined in detail.

Improved Calculations for Viscous and Gravity Displacement in Matrix Blocks in Dual-Porosity Simulators (includes associated papers 17851, 17921, 18017, 18018, 18939, 19038, 19361 and 20174 )

Summary This paper presents two innovations to improve the mathematical representation of viscous and gravity displacement from matrix blocks in dual-porosity simulators. The first improvement accounts for viscous displacement and convective mass transport in matrix blocks caused by potential gradients in the fracture network. Computation time is not significantly increased compared with current dual-porosity simulators. The second improvement provides two computationally efficient methods to refine simulation of gravity displacement in the matrix. Examples show that the improvements are highly desirable in many situations. Introduction Simulation of dual-porosity reservoirs has received much attention over the last decade.1–8 The most widely accepted and practical approach has been to treat fractures as the continuum and matrix as discontinuous source or sink terms embedded in the fracture network. In a finite-difference approximation of reservoir flow equations, all rnatrix blocks in a given grid cell are assumed to behave identically. The matrix has high porosity and low permeability; the fractures have very high permeability and low PV. Viscous displacement in matrix blocks caused by pressure (potential) gradient in the fracture network is generally neglected. This paper describes a practical way to correct this problem, and discusses when this correction is important. In the state-of-the-art dual-porosity simulators, capillary/gravity interaction between fractures and matrix blocks is not rigorously accounted for. Saidi,2 Litvak,3 and Sonier et al.4 simulated gravity effects by segregating all phases in each computational grid and in each timestep. This was done by simple material balance without accounting for the time-dependent nature of gravity segregation as affected by capillary pressure and the vertical transmissibility of the matrix. In this paper we present an example that shows that these approaches can lead to optimistic oil recoveries. Gilman5 and Pruess and Narasimhan6 have reported a grid refinement scheme to calculate heat transfer and fluid movement in the matrix blocks of a fractured reservoir more accurately. In this paper, we show that this approach of matrix-block grid refinement improves the calculation of gravity displacement. We also compare dual-porosity and dual-permeability concepts reported by Hill and Thomas.7 The latter concept allows both matrix-to-matrix and fracture-to-fracture flow between gridblocks. Dual-permeability models require much more computing time than the dual-porosity models in which the matrix is discontinuous. In this paper, we used the dual-permeability concept to develop a computationally efficient algorithm to account accurately for gravity effects both in the fracture and the matrix. We also show to what extent the more relaxed assumptions used in current dual-porosity simulators and the new ideas presented in this paper are compatible with results from fine-grid simulations that more accurately describe the fracture and matrix network. Flow Equations in Dual-Porosity Simulators The basic flow equations are represented in the same manner as those used by Gilman and Kazemi.8 For simplicity, we consider only the equations for immiscible flow of insoluble phases. The inclusion of solubility of one phase into another phase does not affect our presentation. The finite-difference form for the fracture isEquation 1 and for the matrix,Equation 2 where tamaf is the rate of fluid transfer between matrix and fracture in a gridblock and is given byEquation 3 The subscript a represents water, oil, and gas phases. Fracture transmissibility, Taf, in the x direction is given byEquation 4 Similar equations are defined for the y and z directions. Upstream mobilities are used in the equations. The equations with our new modifications will be presented later. For the simulations presented in this paper, single-point upstream relative permeability, without any modifications, is used for finite-matrix flow in Eq. 3 so that the dual-porosity models can be compared with fine-grid single-porosity models that use separate nodes for the fracture and matrix.

Physical Principles of Oil Production: Reply to Review

Previous articleNext article No AccessDiscussionPhysical Principles of Oil Production: Reply to ReviewMorris MuskatMorris Muskat Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The Journal of Geology Volume 59, Number 5Sep., 1951 Article DOIhttps://doi.org/10.1086/625900 Views: 6Total views on this site Citations: 2Citations are reported from Crossref PDF download Crossref reports the following articles citing this article:Rong-Chao Cheng, Ming-Ming Chu, Jie Zhang, Hui-Jiun Hao, Hai Lin, Qiang Tan, Yong-Cun Feng An Integrated Method for Evaluating Lost Circulation Risk for Shale Gas Drilling in Sichuan Basin, IOP Conference Series: Earth and Environmental Science 861, no.66 (Oct 2021): 062075.https://doi.org/10.1088/1755-1315/861/6/062075Mohsen Safari-Beidokhti, Abdolnabi Hashemi, Reza Abdollahi, Hamed Hematpur, Hamid Esfandyari, Mohammad Yazdi Numerical Well Test Analysis of Condensate Dropout Effects in Dual-Permeability Model of Naturally Fractured Gas Condensate Reservoirs: Case Studies in the South of Iran, Mathematical Problems in Engineering 2021 (May 2021): 1–10.https://doi.org/10.1155/2021/9916914