A Fully Implicit, Compositional, Parallel Simulator for IOR Processes in Fractured Reservoirs

SPE Journal ◽  
2007 ◽  
Vol 12 (03) ◽  
pp. 367-381 ◽  
Author(s):  
Reza Naimi-Tajdar ◽  
Choongyong Han ◽  
Kamy Sepehrnoori ◽  
Todd James Arbogast ◽  
Mark A. Miller
Keyword(s):  
Fractured Reservoirs ◽  
Naturally Fractured Reservoirs ◽  
Dual Porosity ◽  
Fractured Reservoir ◽  
Matrix Fracture ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
Dual Porosity Model ◽  
The Matrix ◽  
Porosity Model

Summary Naturally fractured reservoirs contain a significant amount of the world oil reserves. A number of these reservoirs contain several billion barrels of oil. Accurate and efficient reservoir simulation of naturally fractured reservoirs is one of the most important, challenging, and computationally intensive problems in reservoir engineering. Parallel reservoir simulators developed for naturally fractured reservoirs can effectively address the computational problem. A new accurate parallel simulator for large-scale naturally fractured reservoirs, capable of modeling fluid flow in both rock matrix and fractures, has been developed. The simulator is a parallel, 3D, fully implicit, equation-of-state compositional model that solves very large, sparse linear systems arising from discretization of the governing partial differential equations. A generalized dual-porosity model, the multiple-interacting-continua (MINC), has been implemented in this simulator. The matrix blocks are discretized into subgrids in both horizontal and vertical directions to offer a more accurate transient flow description in matrix blocks. We believe this implementation has led to a unique and powerful reservoir simulator that can be used by small and large oil producers to help them in the design and prediction of complex gas and waterflooding processes on their desktops or a cluster of computers. Some features of this simulator, such as modeling both gas and water processes and the ability of 2D matrix subgridding are not available in any commercial simulator to the best of our knowledge. The code was developed on a cluster of processors, which has proven to be a very efficient and convenient resource for developing parallel programs. The results were successfully verified against analytical solutions and commercial simulators (ECLIPSE and GEM). Excellent results were achieved for a variety of reservoir case studies. Applications of this model for several IOR processes (including gas and water injection) are demonstrated. Results from using the simulator on a cluster of processors are also presented. Excellent speedup ratios were obtained. Introduction The dual-porosity model is one of the most widely used conceptual models for simulating naturally fractured reservoirs. In the dual-porosity model, two types of porosity are present in a rock volume: fracture and matrix. Matrix blocks are surrounded by fractures and the system is visualized as a set of stacked volumes, representing matrix blocks separated by fractures (Fig. 1). There is no communication between matrix blocks in this model, and the fracture network is continuous. Matrix blocks do communicate with the fractures that surround them. A mass balance for each of the media yields two continuity equations that are connected by matrix-fracture transfer functions which characterize fluid flow between matrix blocks and fractures. The performance of dual-porosity simulators is largely determined by the accuracy of this transfer function. The dual-porosity continuum approach was first proposed by Barenblatt et al. (1960) for a single-phase system. Later, Warren and Root (1963) used this approach to develop a pressure-transient analysis method for naturally fractured reservoirs. Kazemi et al. (1976) extended the Warren and Root method to multiphase flow using a 2D, two-phase, black-oil formulation. The two equations were then linked by means of a matrix-fracture transfer function. Since the publication of Kazemi et al. (1976), the dual-porosity approach has been widely used in the industry to develop field-scale reservoir simulation models for naturally fractured reservoir performance (Thomas et al. 1983; Gilman and Kazemi 1983; Dean and Lo 1988; Beckner et al. 1988; Rossen and Shen 1989). In simulating a fractured reservoir, we are faced with the fact that matrix blocks may contain well over 90% of the total oil reserve. The primary problem of oil recovery from a fractured reservoir is essentially that of extracting oil from these matrix blocks. Therefore it is crucial to understand the mechanisms that take place in matrix blocks and to simulate these processes within their container as accurately as possible. Discretizing the matrix blocks into subgrids or subdomains is a very good solution to accurately take into account transient and spatially nonlinear flow behavior in the matrix blocks. The resulting finite-difference equations are solved along with the fracture equations to calculate matrix-fracture transfer flow. The way that matrix blocks are discretized varies in the proposed models, but the objective is to accurately model pressure and saturation gradients in the matrix blocks (Saidi 1975; Gilman and Kazemi 1983; Gilman 1986; Pruess and Narasimhan 1985; Wu and Pruess 1988; Chen et al. 1987; Douglas et al. 1989; Beckner et al. 1991; Aldejain 1999).

A Novel Multirate Dual-Porosity Model for Improved Simulation of Fractured and Multiporosity Reservoirs

SPE Journal ◽  
2013 ◽  
Vol 18 (04) ◽  
pp. 670-684 ◽  
Author(s):  
S.. Geiger ◽  
M.. Dentz ◽  
I.. Neuweiler
Keyword(s):  
Fractured Reservoirs ◽  
Dual Porosity ◽  
Transfer Rates ◽  
Wide Range ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
Dual Porosity Model ◽  
The Matrix ◽  
Water Cut ◽  
Porosity Model

Summary A major part of the world's remaining oil reserves is in fractured carbonate reservoirs, which are dual-porosity (fracture-matrix) or multiporosity (fracture/vug/matrix) in nature. Fractured reservoirs suffer from poor recovery, high water cut, and generally low performance. They are modeled commonly by use of a dual-porosity approach, which assumes that the high-permeability fractures are mobile and low-permeability matrix is immobile. A single transfer function models the rate at which hydrocarbons migrate from the matrix into the fractures. As shown in many numerical, laboratory, and field experiments, a wide range of transfer rates occurs between the immobile matrix and mobile fractures. These arise, for example, from the different sizes of matrix blocks (yielding a distribution of shape factors), different porosity types, or the inhomogeneous distribution of saturations in the matrix blocks. Thus, accurate models are needed that capture all the transfer rates between immobile matrix and mobile fracture domains, particularly to predict late-time recovery more reliably when the water cut is already high. In this work, we propose a novel multi-rate mass-transfer (MRMT) model for two-phase flow, which accounts for viscous-dominated flow in the fracture domain and capillary flow in the matrix domain. It extends the classical (i.e., single-rate) dual-porosity model to allow us to simulate the wide range of transfer rates occurring in naturally fractured multiporosity rocks. We demonstrate, by use of numerical simulations of waterflooding in naturally fractured rock masses at the gridblock scale, that our MRMT model matches the observed recovery curves more accurately compared with the classical dual-porosity model. We further discuss how our multi-rate dual-porosity model can be parameterized in a predictive manner and how the model could be used to complement traditional commercial reservoir-simulation workflows.

Unsteady-State Behavior of Naturally Fractured Reservoirs

1965 ◽  
Vol 5 (01) ◽  
pp. 60-66 ◽  
Author(s):  
A.S. Odeh
Keyword(s):  
Fractured Reservoirs ◽  
Naturally Fractured Reservoirs ◽  
Unsteady State ◽  
Fractured Reservoir ◽  
State Behavior ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
The Matrix ◽  
Porosity And Permeability ◽  
Bulk Volume

Abstract A simplified model was employed to develop mathematically equations that describe the unsteady-state behavior of naturally fractured reservoirs. The analysis resulted in an equation of flow of radial symmetry whose solution, for the infinite case, is identical in form and function to that describing the unsteady-state behavior of homogeneous reservoirs. Accepting the assumed model, for all practical purposes one cannot distinguish between fractured and homogeneous reservoirs from pressure build-up and/or drawdown plots. Introduction The bulk of reservoir engineering research and techniques has been directed toward homogeneous reservoirs, whose physical characteristics, such as porosity and permeability, are considered, on the average, to be constant. However, many prolific reservoirs, especially in the Middle East, are naturally fractured. These reservoirs consist of two distinct elements, namely fractures and matrix, each of which contains its characteristic porosity and permeability. Because of this, the extension of conventional methods of reservoir engineering analysis to fractured reservoirs without mathematical justification could lead to results of uncertain value. The early reported work on artificially and naturally fractured reservoirs consists mainly of papers by Pollard, Freeman and Natanson, and Samara. The most familiar method is that of Pollard. A more recent paper by Warren and Root showed how the Pollard method could lead to erroneous results. Warren and Root analyzed a plausible two-dimensional model of fractured reservoirs. They concluded that a Horner-type pressure build-up plot of a well producing from a factured reservoir may be characterized by two parallel linear segments. These segments form the early and the late portions of the build-up plot and are connected by a transitional curve. In our analysis of pressure build-up and drawdown data obtained on several wells from various fractured reservoirs, two parallel straight lines were not observed. In fact, the build-up and drawdown plots were similar in shape to those obtained on homogeneous reservoirs. Fractured reservoirs, due to their complexity, could be represented by various mathematical models, none of which may be completely descriptive and satisfactory for all systems. This is so because the fractures and matrix blocks can be diverse in pattern, size, and geometry not only between one reservoir and another but also within a single reservoir. Therefore, one mathematical model may lead to a satisfactory solution in one case and fail in another. To understand the behavior of the pressure build-up and drawdown data that were studied, and to explain the shape of the resulting plots, a fractured reservoir model was employed and analyzed mathematically. The model is based on the following assumptions:1. The matrix blocks act like sources which feed the fractures with fluid;2. The net fluid movement toward the wellbore obtains only in the fractures; and3. The fractures' flow capacity and the degree of fracturing of the reservoir are uniform. By the degree of fracturing is meant the fractures' bulk volume per unit reservoir bulk volume. Assumption 3 does not stipulate that either the fractures or the matrix blocks should possess certain size, uniformity, geometric pattern, spacing, or direction. Moreover, this assumption of uniform flow capacity and degree of fracturing should be taken in the same general sense as one accepts uniform permeability and porosity assumptions in a homogeneous reservoir when deriving the unsteady-state fluid flow equation. Thus, the assumption may not be unreasonable, especially if one considers the evidence obtained from examining samples of fractured outcrops and reservoirs. Such samples show that the matrix usually consists of numerous blocks, all of which are small compared to the reservoir dimensions and well spacings. Therefore, the model could be described to represent a "homogeneously" fractured reservoir. SPEJ P. 60ˆ

Fractured Reservoir Simulation

1983 ◽  
Vol 23 (01) ◽  
pp. 42-54 ◽  
Author(s):  
L. Kent Thomas ◽  
Thomas N. Dixon ◽  
Ray G. Pierson
Keyword(s):  
Fractured Reservoirs ◽  
Fracture Flow ◽  
Fracture System ◽  
Fractured Reservoir ◽  
Three Phase ◽  
Matrix Fracture ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
The Matrix ◽  
Saturation History

Abstract This paper describes the development of a three-dimensional (3D), three-phase model for simulating the flow of water, oil, and gas in a naturally fractured reservoir. A dual porosity system is used to describe the fluids present in the fractures and matrix blocks. Primary flow present in the fractures and matrix blocks. Primary flow in the reservoir occurs within the fractures with local exchange of fluids between the fracture system and matrix blocks. The matrix/fracture transfer function is based on an extension of the equation developed by Warren and Root and accounts for capillary pressure, gravity, and viscous forces. Both the fracture flow equations and matrix/fracture flow are solved implicitly for pressure, water saturation, gas saturation, and saturation pressure. We present example problems to demonstrate the utility of the model. These include a comparison of our results with previous results: comparisons of individual block matrix/fracture transfers obtained using a detailed 3D grid with results using the fracture model's matrix/fracture transfer function; and 3D field-scale simulations of two- and three-phase flow. The three-phase example illustrates the effect of free gas saturation on oil recovery by waterflooding. Introduction Simulation of naturally fractured reservoirs is a challenging task from both a reservoir description and a numerical standpoint. Flow of fluids through the reservoir primarily is through the high-permeability, low-effective-porosity fractures surrounding individual matrix blocks. The matrix blocks contain the majority of the reservoir PV and act as source or sink terms to the fractures. The rate of recovery of oil and gas from a fractured reservoir is a function of several variables, included size and properties of matrix blocks and pressure and saturation history of the fracture system. Ultimate recovery is influenced by block size, wettability, and pressure and saturation history. Specific mechanisms pressure and saturation history. Specific mechanisms controlling matrix/fracture flow include water/oil imbibition, oil imbibition, gas/oil drainage, and fluid expansion. The study of naturally fractured reservoirs has been the subject of numerous papers over the last four decades. These include laboratory investigations of oil recovery from individual matrix blocks and simulation of single- and multiphase flow in fractured reservoirs. Warren and Root presented an analytical solution for single-phase, unsteady-state flow in a naturally fractured reservoir and introduced the concept of dual porosity. Their work assumed a continuous uniform porosity. Their work assumed a continuous uniform fracture system parallel to each of the principal axes of permeability. Superimposed on this system was a set of permeability. Superimposed on this system was a set of identical rectangular parallelopipeds representing the matrix blocks. Mattax and Kyte presented experimental results on water/oil imbibition in laboratory core samples and defined a dimensionless group that relates recovery to time. This work showed that recovery time is proportional to the square root of matrix permeability divided by porosity and is inversely proportional to the square of porosity and is inversely proportional to the square of the characteristic matrix length. Yamamoto et al. developed a compositional model of a single matrix block. Recovery mechanisms for various-size blocks surrounded by oil or gas were studied. SPEJ P. 42

Pressure-Transient Behavior of Continuously and Discretely Fractured Reservoirs

2014 ◽  
Vol 17 (01) ◽  
pp. 82-97 ◽  
Author(s):  
Fikri Kuchuk ◽  
Denis Biryukov
Keyword(s):  
Fractured Reservoirs ◽  
Flow Regimes ◽  
Transient Behavior ◽  
Naturally Fractured Reservoirs ◽  
Dual Porosity ◽  
Pressure Transient ◽  
Naturally Fractured ◽  
Dual Porosity Model ◽  
Well Tests ◽  
Porosity Model

Summary Fractures are common features of many well-known reservoirs. Naturally fractured reservoirs contain fractures in igneous, metamorphic, and sedimentary formations. Faults in many naturally fractured carbonate reservoirs often have high-permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected-network fracture system). New semianalytical solutions are used to understand the pressure behavior of naturally fractured reservoirs containing a network of discrete and/or connected (continuous) finite- and infinite-conductivity fractures. We present an extensive literature review of the pressure-transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual-porosity model is a fictitious homogeneous porous medium because it does not contain any fractures. Second, by use of the new solutions, we show that for most naturally fractured reservoirs, the Warren and Root (1963) dual-porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure-transient well tests because it does not capture the behavior of these reservoirs. We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual-porosity model. These examples exhibit very diverse pressure behaviors of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution and on their intensity and conductivity. We show these flow regimes with our new model for discretely and continuously fractured reservoirs. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual-porosity reservoir-model behavior. If we interpret them by use of this dual-porosity model, then the estimated permeability, skin factor, interporosity flow coefficient (λ), and storativity ratio (ω) will not represent the actual reservoir parameters.

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