Convection in Fractured Reservoirs Numerical Calculation of Convection in a Vertical Fissure, Including the Effect of Matrix-Fissure Transfer

1976 ◽  
Vol 16 (05) ◽  
pp. 281-301 ◽  
Author(s):  
D.W. Peaceman
Keyword(s):  
Fractured Reservoirs ◽  
Saturation Pressure ◽  
Fractured Reservoir ◽  
Density Inversion ◽  
Dissolved Gas ◽  
Rectangular Shape ◽  
Pressure Depletion ◽  
Matrix Blocks ◽  
The Matrix ◽  
Transfer Parameters

Abstract A fractured reservoir undergoing pressure depletion, evolution of gas at the top of the oil zone leads to an unstable density inversion in the fissures. The resulting convection brings heavy oil into contact with matrix blocks containing light oil, and results in the transfer of dissolved gas between matrix and fissure in the undersaturated region of the oil zone. To provide a better understanding of this process, numerical solutions were obtained to the differential equations that describe convection in a vertical fissure and include the matrix-fissure transfer. The numerical procedure is an extension of the method of characteristics for miscible displacement problems in two dimensions. The effect of gas evolution in the upper portion of the oil zone is also included in the numerical model. Calculations for a vertical fissure of rectangular shape, with an initial sinusoidal perturbation of an in verse density gradient, show an initial exponential growth of the perturbation that agrees well with that predicted from mathematical theory. Calculations predicted from mathematical theory. Calculations for a vertical fissure having a similar, but slightly tilted, rectangular shape and with an initial, correspondingly tilted, inverse density gradient, show that the effect of the matrix-fissure transfer parameters on the time for overturning can be parameters on the time for overturning can be correlated quite well by curves obtained from mathematical perturbation theory. The most realistic calculations were carried out for the same vertical fissure having a slightly tilted rectangular shape, with declining pressure at the apex and gas evolution included in the gassing zone. The relative saturation-pressure depression in the fissure below the gassing zone can be characterized as increasing as the square of the time, following an incubation period. For practical ranges of the matrix-fissure period. For practical ranges of the matrix-fissure transfer parameters investigated, it is concluded that saturation-pressure depression will be significant. A preliminary correlation for this Pb depression was obtained. The fissure thickness was found to affect the Pb depression significantly. Introduction Most fractured reservoirs of commercial interest are characterized by the existence of a system of high-conductivity fissures together with a large number of matrix blocks containing most of the oil. It has been recognized for some time that analysis of the behavior of a fractured reservoir must involve an understanding of the performance of single matrix blocks under the various environmental conditions that can exist in the fissures. However, it has only recently been recognized that a similar need exists for a better understanding of convective mixing taking place within the oil-filled portion of the fissure system. In a fractured reservoir undergoing pressure depletion, gas will be evolved at all points of the reservoir where the oil pressure has declined below the original saturation pressure. This depth interval is referred to as the gassing zone. Because of the high conductivity of the fracture, the gas in the fissures will segregate rapidly from the oil before reaching the producing wells and most, if not all, of it will join the expanding gas cap. At a sufficient depth, however, the oil pressure will still be higher than the saturation pressure, and the oil there remains in an undersaturated condition. In the gassing zone, gas evolves from the oil in both the fissures and the matrix. The oil left behind in the fissures within the gassing zone contains less dissolved gas and is heavier than the oil below it in the undersaturated zone. This density inversion can result in considerable convection within the highly conductive fissures. As a result of this convection, heavy oil containing less gas is transported downward through the fissures, placing it in contact with matrix blocks containing lighter oil with more dissolved gas. Transfer of dissolved gas from matrix to fissure takes place owing to molecular diffusion through the porous matrix rock; and even more transfer takes place owing to local convection within the matrix block induced by the density contrast between fissure and matrix oil. SPEJ p. 281

Convection in Fractured Reservoirs - The Effect of Matrix-Fissure Transfer on the Instability of a Density Inversion in a Vertical Fissure

1976 ◽  
Vol 16 (05) ◽  
pp. 269-280 ◽  
Author(s):  
D.W. Peaceman
Keyword(s):  
Fractured Reservoirs ◽  
Saturation Pressure ◽  
High Conductivity ◽  
Fractured Reservoir ◽  
Density Inversion ◽  
Dissolved Gas ◽  
Rate Of Growth ◽  
Pressure Depletion ◽  
Matrix Blocks ◽  
The Matrix

Abstract In a fractured reservoir undergoing pressure depletion, evolution of gas at the top of the oil zone leads to an unstable density inversion in the fissures. The resulting convection brings heavy oil into contact with matrix blocks containing light oil, and results in the transfer of dissolved gas between matrix and fissure in the undersaturated region of the oil zone. To provide a better understanding of dis process, an earlier perturbation analysis of a density inversion in a vertical fissure has been extended to include the matrix-assure transfer. It was found that matrix-fissure transfer does not affect the stability or instability of a density inversion, nor does it affect the spacing of density angers or the size of convection cells. A quantitative expression for the rate of growth of unstable density fingers was derived. The effect of matrix-fissure transfer is always to reduce the rate of growth. For practical reservoir cases, while any density inversion should be highly unstable, matrix-fissure transfer can be expected to cause a significant reduction in the linger growth rate. Introduction Most fractured reservoirs of commercial interest are characterized by the existence of a system of high-conductivity fissures together with a large number of matrix blocks containing most of the oil. It has been recognized for some time that the analysis of the behavior of a fractured reservoir must involve an understanding of the performance of single matrix blocks under the various environmental conditions that can exist in the fissures. However, only recently has it been recognized that a similar need exists for a better understanding of the convective mixing that probably takes place within the oil-filled portion of the fissure system. In a fractured reservoir undergoing pressure depletion, gas will be evolved at all points of the reservoir where the oil pressure has declined below the original saturation pressure. This depth interval is referred to as the gassing zone. Because of the high conductivity of the fracture, the gas in the fissures will segregate rapidly from the oil before reaching the producing wells and most, if not all, of it will join the expanding gas cap. At a sufficient depth, however, the oil pressure will still be higher than the saturation pressure, and the oil there remains in an undersaturated condition. (See Fig. 1.) In the gassing zone, gas evolves from the oil in both the fissures and the matrix. The oil left behind in the fissures within the gassing zone contains less dissolved gas and is heavier than the oil below it in the undersaturated zone. SPEJ P. 269

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

The "Checker Model," An Improvement in Modeling Naturally Fractured Reservoirs With a Tridimensional, Triphasic, Black-Oil Numerical Model

1985 ◽  
Vol 25 (05) ◽  
pp. 743-756 ◽  
Author(s):  
Dimitrie Bossie-Codreanu ◽  
Paul R. Bia ◽  
Jean-Claude Sabathier
Keyword(s):  
Transfer Functions ◽  
Fractured Reservoirs ◽  
Fracture Network ◽  
Fractured Reservoir ◽  
Local Flow ◽  
Fractured Medium ◽  
Black Oil ◽  
Matrix Blocks ◽  
The Matrix ◽  
Conventional Models

Abstract This paper describes an approach to simulating the flow of water, oil, and gas in fully or partially fractured reservoirs with conventional black-oil models. This approach is based on the dual porosity concept and uses a conventional tridimensional, triphasic, black-oil model with minor modifications. The basic feature is an elementary volume of the fractured reservoir that is simulated by several model cells; the matrix is concentrated into one matrix cell and tee fractures into the adjacent fracture cells. Fracture cells offer a continuous path for fluid flows, while matrix cello are discontinuous ("checker board" display). The matrix-fracture flows are calculated directly by the model. Limitations and applications of this approximate approach are discussed and examples given. Introduction Fractured reservoir models were developed to simulate fluid flows in a system of continuous fractures of high permeability and low porosity that surround discontinuous, porous, oil-saturated matrix blocks of much lower permeability but higher porosity. The use of conventional models that permeability but higher porosity. The use of conventional models that actually simulate the fractures and matrix blocks is restricted to small systems composed of a limited number of matrix blocks. The common approach to simulating a full-field fractured reservoir is to consider a general flow within the fracture network and a local flow (exchange of fluids) between matrix blocks and fractures. This local flow is accounted for by the introduction of source or sink terms (transfer functions). In this formulation, the model is not directly predictive because the source term (transfer function) is, in fact, entered data and is derived from outside the model by one of the following approaches:analytical computation,empirical determination (laboratory experiments), ornumerical simulation of one or several matrix blocks on a conventional model. To derive these transfer functions, imposing some boundary conditions is necessary. Unfortunately, it is generally impossible to foresee all the conditions that will arise in a, matrix block and its surrounding fractures during its field life. It would be helpful, therefore, to have a model that is able to compute directly the local flows according to changing conditions. However, to have low computing times, it is necessary to use an approximate formulation and, thus, to adjust some parameters to match results that are externally (and more accurately) derived in a few basis, well-defined conditions. By current investigative techniques, only a very general description of the matrix blocks and fissures can be obtained, so our knowledge of local flows is very approximate. This paper presents a modeling procedure that is an approximate but helpful approach to the simulation of fractured reservoirs and requires a few, simple modifications of conventional black-oil mathematical models. Review of the Literature Numerous papers related to single- and multiphase flow in fractured porous media have been published over the last three decades. On the basis of data from fractured limestone and sand-stone reservoirs, fractured reservoirs are pictured as stacks of matrix blocks separated by fractures (Figs. 1 and 2). The fractured reservoirs with oil-saturated matrices usually are referred to as "double porosity" systems. Primary porosity is associated with matrix blocks, while secondary porosity is associated with fractures. The porosity of the matrices is generally much greater than that of the fractures, but permeability within fractures may be 100 and even over 10,000 times higher permeability within fractures may be 100 and even over 10,000 times higher than within the matrices. The main difference between flow in a fractured medium and flow in a conventional porous system is that, in a fractured medium, the interconnected fracture network provides the main path for fluid flow through the reservoir, while local flows (exchanges of fluids) occur between the discontinuous matrix blocks and the surrounding fractures. Matrix oil flows into the fractures, and the fractures carry the oil to the wellbore. For single-phase flow, Barenblatt et al constructed a formula based on the dual porosity approach. They consider the reservoir as two overlying continua, the matrices and the fractures. SPEJ p. 743

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

Simulation of Naturally Fractured Reservoirs With Semi-Implicit Source Terms

1977 ◽  
Vol 17 (03) ◽  
pp. 201-210 ◽  
Author(s):  
R.H. Rossen
Keyword(s):  
Fractured Reservoirs ◽  
Complex Problem ◽  
Fractured Reservoir ◽  
Simulation Techniques ◽  
Fracture Simulation ◽  
Reservoir Systems ◽  
Matrix Blocks ◽  
The Matrix ◽  
Matrix Rock ◽  
Source Sink

Abstract Conventional reservoir simulation techniques prove to be inadequate when applied directly to the prove to be inadequate when applied directly to the study of fractured reservoir systems. Such systems are characterized by extremes in porosity, permeability, and saturation. The vast bulk of the permeability, and saturation. The vast bulk of the reservoir volume is occupied by relatively low-permeability, disjoint matrix blocks of various sizes surrounded by a small volume of high-permeability, interconnected fracture space. Our approach to this complex problem bas been to treat the matrix blocks as source and sink terms in an otherwise conventional simulation that models only the fracture system. The source/sink terms are functions of matrix rock and fluid properties with fracture saturation and pressure defining the boundary conditions. These functions are derived either by history-matching simulations or independently by laboratory experiments or single matrix-block simulation. This basic concept of a source/sink treatment is not unique to this work. However, the numerical formulation and implementation of these terms in the fracture simulation offers significant advantages over existing modeling procedures. The fundamental advantage of our approach is that these source terms are handled semi-implicitly in both the pressure and saturation calculations involved in pressure and saturation calculations involved in the fracture simulation. This avoids instability problems that are inherent in a sequential problems that are inherent in a sequential fracture-matrix solution and links more closely the behavior of the matrix and the fracture. Special techniques are developed for modeling the effects of fluid contact movement within a large simulation grid block and for treating receding gas-oil and water-oil contacts. Hysteresis effects are included in matrix blocks that begin to imbibe oil after drainage has begun. Introduction Conventional reservoir simulation techniques are not capable of adequately modeling large, naturally fractured reservoir systems. Extreme discontinuities in porosity, permeability, and saturation exist throughout the reservoir. Most of the fluids are found in very low-permeability, disjoint matrix blocks of various sizes, while most of the fluid mobility is in a small volume of high-permeability, interconnected fracture space. The distribution of fluids within the fracture is usually governed primarily by gravity segregation while the behavior of the individual matrix blocks depends on pressure, fluid environment, and matrix fluid saturation. Fluids can move readily throughout the reservoir in the fracture space, but fluids that reside in matrix rock must enter the fracture to move any great distance. The behavior of individual matrix blocks in response to various drive mechanisms has been studied experimentally by Crawford and Yazdil and has been simulated in two dimensions by Kleppe and Morse and Yamamoto et al. Other investigators have studied the single-phase pressure behavior of fractured reservoirs and its effect on pressure buildup curves. Kazemi investigated single-phase flow in a radial reservoir dominated by horizontal fractures. Closman simultaneously solved equations for flow between matrix and fracture and for flow along the fracture planes for a radial aquifer to develop relationships similar to those developed by van Everdingen and Hursts for water influx. Simulation of an entire reservoir system with multiple phases further complicates the problem and makes additional simulator modifications necessary. Asfari and Witherspoon have developed a modeling approach for reservoirs with a regular pattern of noncommunicating vertical fractures by pattern of noncommunicating vertical fractures by assigning constant pressures along each fracture. Several investigators have applied finiteelement techniques to the fracture-matrix flow problem. problem. Our approach to this complex problem has been to model the flow in the fracture system and to treat fluid transfer to and from the matrix much as injection and production are modeled in conventional simulators. Transfer of fluid into the fracture will be represented by a "source" term and transfer from the fracture to the matrix will be represented by a "sink" (or negative source) term. SPEJ P. 201

Theory of Waterflooding in Fractured Reservoirs

1978 ◽  
Vol 18 (02) ◽  
pp. 117-122 ◽  
Author(s):  
Abraham de Swaan
Keyword(s):  
Analytical Approach ◽  
Laboratory Experiments ◽  
Fractured Reservoirs ◽  
Oil Production ◽  
Fractured Reservoir ◽  
Relative Permeabilities ◽  
Water Imbibition ◽  
Matrix Blocks ◽  
The Matrix ◽  
Numerical Formulation

Abstract This paper presents a new theory of the incompressible flow of two fluids (water displacing oil) in a fractured porous material composed of two distinct media - matrix blocks of low transmissibility embedded in a highly transmissible medium. This general description includes heterogeneous porous media not necessarily of the fractured type. The theory accounts for an important fact not considered in framer analytical model found in the literature. The blocks downstream in a reservoir subject to waterflood are exposed to a varying water saturation resulting from the water imbibition of the upstream blocks. Expressions for the water-oil ratio and the cumulative-oil production are derived, allowing a complete economic evaluation of a fractured-reservoir waterflood project. Comparison of experimental curves reported in the literature with curves obtained using this theory show a good fit. Introduction Imbibition is a most important mechanism of oil production in the waterflooding of fractured production in the waterflooding of fractured reservoirs. Using the action of capillary forces, it allows the recovery of oil from the interior of blocks that cannot be reached by the externally applied gradients of the waterflood. Previous papers assume a function to describe the time rate of exchange of oil and water for a single matrix block. In a lineal reservoir, a water table advances as water is injected with the matrix blocks progressively exposed to water, depending on their position. The oil released by the matrix blocks is assumed transferred instantly to the water-oil interphase,. In this way, the oil production is an added function of individual block contributions. An analytical approach to this problem, and a numerical model, use the problem, and a numerical model, use the simplifying assumption of a water front. This may be a sound description in the presence of vertical high-transmissivity fractures where oil may segregate readily, but in fractures with a discrete transmissivity, it is expected that water imbibition and the simultaneous release of oil by these blocks will give rise to a varying saturation in the fractures that will affect the imbibition rates of the downstream blocks. Braester's analytical approach assumes relative permeabilities of both wetting and nonwetting permeabilities of both wetting and nonwetting phases, intermediate between the fracture's and the phases, intermediate between the fracture's and the matrix's relative permeabilities; these intermediate permeabilities are impossible to measure. The permeabilities are impossible to measure. The model also uses an approximation of the fluid interchange between fractures and blocks. The model may be used for predictions after finding parameters to match observed oil and water parameters to match observed oil and water productions. productions. Kleppe and Morse conducted laboratory experiments on matrix blocks surrounded by fractures and numerical simulations (with rather coarse numerical grids) of Braester's laboratory system. Their numerical simulation computations agree well with the experimental results. This numerical formulation is exact or causalistic; capillary pressures and relative permeabilities are computed pressures and relative permeabilities are computed at every grid block. Their experimental and numerical results are used to test the theory presented here. presented here. Another numerical formulation assumes an approximation for the fluid interchange between fractures and matrix blocks. This approximate formulation did not try to reproduce the exact formulation results of Kleppe and Morse, nor their laboratory experiments. The theory presented here analitically accounts for varying saturations in the fractures by introducing a convolution. A somewhat similar approach -was used successfully to describe the transient one-phase flow in a fractured reservoir. THEORY An outline of the subject theory (developed in the Appendix) includes the following assumed mechanisms and their corresponding mathematical expressions. SPEJ P. 117

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).

Matrix Refinement in Mass Transport Across Fracture-Matrix Interface: Application to Improved Oil Recovery in Fractured Reservoirs

2021 ◽  
Author(s):  
Sarah Abdullatif Alruwayi ◽  
Ozan Uzun ◽  
Hossein Kazemi
Keyword(s):  
Fractured Reservoirs ◽  
Block Size ◽  
Grid Cell ◽  
Matrix Interface ◽  
Unsteady State ◽  
Matrix Block ◽  
Matrix Blocks ◽  
The Matrix ◽  
Saturation Distribution ◽  
State Pressure

Abstract In this paper, we will show that it is highly beneficial to model dual-porosity reservoirs using matrix refinement (similar to the multiple interacting continua, MINC, of Preuss, 1985) for water displacing oil. Two practical situations are considered. The first is the effect of matrix refinement on the unsteady-state pressure solution, and the second situation is modeling water-oil, Buckley-Leverett (BL) displacement in waterflooding a fracture-dominated flow domain. The usefulness of matrix refinement will be illustrated using a three-node refinement of individual matrix blocks. Furthermore, this model was modified to account for matrix block size variability within each grid cell (in other words, statistical distribution of matrix size within each grid cell) using a discrete matrix-block-size distribution function. The paper will include two mathematical models, one unsteady-state pressure solution of the pressure diffusivity equation for use in rate transient analysis, and a second model, the Buckley-Leverett model to track saturation changes both in the reservoir fractures and within individual matrix blocks. To illustrate the effect of matrix heterogeneity on modeling results, we used three matrix bock sizes within each computation grid and one level of grid refinement for the individual matrix blocks. A critical issue in dual-porosity modeling is that much of the fluid interactions occur at the fracture-matrix interface. Therefore, refining the matrix block helps capture a more accurate transport of the fluid in-and-out of the matrix blocks. Our numerical results indicate that the none-refined matrix models provide only a poor approximation to saturation distribution within individual matrices. In other words, the saturation distribution is numerically dispersed; that is, no matrix refinement causes unwarranted large numerical dispersion in saturation distribution. Furthermore, matrix block size-distribution is more representative of fractured reservoirs.

scholarly journals Geomechanical Response of Fractured Reservoirs

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 70 ◽  
Author(s):  
Ahmad Zareidarmiyan ◽  
Hossein Salarirad ◽  
Victor Vilarrasa ◽  
Silvia De Simone ◽  
Sebastia Olivella
Keyword(s):  
Oil Recovery ◽  
Shear Failure ◽  
Numerical Models ◽  
Fractured Reservoirs ◽  
Fluid Pressure ◽  
Carbonate Reservoir ◽  
Improved Oil Recovery ◽  
Matrix Blocks ◽  
The Matrix ◽  
Pressure Changes

Geologic carbon storage will most likely be feasible only if carbon dioxide (CO2) is utilized for improved oil recovery (IOR). The majority of carbonate reservoirs that bear hydrocarbons are fractured. Thus, the geomechanical response of the reservoir and caprock to IOR operations is controlled by pre-existing fractures. However, given the complexity of including fractures in numerical models, they are usually neglected and incorporated into an equivalent porous media. In this paper, we perform fully coupled thermo-hydro-mechanical numerical simulations of fluid injection and production into a naturally fractured carbonate reservoir. Simulation results show that fluid pressure propagates through the fractures much faster than the reservoir matrix as a result of their permeability contrast. Nevertheless, pressure diffusion propagates through the matrix blocks within days, reaching equilibrium with the fluid pressure in the fractures. In contrast, the cooling front remains within the fractures because it advances much faster by advection through the fractures than by conduction towards the matrix blocks. Moreover, the total stresses change proportionally to pressure changes and inversely proportional to temperature changes, with the maximum change occurring in the longitudinal direction of the fracture and the minimum in the direction normal to it. We find that shear failure is more likely to occur in the fractures and reservoir matrix that undergo cooling than in the region that is only affected by pressure changes. We also find that stability changes in the caprock are small and its integrity is maintained. We conclude that explicitly including fractures into numerical models permits identifying fracture instability that may be otherwise neglected.

Gravity-Drainage and Oil-Reinfiltration Modeling in Naturally Fractured Reservoir Simulation

2009 ◽  
Vol 12 (03) ◽  
pp. 380-389 ◽  
Author(s):  
Juan Ernesto Ladron de Guevara-Torres ◽  
Fernando Rodriguez-de la Garza ◽  
Agustin Galindo-Nava
Keyword(s):  
Reservoir Simulation ◽  
Fractured Reservoirs ◽  
Dual Porosity ◽  
Gravity Drainage ◽  
Fractured Reservoir ◽  
Fluid Exchange ◽  
Production Forecast ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
Refined Grid

Summary The gravity-drainage and oil-reinfiltration processes that occur in the gas-cap zone of naturally fractured reservoirs (NFRs) are studied through single porosity refined grid simulations. A stack of initially oil-saturated matrix blocks in the presence of connate water surrounded by gas-saturated fractures is considered; gas is provided at the top of the stack at a constant pressure under gravity-capillary dominated flow conditions. An in-house reservoir simulator, SIMPUMA-FRAC, and two other commercial simulators were used to run the numerical experiments; the three simulators gave basically the same results. Gravity-drainage and oil-reinfiltration rates, along with average fluid saturations, were computed in the stack of matrix blocks through time. Pseudofunctions for oil reinfiltration and gravity drainage were developed and considered in a revised formulation of the dual-porosity flow equations used in the fractured reservoir simulation. The modified dual-porosity equations were implemented in SIMPUMA-FRAC (Galindo-Nava 1998; Galindo-Nava et al. 1998), and solutions were verified with good results against those obtained from the equivalent single porosity refined grid simulations. The same simulations--considering gravity drainage and oil reinfiltration processes--were attempted to run in the two other commercial simulators, in their dual-porosity mode and using available options. Results obtained were different among them and significantly different from those obtained from SIMPUMA-FRAC. Introduction One of the most important aspects in the numerical simulation of fractured reservoirs is the description of the processes that occur during the rock-matrix/fracture fluid exchange and the connection with the fractured network. This description was initially done in a simplified manner and therefore incomplete (Gilman and Kazemi 1988; Saidi and Sakthikumar 1993). Experiments and theoretical and numerical studies (Saidi and Sakthikumar 1993; Horie et al. 1998; Tan and Firoozabadi 1990; Coats 1989) have allowed to understand that there are mechanisms and processes, such as oil reinfiltritation or oil imbibition and capillary continuity between matrix blocks, that were not taken into account with sufficient detail in the original dual-porosity formulations to model them properly and that modify significantly the oil-production forecast and the ultimate recovery in an NFR. The main idea of this paper is to study in further detail the oil reinfiltration process that occurs in the gas invaded zone (gas cap zone) in an NFR and to evaluate its modeling to implement it in a dual-porosity numerical simulator.

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