scholarly journals Analytical Solutions for Multiple Matrix in Fractured Reservoirs: Application to Conventional and Unconventional Reservoirs

SPE Journal ◽  
2013 ◽  
Vol 18 (05) ◽  
pp. 969-981 ◽  
Author(s):  
Mehmet A. Torcuk ◽  
Basak Kurtoglu ◽  
Najeeb Alharthy ◽  
Hossein Kazemi
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Control Volume ◽  
Fractured Reservoirs ◽  
Unconventional Reservoirs ◽  
Unconventional Reservoir ◽  
Flow Equations ◽  
Enhanced Production ◽  
Matrix Blocks ◽  
The Matrix

Summary In this paper, we present a new method to model heterogeneity and flow channeling in petroleum reservoirs—especially reservoirs containing interconnected microfractures. The method is applicable to both conventional and unconventional reservoirs where the interconnected microfractures form the major flow path. The flow equations, which could include flow contributions from matrix blocks of various size, permeability, and porosities, are solved by the Laplace-transform analytical solutions and finite-difference numerical solutions. The accuracy of flow from and into nanodarcy matrix blocks is of great interest to those dealing with unconventional reservoirs; thus, matrix flow equations are solved by use of both pseudosteady-state (PSS) and unsteady state (USS) formulations and the results are compared. The matrix blocks can be of different size and properties within the representative elementary volume (REV) in the analytical solutions, and within each control volume (CV) in the numerical solutions. Although the analytical solutions were developed for slightly compressible rock/fluid linear systems, the numerical solutions are general and can be used for nonlinear, multiphase, multicomponent flow problems. The mathematical solutions were used to analyze the longterm and short-term performances of two separate wells in an unconventional reservoir. It is concluded that matrix contribution to flow is very slow in a typical low-permeability unconventional reservoir and much of the enhanced production is from the fluids contained in the microfractures rather than in the matrix. In addition to field applications, the mathematical formulations and solution methods are presented in a transparent fashion to allow easy usage of the techniques for reservoir and engineering applications.

Numerical Simulation of Water-Oil Flow in Naturally Fractured Reservoirs

1976 ◽  
Vol 16 (06) ◽  
pp. 317-326 ◽  
Author(s):  
H. Kazemi ◽  
L.S. Merrill ◽  
K.L. Porterfield ◽  
P.R. Zeman
Keyword(s):  
Finite Difference ◽  
Fractured Reservoirs ◽  
Naturally Fractured Reservoirs ◽  
Phase Flow ◽  
Two Phase ◽  
Flow Equations ◽  
Difference Grid ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
The Matrix

Abstract A three-dimensional, multiple-well, numerical simulator for simulating single- or two-phase flow of water and oil is developed for fractured reservoirs. The simulator equations are two-phase flow extensions of the single-phase flow equations derived by Warren and Root. The simulator accounts for relative fluid mobilities, gravity force, imbibition, and variation in reservoir properties. The simulator handles uniformly and nonuniformly properties. The simulator handles uniformly and nonuniformly distributed fractures and for no fractures at all. The simulator can be used to simulate the water-oil displacement process and in the transient testing of fractured reservoirs. The simulator was used on the conceptual models of two naturally fractured reservoirs: a quadrant of a five-spot reservoir and a live-well dipping reservoir with water drive. These results show the significance of imbibition in recovering oil from the reservoir rock in reservoirs with an interconnected fracture network. Introduction Numerical reservoir simulators are being used extensively to simulate multiphase, multicomponent flow in "single-porosity" petroleum reservoirs. Such simulators generally cannot be used to petroleum reservoirs. Such simulators generally cannot be used to study flow behavior in the naturally fractured reservoirs that are usually classified as double-porosity systems. In the latter, one porosity is associated with the matrix blocks and the other porosity is associated with the matrix blocks and the other represents that of the fractures and vugs. If fractures provide the main path for fluid flow from the reservoir, then usually the oil from the matrix blocks flows into the fracture space, and the fractures carry the oil to the wellbore. When water comes in contact with the oil zone, water may imbibe into the matrix blocks to displace oil. Combinations of large flow rates, low matrix permeability, and weak imbibition may result in water fingering permeability, and weak imbibition may result in water fingering through the fractures into the wellbore. Once fingering of water occurs, the water-oil ratio may increase to a large value. None of the published theoretical work on multiphase flow in naturally fractured systems has been applied directly to the simulation of a reservoir as a whole. Usually, only a segment of the reservoir was simulated, and the results were extrapolated to the entire reservoir. To simulate a reservoir as a whole, we have developed a mathematical formulation of the flow problem that has been programmed as a three-dimensional, compressible, water-oil reservoir simulator. The simulator equations are two-phase flow extensions of the single-phase flow equations derived by Warren and Root. The theory is based on the assumption of double porosity at each point in a manner that the fractures form a continuum filled by the noncontinuous matrix blocks. In other words, the fractures are the boundaries of the matrix blocks. The flow equations are solved by a finite difference method. A typical finite-difference grid cell usually contains one or several matrix blocks. In this case, all the matrix blocks within the finite-difference grid cell have the same pressure and saturation. Gravity segregation within individual matrix blocks is not calculated, but the over-all gravity segregation from one grid cell to another is accounted for. In many practical problems, this approximation is acceptable. In some situations, a matrix block encloses several finite-difference grid cells. In this case, the gravity segregation within the matrix block is calculated. To include heterogeneity, a redefinition of local porosities and permeabilities provides a method for simulating situations where part of the reservoir is fractured and where part is not fractured. The above description points to the complexity of the situations that one encounters. Therefore, the judicious choice of the number of finite difference grid cells with respect to the number of matrix blocks becomes a critical engineering decision. Later sections will provide insight to alleviate such decisions. SPEJ P. 317

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

scholarly journals Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: An Analytical Approach

2020 ◽  
pp. 106-118
Author(s):  
Abu Bakar Siddique ◽  
Tariq A. Khraishi
Keyword(s):  
Analytical Solutions ◽  
Analytical Approach ◽  
Numerical Solutions ◽  
Linear Equations ◽  
Symmetric Matrices ◽  
Matrix Equations ◽  
Eigenvalues And Eigenvectors ◽  
Engineering Problems ◽  
The Matrix ◽  
Research Problems

Research problems are often modeled using sets of linear equations and presented as matrix equations. Eigenvalues and eigenvectors of those coupling matrices provide vital information about the dynamics/flow of the problems and so needs to be calculated accurately. Analytical solutions are advantageous over numerical solutions because numerical solutions are approximate in nature, whereas analytical solutions are exact. In many engineering problems, the dimension of the problem matrix is 3 and the matrix is symmetric. In this paper, the theory behind finding eigenvalues and eigenvectors for order 3×3 symmetric matrices is presented. This is followed by the development of analytical solutions for the eigenvalues and eigenvectors, depending on patterns of the sparsity of the matrix. The developed solutions are tested against some examples with numerical solutions.

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.

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

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.

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

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

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

Problems in Characterization of Naturally Fractured Reservoirs From Well Test Data

1985 ◽  
Vol 25 (03) ◽  
pp. 445-450 ◽  
Author(s):  
I. Ershaghi ◽  
R. Aflaki
Keyword(s):  
Test Data ◽  
Transition Period ◽  
Fractured Reservoirs ◽  
Naturally Fractured Reservoirs ◽  
Late Time ◽  
Time Data ◽  
Straight Line ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
The Matrix

Abstract This paper presents a critical analysis of some recently published papers on naturally fractured reservoirs. These published papers on naturally fractured reservoirs. These publications have pointed out that for a publications have pointed out that for a matrix-to-fracture-gradient flow regime, the transition portion of pressure test data on the semilog plot develops a portion of pressure test data on the semilog plot develops a slope one half that of the late-time data. We show that systems under pseudosteady state also may develop a 1:2 slope ratio. Examples from published case studies are included to show the significant errors associated with the characterization of a naturally fractured system by using the 1:2 slope concept for semicomplete well tests. Introduction Idealistic models of the dual-porosity type often have been recommended for interpretation of a well test in naturally fractured reservoirs. The evolutionary aspects of these models have been reviewed by several authors. Gradual availability of actual field tests and recent developments in analytical and numerical solution techniques have helped to create a better understanding of application and limitation of various proposed models. Two important observations should be made here. First, just as it is now recognized that classical work published by Warren and Root in 1963 was not the end of the line for interpretation of the behavior of naturally fractured systems, the present state of knowledge later may be considered the beginning of the technology. Second, parallel with the ongoing work by various investigators who progressively include more realistic assumptions in their progressively include more realistic assumptions in their analytical modeling, one needs to ponder the implication of these findings and point out the inappropriate impressions that such publications may precipitate in the mind of practicing engineers. practicing engineers. This paper is intended to scrutinize statements published in recent years about certain aspects of the anticipated transition period developed on the semilog plot of pressure-drawdown or pressure-buildup test data. pressure-drawdown or pressure-buildup test data. The Transition Period In the dual-porosity models published to date, a naturally fractured reservoir is assumed to follow the behavior of low-permeability and high-storage matrix blocks in communication with a network of high-permeability and low-storage fractures. The difference among the models has been the assumed geometry of the matrix blocks or the nature of flow between the matrix and the fracture. However, in all cases, it is agreed that a transition period develops that is strictly a function of the matrix period develops that is strictly a function of the matrix properties and matrix-fracture relationship. Fig. 1 shows properties and matrix-fracture relationship. Fig. 1 shows a typical semilog plot depicting the transition period and the parallel lines. Estimation of Warren and Root's proposed and to characterize a naturally fractured proposed and to characterize a naturally fractured system requires the development of the transition period. The Warren and Root model assumes a set of uniformly distributed matrix blocks. Furthermore, the flow from matrix to fracture is assumed to follow a pseudosteady-state regime. Under such conditions, in theory, this period is an S-shaped curve with a point of inflection. Uldrich and Ershaghi developed a technique to use the coordinates of this point of inflection for estimating and under conditions where either the early- or the late-time straight lines were not available. Kazemi and de Swann presented alternative approaches to represent naturally fractured reservoirs. They assumed a geometrical configuration consisting of layered matrix blocks separated by horizontal fractures. Their observation was that for such a system the transition period develops as a straight line with no inflection point. Bourdet and Gringarten identified a semilog straight line during the transition period for unsteady-state matrix-fracture flow. Recent work by Streltsova and Serra et al emphasized the transient nature of flow from matrix to fracture and pointed out the development of a unique slope ratio. These authors, later joined by Cinco-L. and Samaniego-V., stated that under a transient flow condition, the straight-line shape of the transition period develops a slope that is numerically one-half the slope of the parallel straight lines corresponding to the early- or late-time data. It was further pointed out that the transient flow model is a more realistic method of describing the matrix-fracture flow. As such, they implied that in the absence of wellbore-storage-free early-time data, or late-time data in the case of limited-duration tests, one may use the slope of the transition straight line and proceed with the estimation of the reservoir properties. Statement of the Problem The major questions that need to be addressed at this time are as follows. SPEJ P. 445

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