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.

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

Bivariate Log-Normal Distribution of Stimulated Matrix Permeability and Block Size in Fractured Reservoirs: Proposing New Multilinear-Flow Regime for Transient-State Production

SPE Journal ◽  
2018 ◽  
Vol 23 (04) ◽  
pp. 1316-1342 ◽  
Author(s):  
Salam Al-Rbeawi
Keyword(s):  
Normal Distribution ◽  
Flow Regime ◽  
Fractured Reservoirs ◽  
Block Size ◽  
Analytical Models ◽  
Matrix Permeability ◽  
Matrix Block ◽  
The Matrix ◽  
Log Normal ◽  
Log Normal Distribution

Summary This paper investigates the impacts of varied stimulated matrix permeability and matrix-block size on pressure behaviors and flow regimes of hydraulically fractured reservoirs using bivariate log-normal distribution. The main objective is assembling the variance in these two parameters to the analytical models of pressure and pressure derivative considering different porous-media petrophysical properties, reservoir configurations, and hydraulic-fracture (HF) characteristics. The motivation is eliminating the long-run discretization treatment in the porous media required by applying analytical models to describe the variance in the previously discussed parameters with the distance between HFs. Several analytical models for pressure response were generated in this study for hydraulically fractured reservoirs with rectangular-shaped drainage areas. These models take into account the change in stimulated matrix permeability from the maximum value close to the HF face to the minimum value at the so-called no-flow boundary between fractures. They also consider the change in the matrix-block size, corresponding to the change in the induced-fracture density (number of fractures per foot of length), from the minimum value close to the fracture face to the maximum value at the no-flow boundary. Bivariate log-normal distribution was used to describe the change in the stimulated matrix permeability and matrix-block size. The formations of interest are composed of stimulated reservoir volume (SRV), where the matrix is stimulated by the fracturing process, and unstimulated reservoir volume (USRV), where the stimulation process does not affect the matrix. The outcomes of this study can be summarized as Generating new analytical models for pressure and pressure derivative in hydraulically fractured reservoirs that consider the change in stimulated matrix-block size and permeability using bivariate log-normal distribution Understanding the effect of using the probability-density function (PDF) of matrix-block size and permeability in the pressure distribution of different reservoirs Observing the new multilinear-flow regime that develops at intermediate production time and represents several simultaneous linear-flow regimes inside HFs, SRV, and USRV Developing analytical models for the new multilinear-flow regime Studying the effects of petrophysical properties of HFs, induced fractures, and matrix as well as reservoir size and configuration on pressure behavior The most interesting points in this study are The applicability of bivariate log-normal distribution for describing the variance and nonuniform distribution of matrix-block size and permeability. The large variance in the matrix-block size and permeability causes significant decrease in wellbore-pressure drop. Small value of standard deviation of matrix-block size and permeability indicates the possibilities for significant decrease in wellbore pressure drop. The means of matrix-block size and permeability may not have significant effects on reservoir-pressure distribution. The new multilinear-flow regime is characterized by a one-eighth slope on the pressure-derivative curve and is seen always after HF linear flow and before boundary-dominated flow regime. Multilinear-flow regime develops to bilinear-flow regime with a one-quarter slope for uniform distribution of equal matrix-block size and permeability.

Revisiting the Dual-Porosity/Dual-Permeability Modeling of Unconventional Reservoirs: The Induced-Interporosity Flow Field

SPE Journal ◽  
2015 ◽  
Vol 20 (01) ◽  
pp. 124-141 ◽  
Author(s):  
Gorgonio Fuentes-Cruz ◽  
Peter P. Valkó
Keyword(s):  
Hydraulic Fracture ◽  
Block Size ◽  
Well Performance ◽  
Fracture Permeability ◽  
Permeability Ratio ◽  
Linear Flow ◽  
Matrix Block ◽  
Matrix Blocks ◽  
The Matrix ◽  
Shale Reservoirs

Summary This work presents a new approach to account for variable matrix-block size in the well performance of unconventional shale reservoirs. In contrast to the standard models that consider either a fixed matrix-block size or a stochastic distribution of sizes with no particular dependence on the spatial location, in this model we consider the case when the characteristic length of the blocks depends on the distance from the main hydraulic-fracture plane. In particular, we assume that, as a result of the hydraulic-fracturing treatment, the density of microfractures (natural and induced) is high near the hydraulic-fracture face, but gradually decreases away from it. In other words, the matrix-block size is an increasing function of the distance from the fracture face. We show that the boost of the contact area between the matrix blocks and the microfractures can be the determinant feature of hydrocarbon production from shale reservoirs, even when the natural and induced microfractures do not have uniform density throughout the whole stimulated reservoir volume. In addition, the elongated linear flow (the consensus characteristic signature in the well performance of shale systems) may be the result of an induced interporosity flow when the matrix-/fracture-permeability ratio is small. The duration of this flow regime increases if the effective size of the matrix-block distribution increases. On the other hand, the linear flow can be the result of the total-system response when the matrix-/fracture-permeability ratio is high; in this case, the beginning of the linear flow is strongly affected by the small matrix blocks near the hydraulic-fracture faces. The resulting mathematical model in Laplace space has a fundamentally different structure compared with standard dual-porosity models because of the spatial dependence of the parameters characterizing the interporosity flow. In this work, we develop the Airy-spline scheme, a new technique to calculate the pressure solution in an efficient way. Closed-form approximate formulae in the time domain are also provided, revealing that during the induced linear-interporosity-flow period, the production behavior is controlled by the logarithmic mean of the minimum and maximum matrix-block size. A field example from the Barnett shale is presented to illustrate the use of the new model.

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

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