Dual-porosity Systems Research Articles

Insights into pore-scale controls on mudstone permeability through resedimentation experiments

At a given porosity, mudstone permeability increases by an order of magnitude for clay contents ranging from 57% to 36% (<2 μm). This increase in vertical permeability results from a dual-porosity system that develops through three mechanisms: (1) silt bridging preserves large pore throats, (2) stress bridges inhibit clay particle alignment, and (3) local clay particle compression within stress bridges alters pore throat size distribution. Uniaxial consolidation experiments on resedimented clay-silt mixtures illuminate how permeability varies as a function of clay fraction during burial. Backscattered electron microscope images show that silty mixtures have larger pore throats and fewer aligned clay particles than do more clay-rich mixtures. We describe the permeability of clay-silt mixtures with a geometric mean model. Our method provides a promising framework for modeling of mudstone permeability as a function of clay fraction and porosity. How permeability and consolidation evolve during burial affects the ability of mudstones to seal CO2 and hydrocarbons in the subsurface, how mudstones behave as gas reservoirs, and under what conditions mudstones will be overpressured. Dual-porosity systems have fundamentally different transient flow and solute transport behaviors.

Derivation of the Kazemi–Gilman–Elsharkawy Generalized Dual Porosity Shape Factor

Kazemi et al. (SPE Reserv Eng 7(2):219–227, 1992) suggested an empirical matrix-fracture transfer function, verified based on experimental data of Mattax and Kyte (Trans AIME 225(15):177–184, 1962), to model fluid flow in naturally fractured dual porosity petroleum reservoirs using a dual-porosity numerical simulator. Their generalized shape factor should be valid for all possible irregular matrix blocks. The factor is calculated based on the volume of the matrix block, the surface open to flow in all directions and the distances of these surfaces to the centre of the matrix block. The summation is done over all open surfaces of a matrix block. Kazemi et al. (1992) showed that for rectangles and cylinders the formula reduces to the well-known forms of the shape factor. By the time, many authors indicated the validity of the formula, but no theoretical proof was offered for that so far. This study derives the Kazemi et al. (1992) shape factor using control volume finite difference discretization on the fracture-matrix dual continuum. The matrix blocks are handled as Voronoi polyhedra. The derivation is given for both isotropic and tensorial matrix permeability. Based on this derivation the authors conclude that the Kazemi et al. (SPE Reserv Eng 7(2):219–227, 1992) formula is exact under pseudo-steady-state conditions within the dual continuum mathematical concept of natural fractured dual porosity systems.

From computed microtomography images to resistivity index calculations of heterogeneous carbonates using a dual-porosity pore-network approach: Influence of percolation on the electrical transport properties

Standard reservoir evaluations are based on Archie's law relating the average water saturation to the average electrical resistivity by R(ind) = S(w)(-2). However, especially in the case of complex heterogeneous carbonates, deviation from Archie's law is observed and generally attributed to factors affecting the percolation or disconnectedness of the different phases (wetting films, microporosity, macropores) assuring electrical conductance. Pore-network models (PNM's) in combination with high-resolution computed microtomography (μ-CT) constitute a very effective tool to investigate the influence of the geometry and topology of the porous media on the spatial distribution of the conductive phase, and therefore on the shape of the resistivity index curve. An extended version of the classical PNM applicable to dual-porosity systems is presented. It combines the classical pore-network modeling applied on the macroporous space with the macroscopic properties of the microporous phase, supposing that the two pore systems act in parallel. Three-dimensional images provide information on the connectedness of the microporous phase, which is then included in the simulations. Electrical behavior of sandstone and two carbonates presenting distinct resistivity index curves were simulated and compared to measurements. Both Archie and "non-Archie" behavior were correctly reproduced, and the curve shape was explained considering percolation of the different phases.

Matrix–fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media

The matrix–fracture transfer shape factor is one of the important parameters in the modeling of fluid flow in fractured porous media using a dual-porosity concept. Warren and Root [36] introduced the dual-porosity concept and suggested a relation for the shape factor. There is no general relationship for determining the shape factor for a single-phase flow of slightly compressible fluids. Therefore, different studies reported different values for this parameter, as an input into the flow models. Several investigations have been reported on the shape factor for slightly compressible fluids. However, the case of compressible fluids has not been investigated in the past. The focus of this study is, therefore, to find the shape factor for the single-phase flow of compressible fluids (gases) in fractured porous media. In this study, a model for the determination of the shape factor for compressible fluids is presented; and, the solution of nonlinear gas diffusivity equation is used to derive the shape factor. The integral method and the method of moments are used to solve the nonlinear governing equation by considering the pressure dependency of the viscosity and isothermal compressibility of the fluid. The approximate semi-analytical model for the shape factor presented in this study is verified using single-porosity, fine-grid, numerical simulations. The dependency of the shape factor on the gas specific gravity, pressure and temperature are also investigated. The theoretical analysis presented improves our understanding of fluid flow in fractured porous media. In addition, the developed matrix–fracture transfer shape factor can be used as an input for modeling flow of compressible fluids in dual-porosity systems, such as naturally fractured gas reservoirs, coalbed methane reservoirs and fractured tight gas reservoirs.

Applying Linear-Flow Analysis to Shale-Gas Wells - Field Cases

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 130370, ’Application of Linear-Flow Analysis to Shale-Gas Wells - Field Cases,’ by Hasan A. Al-Ahmadi, SPE, Anas M. Almarzooq, SPE, and R.A. Wattenbarger, SPE, Texas A&amp;M University, prepared for the 2010 SPE Unconventional Gas Conference, Pittsburgh, Pennsylvania, 23-25 February. The paper has not been peer reviewed. Tight gas wells behave as if controlled by transient linear flow. The “half-slope” on a type curve indicates this behavior, which enables determining certain reservoir parameters. Linear-flow behavior has been observed in shale-gas wells also, but these wells tend to exhibit a significant “skin effect” that is less common in tight gas wells. This skin effect can mask early linear behavior but may be accounted for with a modified equation. Introduction Shale-gas reservoirs are being developed with horizontal wells and multistage fracturing. These wells produce in a transient-linear-flow manner, and, in some cases, this flow regime could last for years and might be the only flow regime available for analysis. A half-slope on the log-log graph of gas rate vs. time or a straight line on the square-root-of-time graph characterizes this behavior. Fig. 1 shows an example of a shale-gas well in transient linear flow for more than 2 years. The approach used in this study was to assume that linear-transient-flow drainage out of matrix blocks controls production. However, unlike tight gas wells, shale-gas wells tend to exhibit a significant skin effect, which masks the early linear behavior and which must be accounted for with a modified equation. Gas adsorption is not accounted for in this model because most of the data are in the transient-flow regime in which gas-desorption effects are negligible. Occurrence of Linear Flow Fig. 1 shows a log-log graph and a square-root-of-time graph for daily production from Well 114. Transient linear flow is shown as a half-slope on the log-log graph and as a straight line on the square-root-of-time graph. However, the early part of these curves, before 200 days, does not seem to represent transient linear flow. In earlier work, it was thought that the early departures from a half-slope might represent bilinear flow in a dual-porosity reservoir. A more-plausible explanation and analysis were determined with this study. Dual-Porosity Linear-Flow Model An ideal shale-gas well would produce from a rectangular dual-porosity reservoir—a system of fractures with matrix blocks flowing into the fractures, with the reservoir not extending beyond the fracture system. Thus, this system is a linear dual-porosity system, and solutions have been presented as Laplace-domain solutions. Two conceptual models are shown in Fig. 2. They are equivalent in the sense that they both represent dual-porosity linear systems. Model 1 is a linear, dual-porosity “transient-slab model.” The dominant fracture system in Model 1 is hydraulic fractures emanating from equally spaced perforation clusters in the wellbore. The matrix blocks in Model 1 are treated as homogeneous, although they may contain natural fractures. The main calculation advantage of Model 1 is knowledge of fracture spacing, L1, because it depends on the perforation-cluster spacing.

Shape factor in the drawdown solution for well testing of dual-porosity systems

One of the important parameters in existing commercial dual-porosity reservoir simulators is matrix–fracture shape factor, which is customarily obtained by assuming a constant pressure at the matrix–fracture boundary. In his work, Chang [1,2] addressed the impact of boundary conditions at the matrix–fracture interface and presented analytical solutions for the transient shape factor and showed that for a slab-shaped matrix block a constant pressure boundary condition leads to an asymptotic (long-time) shape factor of π 2/ L 2, and that a constant volumetric flux leads to an asymptotic shape factor of 12/ L 2. In a recent paper [3], we reconfirmed Chang’s [1,2] results using a Laplace transform approach. In this study, we extend our previous analysis and use infinite-acting radial and linear dual-porosity models, where the boundary condition is chosen at the wellbore, as opposed to at the matrix boundary. The coupled equations for fracture and matrix are solved analytically, taking into account the transient exchange between matrix and fracture. The analytical solution that invokes the time dependency of fracture boundary condition under constant rate is then used to calculate the transient shape factors. It is shown that, for a well producing at constant rate from a naturally fractured reservoir, the appropriate value of stabilized shape factor is 12/ L 2. This contrasts with the commonly used shape factor for a slab-shaped matrix block that is subject to a constant pressure boundary condition, which is π 2/ L 2. The errors in the matrix–fracture exchange term in a dual-porosity model associated with the use of a shape factor derived based on constant pressure boundary condition at the matrix boundary are then evaluated.

A finite element approach for modelling single-phase compressible flow in dual porosity systems

Fluid flow in a rock formation that contains a network of fractures occurs through two coupled systems, the fracture network and the blocks of porous and permeable rock matrix. Dynamic modelling is challenging when low permeability of the rock matrix creates transient effects that can persist for long periods. In this paper, we describe a finite element (FE) based algorithm developed to model the flow of highly compressible gas through a fractured reservoir. The fracture network is represented by a two-dimensional (2D) FE mesh, coupled to a number of customised strings of one-dimensional (1D) elements representing individual matrix blocks. The use of multiple FEs to model each matrix block replaces the transfer function and facilitates accurate modelling of transient effects in the matrix blocks while still honouring the fully compressible nature of gas. We use the algorithms to study some of the peculiarities of gas flow through fractured reservoirs with low matrix permeability. Matrix block geometry is shown to be an important parameter requiring accurate representation, particularly under transient flow conditions. We show how different geometric shapes can be reduced to 1D representations while still retaining much of the essential geometric information. It is shown that where transient effects occur, neglecting to capture them in the model can result in large errors. A conceptual system is suggested for classifying fractured reservoirs into six categories, each with a preferred modelling approach determined by the values of matrix permeability and fracture density. Finally, a case study is presented of a gas field producing from a low permeability fractured carbonate via two wells. A history match is successfully accomplished due to the flexible meshing capability of the finite element method and the transient modelling capabilities of the algorithms.

Integrated dip data, seismic and potential field data interpretation for fault analysis of the Elk/Antelope gas field

The Elk/Antelope gas field was discovered during drilling of the Elk-1 and Elk-4 wildcat wells in the eastern Papuan Fold Belt, Papua New Guinea. The gas field is located in a frontier greenfields area with limited wells, regional scale potential field data datasets and 2D seismic datasets of varying quality. The Elk structure is a fractured carbonate reservoir while the Antelope structure is potentially a reefal carbonate reservoir with fracture control on permeability. An important consideration in reservoir modelling of dual porosity systems is the effect of faults on performance and storage of the reservoir. Key factors are the fault damage zone width and its affect on permeability. The integrated interpretation of dip data, Seismic and potential field data has been conducted with the aim of identification of faults in the Elk/Antelope reservoir and analysis of their likely effect on reservoir performance.

Decline-Curve Analysis of Fractured Reservoirs With Fractal Geometry

Summary Evaluation of reservoir parameters through well-test and decline-curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely, but it can be found in the distribution of fractures within a naturally fractured reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected. However, evidence from outcrops, well logging, production-behavior studies, and the dynamic behavior observed in these systems, in general, indicate the above assumptions are not representative of these systems. Fractal theory considers a nonuniform distribution of fractures and the presence of fractures at different scales; thus, it can contribute to explain the behavior of many fractured reservoirs. The objective of this paper is to investigate the production-decline behavior in an NFR exhibiting single and double porosity with fractal networks of fractures. The diffusion equations used in this work are a fractal-continuity expressions presented in previous studies in the literature and a more recent generalization of these equations, which includes a temporal fractional derivative. The second objective is to present a combined analysis methodology, which uses transient-well-test and bound-ary-dominated-decline production data to characterize an NFR exhibiting fractures, depending on scale. Several analytical solutions for different diffusion equations in fractal systems are presented in Laplace space for both constant-wellbore-pressure and pressure-variable-rate inner-boundary conditions. Both single- and dual-porosity systems are considered. For the case of single-porosity reservoirs, analytical solutions for different diffusion equations in fractal systems are presented. For the dual-porosity case, an approximate analytical solution, which uses a pseudosteady-state matrix-to-fractal fracture-transfer function, is introduced. This solution is compared with a finite-difference solution, and good agreement is found for both rate and cumulative production. Short- and long-time approximations are used to obtain practical procedures in time for determining some fractal parameters. Thus, this paper demonstrates the importance of analyzing both transient and boundary-dominated flow-rate data for a single-well situation to fully characterize an NFR exhibiting fractal geometry. Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation consistently explains the peculiar behavior observed in some real production-decline curves.

Estimation of Storativity Ratio in a Layered Reservoir With Crossflow

Summary This paper presents a practical method to estimate the storativity ratio of a dual-permeability layered reservoir with crossflow from pressure-transient data. The method uses an analytical formula for the storativity ratio in terms of the separation between the two semilog straight lines on the pressure vs. log-time plot, similar to the method used for dual-porosity systems. Knowing the storativity ratio from a well test allows individual-layer properties to be estimated if the layer flow rates are available from production logs. Demonstrations of the method to estimate the storativity ratio and individual-layer properties are presented by examples. Comparison of the results with those obtained from the existing techniques is also provided to highlight the accuracy of the proposed technique.

Decline-Curve Analysis Considering Nonlaminar Flow in Dual-Porosity Systems

Summary Most of the theoretical work on decline-curve analysis with non-laminar flow in the reservoir considers either a dry gas or liquid flow in a homogeneous system. The purpose of this work is to analyze decline curves considering the non-Darcy flow effect in the reservoir for a slightly-compressible-liquid flow in dual-porosity systems at constant wellbore pressure. The transient and boundary-dominated flow periods are examined by means of computer results generated with a finite-difference simulator. Analytical expressions of velocity and wellbore rate are presented for the first time in this paper. These equations contain the laminar-flow solution as a particular case. A method is proposed to identify the presence of inertial effects by plotting d[log(q)]/d[log(t)] against log(t). The method for area calculation under laminar conditions produces a good approximation for a reservoir under nonlaminar flow. However, the presence of inertial effects distorts the shape of decline curves, resulting in erroneous estimates of wellbore and reservoir parameters by type-curve analysis with laminar decline curves. The utility of the methodology presented in this work is illustrated with synthetic examples.

Streamline simulation of counter-current imbibition in naturally fractured reservoirs

Describing fluid transport in naturally fractured reservoirs entails additional challenge because of the complicated physics arising from matrix–fracture interactions. In this paper, the streamline-based simulation is generalized to describe fluid transport in naturally fractured reservoirs through a dual-media approach. The fractures and matrix are treated as separate continua that are connected through a transfer function, as in conventional finite difference simulators for modeling fractured systems. The transfer functions that describe fluid exchange between the fracture and matrix system can be implemented easily within the framework of the current single-porosity streamline models. In particular, the streamline time of flight concept is utilized to develop a general dual-porosity, dual-permeability system of equations for water injection in naturally fractured reservoirs. The saturations equations are solved using an operator splitting approach that involves ‘convection’ along streamline followed ‘matrix–fracture’ exchange calculations on the grid. The proposed formulation reduces to the commonly used dual-porosity model when the flow in the matrix is considered negligible. For modeling matrix–fracture interactions, two different transfer functions are examined: an empirical transfer function (ETF) and a conventional transfer function (CTF). The ETF allows for analytical solution of the saturation equation for dual porosity systems and is used to validate numerical implementation. Results obtained using the CTF are compared with a commercial finite-difference simulator (ECLIPSE) for waterflooding in five-spot and nine-spot patterns. The streamline approach shows close agreement in terms of recovery histories and saturation profiles with a marked reduction in numerical dispersion and grid orientation effects. Finally, an examination of the scaling behavior of the computation time indicates that the streamline approach is likely to result in significant savings for large-scale field applications.

Streamline simulation of counter-current imbibition in naturally fractured reservoirs

Describing fluid transport in naturally fractured reservoirs entails additional challenge because of the complicated physics arising from matrix–fracture interactions. In this paper, the streamline-based simulation is generalized to describe fluid transport in naturally fractured reservoirs through a dual-media approach. The fractures and matrix are treated as separate continua that are connected through a transfer function, as in conventional finite difference simulators for modeling fractured systems. The transfer functions that describe fluid exchange between the fracture and matrix system can be implemented easily within the framework of the current single-porosity streamline models. In particular, the streamline time of flight concept is utilized to develop a general dual-porosity, dual-permeability system of equations for water injection in naturally fractured reservoirs. The saturations equations are solved using an operator splitting approach that involves ‘convection’ along streamline followed ‘matrix–fracture’ exchange calculations on the grid. The proposed formulation reduces to the commonly used dual-porosity model when the flow in the matrix is considered negligible. For modeling matrix–fracture interactions, two different transfer functions are examined: an empirical transfer function (ETF) and a conventional transfer function (CTF). The ETF allows for analytical solution of the saturation equation for dual porosity systems and is used to validate numerical implementation. Results obtained using the CTF are compared with a commercial finite-difference simulator (ECLIPSE) for waterflooding in five-spot and nine-spot patterns. The streamline approach shows close agreement in terms of recovery histories and saturation profiles with a marked reduction in numerical dispersion and grid orientation effects. Finally, an examination of the scaling behavior of the computation time indicates that the streamline approach is likely to result in significant savings for large-scale field applications.

On using particle tracking methods to simulate transport in single-continuum and dual continua porous media

The use of particle tracking methods to predict transport in single continuum and dual continua (mobile–immobile systems and fractured media) is studied. The accuracy of the particle tracking methods with different interpolation and tracking techniques is evaluated, and its transport predictions are compared to analytical solutions, finite element solutions (e.g. SUTRA) and finite difference solution (e.g. MT3D). For a two-dimensional problem with homogeneous conductivity and pulse injection of contaminant, the particle tracking solution matches the analytical solution better than those using standard finite difference and finite element techniques, which suffer from numerical dispersion. Furthermore, the particle tracking method accurately predicts the mean and variance of the stochastic concentration distribution and compares favorably with MT3DMS that employs a total variance diminishing technique for discretizing the advection term. For modeling matrix diffusion in fractured media and mass transfer in dual porosity (mobile–immobile) systems, two approaches are studied and compared. A semi-analytical approach is compared to a particle tracking technique that accounts for matrix diffusion using particle transfer probabilities. An empirical relationship that can be used to map the governing parameter of the semi-analytical approach to the corresponding particle transfer probability is derived. The similarities and differences between these two techniques and their suitability for practical applications are also discussed.

Determination of Fractal Parameters of Fracture Networks Using Pressure-Transient Data

SummaryThe objective of this paper is to investigate the transient pressure behavior of naturally fractured reservoirs with fractal characteristics. This work is based on the findings of previous studies, which have shown that the networks of fractures in some reservoirs are fractals. Thus, using this assumption, an approximate analytical solution for dual-porosity systems is derived in the Laplace space.The approximate solution presented in this work uses a pseudosteady-state matrix-to-fractal fracture transfer function. This solution is compared with a finite-element solution, and good agreement is found. Short- and long-time approximations are used to obtain procedures in time to determine some fractal parameters. These approximations are compared to the appropriate expressions when an unsteady-state matrix-to-fractal fracture transfer function is used.Synthetic and field examples are presented to illustrate the methodology proposed in this work.This paper also presents an analytical solution for the pseudosteady-state flow period and demonstrates the importance of analyzing both transient and pseudosteady-state flow-pressure data for a single-well situation to fully characterize a naturally fractured reservoir with a fractal geometry.

Non‐Isotropic structured fibrous media: The permeability of arrays of fiber bundles of elliptical cross section

AbstractA computational analysis of viscous flow across hexagonal arrays of fiber bundles of elliptical cross section, each consisting of up to 350 individual filaments, is carried out using the Boundary Element Method. Such arrays are non‐isotropic dualporosity systems, characterized by the aspect ratio (λ) of the fiber bundles, their inter‐ (ϕi) and intra‐tow (ϕt) porosities as well as by the number (Nf) of filaments within each bundle. The results, in terms of the hyraulic permeability (K) of these arrays, are compared to earlier analytical models and a good agreement is found. A dimensionless correlation, earlier proposed for arrays of circular cross section, is used to reduce the computed permeabilities for bundles of aspect ratio λ = 2 and λ = 3 on a single master curve. This correlation offers a simple model for the calculation of the permeability of such dual porosity systems form knowledge of ϕi, ϕt, λ and Nf. By considering bulk flow along the two principal directions of the fiber bundle, we also compare the two principal permeabilities of such arrays.

Development of Permeability Models for Saturated Fluid Flow across Arrays of Fibre Clusters

Dual porosity fibrous media are important in a number of applications, ranging from bioreactor design and transport in living systems to composites manufacturing. In the present study we are concerned with the development of predictive models for the hydraulic permeability ( Kp) of various arrays of fibre bundles. For this we carry out extensive computations for viscous flow through arrays of fibre bundles using the Boundary Element Method (BEM) implemented on a multi-processor computer. Up to 350 individual filaments, arranged in square or hexagonal packing within bundles, which are also arranged in square of hexagonal packing, are included in each simulation. These are simple but not trivial models for fibrous preforms used in composites manufacturing – dual porosity systems characterised by different inter- and intra-tow porosities. The way these porosities affect the hydraulic permeability of such media is currently unknown and is elucidated through our simulations. Following numerical solution of the governing equations, ( Kp) is calculated from the computed flowrate through Darcy's law and is expressed as function of the inter- and intra-tow porosities (φ, φt) and of the filament radius ( Rf). Numerical results are also compared to analytical models. The latter form the starting point in the development of a dimensionless correlation for the permeability of such dual porosity media. It is found that the numerically computed permeabilities follow that correlation for a wide range of φ i, φt and Rf.

Relative Permeability in Dual Porosity Porous Media

Abstract Reliable relative permeability data is an essential input parameter in reservoir engineering, most significantly in the area of reservoir simulation of dual porosity systems. However, measurements of relative permeability do not work well, because of laboratory limitations. Also, reservoir core samples are generally extracted from zones where induced or natural fractures are absent. Obviously, data obtained from these cores may not reflect the real behaviour of naturally fractured reservoirs. Because of this laboratory limitation, many commercial reservoir simulators neglect the effect of dual porosity on relative permeability, and implicitly assume that relative permeability is a straight line. This assumption in naturally fractured systems may lead to erroneous results. The purpose of this study is to rectify these shortcomings by laboratory experimental contributions. Thus, the main objectives are: (a) to perform special core analyses on Berea outcrop core samples as a model of rock; (b) to simulate fracture opening by cutting these samples in such a way as to get different fracture apertures; (c) to investigate the effect of dual porosity on the shape of capillary pressure curves; and (d) to evaluate the effect of fracture opening on both absolute and relative permeability. A good correlation between absolute permeability and fracture aperture is obtained. The effect of dual porosity is observed clearly on capillary pressure curves. Unsteady-state tests could not be used to measure relative permeability on these specially prepared core samples. This is due to the fact that fractures become the easiest pathway for water flow, which results in high residual oil saturation in the matrix. However, the centrifuge technique test is run with success because both matrix and fracture are subjected to the centrifuge field. These findings are extended to an actual naturally fractured reservoir (NFR) in Algeria. The Tin Fouye Tabankort (TFT) reservoir is selected as a prototype of an Algerian NFR. Availability of naturally fractured cores and published data are the principal reasons for this selection. A discussion of TFT natural fracture indicators is presented, including core observations, well test analysis, and borehole imager tools. Displacement tests are conducted on a full diameter core in order to solve the laboratory limitations, and to obtain representative data of relative permeability. These laboratory tests indicate the existence of a good correlation between permeability and fracture opening. The correlation is used to estimate the aperture of natural fractures in the TFT reservoir. This study may also lead to the development of a laboratory technique for determining systematically the fracture intensity index. Background Multiphase flow of fluids through porous media can be related to the relative permeability of each phase. These flow properties are the composite effect of several petrophysical parameters, including: pore geometry, wettability, fluid distribution, and saturation history. This concept of relative permeability is utilized extensively in reservoir engineering for understanding and predicting many physical phenomena that occur during reservoir exploitation. According to Rossen and Kaumar(1), most reservoir simulators neglect the effect of the dual porosity on relative permeability. However conventional fractured-reservoir simulators assume that straight-line relative permeability curves apply within the fracture pore space.

An Integrated Approach to the Determination of Permeability Tensors for Naturally Fractured Reservoirs

Abstract The use of permeability tensors is required when modelling fluid flow in anisotropic and heterogeneous reservoirs presenting multiple zones of directional permeability, or those categorized as naturally fractured reservoirs. A general procedure for characterizing complex reservoirs utilizing their permeability tensor is being developed by integrating data and methods from different disciplines. Permeability tensors for geologically defined fracture patterns are derived, and finally these small-scale descriptors are incorporated into a reservoir simulation program capable of handling full tensor permeabilities. The application and convenience of the method presented in this paper is illustrated with a field example from a naturally fractured reservoir. Introduction For many years, substantial research has been conducted in the areas of geosciences and engineering in order to characterize naturally fractured reservoirs. Numerous approaches have been presented to properly overcome this difficult task. Geoscientists have focused their research towards understanding the process of fracturing (rock mechanics), and the subsequent description of fracture characteristics such as density and orientation. Engineers, on the other hand, have focused their attention to the description of the fluid flow in the fracture systems, and in the development of accurate models (reservoir simulators) to reproduce the history, and predict the hydrocarbon production, for these complex systems. One of the first matrix-fracture models was presented by Warren and Root(1), who presented an idealized sugar cube model with two classes of porosity, a primary porosity that is intergranular, and a secondary porosity that is induced by fractures. Even though the sugar cube model has been widely accepted as the forerunner of the modern interpretation of dual-porosity systems, its limitations in describing the behaviour of some complex fractured reservoir systems have been observed. It is now well known that almost all fracture systems are much more complicated than the suggested model of three orthogonal sets of uniform fractures. One of the most important factors that has been identified as a necessary addition to improve the overall description of such complex reservoirs is the definition of a nine-component permeability tensor for the fracture system. This tensor is used to model fluid flow in complex reservoirs with multiple zones of directional permeability, where the orientation and magnitude of the principal permeabilities may vary between different zones in the reservoir. Snow(2, 3) studied the convenience of using mathematical equivalents of parallel plate openings to simulate fractures dispersed in orientation, distributed in aperture, and of arbitrary spacing. Models for fractured media which contained any number of planar conductors of any orientation and any fine aperture were presented. A key assumption was that all of the conduits have smooth parallel plane walls of indefinite extent (infinite fractures), and an rbitrary aperture. As a result, a permeability tensor could be obtained by superposition of contributions due to the fractures, and due to the permeable matrix. Long et al.(4, 5) addressed the more realistic scenario of finite or discrete fracture systems, where properties such as shape, orientation and location of the fractures in an impermeable matrix were considered to be random variables.

Micromechanical computational modeling of secondary consolidation and hereditary creep in soils

A computational modeling of a fluid-saturated deformable porous media characterized by two levels of hydrodynamics (flow in micro and macropores) is proposed based on a micromechanical analysis of dual porosity systems, i.e., media locally characterized by a porous matrix composed of permeable cells containing micropores and the surrounding system of macropores, void spaces or bulk flow paths (e.g., fissured rock or aggregated soil). The homogenization technique is applied to upscale the constitutive and geometric information available in the fine structure to the field scale leading to a microstructure model of dual porosity type, wherein the poroelastic cells act as distributed sources/sinks of mass and momentum to the global macroscopic medium. The theory provides a rigorous derivation of some secondary compression and hereditary creep effects in soils due to the delayed drainage of the fluid within the micropores under consolidation. Application of the Green's function method reduces the dual porosity system to a single-porosity viscoelastic integrodifferential system of Volterra type in which the constitutive law for the macroscopic stress tensor is given in terms of an hereditary integral with memory. A two-level finite element method is proposed to solve the coupled micro–macro governing equations of dual porosity type. Numerical experiments are performed showing the strong potential of the proposed formulation in solving consolidation problems with microstructure.