Conventional Dual-porosity Models Research Articles

An effective stress-dependent dual-fractal permeability model for coal considering multiple flow mechanisms

Coal permeability is significantly affected by the multi-scale pore-fracture size distribution. More importantly, the pore-fracture size is changed by the effective stress, swelling/shrinkage under the influence of gas sorption, and different flow mechanisms. In conventional dual-porosity models, these effects are normally studied separately and the impacts of heterogeneous structure on permeability are neglected. In this study, a dual-fractal permeability model was proposed to quantitatively investigate the impacts of coal internal structure on the permeability. In the improved permeability model, the fractal dimension and the pore-fracture size of the coal are linked with porosity, which are dependent on the evolution of effective stress and matrix shrinkage. Besides, multiple flow mechanisms were also incorporated into the model. The proposed model was verified with field data and achieved a good agreement. The sensitivity studies and model results indicate that: (1) pore/fracture size distributions affect the contribution ratios of versatile flow regimes and to total gas flux and the total permeability; (2) fractal dimensions of matrix and fracture systems increase with the decline of pore pressure; (3) increment of matrix permeability is jointly decided by the transition of flow regime and effective stress, while the fracture permeability is dominated by effective stress.

Pressure Transient Behaviors of Discretely Fractured Reservoirs Using a Numerical Discrete Fracture Model

Summary The conventional dual-porosity model (Warren and Root 1963) may not apply to naturally fractured reservoirs (NFRs), which have poorly connected fractures. To narrow this gap, a new discrete fracture model (DFM)-based numerical well-testing model is developed for pressure transient analysis in vertical wells interacting with natural fractures (NFs). The numerical model is based on a DFM and unstructured perpendicular bisector (PEBI) grid system. The accuracy and practicality of the proposed model have been demonstrated by model verifications with a commercial numerical software. The results show that the flow regimes of the vertical well interacting with NFs can be divided into wellbore storage and skin effects, bilinear flow, linear flow, radial flow, NF effect, and boundary-dominated flow. This is the radial flow of the formation before pressure propagates to NFs, which is virtually quite different from that in the conventional dual-porosity model. However, there are no bilinear and linear flow stages in the vertical well interacting with no NFs. It is found that the vertical well interacting with NFs has a lower pressure depletion. It is also found that the “V-shape” caused by the NF effect in the pressure derivative curve becomes deeper when there are more NFs, longer NFs, and higher fracture conductivity. Furthermore, the “V-shape” appears earlier, and the duration of the NF effect is longer as the number of NFs increases. Besides, with the decrease of the distance between the fracture and the well, the impacts of NFs on pressure transient behaviors of the vertical well are more significant. This work provides a meaningful way to understand the pressure transient behaviors of discrete NFs.

A novel method for calculating polymer adsorption in fractures through a dual porosity model

Increasing demand for energy and limited capacity of available resources of fossil fuels have drawn attention to enhanced oil recovery for maximum use of the available resources. Among different enhanced recovery methods, polymer injection maintains the advantages of water flooding, including the simplicity of this technique, while eliminating some of its drawbacks such as high mobility of the injected water. Therefore, mechanisms involved in the flow of polymer in the reservoirs need to be well understood and modeled. An important phenomenon affecting the flow of polymer in the oil reservoirs is polymer adsorption on reservoir rock. In fractured reservoirs, the polymer is adsorbed both in the matrix and the fracture. Miscalculation of the amount of adsorbed polymer results in the miscalculation of the fracture permeability on one hand, and concentration of dissolved polymer on the other. This, in turn, leads to over- or under-estimation of polymer injection performance in fractured reservoirs. An important tool for simulating fractured reservoirs is dual-porosity models. The models, however, may not correctly estimate polymer adsorption in fractures. This is attributable to the assumptions used in the dual-porosity model and the nature of adsorption. The present study attempts to improve the way this phenomenon is modeled by dual-porosity models. A new parameter called “pseudo-density” is defined here to replace the density of the fracture cells in the dual-porosity model for polymer adsorption. Then, a hypothetical reservoir is simulated using this new parameter, and the results of the two simulations, one with density and the other with pseudo-density, are compared in terms of three parameters, including the polymer adsorbed in the reservoir, the polymer dissolved in the aqueous phase, and the polymer produced in the reservoir. The results show that using pseudo-density instead of rock density decreases the required polymer amount to reach the maximum polymer concentration. Consequently, polymer solution viscosity and permeability reduction are increased. Furthermore, it results in increasing the polymer production compared to the conventional dual-porosity models.

A semi-analytical two-phase flow model of fractured horizontal well with complex fracture networks in natural fractured reservoirs

Complex fracture networks are easily developed along the horizontal wellbore during hydraulic fracturing. The water phase increases the seepage resistance of oil in natural fractured reservoir. The flow regimes become more intricate due to the complex fractures and the occurrence of two-phase flow. Therefore, a semi-analytical two-phase flow model is developed based on the assumption of orthogonal fracture networks to describe the complicate flow regimes. The natural micro-fractures are treated as a dual-porosity system and the hydraulic fracture with complex fracture networks are characterized explicitly by discretizing the fracture networks into multiple fracture segments. The model is solved according to Laplace transformation and Duhamel superposition principle. Results show that seven possible flow regimes are described according to the typical curves. The major difference between the vertical fractures and the fracture networks along the horizontal wellbore is the fluid “feed flow” behavior from the secondary fracture to the main fracture. A natural fracture pseudo-radial flow stage is added in the proposed model comparing with the conventional dual-porosity model. The water content has a major effect on the fluid total mobility and flow capacity in dual-porosity system and complex fracture networks. With the increase of the main fracture number, the interference of the fractures increases and the linear flow characteristics in the fracture become more obvious. The secondary fracture number has major influence on the fluid feed capacity from the secondary fracture to the main fracture. The elastic storativity ratio mainly influences the fracture flow period and inter-porosity flow period in the dual-porosity system. The inter-porosity flow coefficient corresponds to the inter-porosity flow period of the pressure curves. This work is significantly important for the hydraulic fracture characterization and performance prediction of the fractured horizontal well with complex fracture networks in natural fractured reservoirs.

Impact of equilibration time lag between matrix and fractures on the evolution of coal permeability

In conventional dual porosity models, the interactions between matrix and fractures are normally characterized through two equilibrium systems within the same REV (representative elementary volume). This pseudo-steady approach cannot capture the true impact of the non-equilibrium period within each system since it ignores the true transient nature of the fracture-matrix interaction. In this study, a conventional dual porosity model is extended to include the impact of equilibration time lag between matrix and fractures caused by their contrasting properties. To incorporate this important mechanism, the matrix REV is divided into two sub-REVs by using the MINC concept (Multiple Interacting Continua). The time lag effect is defined as a function of the difference between the strain in the matrix REV and that in the sub-matrix REV, and incorporated into a coal permeability model. Consequently, the coal permeability evolves also from initial to final equilibrium. Conventional dual porosity/permeability models represent two end points (initial and final equilibrium) while this new permeability model represents the evolution of coal permeability between these two end points. The model is verified against experimental observations of the evolution of coal permeability under a constant effective stress that extend for more than 80 days. If effective stress remains unchanged, conventional permeability models predict no permeability changes while our new model predicts that coal permeability evolves as a function of time from initial to ultimate equilibrium. Our results suggest that the impact of matrix strain variations on the evolution of coal permeability is significant and should not be ignored.

A semianalytical well-testing model of fracture-network horizontal wells in unconventional reservoirs with multiple discretely natural fractures

Microseismic data shows that some unconventional reservoirs comprise well-developed natural fractures and complex hydraulic fracture networks. It is neither practical nor advantageous to simulate a huge number of natural and hydraulic fractures with numerical models. Given that the conventional dual-porosity models are not applicable to the highly discrete natural fractures, the paper develops a semianalytical well testing model for horizontal wells with hydraulic fracture networks and randomly-distributed discretely natural fractures.The proposed model has the capability to analyze the pressure behaviors by considering complex fracture networks and isolated natural fractures rapidly and efficiently. The model includes diffusivity equations in three domains: (1) matrix, (2) discretely natural fractures, and (3) hydraulic fracture networks. The pressure transient solution of these diffusivity equations is obtained by using Laplace transforms and superposition principle. We verify the presented model by performing a case study with a numerical simulator for complex natural fractures.It is found that there are some interesting flow behaviors for fracture-network horizontal well with discretely natural fractures like bilinear flow, “V-shape” caused by fluid supply, pseudo boundary-dominated flow, impact of natural fractures, etc. The pseudo boundary-dominated flow provides us the information about how large the area covered by hydraulic fracture networks. The impact of natural fracture shows the parameters of natural fractures. This work provides a good understanding of transient pressure behaviors in unconventional reservoirs and guidelines for the producer optimize field development and well economics.

Micro-scale investigation on coupling of gas diffusion and mechanical deformation of shale

The nanopore structure usually exhibit complex geometry in the shale formation. The interactions between gas and pore structure in such heterogeneous property impacts the properties of shale gas transport in micro-scale. The precise description of this shale gas flow processes in detail is impossible if the micropore is not properly characterized. Thus, this study provides a simulation approach to model the complex geometry of nanopore structures in the shale formation. Based on SEM image segmentation of the shale matrix, the geometry of three compositions (nanopore, kerogen, and matrix) are explicitly simulated. Mass storage, transport mechanisms, and geomechanical properties are fully modeled in the micro-scale model. It demonstrates that the conventional dual porosity model fails to capture the storage and transport mechanisms in micro-scale by comparison with the micro-scale model. The simulation results reveal that stress-induced decrease of the diffusion coefficient is both time-dependent and space-dependent. The reduction of the diffusion coefficient can significantly cut down the adsorbed gas production in kerogen. It results in the low recovery rate of shale gas that a large proportion of adsorbed gas is unable to liberate. Moreover, the later stage of gas production is depended on the supply of adsorbed gas in kerogen.

General Multi-Porosity simulation for fractured reservoir modeling

In the area of fractured reservoir modeling, conventional Dual Porosity Models is challenged to flexibly simulate more than two porosity systems and it is also difficult to capture the transient fluid transfer between matrix and fracture. This work aims to solve those problems in fractured reservoir simulation. The newly introduced Multi-Porosity Model honors any number of porosity types with different properties, such as permeability, porosity and wettability thus achieving significant improvements over conventional Dual-Porosity Models. Arbitrary connections for intra-porosity flow and inter-porosity flow are incorporated into the design to allow for the convenient transformation between Multi-Porosity and Multi-Permeability formulations. The addition of a flexible subdivision in each porosity system has allowed us to characterize the transient flow for inter-porosity flow. Due to the low permeability in the matrix, transient flow between matrix and fracture dominates, and thus matrix spatial subdivision is necessary to accurately capture the dynamics. The formulation is designed to allow the proposed scheme to be generalized. To quantify the improvements available with the new formulation, several typical Multi-Porosity Models are compared with Fine-Grid Single-Porosity Models. Consistent results have been obtained with those similar cases from literature, and the robustness and efficiency of the model is validated. Besides, the extra matrix subdivision is proven to accurately capture the transient fluid transfer between matrix and fracture.

Numerical modeling on the sensitivity of directional dependent interface heat transfer on thermal transport in a coupled fracture-matrix system

The conventional dual-porosity model has been modified by considering the heat exchange term at the fracture-matrix interface in the governing equation for thermal transport within the low permeable rock-matrix as against its conventional consideration within the high permeable fracture. A finite volume numerical model has been developed in order to analyze the influence of the source/sink term which defines the heat transfer at the fracturematrix interface. The comparison of the spatial distribution for temperature within the fracture and within the reservoir matrix for two different models, (1) conventional model in which the source/sink heat transfer term included in the equation for thermal transport within the fracture; (2) proposed model in which the source/sink heat transfer term included in the equation for thermal transport within the rock-matrix, have been performed. In addition, the sensitivity of the reservoir matrix thermal conductivities, both horizontal and vertical, on thermal energy extraction from the reservoir matrix has also been analyzed using the proposed model. Numerical results suggest that the estimation of temperature distribution in the fracture and rock-matrix and thus quantifying the heat extraction from the reservoir matrix is underestimated in a fracture-matrix system by using the conventional thermal transport model. It has been also observed that the temperature distribution obtained in the fracture and the rock-matrix by considering the heat transfer term in the thermal transport equation within the fracture shows significant variation from the temperature distribution obtained by considering the heat transfer term in the equation for thermal transport within the rock-matrix.

Beyond dual-porosity modeling for the simulation of complex flow mechanisms in shale reservoirs

The state of the art of modeling fluid flow in shale reservoirs is dominated by dual-porosity models which divide the reservoirs into matrix blocks that significantly contribute to fluid storage and fracture networks which principally control flow capacity. However, recent extensive microscopic studies reveal that there exist massive micro- and nano-pore systems in shale matrices. Because of this, the actual flow mechanisms in shale reservoirs are considerably more complex than can be simulated by the conventional dual-porosity models and Darcy’s law. Therefore, a model capturing multiple pore scales and flow can provide a better understanding of the complex flow mechanisms occurring in these reservoirs. This paper presents a micro-scale multiple-porosity model for fluid flow in shale reservoirs by capturing the dynamics occurring in three porosity systems: inorganic matter, organic matter (mainly kerogen), and natural fractures. Inorganic and organic portions of shale matrix are treated as sub-blocks with different attributes, such as wettability and pore structures. In kerogen, gas desorption and diffusion are the dominant physics. Since the flow regimes are sensitive to pore size, the effects of nano-pores and micro-pores in kerogen are incorporated into the simulator. The multiple-porosity model is built upon a unique tool for simulating general multiple-porosity systems in which several porosity systems may be tied to each other through arbitrary connectivities. This new model allows us to better understand complex flow mechanisms and eventually is extended into the reservoir scale through upscaling techniques. Sensitivity studies on the contributions of the different flow mechanisms and kerogen properties give some insight as to their importance. Results also include a comparison of the conventional dual-porosity treatment and show that significant differences in fluid distributions and dynamics are obtained with the improved multiple-porosity simulation.

Fractured-Reservoir Modeling and Interpretation

SummaryFractures are common features of many well-known reservoirs. Naturally fractured reservoirs (NFRs) consist of fractures in igneous, metamorphic, and sedimentary rocks (matrix). Faults in many naturally fractured carbonate reservoirs often have high-permeability zones and are connected to numerous fractures with varying conductivities. In many NFRs, faults and fractures frequently have discrete distributions rather than connected-fracture networks. Because faulting often creates fractures, faults and fractures should be modeled together. Accurately modeling NFR pressure-transient behavior is important in hydrogeology, the earth sciences, and petroleum engineering, including groundwater contamination to shale gas and oil reservoirs. For more than 50 years, conventional dual-porosity-type models, which do not include any fractures, have been used for modeling fluid flow in NFRs and aquifers. They have been continuously modified to add unphysical matrix-block properties such as matrix skin factor.In general, fractured reservoirs are heterogeneous at different length scales. It is clear that even with millions of gridblocks, numerical models may not be capable of accurately simulating the pressure-transient behavior of continuously and discretely NFRs containing variable-conductivity fractures. The conventional dual-porosity-type models are obviously an oversimplification; their serious limitations for interpreting well-test data from NFRs are discussed in detail. These models do not include wellbore-intersecting fractures, even though they dominate the pressure behavior of NFRs for a considerable length of testing time. Fracture conductivities of unity to infinity dominate transient behavior of both continuously and discretely fractured reservoirs, but again, dual-porosity models do not contain any fractures. Our fractured-reservoir model is capable of treating thousands of fractures that are periodically or arbitrarily distributed with finite- and/or infinite conductivities, different lengths, densities, and orientations.Appropriate inner-boundary conditions are used to account for wellbore-intersecting fractures and direct wellbore contributions to production. Wellbore-storage and skin effects in bounded and unbounded systems are included in the model. Three types of damaged-skin factors that may exist in wellbore-intersecting fracture(s) are specified. With this highly accurate model, the pressure-transient behavior of conventional dual-porosity-type models are investigated, and their limitations and range of applicability are identified. The behavior of the triple-porosity models is also investigated. It is very unlikely that triple-porosity behavior is caused by the local variability of matrix properties at the microscopic level. Rather, it is caused by the spatial variability of conductivity, length, density, and orientation of the fracture distributions. Finally, we have presented an interpretation of a field-buildup-test example from an NFR by use of both conventional dual-porosity models and our fractured-reservoir model.A substantial part of this paper is a review and discussion of the earlier work on NFRs, including the authors’ work.

Pressure Transient Behavior of Horizontal Wells Intersecting Multiple Hydraulic Fractures in Naturally Fractured Reservoirs

In this study, we present a mesh-free semi-analytical technique for modeling pressure transient behavior of continuously and discretely hydraulically and naturally fractured reservoirs for a single-phase fluid. In our model, we consider a 3D reservoir, where each fracture is explicitly modeled without any upscaling or homogenization as required for dual-porosity media. Fractures can have finite or infinite conductivities, and the formation (matrix) is assumed to have a finite permeability. Our approach is based on the boundary element method. The method has advantages such as the absence of grids and reduced dimensionality. It provides continuous rather than discrete solutions. The uniform-pressure boundary condition over the wellbore is used in our mathematical model. This is the true physical boundary condition for any type of well, whether fractured or not, provided that the friction pressure drop in the wellbore is small and the fluid is Newtonian. The method is sufficiently general to be applied to many different well geometries and reservoir geological settings, where the spatial domain may include arbitrary fracture and/or fault distribution, a number of horizontal wells with and without hydraulic fractures, and different types of outer boundaries. The model also applies to multistage hydraulically fractured horizontal wells in homogenous reservoirs. More specifically, it is applied to investigate the pressure transient behavior of horizontal wells in continuously and discretely naturally fractured reservoirs, including multistage hydraulically fractured horizontal wells. A number of solutions have been published in the literature for horizontal wells in naturally fractured reservoirs using the conventional dual-porosity models that are not applicable to many of these reservoirs that contain horizontal wells with multiple fractures. Most published solutions for fractured horizontal wells in homogenous and naturally fractured reservoirs ignore the presence of the wellbore and the contribution to flow from the formation directly into the unfractured horizontal sections of the wellbore. Therefore, some of the flow regimes from these solutions are incorrect or do not exist, such as fracture-radial flow regime. In our solutions, all or some of multistage hydraulic fractures may intersect the natural fractures, which is very important for shale gas and oil reservoir production. The number and type of fractures (hydraulic or natural) intersecting the wellbore and with each other are not limited in both homogeneous and naturally fractured reservoirs. Our solutions are compared with a number of existing solutions published in the literature. Example diagnostic derivative plots are presented for a variety of horizontal wells with multiple fractures in homogenous and naturally fractured reservoirs.

Evaluation of Engineered Geothermal Systems as a Heat Source for Oil Sands Production in Northern Alberta

This paper evaluates the application of geothermal energy by numerically modeling the heat extraction that would result from the injection of cold water into an artificially fractured hot dry rock (HDR). The HDR that would be utilized in Alberta is expected to be granite with a network of pre-existing natural fractures. However, to ensure a continued flow of injected water from the reservoir to the production wells, creation of additional fractures is required. Thus, the properties of these fractures are of prime importance to the efficiency of geothermal energy production. The fracture networks for the simulations were created using a numerical code and were converted into a grid format to be used in a commercial thermal simulator. A new approach to embed a complex fracture system into the numerical model was applied. Various properties of the fractures such as aperture, length, and spacing were changed and their absolute and relative effects on energy production were quantified and the results are presented in this paper. This modeling technique was also verified by comparison with the conventional dual porosity model and by performing a history match with real field data obtained from literature. The applicability of this approach to provide heat for oil sands extraction was investigated using the volumes of water currently needed in northern Alberta. Based on these constraints, numerical simulations were run to evaluate the optimum well spacing that would be required using a three-well configuration. In this simulation, the fracture parameters (density and aperture) were kept fixed assuming that they are not affected by cold water injection. The results of this study suggest that geothermal energy has a potential to be a sustainable form of thermal energy for oil sands extraction in northern Alberta.