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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.

Abstract The gridless analytical and semianalytical methodologies can provide credible solutions for describing the well performance of the fracture networks in a homogeneous reservoir. Reservoir heterogeneity, however, is common in unconventional reservoirs, and the productivity can vary significantly along the horizontal wells drilled for producing such reservoirs. It is oversimplified to treat the entire reservoir matrix as homogeneous if there are regions with extremely nonuniform properties in the reservoir. However, the existing analytical and semianalytical methods can only model simple cases involving matrix heterogeneity, such as composite, layered, or compartmentalized reservoirs. A semianalytical methodology, which can model fracture networks in heterogeneous reservoirs, is still absent; in this study, we propose a decomposed fracture network model to fill this gap. We discretize a fractured reservoir into matrix blocks that are bounded by the fractures and/or the reservoir boundary and upscale the local properties to these blocks; therefore, a heterogeneous reservoir can be represented with these blocks that have nonuniform properties. To obtain a general flow equation to characterize the transient flow in the blocks that may exhibit different geometries, we approximate the contours of pressure with the contours of the depth of investigation (DOI) in each block. Additionally, the borders of each matrix block represent the fractures in the reservoir; thus, we can characterize the configurations of complex fracture networks by assembling all the borders of the matrix blocks. This proposed model is validated against a commercial software (Eclipse) on a multistage hydraulic fracture model and a fracture network model; both a homogeneous case and a heterogeneous case are examined in each of these two models. For the heterogeneous case, we assign different permeabilities to the matrix blocks in an attempt to characterize the reservoir heterogeneity. The calculation results demonstrate that our new model can accurately simulate the well performance even when there is a high degree of permeability heterogeneity in the reservoir. Besides, if there are high-permeability regions existing in the fractured reservoir, a BDF may be observed in the early production period, and formation linear flow may be indistinguishable in the early production period because of the influence of reservoir heterogeneity.

Spontaneous imbibition is an important mechanism of oil recovery from fractured reservoirs and unconventional reservoirs. Oil is produced by combining co- and counter-current imbibition when the matrix blocks was partially covered by water. In this paper, we focused on the effect of viscosity ratios on oil production by spontaneous imbibition and established the numerical model for one-dimensional linear imbibition with TEO-OW boundary conditions, which was validated by the experimental data. The effect of viscosity ratio on co- and counter-current imbibition is investigated and scaling result of the imbibition recovery curve for wide range of viscosity ratio using the conventional scaling equation was tested, which indicates that the close correlation was achieved only when oil-water viscosity ratios are higher. Then, a modified scaling equation was developed based on the piston-like assumption for one-dimensional co-current imbibition and close correlation of imbibition recovery curves was achieved when viscosity ratios are lower. Finally, correlation of imbibition recovery curves was improved for wide range of viscosity ratios by combining conventional and modified scaling equation. Results show that since the shape of imbibition recovery curves is not similar for different viscosity ratios, it is difficult to obtain the perfect correlation using the constant viscosity term.

AbstractGravity drainage is known as the controlling mechanism of oil recovery in naturally fractured reservoirs. The efficiency of this mechanism is controlled by block-to-block interactions through capillary continuity and/or reinfiltration processes. In this study, at first, several free-fall gravity drainage experiments were conducted on a well-designed three-block apparatus and the role of tilt angle, spacers’ permeability, wettability and effective contact area (representing a different status of the block-to-block interactions between matrix blocks) on the recovery efficiency were investigated. Then, an experimental-based numerical model of free-fall gravity drainage process was developed, validated and used for monitoring the saturation profiles along with the matrix blocks. Results showed that gas wetting condition of horizontal fracture weakens the capillary continuity and in consequence decreases the recovery factor in comparison with the original liquid wetting condition. Moreover, higher spacers’ permeability increases oil recovery at early times, while it decreases the ultimate recovery factor. Tilt angle from the vertical axis decreases recovery factor, due to greater connectivity of matrix blocks to vertical fracture and consequent channelling. Decreasing horizontal fracture aperture decreases recovery at early times but increases the ultimate recovery due to a greater extent of capillary continuity between the adjacent blocks. Well match observed between the numerical model results and the experimental data of oil recovery makes the COMSOL multiphysics model attractive for application in multi-blocks fractured systems considering block-to-block interactions. The findings of this research improve our understanding of the role of different fracture properties on the block-to-block interactions and how they change the ultimate recovery of a multi-block system.

Summary Double-porosity/naturally fractured reservoir models have traditionally been used to represent the flow and pressure behavior for highly fractured carbonate reservoirs. Given that unconventional reservoirs such as shale-oil/gas reservoirs might not be considered to be multiporosity media, the use of the traditional/classical “double-porosity” models might not be adequate (or appropriate). The recent development of anomalous diffusion models has opened the possibility of adapting double-porosity models to estimate reservoir (and related) parameters for unconventional reservoirs. The primary objective of this work is to develop and demonstrate analytical reservoir models that provide (possible) physical explanations for the anomalous diffusion phenomenon. The models considering anomalous diffusion in reservoirs with Euclidean shape are developed using a convolved (i.e., time-dependent) version of Darcy's law. The use of these models can yield a power-law (straight-line) behavior for the pressure and/or rate performance, similar to the fractal reservoir models. The main advantage of using anomalous diffusion models compared with models considering fractal geometry is the reduction from two parameters (i.e., the fractal dimension and the conductivity index) to only one parameter (i.e., the anomalous diffusion exponent). However, the anomalous diffusion exponent does not provide information regarding the geometry or spatial distribution of the reservoir properties. To provide an alternative explanation for the anomalous diffusion phenomenon in petroleum reservoirs, we have developed double-porosity models considering matrix blocks with fractal geometry and fracture networks with either radial or fractal fracture networks. The flows inside the matrix blocks and the fractal fracture network assume that Darcy’s law is valid in its space-dependent (fractal) form, whereas the classical version of Darcy’s law is assumed for the radial-fracture-network case. The transient interporosity transfer is modeled using the classical convolution schemes given in the literature. We have defined the matrix blocks to be “infinite-acting” to represent the nano/micropermeability of shale reservoir. For the system defined by a fractal fracture network and infinite-acting fractal matrix blocks, we have investigated the influence of the fractal parameters (both matrix and fracture network) in the pressure- and rate-transient performance behaviors. We have defined the flow periods that can be observed in these sorts of systems and we have developed analytical solutions for pressure-transient analysis. We demonstrate that the use of the convolved version of Darcy’s law results in a model very similar to the diffusivity equation for double-porosity systems (which incorporates transient interporosity flow). In performing this work, we establish the following observations/conclusions derived from our new solutions: We find that the assumption of a well producing at variable rate (time-dependent inner-boundary condition) has a more-significant effect on the pressure (and derivative) functions and obscures the effects of the properties of the reservoir. We demonstrate that the anomalous-diffusion-phenomena model proposed for unconventional reservoirs can be directly related to the multiporosity concept model. Pressure and pressure-derivative responses can be used in the diagnosis of flow periods and in the evaluation/estimation of reservoir parameters in unconventional reservoirs.

It is of current concern for the Permian-Carboniferous reservior of the Usinskoye field to develop low-permeable matrix blocks of carboniferous reservoirs, which contain major reserves of high-viscosity oil. To increase effectiveness of the currently used thermal oil recovery methods, the authors suggest using carbon dioxide as a reservoir stimulation agent. Due to a high mobility in its supercritical condition, СО2 is, theoretically, able to penetrate matrix blocks, dissolve in oil and, additionally, decrease its viscosity. Thus, СО2 applications together with a heat carrier could increase effectiveness of the high-viscosity oil recoveries and improve production parameters of the Permian-Carboniferous reservior of the Usinskoye field. During carbon dioxide injections, including combinations with various agents, some additional oil production is possible due to certain factors. Determination of the influencing factors and detection of the most critical ones is possible in laboratory tests. So, laboratory studies entail the key stage in justification of the technology effectiveness. The paper deals with describing the laboratory facilities and methodologies based on reviews of the best world practice and previous laboratory researches. These aim at evaluating effectiveness of thermal, gas and combined oil recovery enhancement methods. In particular, the authors explore experimental facilities and propose methodology to perform integrated researches of the combined heat carrier and carbon dioxide injection technology to justify the effective super-viscous oil recovery method.

Diffusion kinetics is widely acknowledged to dominate gas flow in coal matrix blocks. Knowledge of this topic is important for ongoing coalbed methane recovery and CO2-enhanced coalbed methane production. Because laboratory diffusivity measurements are normally conducted on powdered coals, it is unclear how representative the results are for coalbeds. Investigations into the effects of particle size on gas diffusivity can provide insights into the in situ diffusivity of the coal matrix. This paper presents measured CH4 desorption data in two Chinese anthracites (one brittle, one hard) having different particle sizes, to investigate the effects of particle size on diffusion kinetics. The experimental data were fitted by both the unipore (UP) and bidisperse (BD) models. The BD model agreed better with the measured data than the UP model, especially for the brittle coal. This indicated that the brittle coal was more abundant in macropores than the hard coal. Diffusivity in the hard coal decreased with increasing particle size but varied stochastically within a small value range in the brittle coal as the particle size increased. The diffusivity of the brittle coal, with its higher vitrinite content and lower inertinite content, was greater compared with the hard coal. This was inconsistent with reported data in which vitrinite had a smaller diffusivity than inertinite. This anomalous phenomenon may be caused by the generation of comparatively more macropores during grinding in the brittle coal. These results indicate that the effects of particle size on diffusivity may be coal-dependent, and further, the effects of particle size are influenced by other factors, including coal structure.

AbstractDescribing matrix–fracture interaction is one of the most important factors for modeling natural fractured reservoirs. A common approach for simulation of naturally fractured reservoirs is dual-porosity modeling where the degree of communication between the low-permeability medium (matrix) and high-permeability medium (fracture) is usually determined by a transfer function. Most of the proposed matrix–fracture functions depend on the geometry of the matrix and fractures that are lumped to a factor called shape factor. Unfortunately, there is no unique solution for calculating the shape factor even for symmetric cases. Conducting fine-scale modeling is a tool for calculating the shape factor and validating the current solutions in the literature. In this study, the shape factor is calculated based on the numerical simulation of fine-grid simulations for single-phase flow using finite element method. To the best of the author’s knowledge, this is the first study to calculate the shape factors for multidimensional irregular bodies in a systematic approach. Several models were used, and shape factors were calculated for both transient and pseudo-steady-state (PSS) cases, although in some cases they were not clarified and assumptions were not clear. The boundary condition dependency of the shape factor was also investigated, and the obtained results were compared with the results of other studies. Results show that some of the most popular formulas cannot capture the exact physics of matrix–fracture interaction. The obtained results also show that both PSS and transient approaches for describing matrix–fracture transfer lead to constant shape factors that are not unique and depend on the fracture pressure (boundary condition) and how it changes with time.