Matrix Block Size Research Articles

Analytic representative element rate decline models for naturally fractured reservoir depletion

Representative single anisotropic matrix block 2D Green’s function models for depletion through fully-penetrating, vertical fractures through different numbers of fracture faces are constructed that analytically capture both fracture and block depletion with fracture-matrix mass transfer. The 1D Green’s function for a fracture system is likewise solved in terms of the time evolution of average fracture pressure. While transient average pressure values are not inherently measurable, they are transformed into cumulative production or instantaneous flowrate values, thus producing new rate decline model functional forms. Primary variables in assembling the interacting systems model are the volume ratio, Vf /Vm, permeability ratio, kf /kx, and geometry, (a/b)(ky/kx), with the last term accounting for both block shape and permeability anisotropy. We construct interacting systems models in terms of various ratios of Vf /Vm, and kf /kx for three fracture architecture prototypes: representative matrix blocks depleted by 4, 2, or 1 contacting fractures. The single matrix block models can be migrated to ones for heterogeneous systems using superposition and matrix block distributions, as demonstrated with a binary distribution of block sizes with variable fractions. Analytic solutions for rate decline problems can be used to understand the production signatures of naturally fractured reservoirs and interpretation of fracture volume fraction, permeability ratio, average matrix block size, and measures of heterogeneity.

A Simple Analytical Model for Oil Production from Partially Fractured Reservoirs to Estimate Size of Finite Fracture Networks

Summary Most oil reservoirs are partially fractured, characterized by finite fracture networks (FFNs) in a sea of isolated fractures. It is necessary to determine size and shape of each FFN explicitly for reservoir simulation. FFN size is correlated with fracture connectivity, which is a function of fracture density, length, and angular scatter. Oil production from FFNs exhibits a long-term dual-porosity behavior. The initial fast rate (Phase I) represents depletion of matrix within FFN, and the subsequent gradual decline phase represents radial flow from the matrix outside the FFN perimeter. Thus, FFN size can be calculated from the cumulative oil production from Phase I, taking into account the pore volume, oil compressibility, and pressure decline. It is not always possible to identify the dual-porosity behavior by visual inspection. A mathematical model is needed to estimate FFN size. For this purpose, a set of three fundamental equations are derived for production rate, cumulative production, and pressure as a function of time. The model is a modified and simplified version of material balance equations with easy analytical solution. It is designed for fractured reservoirs with layer-bound fractures. Production is single-phase black oil under depletion drive. The analytical model was tested on four vertical wells. The unknown parameters such as FFN size, size of well drainage area, and fracture aperture are adjusted until an optimum fit to actual production data is obtained. FFN elliptical shape is estimated from average fracture strike and strike standard deviation. The results are validated by FFN size, fracture length, and aperture measurements from borehole images. The results are approximate but sufficient for preliminary mapping of FFNs with location and size and other critical attributes including fracture drainage area, matrix block size, fracture aperture, and permeability in partially fractured reservoirs.

Evaluation of the effects of homogenizing matrix block sizes on the simulation of naturally fractured reservoirs

A vast amount of recoverable oil is buried in naturally fractured reservoirs. Due to the heterogeneous nature of matrix and fracture systems, there are many complexities in the simulation and production of these kinds of reservoirs. Fractures in naturally fractured reservoirs may be caused by various natural factors such as tectonic forces. Fracture distribution does not have a systematic pattern in naturally fractured reservoirs, and fractures are scattered across the formation. For the sake of simplicity, uneven distribution of fractures is not completely considered within practical reservoir simulation methodologies such as the commonly used dual-porosity approach. Using the dual-porosity methods, naturally fractured reservoirs are divided into simulation cells consisting of several matrix blocks. Against the nature of naturally fractured reservoirs, in the dual-porosity method, the fractures are often uniformly spread throughout the entire reservoir, and the sizes of matrix blocks inside the simulation cells are considered equal. For many years the assumption of homogenized matrix blocks in simulation cells has been used in the industry without proper evaluation of its effect on simulation results. This work examines the assumption of uniform matrix block size on the simulation results of naturally fractured reservoirs. An approach was proposed to create a simulation cell with randomly distributed matrix blocks to mimic the actual condition of naturally fractured reservoirs. Furthermore, realistic simulation approaches were utilized to investigate the oil recovery factor of different non-homogenized simulation cells during gravity drainage and imbibition processes. Also, the recovery factor performances of non-homogenized simulation cells were compared with simulation cells with uniform matrix block sizes created with different homogenization methods. An examination of the results revealed that a significant amount of inaccuracy in the simulation results could be caused by homogenization. Moreover, two types of homogenization approaches were used. In the first approach, a simulation cell with homogenized matrix block sizes based on the arithmetic mean of variable size matrix blocks was built. Homogenized matrix blocks in the second approach were built according to the arithmetic mean size of matrix blocks in different x, y, z directions. The first homogenization method resulted in less error compared to the second one. As compared with the gas invaded zone, simulation results of the water invaded zone are more error-prone. The findings of this work can help for better understanding the homogenization effect and prove that disregarding the heterogeneity of matrix blocks may lead to severe errors in the simulation results of naturally fractured reservoirs. The outcome also highlights the need for a more precise modeling and simulation approach for naturally fractured reservoirs.

Experimental Study on Oil Drop Discharge Behavior during Dynamic Imbibition in Tight Oil Sandstone with Aid of Surfactant

The pore and throat structure of tight oil reservoir cores is complex, and the resistance of oil drop to discharge from the core is very high during dynamic imbibition. Surfactant has good ability in interfacial tension reduction and wettability reversal. It can reduce oil drop discharge resistance and enhance oil recovery effectively during dynamic imbibition in tight reservoirs. Here, we first analyzed the pore throat structure and mineral composition of tight core, and then the oil drop visualization instrument was used to study the discharge behavior of oil drop during dynamic imbibition. The oil drop discharge form was analyzed, and the influence of various factors on the oil drop discharge behavior was explored, and then the dynamic imbibition performance of surfactant in tight cores was obtained. The core throat diameter was mainly distributed in 0.07–1.1 μm, and the hydrophilic mineral content in core reached 50.8%. In the case of fluid flow in fracture, the oil drop discharge from near fracture matrix was faster, and its growth rate in height and width was faster than that without external fluid flow. Within a certain range, with the increase of IFT, the rate of core imbibition increased gradually. When the IFT increased from 0.32 mN/m to 0.59 mN/m, the oil drop rapture time decreased by 66.3%. The growth rate of oil drop discharged from the top and side of the core was faster than that from the bottom surface. Furthermore, it was easier to discharge. With the core thickness reduced by half, the rapture time of oil drop was reduced by 74.7%. For tight reservoirs, hydraulic fracturing can create more fracture surfaces and reduce the size of matrix blocks, which contributes to reduce the oil drop discharge resistance during imbibition and improve the oil recovery. This study provides a basis for surfactant to improve dynamic imbibition and oil production performance of tight sandstone oil reservoir.

Investigation of rock properties distribution effect on pressure transient analysis of naturally fractured reservoirs

Naturally fractured reservoirs (NFRs) are considered as one of the main resources of oil and gas around the world. The nature of fractures in this kind of reservoirs splits these reservoir rocks into two parts; matrix and fracture. In these two media, geometrical properties like matrix block size and also petrophysical properties such as porosity and permeability cannot considered to be constant and they usually contain heterogeneity in extent of the reservoir. In this study, a statistical distribution concept is used to be able to model variance of properties including matrix block size, matrix and fracture permeability, and matrix and fracture porosity as independent statistical variables. For the purpose of investigating effect of the five aforementioned properties distribution on pressure transient analysis (PTA) plots, five distinct cases of property distributions are considered with their corresponding precise statistical modeling applied on governing flow equation of a dual-porosity base case reservoir model. Abovementioned distributed parameters are included in semi-analytically derived dual porosity reservoir models and results are compared to their corresponding constant-value response. In each case the constant parameter is the median of corresponding statistical distribution. The statistical approach is validated using numerical modeling and field data applications are also presented. Different unimodal and bimodal matrix properties (block size, permeability, and porosity) distribution have slight to intense effects on PTA plots of an NFR. In matrix block size and porosity distribution cases, lower values, and in matrix permeability distribution case, higher values of the range dominate the pressure behavior of the fractured system. On the other hand, although fracture permeability and porosity distribution can shift PTA plots of the model but the effect is not significant and both fracture permeability and porosity distribution can be represented by constant representative values instead of a statistical model. It is worth mentioning that all matrix properties bimodal distribution has a common signature on PTA plots which is similar to the triple porosity reservoir response (i.e. the presence of double-bump descent on pressure derivative plot).

A process-based coal swelling model: Bridging the gaps between localized swelling and bulk swelling

Injection of CO2 and coal seam gas production induce coal swelling/shrinkage. Laboratory tests indicate that coal swelling may go through several stages from localized swelling at fracture surfaces to bulk rock swelling. The coal Langmuir swelling strain constant normally ranges from 0 to 0.04. This study provides a process-based coal swelling model that covers the whole swelling procedure. We account for three swelling strains, videlicet, the matrix swelling strain, fracture swelling strain, and bulk rock swelling strain. Dynamic partial separation coefficients are employed to quantify the contributions of the matrix swelling strain to bulk swelling and the fracture (cleat) aperture reduction, bridge the gaps between localized swelling and bulk swelling, and link the three types of strains. This model is inserted into a permeability model and verified against permeability measurement and rock swelling data. The three samples used for verification are bituminous coals from the Illinois Basin (depth of ~ 200 m) in the US and Henan Province in China (depth of ~ 310 m). In essence, the swelling model is also applicable to other coal types with a linear relationship between the matrix swelling strain and adsorbed gas content. By matching with long-term permeability measurement data, multiple permeability evolution stages are observed due to the swelling transition. Sensitivity analyses show that the matrix block size, matrix swelling properties, and diffusivity control the duration, occurrence, and magnitude of matrix swelling’s contribution to local and bulk swelling during swelling transition, while rock bridge swelling properties have a marginal influence on coal swelling.

Carbonate reservoir characterization: an integrated approach

Estimation of fluid and rock properties of a hydrocarbon reservoir is always a challenging matter; especially, it is true for heterogeneous carbonate reservoirs. Petrophysical logs and laboratory activities are common methods for characterizing a hydrocarbon reservoir. This method in conjunction with geostatistical methods is applied to relatively homogeneous sandstone reservoirs or matrix media of dual-porosity heterogeneous carbonate reservoirs. To estimate properties of fracture system of a dual-media carbonate reservoir, outcrop properties, electrical borehole scans, fractal discrete fracture network and analogy with other reservoirs are common methods. Results which obtained from these methods describe the reservoir in a static way and could be relied on them for very small portion of the reservoir. In this paper, a dynamic procedure for describing reservoir features is proposed in order to enhance the conventional reservoir characterization methods. This method utilizes the reported production data for a specific period of time in conjunction with rock and fluid properties to estimate drainage radius of the well and matrix block height, porosity and width of fracture in the estimated drainage radius. The presented method is elaborated through its application in a real case. By this method, one can generate maps of matrix block size and fracture width and porosity throughout the reservoir. Also, it could be a powerful tool for estimation of effectiveness of acid/hydraulic fracturing activity.

The performance of complex-structure fractured reservoirs considering natural and induced matrix block size, shape, and distribution

This paper introduces an analytical approach for the performance of the fractured reservoir considering the impact of the matrix block size, shape, and distribution. The objective is better understanding the roles of the variable interporosity flow systems, fracture flowing systems, and reservoir matrix heterogeneity in the responses of the wellbore pressure drop, flow rate, cumulative production, and productivity index. This understanding could help to eliminate the uncertainties and increase the accuracy in characterizing the conventional and unconventional reservoirs where very complex structures of a significant disorder in the matrix block size, shape and distribution could be existed in the porous media.The study uses the well documented multilinear flow model to describe the pressure distribution in the naturally fractured reservoirs depleted by multiple hydraulic fractures considering different matrix block sizes, shapes, and distributions. The analytical models of the flow rate, cumulative production, and the productivity index are also developed in this study from the multilinear flow model. The reservoirs of interest are assumed to have stimulated porous media extending between hydraulic fracture and unstimulated volume extending beyond the fracture tips toward the reservoir boundary. The unstimulated reservoir volume is assumed consisting of uniformly distributed single matrix block size and shape because there is no impact for the hydraulic fracturing process. While the stimulated reservoir volume is considered consisting of a variable matrix block size, shape, and distribution because of the impact of hydraulic fracture propagation. The petrophysical properties of the two porous media, as well as the storativity and interporosity flow coefficient, are assumed functions of the matrix characteristics in the two volumes. The matrix block dimensions (radius or length) are correlated exponentially with the distance from the hydraulic fracture face toward the no-flow boundary between these fractures. Three fracture-matrix fluid flux functions are developed for the reservoirs where the matrix blocks could have a spherical, slab, and cylindrical configuration.The study has reached several conclusions. A positive impact on the reservoir performance is seen when the matrix block size becomes small or when the small size matrix blocks control the distribution of the blocks in the porous media. Reservoir performance is enhanced also when there are several matrix block sizes compared to a single size or very few sizes. It is observed that the reservoirs with spherical blocks may give better performance than the reservoir with slab or cylindrical matrix blocks. In terms of matrix block size and distribution, heterogeneous reservoirs may have a higher productivity index than homogenous reservoirs. The variable matrix block size and distribution in the stimulated reservoir volume is more beneficial than the unstimulated volume. The novel point presented in this paper is developing several analytical models for the wellbore pressure drop, flow rate, cumulative production, and productivity index of hydraulically fractured reservoirs with a consideration given to the matrix block size, shape, and distribution.

Fluid flow in fractured reservoirs: Estimation of fracture intensity distribution, capillary diffusion coefficient and shape factor from saturation data

Naturally fractured carbonate reservoirs which possess most of the remaining oil in the world, can fulfill the needs of widely nowadays energy demand. Most of the oil remains in the matrix block during production from naturally fractured reservoirs. Water flooding is a technique implemented in naturally fractured reservoirs to recover additional amounts of oil from the matrix blocks. In this paper, for the first time, an analytical model for water flooding in the naturally fractured carbonate reservoirs is developed that describes the transient behavior of the imbibition mechanism in the dual porosity models. Using the fracture-matrix fluid transfer of the dual porosity model, the presented analytical model takes the account of viscous-capillary and capillary imbibition mechanisms in fracture and matrix system, respectively. The analytical solution is validated using both laboratory data and a numerical solution. In order to consider the effect of the heterogeneity in fractured reservoirs, the proposed analytical model is developed for various distributions of block sizes. In addition, a graphical method is developed to estimate capillary diffusion coefficient and matrix block size distribution based on the water saturation data in the monitoring well during water injection. The monitoring well is used to monitor and evaluate reservoir during water flooding. The proposed methodology which is based on water saturation data in the monitoring well, does not call for a specific test in the reservoir.Moreover, the relating shape factor is calculated with the aid of dual porosity model equipped with variable fracture intensity distribution. The findings imply that the fluid transfer is highly dependent on the distribution of matrix block size.

Modeling of Matrix/Fracture Transfer with Nonuniform-Block Distributions in Low-Permeability Fractured Reservoirs

Summary Unconventional shale–gas and tight oil reservoirs are commonly naturally fractured, and developing these kinds of reservoirs requires stimulation by means of hydraulic fracturing to create conductive fluid–flow paths through open–fracture networks for practical exploitation. The presence of the multiscale–fracture network, including hydraulic fractures, stimulated and nonstimulated natural fractures, and microfractures, increases the complexity of the reservoir simulation. The matrix–block sizes are not uniform and can vary in a very wide range, from several tens of centimeters to meters. In such a reservoir, the matrix provides most of the pore volume for storage but makes only a small contribution to the global flow; the fracture supplies the flow, but with negligible contributions to reservoir porosity. The hydrocarbon is mainly produced from matrix/fracture interaction. So, it is essential to accurately model the matrix/fracture transfers with a reservoir simulator. For the fluid–flow simulation in shale–gas and tight oil reservoirs, dual–porosity models are widely used. In a commonly used dual–porosity–reservoir simulator, fractures are homogenized from a discrete–fracture network, and a shape factor based on the homogenized–matrix–block size is applied to model the matrix/fracture transfer. Even for the embedded discrete–fracture model (EDFM), the matrix/fracture interaction is also commonly modeled using the dual–porosity concept with a constant shape factor (or matrix/fracture transmissibility). However, in real cases, the discrete–fracture networks are very complex and nonuniformly distributed. It is difficult to determine an equivalent shape factor to compute matrix/fracture transfer in a multiple–block system. So, a dual–porosity approach might not be accurate for the simulation of shale-gas and tight oil reservoirs because of the presence of complex multiscale–fracture networks. In this paper, we study the multiple–interacting–continua (MINC) method for flow modeling in fractured reservoirs. MINC is commonly considered as an improvement of the dual–porosity model. However, a standard MINC approach, using transmissibilities derived from the MINC proximity function, cannot always correctly handle the matrix/fracture transfers when the matrix–block sizes are not uniformly distributed. To overcome this insufficiency, some new approaches for the MINC subdivision and the transmissibility computations are presented in this paper. Several examples are presented to show that using the new approaches significantly improves the dual–porosity model and the standard MINC method for nonuniform–block–size distributions.

A New Technique for Quantifying Pressure Interference in Fractured Horizontal Shale Wells

Summary Pressure communication is commonly observed in fractured horizontal shale wells, particularly at early times when wells are placed on production. In this paper we present a new technique, based on the diffusion exponent from the power-law model, to quantify connectivity in multistage-hydraulic-fractured wells with complex fracture networks. In addition to explaining the theory and analysis techniques, we present examples using measured bottomhole pressure (BHP) from the Permian Basin Wolfcamp Shale that illustrate the utility of this technique to better understand the relationship between completion size, well spacing, and well performance. Using the concept of anomalous diffusion, Chen and Raghavan (2015) developed a 1D, fractional-order, transient diffusion equation to model fluid flow in complex geological media. They showed that anomalous diffusion, which can be caused by heterogeneities in the matrix or the fracture system, exhibits a power-law behavior. In addition to Chen and Raghavan (2015), Acuña (2016) demonstrated that variations in matrix block sizes, fracture conductivity, and drainage shape also exhibit power-law behavior. While the approach from these two studies is somewhat different, they each demonstrated that a generalized power-law model is often more appropriate than traditional linear- or radial-flow pressure-transient analysis techniques for unconventional shale reservoirs. Further, each work shows that the power-law response can be related to some form of heterogeneity in the drainage volume. While traditional techniques for estimating well interference have been previously developed and applied in conventional reservoirs, in this paper we focus on quantifying the magnitude of pressure interference (MPI) in unconventional reservoirs, which commonly demonstrate a generalized, power-law pressure response that is different from radial or linear flow. The examples presented in this paper are for wells from the Permian Basin Wolfcamp Shale. Under the framework of power-law behavior, our technique involves plotting pressure-interference-test (PIT) data in terms of the Chow pressure group (CPG), which enables us to define an indicator of connectivity reflecting temporal and spatial effects. On each test, we derive a diffusion exponent reflective of the MPI. We will show among other things that multiple PITs over time often indicate degrading connectivity between wells. From PIT analyses in Permian Basin Wolfcamp Shale, we were able to establish a relationship between MPI and well spacing. The first example demonstrates analyses of PITs between wells during the production phase and also shows how connectivity between wells diminishes over time. A second example applies the same analysis techniques to quantify interwell connectivity during the post-stimulation phase by analyzing a pressure falloff (PFO) after communication with other wells. A third example illustrates an application of desuperposition to remove the effect of a power-law pressure trend (PT) on interference tests. Techniques to analyze PITs assuming radial or linear flow have been previously developed; however, Raghavan and Chen (2018) showed that apparent radial or linear flow could exist under anomalous diffusion for heterogeneous reservoirs. In this work, we present a technique for analyzing power-law PIT data, which is typical of most horizontally fractured shale wells. This model is a unique approach to understanding flow behavior, quantifying well interference, and analyzing and predicting well performance in unconventional reservoirs. Our examples, which are based on high-quality BHP gauge data, show how this technique could shorten the cycle time for operators to determine the well spacing for a given completion design.

A new analytical model for flow in acidized fractured-vuggy porous media

Acidizing is one of the widely used technologies that makes the development of naturally fractured-vuggy reservoirs effective. During the process of acidizing, carbonate minerals are dissolved by hydrochloric acid, which can create high conductivity channels and wormholes to connect fractures and pores. In this work, a new analytical model, incorporating the heterogeneity of the pore networks into acidizing region, is proposed to study the flow characteristics in acidized fractured-vuggy reservoirs. The model is coupled by an acidized inner region and a conceptualized outer region of common triple medium. The porosity and permeability of inner region, which are rather heterogeneous and disordered when observed at different length scales, can be well addressed by fractal theory. The properties of the outer region can be described with three basic parameters: the matrix block size LM, the space interval of fracture LF and the radius of the vug Lv. Results show that the flow characteristic curves can be characterized by six flow stages (i.e. wellbore storage stage, radial flow stage in the interior region, fracture-vug inter-porosity flow stage, transition flow stage, fracture-matrix inter-porosity flow stage and external boundary response stage). It can be applied to estimate reservoir parameters for uncertainty reduction using inverse modeling.

Revisiting the Analytical Solutions of Heat Transport in Fractured Reservoirs Using a Generalized Multirate Memory Function

AbstractNumerous analytical solutions have been developed for modeling thermal perturbations to underground formations caused by deep‐well injection of fluids. Each solution has been derived for a specific boundary value problem and a simplified flow network with one set of parallel fractures. In this paper, new generalized solutions G*(x, s) are developed using (existing) global transfer functions and a new memory function g*(s), where x and s are the space and Laplace variable. The memory function represents the solutions of conductive heat exchange between fractures and matrix blocks and between fractured aquifers and unfractured aquitards. The memory function is developed to account for multirate exchange induced by different shapes, sizes, properties, and volumetric fractions of matrix blocks bounded by multiple sets of orthogonal fractures with different spacing. The global transfer functions represent the fundamental solutions to convective, convective‐conductive, and convective‐dispersive heat transport in fractures (or aquifers) without exchange and are available for various (1‐D linear, 1‐D radial, 2‐D dipole, and single‐well injection‐withdrawal) flow fields. The new solutions with exchange are developed using , thereby greatly simplifying solution development in a novel way, where ϑ and B*(s) are a fracture‐matrix scaling factor and the boundary condition function. The new solutions are applied to several example problems, showing that heat transport in fractured aquifers is significantly impacted by (1) thermal dispersion in fractures that is rarely considered, (2) multirate heat exchange with a wide range of size and anisotropy of rectangular matrix blocks, and (3) heat exchange between aquifers and aquitards.

Upscaling of CO2 injection in a fractured oil reservoir

The purpose of this study is to present the numerical evaluation of the upscaling of secondary and tertiary CO2 flooding at reservoir pressure and temperature. In all simulation cases, CO2 is injected from the top of the fracture in a fracture-matrix block system and displaces oil toward the bottom of the block. Our numerical models provide comprehensive sensitivity analyses that are proved to be important for the upscaling of CO2 injection from lab to reservoir scale. In addition, we address the scale dependency of the diffusion mechanism in a large matrix-fracture domain.In our earlier studies, we have conducted a significant number of reservoir-condition CO2 flood experiments by employing fractured core systems and North Sea reservoir oil. Our experimental and numerical studies are proved to be useful to investigate the complex physical phenomena affecting the efficiency of CO2 injection in oil recovery from a fracture reservoir. The height of the core plugs used in our experiments varies from 7 to 28 cm while the fracture spacing varies between ∼4 cm and 12 cm. In all cases, we find the important role of diffusion mechanism in increasing oil recovery in a fracture-matrix system at the lab scale. However, the field matrix block size is often 1-to-2 orders of magnitude larger than that in the lab scale. Recovery mechanisms are significantly affected by the matrix dimension. For instance, the impact of diffusion mechanism on oil recovery may alter with the size of the matrix block and the fracture spacing. This highlights the importance of upscaling from lab to the field scale that is addressed in this study.We employ a compositional reservoir simulation with a developed equation of state (EOS) to model CO2 injection in a fracture-matrix system at reservoir conditions. In all the numerical simulations, the single-porosity (SP) model consists of a single axial fracture with a typical fracture aperture of 1 mm. The SP model also contains a single matrix block that feeds the producer through fracture next to it. The dimension of the fracture-matrix block varies in the SP models to account for the scale dependency of the results. In our modeling framework, we assume a network of open fractures that are homogenously distributed in the reservoir and links the oil in the matrix to the production wells. We initialize the matrix with the North-Sea-Chalk-Field (NSCF) live oil and the fracture with the injected gas (CO2). The whole displacement process is operated at the constant reservoir conditions of 3750 psia and 110 °C, representing typical NSCF reservoir conditions. Prior to conducting simulations, we successfully tune EOS parameters against the extensive PVT data, sampled from a specific fractured NSCF reservoir. The tuned EOS model assures the proper estimation of the complex phase and volumetric properties for CO2 and oil mixtures at reservoir conditions. In addition, the multi-component diffusion coefficients that are tuned using experimental data are employed as input parameters for the compositional simulations.We study the effect of several key parameters on oil recovery; e.g. matrix block size, fracture spacing, CO2 injection rate, density difference, vaporization and the diffusion. The results show that the mass transport is mainly dominated by diffusion in the lab scale which is not the case for the large matrix block size. Finally, we test the accuracy of the dual-porosity (DP) model against the SP model at various displacement conditions. Results show that when diffusion and vaporization are significant the DP model fails to predict the results of the SP reservoir model.Our findings are an important step towards modeling the tertiary-CO2 flooding in an actual fracture-chalk system. We also provide some important inputs that are necessary for upscaling tertiary-CF from a lab-scale into a field-scale reservoir model.

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

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.

Onset of density-driven instabilities in fractured aquifers.

Linear stability analysis is conducted to study the onset of density-driven convection involved in solubility trapping of CO_{2} in fractured aquifers. The effect of physical properties of a fracture network on the stability of a diffusive boundary layer in a saturated fractured porous media is investigated using the dual porosity concept. Linear stability analysis results show that both fracture interporosity flow and fracture storativity play an important role in the stability behavior of the system. It is shown that a diffusive boundary layer under the gravity field in fractured porous media with lower fracture storativity and/or higher fracture interporosity flow coefficient is more stable. We present scaling relations for the onset of convective instability in fractured aquifers with single and variable matrix block size distribution. These findings improve our understanding of density-driven flow in fractured aquifers and are important in the estimation of potential storage capacity, risk assessment, and storage site characterization and screening.

Generalized analytical solution for gravity drainage phenomena in finite matrix block with arbitrary time dependent inlet boundary condition and variable matrix block size

Fractured carbonate reservoirs constitute a considerable proportion of hydrocarbon reservoirs in the world. In these reservoirs, gravity drainage is one of the dominating oil producing mechanisms that controls oil production depending on the interactions between the upper and lower blocks. Nonetheless, few theoretical studies have investigated the modeling of the reinfiltration process between the blocks.In this study, first, the gravity drainage process is modelled for a 1-D single matrix block by considering gravity and capillary forces, then Laplace transformation is used to solve the governing partial differential equation related to a matrix block with appropriate initial and boundary conditions. Next, the obtained equations are extended to a stack of matrix blocks and the effect of the reinfiltration process is investigated afterward. The inlet oil flow rate from the upper boundary of the blocks is a function of time, and the lower boundary of the blocks is fully saturated with oil. At the initial condition, the matrix block is saturated with oil. Finally, based on the aforementioned saturation equations, oil production rate, cumulative production, gravity drainage mechanism and the effect of the reinfiltration process are studied. The presented analytical solution is compared with previous semi-analytical solutions (Firoozabadi and Ishimoto, 1994), finite difference and finite element numerical techniques.Traditional gravity drainage models do not consider the effect of matrix block size distribution on oil recovery. In this study, a gravity drainage model is proposed to evaluate the oil recovery factor that considers this factor. The solution to the uniform distribution can be extended to more general distributions. Regarding the fact that the strength of gravity drainage mechanism is related to block size, higher fracture intensity (smaller grid blocks) leads to a fairly small decrease in gravity drainage forces such that most of oil within the matrix block stays unchanged. The obtained results from this study show that higher fracture intensity (smaller block height) causes lower oil recovery from matrix in gas invaded zone. The results of analytical model were compared with previous analytical model and experimental data and reasonable match was obtained for predicting final oil recovery.

Fluid flow in fractured reservoirs: Exact analytical solution for transient dual porosity model with variable rock matrix block size

Dual porosity reservoirs consist of two comparatively independent systems of fractures and matrix blocks with high and low permeability values, respectively. Semi-analytical and numerical studies on naturally fractured reservoirs have been already cited in the literature. The present study focuses on investigation of a linear double porosity model of a semi-infinite-acting naturally fractured reservoir using an exact analytical method. Different matrix block size distributions are embedded into a transient model to consider the effects of heterogeneity in fractured formations. In order to take into account transient fracture-matrix exchange, an analytical method is proposed to solve the coupled fracture and matrix equations. Gauss Legendre quadrature is employed to evaluate the double integral of general transient solution. Moreover, the corresponding shape factor was evaluated in a double porosity transient model coupled with variable matrix block size distribution. Results demonstrated that matrix block size distributions strongly affect the fluid transfer during the early time region. Also, the presented model was employed to generate interference test curves which in turn were studied to investigate the impacts of storativity ratio and matrix block size distributions on the fracture-matrix fluid transfer. Results illustrated that a series of obtained pressure data from an observational well along with the proposed model may be considered a robust method in fracture intensity assessment. Fast sensitivity analysis and very efficient computational cost are the benefit of the derived analytical solutions with respect to previous numerical solutions.

Contribution of osmotic transport on oil recovery from rock matrix in unconventional reservoirs

The fluid transport mechanisms in fracture and rock matrix play a critical role in economic viability of hydrocarbon recovery from unconventional reservoirs. Several transport mechanisms, including advection, diffusion, and convection, have varying degrees of contribution to hydrocarbon recovery depending on the transport properties of the formation. The contribution of the concentration gradient driven diffusion, or osmotic transport on hydrocarbon recovery has been numerically investigated in this study. The main objective of this study is to determine when the osmotic transport plays an important role on the mass exchange between the fractures and the rock matrix.In this study, a model for the mass transport between the rock matrix and the fractures has been formulated and validated with the experimental data. The numerical results showed that osmosis is an important force imbibing water into low permeability rock matrix and enhancing the effectiveness of low salinity waterflooding on oil recovery. The imbibition of water into oil-wetted shale matrix is mainly driven by the osmotic transport and wettability alteration. The contribution of the osmotic transport on oil recovery continues a long period of time, typically in a few years. This contribution increases if the membrane efficiency is high and the matrix block size is small. However, shale membrane efficiency is typically less than 10% considerably reducing the oil recovery factor by osmosis to less than 5%. Higher membrane efficiency and lower diffusion coefficient of dissolved ions increase the contribution of osmosis on the oil recovery from shale matrix.

Analytical model for convection-conduction heat transfer during water injection in fractured geothermal reservoirs with variable rock matrix block size

The purpose of management in geothermal reservoirs is to economically enhance the energy recovery. This paper describes a model to analyze the process of cold water injection into a fully penetrating well with a number of n equally spaced horizontal fractures in a fractured geothermal reservoir. Previous models basically focused on the numerical and semi-analytical methods or a combination of two. Here, an analytical model is presented in which the effects of thermal convection and conduction are investigated. This solution has the ability to account the following phenomena: convection transport in fractures and conduction from the matrix block to the fracture. Specified assumptions are made which allow the model to be formulated as two coupled one-dimensional partial differential equations, one for the fracture and one for the matrix block in the direction perpendicular to the fracture. The convection-conduction heat transfer between fracture and matrix is not affected by the matrix block size distribution in dual porosity models. In the present study, a heat transfer model is introduced for a radial system with matrix block size distribution to assess the heat transfer shape factor known as a matrix-fracture exchange coefficient. The results delineated that heat transfer is strongly dependable on matrix block size distribution during the early time region. To calculate the total amount of heat extracted from a geothermal reservoir, two methods were defined and also a conceptual definition for the thermal recovery efficiency was presented. Finally, the proposed model was validated by comparing the results of the analytical solution with a pre-defined numerical method in the literature.