Fracture-matrix System Research Articles

Impact of pressure on tertiary-CO2 flooding in a fractured chalk reservoir

The purpose of this study is to present the numerical and experimental evaluation of tertiary-CO2 flooding (CF) at high operating pressure and temperature conditions. In this study, water flooding is followed by CO2 injection into an outcrop chalk core with a centralized fracture. Our validated numerical models reproduce the results of core flooding experiments.The experiments used an outcrop core which is vertically placed in the core-holder with the total length of 28 cm and the diameter of 12.3 cm. The axial “fracture” is represented by a centralized hole with the diameter of 2.2 cm. We utilize the Wood's Metal technique to initially saturate the chalk core with the North-Sea-Chalk-Field (NSCF) live oil. The core sample is aged to restore the original chalk wettability at the operating conditions. Then, the water flooding (WF) is performed by injecting brine from the bottom of the fracture and producing the oil from the top. After no additional produced oil is observed, the WF is stopped. A “shut-in” period follows, which allows preparing the rig for tertiary-CO2 flooding. CO2 is then injected from the top and the hydrocarbon streams are produced from the bottom of the fracture. The whole core flooding is operated at constant reservoir conditions of 300 bara and 110 °C, which is higher than the initial NSCF reservoir pressure (258 bara and 110 °C). This allows us to investigate the efficiency of the tertiary-CF at a higher operating pressure condition.We employ a compositional reservoir simulation with a developed equation of state (EOS) to model the experiment. An automated history matching procedure is developed to match the experimental results. The modeling workflow is capable of taking into account the significant vaporization effect observed during CF when the rich-CO2 enters the three-phase separator.We accurately model the WF experiment through tuning the oil-water capillary pressure and relative permeability data. The numerical model is capable of reproducing the CF lab results by employing the best match multi-component diffusion coefficients. Moreover, we successfully model the excessive water production during CO2 injection by taking into account the hysteresis effect in water-oil capillary pressure and relative permeability.Comparing these results with our previous work at lower reservoir pressure shows the positive effect of pressure on increasing the efficiency of the tertiary-CF in recovering more oil from a matrix-fracture system. Also, the tuned capillary pressure and relative permeability during WF indicate an active imbibition drive and a strongly water-wet system. Our findings are an important step towards modeling the tertiary-CO2 flooding in an actual fracture-chalk system.

Influence of Fracture Heterogeneity Using Linear Congruential Generator (LCG) on the Thermal Front Propagation in a Single Geothermal Fracture-Rock Matrix System

An implicit finite difference numerical model has been developed to investigate the influence of fracture heterogeneity on the propagation of thermal front in a single horizontal fracture-matrix system. Instead of depending on a complex and data-demanding geostatistical method for a precise representation of fracture aperture, a statistical linear congruential generator (LCG) method was applied in the present study to replicate the unpredictable nature of fracture aperture morphology. The results have been compared with the parallel plate model and simple sinusoidal model. Finally, sensitivity analysis of fracture aperture size and fluid flow rate has been carried out to identify the conditions at which fracture heterogeneity is critical. The results indicate that LCG-aperture enhances the heat transfer between fracture and hot rock matrix compared to the parallel and sinusoidal fractures. Further, the temperature profiles in hot rock indicate that there was a greater loss of heat for the case of LCG-aperture (25% loss) compared to sinusoidal (16%) and parallel plate (8%) apertures. It was found that heterogeneity does not play a major role at small fracture aperture size (≤50 μm) and at low flow rates. However, as fracture aperture size increases, the heterogeneity plays a vital part even at low flow rates.

Mathematical Modeling on Mobility and Spreading of BTEX in a Discretely Fractured Aquifer System Under the Coupled Effect of Dissolution, Sorption, and Biodegradation

This paper presents a numerical model to investigate the migration of BTEX (benzene, toluene, ethylbenzene, and xylene) within a fractured aquifer system at the scale of a single fracture under the coupled effect of various transport processes such as dissolution, sorption, biodegradation. The developed model also considers the influence of equilibrium- and kinetic-controlled sorption scenarios on BTEX transport. Further, the transport characteristics of dissolved BTEX within the fracture–matrix system (FMS) are obtained by carrying out the spatial moment analysis on the concentration profiles simulated. In this study, the spatial moment analysis is conducted to estimate the following transport characteristics of dissolved BTEX: (a) velocity within the fracture, (b) dispersion coefficient within the fracture, (c) dissolved mass within the matrix. In order to investigate the sensitivity of various input parameters, two sensitivity indices are computed based on the variation in the velocity of dissolved BTEX constituent within the fracture (SI_vel) and on the variation in the dissolved mass of BTEX constituents within the matrix (SI_mat). Results from the present simulation study suggest that the sorption and biodegradation reactions influence the concentration distribution of highly soluble BTEX constituents (benzene, toluene) within the FMS significantly. The influence of biodegradation on the migration of BTEX within the FMS is found to be more when it co-occurs with the sorption reaction. The effect of sorption and biodegradation reactions on the mobility and dispersion coefficient of dissolved BTEX constituents within the FMS is found to be significant during the early simulation period.

Numerical modeling of hyperbolic dominant transient fluid flow in saturated fractured rocks using Darcian approach

A numerical model has been developed for simulating fluid flow in a coupled fracture-matrix system by considering a hyperbolic dominant fluid flow equation within the fracture as against the conventional parabolic dominant fluid flow equation. In addition, for the first time a transient fluid mass transfer function has been introduced in the governing equation of fluid flow within the rock-matrix in order to explicitly consider the fluid drainage from rock-matrix into the fracture. The effectiveness in modeling fluid flow using different governing equations developed by implementing the pseudo-steady state and transient models of transfer functions with parabolic and hyperbolic conditions within the fracture and rock-matrix has been discussed in detail. Numerical results suggest that the drainage of fluid from the rock-matrix to the fracture and thus the estimation of its fluid recovery from the reservoir are overestimated in a typical coupled fracture-matrix system using conventional parabolic dominant fluid flow equation. It has been further observed that among the hyperbolic and parabolic dominant mathematical models used to describe the fluid flow through fracture, the recent model proposed by the first author, which is associated with the rate limited fluid mass transfer term within the low permeable rock-matrix yields a significantly varying pressure distribution with reference to the rest of the cases. It is also observed that the resultant pressure distribution within a coupled fracture-matrix system is independent of the pseudo-steady state transfer or transient transfer function, which is used for modeling the fluid interaction at the fracture-matrix interface.

A Green's function method for two-dimensional reactive solute transport in a parallel fracture-matrix system

The reactive solute transport in a single fracture bounded by upper and lower matrixes is a classical problem that captures the dominant factors affecting transport behavior beyond pore scale. A parallel fracture-matrix system which considers the interaction among multiple paralleled fractures is an extension to a single fracture-matrix system. The existing analytical or semi-analytical solution for solute transport in a parallel fracture-matrix simplifies the problem to various degrees, such as neglecting the transverse dispersion in the fracture and/or the longitudinal diffusion in the matrix. The difficulty of solving the full two-dimensional (2-D) problem lies in the calculation of the mass exchange between the fracture and matrix. In this study, we propose an innovative Green's function approach to address the 2-D reactive solute transport in a parallel fracture-matrix system. The flux at the interface is calculated numerically. It is found that the transverse dispersion in the fracture can be safely neglected due to the small scale of fracture aperture. However, neglecting the longitudinal matrix diffusion would overestimate the concentration profile near the solute entrance face and underestimate the concentration profile at the far side. The error caused by neglecting the longitudinal matrix diffusion decreases with increasing Peclet number. The longitudinal matrix diffusion does not have obvious influence on the concentration profile in long-term. The developed model is applied to a non-aqueous-phase-liquid (DNAPL) contamination field case in New Haven Arkose of Connecticut in USA to estimate the Trichloroethylene (TCE) behavior over 40 years. The ratio of TCE mass stored in the matrix and the injected TCE mass increases above 90% in less than 10 years.

Experimental and numerical investigation of tertiary-CO2 flooding in a fractured chalk reservoir

This paper describes tertiary CO2 recovery mechanisms observed in two distinct core flooding experiments. In both experiments, water flooding followed by CO2 injection into an outcrop chalk with matrix-fracture system at reservoir conditions. The objective of this paper is to study the main mechanisms that control the rate of oil recovery during tertiary-CO2 flooding. A series of experiments as well as compositional numerical simulations have been carried out to evaluate the efficiency of the process.In the first experiment (Exp-1), the fractured core is initially saturated with North Sea Chalk Field (NSCF) stock tank oil (STO) and the connate water. Whereas in the second experiment (Exp-2), the core is saturated with NSCF live-oil and the connate water. Once the reservoir condition is established (258 bara, 110 °C), sea water is injected from the bottom of the fracture and the oil is recovered from the top. Once no more oil recovery is observed, the water flooding (WF) is stopped. A “shut-in” period follows which allows preparing the rig for the CO2 flooding (CF). Right after the “shut-in”, CO2 is injected from the top of the fracture and the oil is produced from the bottom. A compositional reservoir simulator with a tuned equation-of-state (EOS) is utilized to model the experimental work. An automated history matching algorithm is developed to fit the experimental data of WF and CF periods.Good agreement between the model and experimental data is obtained. We observe a strong impact of hysteresis on fitting the fluid production during CO2 flooding in Exp-2. The sensitivity analysis shows that tertiary CO2 recovery is affected by the water saturation in the core after the secondary WF. Moreover, the fracture-matrix transport function during tertiary-CO2 flooding is dominated by the diffusion rather than the convective flux or viscous forces.The outcome of this work is an important step towards modeling the tertiary-CO2 flooding in an actual fracture-chalk system. Proper modeling of imbibition and diffusion dominated processes in a chalk system at reservoir conditions has been accomplished.

Influence of Thermal Marangoni convection on the recovery of bypassed oil during immiscible injection

Several phenomena could recover the bypassed oil in matrixes and low permeable areas, among them; Marangoni is the most unfamiliar one in fluid flow model. This mechanism triggers a convective flow from low interfacial tension point to a high interfacial tension point because of variation in solvent concentration, temperature or electrical charge.In this study, experimental and analytical tools have been utilized to investigate the effect of thermal Marangoni flux on the enhancement of oil recovery during immiscible injection in a matrix-fracture system. Results of non-isothermal experiments have been compared with isothermal (as the zero level of Marangoni) to study the presence of Marangoni flow and its direction respect to the fracture location. Results show that not only the Marangoni flux does exist and has significant impact on the recovery of bypassed oil but also, its magnitude is proportional to the absolute value of temperature gradient in phases interface.

Structural regeneration of fracture-vug network in naturally fractured vuggy reservoirs

Naturally fractured vuggy carbonate reservoirs play an important role in the world's oil and gas production. Pore-scale modeling is treated as an important approach for unveiling the flow mechanism in carbonate reservoirs. The realistic regeneration of fractured vuggy reservoirs needs to preserve the geological structures and complex pores morphologies. Nevertheless, regeneration of such reservoirs is a great challenge because of the co-existence of fractures and vugs over a wide range of scales. In this paper, the reference fracture-vug network will be divided into two systems: matrix-fracture and matrix-vug. The two systems are reconstructed in two ways: (I), the matrix-vug and matrix-fracture systems are generated by improved simulated annealing method (ISAM) and multiple point statistics algorithm (MPS), respectively; (II), both of the matrix-vug and matrix-fracture systems are generated by MPS. When applying for ISAM, the multipoint histogram mainly depends on the N-point template. The template is a unit configuration, which is considered as the characteristic of the geometric features. Based on this, we introduce the correlation coefficient matrix to determine an optimal template, and the accuracy of template selection algorithm is also discussed. The optimal template represents essential geological features of the training image, and its size is much smaller than the template size selected by entropy. Therefore, it can improve the computing efficiency while preserving the key properties of the network. Then, a superimposing method is proposed to create the fracture-vug network by combining the matrix-vug with matrix-fracture models. The superimposing method using ISAM (for matrix-vug model generation) and MPS (for matrix-fracture model generation) is called SMIM, while superimposing method using MPS (for both of matrix-vug and matrix-fracture models generation) is called SMM. Finally, the spatial connectivity, local percolation probability and patterns reproduction of the generated models are used to compare with those of the reference model. The results show that the 7-point template selected by the correlation coefficient matrix is enough to get the spatial features of the matrix-vug system. The generated fracture-vug network using the selected template is able to retrieve the patterns of training image (reference model) more accurately than other used templates. The fracture-vug network generated by superimposing method produces better results than the other methods (all named as “direct method”). The network simulated with SMIM algorithm matches the reference model better than SMM algorithm. The probable reason maybe exist in the different generation methods used for matrix-vug systems. The local percolation probability (LPP), treated as a general geometric characterization method, is used to characterize the connectivity of measured units at a given local porosity. The results of the probability analysis show that SMIM is better in depicting the expected patterns of geological heterogeneities and in producing higher degree of connectivity of pore space than the SMM. The pattern reproduction of SMIM model is assessed by scanning the generated model, and the ratio of pattern reproduction number to that of reference model (i.e. the similarity) is around 97.28%. These results may guide the further application of the proposed method to field-scale modeling of naturally fractured vuggy reservoirs.

Acceleration of Gas Flow Simulations in Dual-Continuum Porous Media Based on the Mass-Conservation POD Method

Reduced-order modeling approaches for gas flow in dual-porosity dual-permeability porous media are studied based on the proper orthogonal decomposition (POD) method combined with Galerkin projection. The typical modeling approach for non-porous-medium liquid flow problems is not appropriate for this compressible gas flow in a dual-continuum porous media. The reason is that non-zero mass transfer for the dual-continuum system can be generated artificially via the typical POD projection, violating the mass-conservation nature and causing the failure of the POD modeling. A new POD modeling approach is proposed considering the mass conservation of the whole matrix fracture system. Computation can be accelerated as much as 720 times with high precision (reconstruction errors as slow as 7.69 × 10−4%~3.87% for the matrix and 8.27 × 10−4%~2.84% for the fracture).

Solute transport along a single fracture in a porous rock: a simple analytical solution and its extension for modeling velocity dispersion

A simple and robust solution is developed for the problem of solute transport along a single fracture in a porous rock. The solution is referred to as the solution to the single-flow-path model and takes the form of a convolution of two functions. The first function is the probability density function of residence-time distribution of a conservative solute in the fracture-only system as if the rock matrix is impermeable. The second function is the response of the fracture-matrix system to the input source when Fickian-type dispersion is completely neglected; thus, the effects of Fickian-type dispersion and matrix diffusion have been decoupled. It is also found that the solution can be understood in a way in line with the concept of velocity dispersion in fractured rocks. The solution is therefore extended into more general cases to also account for velocity variation between the channels. This leads to a development of the multi-channel model followed by detailed statistical descriptions of channel properties and sensitivity analysis of the model upon changes in the model key parameters. The simulation results obtained by the multi-channel model in this study fairly well agree with what is often observed in field experiments—i.e. the unchanged Peclet number with distance, which cannot be predicted by the classical advection-dispersion equation. In light of the findings from the aforementioned analysis, it is suggested that forced-gradient experiments can result in considerably different estimates of dispersivity compared to what can be found in natural-gradient systems for typical channel widths.

Modeling of flow and mixing in 3D rough-walled rock fracture intersections

The processes of fluid flow and solute transport through rock fractures are of primary importance in environmental engineering and geosciences. This study presented numerical modeling results of fluid flow and solute transport in a 3D rock fracture-matrix system with an orthogonal intersection of two rough-walled rock fractures. The rough-walled fracture geometry models were built from laser-scanned data of a real rock surface, for a realistic representation of natural rock fracture surface roughness. The fluid flow in the two intersected fractures and solute transport in the fracture-matrix system were simulated by solving the Navier–Stokes equations (NSE) and transport equation in the entire system. The dependence of mixing on Péclet number (Pe) and flow directionality features were analyzed. The results directly visualized important channeling flow patterns that significantly enhanced the solute mixing process at the rough-walled fracture intersection. The illustrated channeling flow and associated impacts on mixing are particularly important in the prediction of solute transport in natural fractured rocks, especially when discrete fracture network (DFN) approach is applied.

Influence of Various Dispersion Coefficients on Contaminant Migration in a Fracture-matrix System with Skin Formation

Subsurface contamination due to transient migration of chemicals through weathered fractures is a serious problem in hard rock terrains. In this paper, a numerical model has been used to analyse the effect of various dispersion coefficients on the movement of contaminants in a fracture-matrix syste m with skin formation. A constant continuous source of contaminants is considered at the inlet of a set of parallel fractures and an implicit finite difference technique has been used to solve the model. The transport of contaminants with constant dispersion coefficients were compared with those subjected to distance and time dependent dispersion. Sensitivity analysis has been conducted to study the effect of different fluid velocities, fracture-skin porosities, fracture-skin diffusion coefficients, fracture aperture, retardation factors, and dispersivity-distance ratio on solute transport mechanism within the system. Results suggest that the solute mass retained in the fracture is higher when time-dependent dispersion coefficients are used which may be due to seasonal conditioning of the rocks. When constant dispersion coefficient is considered, the reduction in the solute mass with increment in the fracture-skin porosity is gradual unlike other cases.The solute mass retained in the fracture increases with increase in dispersivity-distance ratio for time-dependent dispersion coefficients.

Reactive solute transport in a filled single fracture-matrix system under unilateral and radial flows

The study of transport processes in a single fracture is the basis of understanding transport in complex fractured networks. Many single fractures in the field are filled with sediments, and the transport in such filled single fractures has received much less attention up to present. When the fracture is partially filled with sediments, a mobile-immobile approach is considered necessary. This study deals with a coupled three-domain transport problem using mobile and immobile domains to characterize a filled single fracture and a matrix domain to characterize the rock body. Mathematical models are developed for such a coupled three-domain transport problem with new semi-analytical solutions to analyze the spatial-temporal concentration and mass distributions in the fracture and rock matrix with the help of Laplace transforms. This study addresses transport in a filled fracture-matrix system under two different flow conditions: unilateral flow, and radial flow. The new solutions have been tested extensively against previous solutions under various special settings and are proven to be robust and accurate. This study has the following findings: 1) Longitudinal dispersion in the fracture often plays an important role in such a coupled system in unilateral flow, 2) Mass partitions in three domains follow similar patterns in respect to the influence of fracture apertures, mobile/immobile ratios, and first-order mass transfer rates, 3) The system is most sensitive to the dispersivity and least sensitive to the first-order mass transfer rate and the mobile/immobile ratio in the unilateral flow model over a wide range of time scales (if the longitudinal dispersivity and Darcian flow velocity remain constant), 4) The system is most sensitive to the dispersivity, less sensitive to the mobile/immobile ratio, and least sensitive to the first-order mass transfer rate in the radial flow model (if the radial dispersivity and injection rate remain constant).

Modeling of Solute Transport in a 3D Rough-Walled Fracture\u2013Matrix System

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.

Practical simulation of multi-porosity reservoirs through existing reservoir simulator

Complex heterogeneous reservoir exhibits special characteristics, such as multiple storage types, wide range of storage sizes, complicated storage combinations, and various flow behaviors. Multi-continuum model is needed to rigorously simulate this type of reservoir. However most reservoir simulators use dual porous media model, and the theory of the models is mainly focused on matrix-fracture system. Therefore when porous media is more complex than matrix-fracture system, such software cannot be utilized to simulate these reservoirs directly. Some studies have been proposed to solve the problem, but most of them are either at the theoretical level or cannot been used widely in the simulation softwares. The goal of this paper is not to develop new numerical simulation software, but to obtain appropriate ways to improve performance of current simulation software for complex reservoir.Our approach is theoretically based on multi-continuum seepage theory. By comparing the microscopic seepage model and multi-continuum seepage model, we find that the conventional theory becomes improper because of the absence of an important restriction to the interporosity flow. In the conventional way, an N-multiple flow model always can be established based on N types of continuous media. In fact, one more restrictive condition needs to be added to make it valid: all the crossflow between any two types of continuum can be simplified into a source or sink terms in the seepage model. Consequently, the multiplicity of the seepage model is always less or equal to the number of the continuum types. In addition, because different type of dual-medium has different crossflow, it is required to construct their crossflow models respectively and derive corresponding calculation formula.Several steps developed in this paper are as follows on the basis of multi-continuum theory and the restrictions appended: First, reservoir storage is categorized into continuous and non-continuous media according to storage scale characteristics, distribution and intensity. In general, small cavities, small fractures, and pores in the rock are continuous while large-scale fractures, big cavities or caves are non-continuous. The whole reservoir is therefore classified into continuum region and dis-continuum region. Second, in the continuum region, multi-media model and its multiplicity are determined by using the new theory proposed in this paper. Third, calculation formulae are constructed for each type of crossflow. This research takes fractured vuggy reservoirs as an example. A new calculation for the flux transfer is derived for such fracture-vug dual-porosity system. Forth, in the discrete region, flow in a filled non-continuous medium is described by Darcy's law and free flow in large unfilled cavities or caves follows Navier-Stokes equation and an approximate approach to using Darcy flow simulation is provided. Lastly, this paper introduces the methods how to simplify multi-media into two media and how to simulate the complex reservoir by using the existing simulators.The proposed method is applied to a naturally fractured carbonate reservoir with cavities in the Tarim Basin in China. The simulation results illustrate the production mechanisms of reservoir development using water flooding, including water injection and shut-in, stairs-shaped line of water cut with time, high production with low water cut for long time.

Effect of random fracture aperture on the transport of colloids in a coupled fracture-matrix system

A variable aperture model, including the random variation of fracture aperture as against the conventional parallel plate model, has been developed to adequately examine the transport of colloids/suspended particles in a single coupled fracturematrix system. Rather than relying on a complex geostatistical method for an accurate representation of fracture aperture, which requires an enormous field data and resource for its validation, a simple statistical method (linear congruential generator) is implemented in the present article. The random variation of fracture aperture is an honest representation of the unpredictable geometry/ morphology of fracture aperture in comparison with widely applied the conventional parallel plate model or the simple mathematical functions based on fractal theory (self-affine structures). A considerable number of parameters are involved in investigating the extent of penetration of colloids into the rock matrix, which creates complexity and ambiguity in the analysis. To overcome this problem, a single parameter “Maximum Penetration Factor” has been introduced for simple and reliable assessment of diffusion of colloids within the rock matrix. Additionally, a non-dimensional parameter ‘Matrix Mitigation Factor’ has been introduced in the present study, which can provide a means of evaluating the diffusion of suspended particles within the rock matrix when it comes to real time applications like microbial enhanced recovery (MEOR) and chemical enhanced recovery (CEOR) in the petroleum industry (nanoparticles and nanofluids). A semi-implicit finite difference model has been adopted for solving the coupled partial differential equations in the present numerical study. Finally, Neumann and Robinson boundary conditions as a function of time have been applied at the fracture inlet to better represent the field scenario as against the conventional constant source condition (Dirichlet). The model results indicate that there is a difference in concentration between the parallel plate model and random fracture model when it comes to colloidal concentration in the fracture and rock matrix. The variance in concentration is due to the inclusion of variation of the aperture in the variable aperture model, which is absent in the parallel plate model. Additionally, the results suggest that the variable source boundary condition has a significant influence on the transport of colloids in fracture-matrix system. Finally, from the evaluation of the extent of diffusion of colloids into rock matrix, it was concluded that that variable aperture model is associated with more mitigation of colloids compared to the parallel plate model, especially in the case of random fracture.

Numerical simulation of chemical potential dominated fracturing fluid flowback in hydraulically fractured shale gas reservoirs

To find out the impact of chemical potential difference between the low salinity fracturing fluid and the high salinity formation water on fracturing fluid flowback, a chemical potential difference expression of fracturing fluid and formation water was deduced, on this basis, a mathematical model which considers viscous force, capillary force and osmosis pressure driven gas-water flow in matrix-fracture system was built, the flow back performance of fracturing fluid driven by chemical potential difference was simulated, and the formation water saturation and salt concentration profile with flow back time were analyzed. The results show that in the process of flow back, the water molecules in the matrix driven by the chemical potential difference continually migrated to the deeper reservoirs, while salt ions in the matrix constantly spread to the fractures. After 168 h of fracturing-fluid flow back, the migration distance of water was up to 40 cm, and the salt concentration near the fracture surface increased by 0.841%, and the cumulative flowback ratio of the gas well was only 22.1%. The cumulative flowback ratio would be 23.5%, 32.4% and 41.1% respectively, without taking into account the effect of gas absorption, chemical osmosis or capillary imbibition. The capillary imbibition and chemical osmosis seriously hindered the fracturing-fluid flow back, therefore, the two factors should be fully considered in the post-fracturing evaluation of shale gas wells.

Two-phase numerical simulation of discrete fracture model based on multiscale mixed finite element method

Fractured reservoirs are characterized by complex fractures on multiple scales and are quite difficult to model. Numerical simulation of fractured media is usually done based on dual-porosity model and equivalent continuum model. Dual-porosity model treats matrix system and fracture system as two parallel continuous systems coupled by crossflow function. This model is only valid for reservoirs with highly developed fractures. Equivalent continuum model treats fractured media as a continuum media and equivalent absolute permeability tensors for each grid block are calculated to describe the heterogeneity of the reservoir. This model is efficient only when there exist representative element volume and the equivalent permeability are difficult to decide. Both models treat the fractured media as a simplified model and cannot describe the multiscale flows exactly, because they cannot precisely consider the diversion effect of the fractures. Although the discrete fracture network (DFN) model can provide a detailed representation of flow characteristic, traditional numerical method does not suitable for DFN. The major difficulty is the size of the computation. A tremendous amount of computer memory and CPU time are required, and this can exceed limit of today’s computer resources. Upscaling methods are generally used to reduce the computational cost. However, it is not possible to have a priori estimates of the errors that are present when complex flow processes are investigated using coarse models constructed via simplified settings. In this paper, multiscale mixed finite element method (MsMFEM) is proposed to simulate water/oil two phase flow in discrete fracture media. By combining MsMFEM with the discrete fracture model, we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling. The MsMFEM uses a standard Darcy model to approximate pressure and fluxes on a coarse grid. The multiscale basis functions are constructed numerically by solving local differential equations on the fine-scale grid. The advantage of MsMFEM is that the basis functions capable of reflecting information about fractures within elements. Therefore, this method can capture the fine-scale effects on the coarse grid, that is, multiscale method can reduce the computational cost and keep high calculation accuracy at the same time. Traditional numerical methods generally difficult to deal with complex grid element, in this paper, mimetic finite difference (MFD) method is used to construct the multiscale basis functions due to its local conservativeness and applicability of complex grids. Compared with traditional multiscale mixed finite element methods, this method is suitable for arbitrary complex grid system. This paper introduced fundamental principles of the multiscale mixed finite element method and described the numerical scheme of discrete fracture model based on MsMFEM in detail. Then we deduced discrete fracture model computing formulate for the multiscale basis function by using mimetic finite difference method. Oversampling technique is applied to get more accurate small-scale details. IMPES scheme is used in the two-phase flow simulation. Physical experiment is used to prove the validity the multiscale method. The numerical results show that compared with traditional numerical method, the MsMFEM can represent the fine-scale flow in fracture networks exactly and meanwhile has a higher computational efficiency.

Discussion on similarity of recovery curves in scaling of imbibition process in fractured porous media

Spontaneous imbibition is the main production mechanism for water invaded naturally fractured reservoirs. The efficiency of this mechanism depends on different parameters such as fluid, rock and geometry properties of the matrix-fracture system. Upscaling of imbibition recovery curves from laboratory scale tests to field scale matrix blocks is important for properly characterizing and simulation of fluid flow in naturally fractured reservoirs. Distinct scaling equations have been presented in the literature presuming similarity of recovery curves. In the previous works, it was attempted that the recovery curves from different samples are scaled to a single recovery curve by presenting the recovery factors versus dimensionless times. The dimensionless times were obtained by multiplying the recovery times in a constant value. The various existing scaling groups mainly differ in the manner this constant value is defined. This work, proposes a framework to investigate the possibility of having a single scaled recovery curve by multiplying the recovery curve time in a specific constant value. Therefore, the similarity of recovery curves obtained by varying different matrix block properties is studied. Hence, numerous imbibition recovery curves are simulated by employing a commercial numerical reservoir simulator and their similarity is analyzed and calculated using an innovative similarity analysis method. Results of this research show that the proposed methods in the previous studies for scaling of imbibition data could fail where certain matrix block parameters including rock permeability, oil viscosity and oil-water capillary pressure have considerable differences. Therefore, one can say that actually a non-uniform shifting is required for scaling of recovery curves if the certain mentioned properties are different among the samples. It can be concluded that a new scaling method is required that incorporates time-dependent parameters to scale recovery curves to a single recovery curve.

A similarity solution for reaction front propagation in a fracture-matrix system.

We propose a new composite similarity variable, based on which a similarity solution is derived for reaction front propagation in fracture-matrix systems. The similarity solution neglects diffusion/dispersion within the fracture and assumes the existence of a sharp reaction front in the rock matrix. The reaction front location in the rock matrix is shown to follow a linear decrease with distance along the fracture. The reaction front propagation along the fracture is shown to scale like diffusion (i.e. as the square root of time). The similarity solution using the composite similarity variable appears to be applicable to a broad class of reactive transport problems involving mineral reactions in fracture-matrix systems. It also reproduces the solutions for non-reactive solute and heat transport when diffusion/dispersion/conduction are neglected in the fracture. We compared our similarity solution against numerical simulations for nonlinear reactive transport of an aqueous species with a mineral in the rock matrix. The similarity solutions agree very well with the numerical solutions, especially at later times when diffusion limitations are more pronounced.This article is part of the themed issue 'Energy and the subsurface'.