Displacement of Cr(III)–Partially Hydrolyzed Polyacrylamide Gelling Solution in a Fracture in Porous Media

During waterflooding of a fractured formation, water may channel through the fracture or interconnected network of fractures, leaving a large portion of oil bearing rock unswept. One remedial practice is injection of a gelling solution into the fracture. Such placement of a gelling mixture is associated with leak-off from the fracture face into the adjoining matrix. Design of a gel treatment needs understanding of the flow of gelling mixture in and around the fracture. This flow is addressed here for Cr(III)–partially hydrolyzed polyacrylamide formulation through experiments and conceptual model. A fractured slab was used to develop a lab-model, where the flow along the fracture and simultaneous leak-off into the matrix can be controlled. Also, the fracture and matrix properties had to be evaluated individually for a meaningful analysis of the displacement of gelling solution. During this displacement, the gelling fluid leaked off from the fracture into the matrix as a front, resulting in a decreasing velocity (and pressure gradient) along the fracture. With pressure in the fracture held constant with time, the leak-off rate decreased as the viscous front progressed into the matrix. The drop in leak-off rate was rapid during the initial phase of displacement. A simple model, based on the injection of a viscous solution into the dual continua, could explain the displacement of Cr(III)–polyacrylamide gelling mixture through the fractured slab. This study rules out any major complication from the immature gelling fluid, e.g., build-up of cake layer on the fracture face. The model, due to its simplicity may become useful for quick sizing of gel treatment, and any regression-based evaluation of fluid properties in a fracture for other applications.     

Numerical Analysis of Imbibition Front Evolution in Fractured Sandstone under Capillary-Dominated Conditions

Fractures serve as primary conduits having a great impact on the migration of injected fluid into fractured permeable media. Appropriate transport properties such as relative permeability and capillary pressure are essential for successful simulation and prediction of multi-phase flow in such systems. However, the lack of a thorough understanding of the dynamics governing immiscible displacement in fractured media, limits our ability to properly represent their macroscopic transport properties. Previous experimental observations of imbibition front evolution in fractured rocks are examined in the present study using an automated history-matching approach to obtain representative relative permeability and capillary pressure curves. Predicted imbibition front evolution under different flow conditions resulted in an excellent agreement with experimental observations. Sensitivity analyses, in combination with direct experimental observation, allowed exploring the competing effects of relative permeability and capillary pressure on the development of saturation distribution and imbibing front evolution in fractured porous media. Results show that residual saturations are most sensitive to matrix relative permeability to oil, while the ratio of oil and water relative permeability, rock heterogeneity, boundary condition, and matrix–fracture capillary pressure contrast, affect displacement shape, speed, and geometry of the imbibing front.     

On Vaporizing Water Flow in Hot Sub-Vertical Rock Fractures

Water injection into unsaturated fractured rock at above-boiling temperatures gives rise to complex fluid flow and heat transfer processes. Examples include water injection into depleted vapor-dominated geothermal reservoirs, and emplacement of heat-generating nuclear wastes in unsaturated fractured rock. We conceptualize fractures as two-dimensional heterogeneous porous media, and use geostatistical techniques to generate synthetic permeability distributions in the fracture plane. Water flow in hot high-angle fractures is simulated numerically, taking into account the combined action of gravity, capillary, and pressure forces, and conductive heat transfer from the wall rocks which gives rise to strong vaporization. In heterogeneous fractures boiling plumes are found to have dendritic shapes, and to be subject to strong lateral flow effects. Fractures with spatially-averaged homogeneous permeabilities tend to give poor approximations for vaporization behavior and liquid migration patterns. Depending on water flow rates, rock temperature, and fracture permeability, liquid water can migrate considerable distances through fractured rock that is at above-boiling temperatures and be only partially vaporized.     

An Extended-FEM Model for CO2 Leakage Through a Naturally Fractured Cap-Rock During Carbon Dioxide Sequestration

In this paper, a numerical model is developed for the assessment of carbon dioxide transport through naturally fractured cap-rocks during CO₂ sequestration in underground aquifers. The cap-rock contains two types of fracture with different length scales: micro-cracks (fissures) and macro-cracks (faults). The effect of micro-cracks is incorporated implicitly by modifying the intrinsic permeability tensor of porous matrix, while the macro-cracks are modeled explicitly using the extended finite element method (X-FEM). The fractured porous medium is decomposed into the porous matrix and fracture domain, which are occupied with two immiscible fluid phases, water and CO₂. The flow inside the matrix domain is governed by the Darcy law, while the flow within the fracture is modeled using the Poiseuille flow. The mass conservation law is fulfilled for each fluid phase at both porous matrix and fracture domain; moreover, the mass exchange between the matrix and fracture is guaranteed through the integral formulation of mass conservation law. Applying the X-FEM technique, the explicit representation of macro-cracks is modeled by enriching the standard finite element approximation space with an enrichment function. Finally, several numerical examples of CO₂ injection into a brine aquifer below a naturally fractured cap-rock are modeled in order to investigate the effects of cracks’ aperture and orientation as well as the temperature of aquifer and the depth of injection on the leakage pattern of the carbon dioxide through the cap-rock.

     

Oil Displacement for One-Dimensional Three-Phase Flow with Phase Change in Fractured Media

In this article, the numerical simulations for one-dimensional three-phase flows in fractured porous media are implemented. The simulation results show that oil displacement in matrix is dominated by oil–water capillary pressure only under certain conditions. When conditions are changed to decrease the amount of water entering into the fractured media from the boundary of the flow field, water in fracture may be vaporized to superheated steam. In these cases, the appearance of superheated steam in fracture rather than in matrix will decrease the fracture pressure and generate the pressure difference between matrix and fracture, which results in oil flowing from matrix to fracture. Assuming that oil is wetting to steam, the matrix steam–oil capillary pressure will decrease the matrix oil-phase pressure as the matrix steam saturation increases. After the steam–oil capillary pressure finally exceeds the pressure difference due to the appearance of superheated steam in fracture, the oil displacement in matrix will stop. It is also shown that variations of the water relative permeability curve in matrix do not result in different mechanisms for oil displacement in matrix. The simulation results suggest that the amount of liquid water supply from the boundary of flow field fundamentally influence the mechanisms for oil displacement in matrix.     

On the problem of fluid leakoff during hydraulic fracturing

While a hydraulic fracture is propagating, fluid flow and associated pressure drops must be accounted for both along the fracture path and perpendicularly, into the formation that is fractured, because of fluid leakoff. The accounting for the leakoff shows that it is the main factor that determines the crack length. The solved problem is useful for the technology of hydraulic fracturing and a good example of mass transport in a porous medium. To find an effective approach for the solution, the thin crack is represented here as the boundary condition for pore pressure spreading in the formation. Earlier such model was used for heat conduction into a rock massif from a seam under injection of hot water. Of course, the equations have other physical sense and mathematically they are somewhat different. The new plane solution is developed for a linearized form that permits the application of the integral transform. The linearization itself is analogous to the linearization of the natural gas equation using the real gas pseudo-pressure function and where the flux rates are held constant and approximations are introduced only into the time derivatives. The resulting analytical solution includes some integrals that can be calculated numerically. This provides rigorous tracking of the created fracture volume, leakoff volume and increasing fracture width. The solutions are an advance over existing discreet formulations and allow ready calculations of the resulting fracture dimensions during the injection of the fracturing fluid.     

Efficiency of Capillary Imbibition Dominated Displacement of Nonwetting Phase by Wetting Phase in Fractured Porous Media

The critical and optimum injection rates as well as the critical fracture capillary number for an efficient displacement process are determined based on the experimental and numerical modeling of the displacement of nonwetting phase (oil) by wetting phase (water) in fractured porous media. The efficiency of the process is defined in terms of the nonwetting phase displaced from the system per amount of wetting phase injected and per time. Also, the effects of injection rate on capillary imbibition transfer dominated two-phase flow in fractured porous media are clarified by visualizing the experiments. The results reveal that as the injection rate is increased, fracture pattern begins to become an effective parameter on the matrix saturation distribution. As the rate is lowered, however, the system begins to behave like a homogeneous system showing a frontal displacement regardless the fracture configuration.     

Computational Modeling of Fluid Flow through a Fracture in Permeable Rock

Laminar, single-phase, finite-volume solutions to the Navier–Stokes equations of fluid flow through a fracture within permeable media have been obtained. The fracture geometry was acquired from computed tomography scans of a fracture in Berea sandstone, capturing the small-scale roughness of these natural fluid conduits. First, the roughness of the two-dimensional fracture profiles was analyzed and shown to be similar to Brownian fractal structures. The permeability and tortuosity of each fracture profile was determined from simulations of fluid flow through these geometries with impermeable fracture walls. A surrounding permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2 to 2,000 millidarcies, was then included in the analysis. A series of simulations for flows in fractured permeable rocks was performed, and the results were used to develop a relationship between the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which accounts for the fracture geometric properties, is similar to the cubic law; it has the potential to be of use in discrete fracture reservoir-scale simulations of fluid flow through highly fractured geologic formations with appreciable matrix permeability. The observed fluid flow from the surrounding permeable medium to the fracture was significant when the resistance within the fracture and the medium were of the same order. An increase in the volumetric flow rate within the fracture profile increased by more than 5% was observed for flows within high permeability-fractured porous media.     

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.     

One-Dimensional Matrix-Fracture Transfer in Dual Porosity Systems with Variable Block Size Distribution

Most of the developed models for fractured reservoirs assume ideal matrix block size distribution. This assumption may not be valid in reality for naturally fractured reservoirs and possibly lead to errors in prediction of production from the naturally fractured reservoirs especially during a transient period or early time production from the matrix blocks. In this study, we investigate the effect of variable block size distribution on one- dimensional flow of compressible fluids in fractured reservoirs. The effect of different matrix block size distributions on the single phase matrix-fracture transfer is studied using a recently developed semi-analytical approach. The proposed model is able to simulate fluid exchange between matrix and fracture for continuous or discrete block size distributions using probability density functions or structural information of a fractured formation. The presented semi-analytical model demonstrates a good accuracy compared to the numerical results. There have been recent attempts to consider the effect of variable block size distribution in naturally fractured reservoir modeling for slightly compressible fluids with a constant viscosity and compressibility. The main objective of this study is to consider the effect of variable block size distribution on a one-dimensional matrix-fracture transfer function for single-phase flow of a compressible fluid in fractured porous media. In the proposed semi-analytical model, the pressure variability of viscosity and isothermal compressibility is considered by solving the nonlinear partial differential equation of compressible fluid flow in the fractured media. The closed form solution provided can be applied to flow of compressible fluids with variable matrix block size distribution in naturally fractured gas reservoirs.     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture

One of the most popular models that has been applied to predict the fluid velocity inside the fracture with impermeable walls is the cubic law. It highlights that the mean flux along the fracture is proportional to the cubic of fracture aperture. However, for a fractured porous medium, the normal and tangential interface conditions between the fracture and porous matrix can change the velocity profile inside the fracture. In this paper, a correction factor is introduced for flow equation along the fracture by imposing the continuity of normal and tangential components of velocity at the interface between the fracture and porous matrix. As a result, the mean velocity inside the fracture depends not only on the fracture aperture, but also on a set of non-dimensional numbers, including the matrix porosity, the ratio of intrinsic permeability of fracture to that of matrix, the wall Reynolds number, and the ratio of normal velocity on one wall to the other. Finally, the introduced correction factor is employed within the extended finite element method, which is widely used for numerical simulation of fluid flow within the fractured porous media. Several numerical results are presented for the fluid flow through a specimen containing single fracture, in order to investigate the deviation from the cubic law in different case studies.

     

Acid Leakoff Mechanism in Acid Fracturing of Naturally Fractured Carbonate Oil Reservoirs

In acid fracturing, excessive acid leakoff is thought to be the main reason that limits fracture propagation and live acid penetration distance. Since most carbonates are naturally fractured, we developed a new model in this paper to simulate acid leakoff into a naturally fractured carbonate oil reservoir during acid fracturing. Our model incorporates the acid-rock reaction, fracture width variation due to rock dissolution on the fractured surfaces, and fluid flow in naturally fractured carbonate oil reservoirs. Given the information of the reservoir, injected acid, and pressure in the hydraulic facture and the reservoir, the model predicts acid leakoff with time. In this study, we found that acid leakoff mechanism in naturally fractured carbonates is much different from that in reservoirs without natural fractures. Widened natural fractures by acid-rock reaction act as high-conductivity conduits allowing leakoff acid to penetrate deeper into the formation, resulting in serious leakoff. Wide natural fractures have a dominant effect on acid leakoff compared to micro-fractures and matrix.     

An investigation of hydraulic fracturing. Part I: one-phase flow model

The prediction of the growth of a hydraulic fracture in an oil bearing formation based on the injection rate of fluid is valuable in applications of the waterflood technique in secondary oil recovery. In this paper, the problem of hydraulic fracture growth is studied under the assumption of uniform distribution of pressure in the fracture and unidirectional permeating flow in an infinitely large isothermal linearly elastic porous medium saturated with a one-phase incompressible fluid. The condition of plane strain is imposed in the study. A comparison of the constant fracture toughness criterion based on the asymptotic value for large crack growth with the crack tip ductility criterion for an ideally plastic solid under plane strain and small-scale yielding conditions indicates that the effect of ductility of rock on the crack growth is so small that the steady state value of the energy release rate can be reached within a short period of crack growth. Thus we can employ the constant fracture toughness criterion in our study. The analysis includes the effects of both fracture volume increase and leak-off of fluid from the surface of the fracture. A nonlinear singular integro-differential equation can be formulated for the quasi-static hydraulic fracture growth under a prescribed injection rate. It is solved numerically by a modified fourth order Runge-Kutta method.     

LBM simulation of fluid flow in fractured porous media with permeable matrix

To analyze and depict complicated fluid behaviors in fractured porous media with variably permeable matrix, an integrated discrete computational algorithm is proposed based on lattice Boltzmann method (LBM). This paper combines with the external force model and statistical material physics to effectively describe the feature changes while the fluid passes through the fractures within the permeable matrix. As an application example, a two dimensional rock sample is reconstructed using the digital image and characterized with different feature values at each LBM grid to distinguish pores, impermeable and permeable matrix by stating its local physical property. Compared with the conventional LBM, the results demonstrate the advantages of proposed algorithm in modeling fluid flow phenomenon in fractured porous media with variably permeable matrix.     

Modeling Water Leak-off Behavior in Hydraulically Fractured Gas Shale under Multi-mechanism Dominated Conditions

Fracturing-fluid leak-off in fractured gas shale is a complex process involving multiple pore/fluid transports and interactions. However, water leak-off behavior has not been modeled comprehensively by considering the multi-pores and multi-mechanisms in shale with existing simulators. In this paper, we present the development of a comprehensive multi-mechanistic, multi-porosity, and multi-permeability water/gas flow model that uses experimentally determined formation properties to simulate the fracturing-fluid leak-off of hydraulically fractured shale gas wells. The multi-mechanistic model takes into account water transport driven by hydraulic convection, capillary and osmosis, gas transport caused by hydraulic convection, and salt ion transport caused by advection and diffusion. The multi-porosity includes hydraulic fracture millipores, organic nanopores, clay nanopores, and other inorganic micropores. The multi-permeability model accounts for all the important processes in shale system, including gas adsorption on the organics’ surface, multi-mechanistic clay/other inorganic mineral mass transfer, inorganic mineral/hydraulic fracture mass transfer, and injection from a hydraulically fractured wellbore. The dynamic water saturation and pressure profiles within clay and other inorganic matrices are compared, revealing the leak-off behavior of water in rock media with different physicochemical properties. In sensitivity analyses, cases with different clay membrane efficiency, volume proportion of source rock, connate water salinity, and saturation are considered. The impacts of shale properties on water fluxes through wellbore, hydraulic fracture and matrix, and the total injection and leak-off volumes of the well during the treatment of hydraulic fracturing are investigated. Results show that physicochemical properties in both organic and inorganic matrices affect the water leak-off behavior.     

Can Moderately Rarefied Gas Transport Through Round and Flat Tight Channels of Fractured Porous Media be Described Accurately?

We study the generation and flow of foam through rough-walled, fractured marble rocks that mimic natural fracture systems in carbonate reservoirs. Flow was isolated to the fracture network because of the very low rock permeability of the marble samples and foam generated in situ during co-injection of surfactant solution and gas. The foam apparent viscosities were calculated at steady pressure gradients for a range of gas fractions, and similar to foam flow in porous media, we identified two flow regimes for foam flow in fractures: a high-quality flow regime only dependent on liquid velocity and a low-quality flow regime determined by the gas and liquid velocities. Variations in local fluid saturation during co-injection were visualized and quantified using positron emission tomography combined with computed tomography.

     

多层平行裂隙型多孔介质通道内流体流动传热特性

基于Brinkman-extended Darcy模型和局部热平衡模型,对多层平行裂隙型多孔介质通道内的流动传热特性进行研究.获得了多层平行裂隙型多孔介质通道内各区域的速度场、温度场、摩擦系数及努塞尔数解析解,并分析了裂隙层数、达西数、空心率、有效热导率之比等对通道内流动传热特性的影响.结果表明:达西数较小时,通道多孔介质层内会出现不随高度变化的达西速度,此达西速度会随裂隙层数的增加而增大,但却不受各裂隙层下多孔介质层位置变化的影响.增加裂隙层数会减弱空心率对压降的影响,会使通道内流体压降升高,但升高程度会逐渐降低.增大热导率之比或减小空心率会使多裂隙通道内出现阶梯式温度分布,而在较小热导率之比或较大空心率时多裂隙情况下的温度分布曲线会趋于一致.此外,当热导率之比较小时,多层裂隙通道内的传热效果在任何空心率下都要优于单裂隙情况,当热导率之比较大时,存在临界空心率使各裂隙层数通道内的传热效果相同,且多裂隙通道内继续增加裂隙层数对传热强度影响不大.     

Two-dimensional flow in a fractured medium

Transport processes within a liquid-filled fractured reservoir can be modelled using a double-diffusive mechanism in fracture and block. Then it is commonly assumed that the flow in the block is purely one-dimensional (e.g. vertical). Lateral flow within the block will, however, become significant at long times. Avdonin has given an analytic solution for the pressure response in an infinite fissure bounded by two homogeneous half-spaces, allowing vertical flow only in the blocks. We extend this solution to include horizontal flow in the blocks. There are significant qualitative differences between the two cases. In particular, we find that if fluid is injected at a constant rate into the fissure and horizontal flow in the blocks is allowed, then the long-time pressure response of the fissure/block assembly has the same character as that due to a line source in a homogeneous anisotropic porous medium.     

Modeling the effects of capillary pressure with the presence of full tensor permeability and discrete fracture models using the mimetic finite difference method

Capillary dominated flow or imbibition—whether spontaneous or forced—is an important physical phenomena in understanding the behavior of naturally fractured water-driven reservoirs (NFR’s). When the water flows through the fractures, it imbibes into the matrix and pushes the oil out of the pores due to the difference in the capillary pressure. In this paper, we focus on modeling and quantifying the oil recovered from NFR’s through the imbibition processes using a novel fully implicit mimetic finite difference (MFD) approach coupled with discrete fracture/discrete matrix (DFDM) technique. The investigation is carried out in the light of different wetting states of the porous media (i.e., varying capillary pressure curves) and a full tensor representation of the permeability. The produced results proved the MFD to be robust in preserving the physics of the problem, and accurately mapping the flow path in the investigated domains. The wetting state of the rock affects greatly the oil recovery factors along with the orientation of the fractures and the principal direction of the permeability tensor. We can conclude that our novel MFD method can handle the fluid flow problems in discrete-fractured reservoirs. Future works will be focused on the extension of MFD method to more complex multi-physics simulations.