Hydraulic Fracture Propagation with 3-D Leak-off

Hydraulic fracture models typically couple a fracture elasticity model with a geological reservoir model to forecast the rate of fluid leak-off from the propagating fracture. The most commonly used leak-off model is that originally specified by Carter, which involves the assumption that the fracture is embedded within an infinite homogenous porous medium where flow only occurs perpendicular to the fracture plane. The objectives of this paper are: (1) to show that assuming one-dimensional leak-off can lead to erroneous conclusions, (2) to present a robust numerical methodology for simulating three-dimensional leak-off from propagating hydraulic fractures, and (3) to present and compare a new analytical method based on assuming three-dimensional flow of an incompressible fluid through an incompressible porous formation from a circular planar fracture. Provided the fluid and formation compressibility can be ignored within the reservoir flow model, the three-dimensional leak-off from a circular planar fracture can be written in closed-form as a function, which depends linearly on fracture pressure and radial extent. This simple expression for leak-off can be easily coupled to a range of circular fracture elasticity models. As a comparison example, the Carter model, our new function and a three-dimensional numerical model of the full problem are coupled to the PK-radial fracture model. Comparison with the numerical model shows that our new function overestimates fracture growth during intermediate times but accurately predicts both the early and late-time asymptotic behavior. In contrast, the Carter model fails to replicate both the early and late-time asymptotic behavior. Our new function additionally improves on the Carter model by not requiring the evaluation of convolution integrals and allowing easy evaluation of both the spatial leakage flux distribution across the fracture face and the three-dimensional pressure distribution within the porous formation.     

Stress-Sensitive Permeability: Application to Fault Integrity During Gas Production

The objective is to propose a simple theoretical approach and the associated numerical algorithm to capture the permeability evolution within a fractured region in response to a stress perturbation. The stress range of interest is typical of a reversible deformation such that the fractures have varying apertures but constant lengths and densities. It is the permeability evolution from a negligible value characteristic of flows on geological times to values more relevant for gas production which is important for the structural integrity of the fractured region. A simple 1D application related to the sealing capacity of a fault bounding a producing gas reservoir is proposed to illustrate the theory. The stress change on the two sides of the faults is obtained with a 2D finite-element simulation based on the theory of poro-elasticity and considering the fault as a material discontinuity. The 1D flow simulation is done in a second step, and the flux is assumed to occur through the fault thickness from the non-depleted (minus side) to the depleted (plus side) regions. It is shown how the depletion results in the fractures opening in the fault damaged zone close to the minus side and the fracture closure next to the plus side. This evolution could be non-monotonic in time because of the development and the thinning of a boundary layer in the fluid pressure at the plus side. The simulations end once a Coulomb criterion is reached, typically at the minus side of the fault. The presence of a low-permeability core in the fault center does not change these conclusions although a positive effective normal stress is detected in the damaged zone on the minus side of the core prior to the Coulomb criterion activation.     

Enhancing Immiscible Fluid Displacement in Porous Media by Capillary Pressure Discontinuities

Fluid displacement in porous media plays an important role in many industrial applications, including biological filtration, carbon capture and storage, enhanced oil recovery, and fluid transport in fuel cells. The displacement front is unstable, which evolves from smooth into ramified patterns, when the mobility (ratio of permeability to viscosity) of the displacing fluid is larger than that of the displaced one; this phenomenon is called viscous fingering. Viscous fingering increases the residual saturation of the displaced fluid, considerably impairing the efficacy of fluid displacement. It is of practical importance to develop suitable methods to improve fluid displacement. This paper presents an experimental study on applying the discontinuity of capillary pressure to improve immiscible fluid displacement in drainage for which the displacing fluid (air) wets the porous media less preferentially than does the displaced fluid (silicone oil). The concept involves using a heterogeneous packing system, where the upstream region features large pores and small capillary pressure, and the downstream region features small pores and large capillary pressure. The increase in capillary pressure prevents fingering from directly crossing the media interface, thus enhancing the displacement. The experimental apparatus was a linear cell comprising porous media between two parallel plates, and glass beads of 0.6 and 0.125 mm diameter were packed to compose the heterogeneous porous media. The time history of the finger flow was recorded using a video camera. Pressure drops over the model from the inlet to the outlet were measured to compare viscous pressure drops with capillary pressures. The results show that the fluid displacement was increased by the capillary discontinuities. The optimal displacement was determined through linear regression by adjusting the relative length of the large- and small-pore region. The results may assist in the understanding of fingering flow across the boundaries of different grain-sized bands for the gas and oil reservoir management, such as setting the relative location of the injection and production wells. The findings may also serve as a reference for industrial applications such as placing the grain bands in an adequate series to improve the displacement efficacy in biological filtration.     

Local mesh adaptation technique for front tracking problems

A numerical model is developed for the simulation of moving interfaces in viscous incompressible flows. The model is based on the finite element method with a pseudo-concentration technique to track the front. Since a Eulerian approach is chosen, the interface is advected by the flow through a fixed mesh. Therefore, material discontinuity across the interface cannot be described accurately. To remedy this problem, the model has been supplemented with a local mesh adaptation technique. This latter consists in updating the mesh at each time step to the interface position, such that element boundaries lie along the front. It has been implemented for unstructured triangular finite element meshes. The outcome of this technique is that it allows an accurate treatment of material discontinuity across the interface and, if necessary, a modelling of interface phenomena such as surface tension by using specific boundary elements. For illustration, two examples are computed and presented in this paper: the broken dam problem and the Rayleigh–Taylor instability. Good agreement has been obtained in the comparison of the numerical results with theory or available experimental data. © 1998 John Wiley & Sons, Ltd.     

Numerical Homogenization of the Absolute Permeability Using the Conformal-Nodal and Mixed-Hybrid Finite Element Method

The modeling of hydrocarbon reservoirs and of aquifer-aquitard systems can be separated into two activities: geological modeling and fluid flow modeling. The geological model focuses on the geometry and the dimensions of the subsurface layers and faults, and on its rock types. The fluid flow model focuses on quantities like pressure, flux and dissipation, which are related to each other by rock parameters like permeability, storage coefficient, porosity and capillary pressure. The absolute permeability, which is the relevant parameter for steady single-phase flow of a fluid with constant viscosity and density, is studied here. When trying to match the geological model with the fluid flow model, it generally turns out that the spatial scale of the fluid flow model is built from units that are at least a hundred times larger in volume than the units of the geological model. To counter this mismatch in scales, the fine-scale permeabilities of the geological data model have to be upscaled' to coarse-scale permeabilities that relate the spatially averaged pressure, flux and dissipation to each other. The upscaled permeabilities may be considered as complicated averages, which are derived from the spatially averaged flow quantities in such a way that the continuity equation, Darcy's law and the dissipation equation remain valid on the coarse scale. In this paper the theory of upscaling will be presented from a physical point of view aiming at understanding, rather than mathematical rigorousness. Under the simplifying assumption of spatial periodicity of the fine-scale permeability distributions, homogenization theory can be applied. However, even then the spatial distribution of the permeability is generally so intricate that exact solutions of the homogenized permeability cannot be found. Therefore, numerical approximation methods have to be applied. To be able to estimate the approximation error, two numerical methods have been developed: one based on the conventional nodal finite element method (CN-FEM) and the other based on the mixed-hybrid finite element method (MH-FEM). CN-FEM gives an upper bound for the sum of the diagonal components of the homogenized mobility matrix, while MH-FEM gives a lower bound. Three numerical examples are presented.     

Buoyancy Effects on Upward Brine Displacement Caused by CO2 Injection

Upward displacement of brine from deep reservoirs driven by pressure increases resulting from CO₂ injection for geologic carbon sequestration may occur through improperly sealed abandoned wells, through permeable faults, or through permeable channels between pinch-outs of shale formations. The concern about upward brine flow is that, upon intrusion into aquifers containing groundwater resources, the brine may degrade groundwater. Because both salinity and temperature increase with depth in sedimentary basins, upward displacement of brine involves lifting fluid that is saline but also warm into shallower regions that contain fresher, cooler water. We have carried out dynamic simulations using TOUGH2/EOS7 of upward displacement of warm, salty water into cooler, fresher aquifers in a highly idealized two-dimensional model consisting of a vertical conduit (representing a well or permeable fault) connecting a deep and a shallow reservoir. Our simulations show that for small pressure increases and/or high-salinity-gradient cases, brine is pushed up the conduit to a new static steady-state equilibrium. On the other hand, if the pressure rise is large enough that brine is pushed up the conduit and into the overlying upper aquifer, flow may be sustained if the dense brine is allowed to spread laterally. In this scenario, dense brine only contacts the lower-most region of the upper aquifer. In a hypothetical case in which strong cooling of the dense brine occurs in the upper reservoir, the brine becomes sufficiently dense that it flows back down into the deeper reservoir from where it came. The brine then heats again in the lower aquifer and moves back up the conduit to repeat the cycle. Parameter studies delineate steady-state (static) and oscillatory solutions and reveal the character and period of oscillatory solutions. Such oscillatory solutions are mostly a curiosity rather than an expected natural phenomenon because in nature the geothermal gradient prevents the cooling in the upper aquifer that occurs in the model. The expected effect of upward brine displacement is either establishment of a new hydrostatic equilibrium or sustained upward flux into the bottom-most region of the upper aquifer.     

Numerical solutions for the transient flow in the homogenous closed circle reservoirs

There are many fault block fields in China. A fault block field consists of fault pools. The small fault pools can be viewed as the closed circle reservoirs in some case. In order to know the pressure change of the developed formation and provide the formation data for developing the fault block fields reasonably, the transient flow should be researched. In this paper, we use the automatic mesh generation technology and the finite element method to solve the transient flow problem for the well located in the closed circle reservoir, especially for the well located in an arbitrary position in the closed circle reservoir. The pressure diffusion process is visualized and the well-location factor concept is first proposed in this paper. The typical curves of pressure vs time for the well with different well-location factors are presented. By comparing numerical results with the analytical solutions of the well located in the center of the closed circle reservoir, the numerical method is verified.     

Influence of Media Properties on Fluid Flow in Faulted and Fractured Anisotropic Media: A Study Using a Modified Flux-Vector-Based Green Element Method

This article illustrates the use of the modified ‘q-based’ GEM for complex geological problems in anisotropic media involving faults/fractures, by adopting various values of the permeability for both the faults/fractures and the media. The bulk permeabilities are compared and illustrated for different cases. Examples based on generating randomly positioned faults for which the number of faults can vary, but the sum of their lengths is fixed, are investigated. Situations are presented for a number of different total fault lengths and for various physical properties of the faults within a fixed total fault length. For some of the cases, particle traces are shown to better illustrate the behaviour of the flow.     

Emulsion flow through a porous medium with allowance for interphase mass transfer between the components

A new model of the flow of two miscible, mutually-insoluble fluids in a porous medium with the formation of an emulsion and adsorption of the fluid components on the skeleton is proposed. The model takes into account the effect of interphase mass transfer on the emulsion dynamics and the active porosity. A continuous general solution of the one-dimensional model and the problem of breakdown of a discontinuity is constructed. The flow regimes generated in displacement problems which depend on the shape of the adsorption isotherms and the densities of the fluid components are considered. The time dependence of the production rate is constructed for frontal displacement regimes and for displacement regimes with the formation of a zone of mixing (Riemann wave) of the initial reservoir and injected fluids. These functions coincide, at least qualitatively, with the experimental data [1] indicating an initial increase in production rate even against a background of falling reservoir pressure, transition through a maximum, and subsequent decline. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 77–88, January–February, 1997. The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 96-01-00991).     

Transient Pressure Behavior of Reservoirs with Discrete Conductive Faults and Fractures

Fractures and faults are common features of many well-known reservoirs. They create traps, serve as conduits to oil and gas migration, and can behave as barriers or baffles to fluid flow. Naturally fractured reservoirs consist of fractures in igneous, metamorphic, sedimentary rocks (matrix), and formations. In most sedimentary formations both fractures and matrix contribute to flow and storage, but in igneous and metamorphic rocks only fractures contribute to flow and storage, and the matrix has almost zero permeability and porosity. In this study, we present a mesh-free semianalytical solution for pressure transient behavior in a 2D infinite reservoir containing a network of discrete and/or connected finite- and infinite-conductivity fractures. The proposed solution methodology is based on an analytical-element method and thus can be easily extended to incorporate other reservoir features such as sealing or leaky faults, domains with altered petrophysical properties (for example, fluid permeability or reservoir porosity), and complicated reservoir boundaries. It is shown that the pressure behavior of discretely fractured reservoirs is considerably different from the well-known Warren and Root dual-porosity reservoir model behavior. The pressure behavior of discretely fractured reservoirs shows many different flow regimes depending on fracture distribution, its intensity and conductivity. In some cases, they also exhibit a dual-porosity reservoir model behavior.     

Three-dimensional surface water modelling using a mesh adaptive technique

Numerical simulation of open water flow in natural courses seems to be doomed to one- or two-dimensional numerical simulations. Investigations of flow hydrodynamics through the application of three-dimensional models actually have very few appearances in the literature. This paper discusses the development and the initial implementation of a general three-dimensional and time-dependent finite volume approach to simulate the hydrodynamics of surface water flow in rivers and lakes. The slightly modified Navier-Stokes equations, together with the continuity and the water depth equations, form the theoretical basis of the model. A body-fitted time-dependent co-ordinate system has been used in the solution process, in order to accommodate the commonly complex and irregular boundary and bathymetry of natural water courses. The proposed adaptive technique allows the mesh to follow the movement of the water boundaries, including the unsteady free-water surface. The primitive variable equations are written in conservative form in the Cartesian co-ordinate system, and the computational procedure is executed in the moveable curvilinear co-ordinate system. Special stabilizing techniques are introduced in order to eliminate the oscillating behaviour associated with the finite volume formulation. Also, a new and comprehensive approximation for the pressure forces at the faces of a control volume is presented. Finally, results of several tests demonstrate the performance of the finite volume approach coupled with the adaptive technique employed in the three-dimensional time-dependent mesh system.     

Numerical Simulation of Single-Phase and Multiphase Non-Darcy Flow in Porous and Fractured Reservoirs

A numerical method as well as a theoretical study of non-Darcy fluid flow through porous and fractured reservoirs is described. The non-Darcy behavior is handled in a three-dimensional, multiphase flow reservoir simulator, while the model formulation incorporates the Forchheimer equation for describing single-phase or multiphase non-Darcy flow and displacement. The non-Darcy flow through a fractured reservoir is handled using a general dual-continuum approach. The numerical scheme has been verified by comparing its results against those of analytical methods. Numerical solutions are used to obtain some insight into the physics of non-Darcy flow and displacement in reservoirs. In addition, several type curves are provided for well-test analyses of non-Darcy flow to demonstrate a methodology for modeling this type of flow in porous and fractured rocks, including flow in petroleum and geothermal reservoirs.     

Dual Mesh Method for Upscaling in Waterflood Simulation

Detailed geological models typically contain many more cells than can be accommodated by reservoir simulation due to computer time and memory constraints. However, recovery predictions performed on a coarser upscaled mesh are inevitably less accurate than those performed on the initial fine mesh. Recent studies have shown how to use both coarse and fine mesh information during waterflooding simulations. In this paper, we present an extension of the dual mesh method (Verdière and Guérillot, 1996) which simulates water flooding injection using both the coarse and the original fine mesh information. The pressure field is first calculated on the coarse mesh. This information is used to estimate the pressure field within each coarse cell and then phase saturations are updated on the fine mesh. This method avoids the most time consuming step of reservoir simulation, namely solving for the pressure field on the fine grid. A conventional finite difference IMPES scheme is used considering a two phase fluid with gravity and vertical wells. Two upscaling methodologies are used and compared for averaging the coarse grid properties: geometric average and the pressure solve method. A series of test cases show that the method provides predictions similar to those of full fine grid simulations but using less computer time.     

New hybrid Cartesian cut cell/enriched multipoint flux approximation approach for modelling and quantification of structural uncertainties in petroleum reservoirs

Efficient and profitable oil production is subject to make reliable predictions about reservoir performance. However, restricted knowledge about reservoir rock and fluid properties and its geometrical structure calls for history matching in which the reservoir model is calibrated to emulate the field observed history. Such an inverse problem yields multiple history‐matched models, which might result in different predictions of reservoir performance. Uncertainty quantification narrows down the model uncertainties and boosts the model reliability for the forecasts of future reservoir behaviour. Conventional approaches of uncertainty quantification ignore large‐scale uncertainties related to reservoir structure, while structural uncertainties can influence the reservoir forecasts more significantly compared with petrophysical uncertainty. Quantification of structural uncertainty has been usually considered impracticable because of the need for global regridding at each step of history matching process. To resolve this obstacle, we develop an efficient methodology based on Cartesian cut cell method that decouples the model from its representation onto the grid and allows uncertain structures to be varied as a part of history matching process. Reduced numerical accuracy due to cell degeneracies in the vicinity of geological structures is adequately compensated with an enhanced scheme of a class of locally conservative flux continuous methods (extended enriched multipoint flux approximation method or extended EMPFA). The robustness and consistency of the proposed hybrid Cartesian cut cell/extended EMPFA approach are demonstrated in terms of true representation of geological structures influence on flow behaviour. Significant improvements in the quality of reservoir recovery forecasts and reservoir volume estimation are presented for synthetic model of uncertain structures. Copyright © 2015 John Wiley & Sons, Ltd.     

深埋地下洞室断裂型岩爆机理的数值模拟

针对深部岩体中由断层、节理等不连续性结构面引发的岩爆地质灾害, 根据深埋地下隧洞中潜在发震断裂的分布特征和几何形态建立数值分析模型, 采用离散元单元法模拟存在刚性平直断裂的深部围岩的开挖响应, 并分别考察开挖接近并通过断裂附近时围岩应力状态的变化特征。通过探讨断裂的存在对围岩应力状态改变的作用机理, 揭示出断裂型岩爆是开挖面附近一定范围内存在的断裂构造在高应力作用下发生错动, 导致能量突然释放, 对围岩造成强烈冲击作用的结果, 基本与地震的断层“粘滑”机制相类似。     

An embedded approach for immiscible multi‐fluid problems

An embedded formulation for the simulation of immiscible multi‐fluid problems is proposed. The method is particularly designed for handling gas–liquid systems. Gas and liquid are modeled using the Eulerian and the Lagrangian formulation, respectively. The Lagrangian domain (liquid) moves on top of the fixed Eulerian mesh. The location of the material interface is exactly defined by the position of the boundary mesh of the Lagrangian domain. The individual fluid problems are solved in a partitioned fashion and are coupled using a Dirichlet–Neumann algorithm. Representation of the pressure discontinuity across the interface does not require any additional techniques being an intrinsic feature of the method. The proposed formulation is validated, and its potential applications are shown. Copyright © 2015 John Wiley & Sons, Ltd.     

缝洞型油藏三维离散缝洞数值试井模型

缝洞型碳酸盐岩油藏发育着大尺度的溶洞和裂缝,非均质性极强,缝洞型碳酸盐岩油藏问题的研究成为了世界级难题之一.由于大尺度溶洞和裂缝对储层的流体流动起主导作用,因此,基于连续介质理论的双重介质或三重介质模型已不适合其中流体流动的描述.根据大型缝洞分布地质特征,探索性地提出了一种板块组合的复合架构离散缝洞模型描述该类油藏中的流体流动,将三维空间大裂缝用板块描述,溶洞用高渗透率和高孔隙度不规则多面体团块描述.将裂缝面用二维三角形单元离散, 溶洞和基质用三维四面体离散, 利用三维混合单元有限元法对建立的不定常渗流模型进行求解,得到了三维渗流条件下的试井理论曲线及压力场分布.通过对试井理论曲线特征的分析, 获得了各敏感参数对试井曲线的影响规律.通过对1口井的实际测试资料解释结果的分析,并与实际地震反射资料及生产实际资料的对比,发现本文所建立的模型可以较好地反映裂缝和溶洞的地质动态状况,并与实际生产状况具有较好的一致性.这一结果说明了所建模型的正确性以及测试资料分析结果的可靠性.      

Equilibrium of a crack in a porous medium with injection of the filtering liquid

In connection with the exploitation of petroleum deposits, the article discusses the equilibrium of a porous medium with a crack under conditions of plane deformation, with the steady-state filtration of a liquid injected into the porous medium through a crack. It is assumed that the crack, which has initial zero dimensions, can become wider and longer with a rise in the pressure. The displacement of the sides of the crack is determined on the basis of the theory of elasticity, taking account of the deformation properties of a saturated porous medium. The stress and the displacement are expressed in terms of two analytical Muskhelishvili functions and the complex filtration potential. A change in the volume of the porous medium leads to a discontinuity of the displacements at the feed contour, and to distortion in the filtration region. For a circular stratum, the dimensions of the crack and the mass flow rate of the liquid are determined in the first approximation. The region of values of the pressure in which there exists a stable equilibrium state of the open crack and a steady-state flow of the liquid is found.     

Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows

In this work, we present a fast and parallel finite volume scheme on unstructured meshes applied to complex fluid flow. The mathematical model is based on a three-dimensional compressible low Mach two-phase flows model, combined with a linearised ‘artificial pressure’ law. This hyperbolic system of conservation laws allows an explicit scheme, improved by a block-based adaptive mesh refinement scheme. Following a previous one-dimensional work, the useful numerical density of entropy production is used as mesh refinement criterion. Moreover, the computational time is preserved using a local time-stepping method. Finally, we show through several test cases the efficiency of the present scheme on two- and three-dimensional dam-break problems over an obstacle.     

A computational framework for fluid–porous structure interaction with large structural deformation

We study the effect of poroelasticity on fluid–structure interaction. More precisely, we analyze the role of fluid flow through a deformable porous matrix in the energy dissipation behavior of a poroelastic structure. For this purpose, we develop and use a nonlinear poroelastic computational model and apply it to the fluid–structure interaction simulations. We discretize the problem by means of the finite element method for the spatial approximation and using finite differences in time. The numerical discretization leads to a system of non-linear equations that are solved by Newton’s method. We adopt a moving mesh algorithm, based on the Arbitrary Lagrangian–Eulerian method to handle large deformations of the structure. To reduce the computational cost, the coupled problem of free fluid, porous media flow and solid mechanics is split among its components and solved using a partitioned approach. Numerical results show that the flow through the porous matrix is responsible for generating a hysteresis loop in the stress versus displacement diagrams of the poroelastic structure. The sensitivity of this effect with respect to the parameters of the problem is also analyzed.