Effect of Vertical Heterogeneity on Long-Term Migration of CO<Subscript>2</Subscript> in Saline Formations

For deep injection of CO₂ in thick saline formations, the movements of both the free gas phase and dissolved CO₂ are sensitive to variations in vertical permeability. A simple model for vertical heterogeneity was studied, consisting of a random distribution of horizontal impermeable barriers with a given overall volume fraction and distribution of lengths. Analytical results were obtained for the distribution of values for the permeability, and compared to numerical simulations of deep CO₂ injection and convection in heterogeneous formations, using multiple realizations for the permeability distribution. It is shown that for a formation of thickness H, the breakthrough times in two dimensions for deep injection scale as H ² for moderate injection rates. In comparison to heterogeneous shale distributions, a homogeneous medium with equivalent effective vertical permeability has a longer breakthrough time for deep injection, and a longer onset time for convection.     

Residual Trapping Beneath Impermeable Barriers During Buoyant Migration of \mathrm{CO}_2

The presence of impermeable barriers in a reservoir can significantly impede the buoyant migration of $\mathrm{CO}_2$ injected deep into a heterogeneous geological formation. An important consequence of the presence of these impermeable barriers in terms of the long-term storage of $\mathrm{CO}_2$ is the residual trapping that takes place beneath the barriers, which acts to both increase the storage potential of the reservoir and improve the storage security of the $\mathrm{CO}_2$ . Analytical results for the total amount of $\mathrm{CO}_2$ trapped in a reservoir with an uncorrelated random distribution of impermeable barriers are obtained for both two and three-dimensional cases. In two dimensions, it is shown that the total amount of $\mathrm{CO}_2$ contained in this fashion scales as $n^{5/4}$ , where $n$ is the number of barriers in the vertical direction. In three dimensions, the trapped amount scales as $n^c$ , where $5/4 \le c \le 2$ depending on the aspect ratio of the barriers. The analytical two-dimensional results are compared with results of detailed numerical simulations, and good agreement is observed.     

地下岩体的垂向渗透率测试及分析

本文介绍了在真实油井中使用阵列压力计测量油藏薄层垂向渗透率的实验方法。通过注水-停注并多次重复此过程使地层产生压差,并在井中不同深度放置压力计接收压力信号。为分析压力的变化,本文从渗流力学基本方程出发,对其无量纲化并采用Laplace变换及数值反变换技术,得到考虑垂向渗透率的地层压力分布表达式,利用这一表达式,对测试的压力资料进行解释。最后根据非线性回归理论对其中的4只压力计的压力资料进行分析,得到了各层垂向渗透率。     

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.     

Effective properties for flow calculations

In this paper we discuss the background to the problems of finding effective flow properties when moving from a detailed representation of reservoir geology to a coarse gridded model required for reservoir performance simulation. In so doing we synthesize the pictures of permeability and transmissibility and show how they may be used to capture the effects of the boundary conditions on the upscaling. These same concepts are applied to the renormalization method of calculating permeability, to show its promise as an accurate, yet fast method.     

Permeability Changes During the Evolution of a Geothermal Field Due to the Dissolution and Precipitation of Quartz

The changes in permeability and porosity associated with quartz deposition in an evolving geothermal reservoir are investigated. We review the processes associated with permeability changes in a geothermal reservoir and also review recent work on quartz solubility and deposition rates. Porosity and permeability changes are calculated for two reservoir models. The first is a generic model of a Taupo Volcanic Zone geothermal reservoir and the second is based on the model of the deep circulation system at Kakkonda published by Hanano. We find that when a reservoir experiences BPD conditions during part of its lifetime the lower reservoir becomes impermeable and a hydrothermal circulation system is only present in the upper part of the reservoir.     

Geometrical shapes of waterflooded zones in an inhomogeneous reservoir

The geometrical shapes of waterflooded zones are determined for an individual injection well in a powerful reservoir whose structure is vertically inhomogeneous. It is found that with decrease in the permeability from the top of the reservoir to its bottom the side spreading of water injected in the neighborhood of the bottom also decreases appreciably. In the cases of the power and exponential vertical variations in the permeability and voidage analytic expressions for the pressure distribution and the geometrical shapes of volumes are obtained.     

复杂边界非均质渗流场流线分布研究

建立了考虑源(汇)影响的含有不渗透区域复杂边界条件下非均质油藏稳定渗流的数学模型。利用扰动边界元方法求解数学模型,获得了地层中任意一点的压力公式.在此基础上,提出了流线场的生成方法。绘制了考虑非均质性、复杂边界和不渗透区域影响的流线分布图,并分析了流线分布的特征。通过分析表明,渗流场的非均质性和不渗透区域的存在都对流线分布存在较大的影响。利用本文方法产生的流线分布图能够较为直观地反映出油藏流体在注采井间的运动轨迹,为优化井网和注入方案提供了重要依据。     

Reservoir fluid flow through vertical fracture of thick stratum with underlying fluid interface

An exact solution is obtained for the problem of steady-state filtration of a heavy dense incompressible fluid in a thin, infinitely deep, inclined reservoir having a crack of given depth along the reservoir rise. The region of filtration of the lighter liquid (oil) has an impermeable upper boundary in the form of a horizontal fault line. Below the filtration region there is a free boundary, below which lies the region of stationary fluid (bottom water). The interface of the fluids, the fissure profile, and the reservoir fluid flow rate are determined from the solution of the problem on the basis of the given parameters (permeability of the reservoir and of the material filling the fissure, viscosity of the filtering fluid, specific weight of the upper and lower fluids, depth of the fissure, pressure differential between a point at the fissure and a point at the interface of the fluids). In the case when the thin reservoir is a vertical filtering layer, the considered flow is interpreted as the motion of the reservoir fluid through a vertical fissure of a thick reservoir (half-space) in the presence of an underlying fluid interface. The problem is solved in finite form with the aid of known analytic functions using integrals of the Cauchy type. The fundamental solution is first found of the special problem of flow with a point singularity. The fundamental solution is also of independent importance as an extension of the solution of certain known problems [1–4].     

尼泊尔波迪·科西水利枢纽三维渗流分析

针对尼泊尔波迪·科西水电工程首部枢纽布置和软土坝基的岩性条件和渗透特征 ,通过对混凝土防渗墙深度、灌浆帷幕长度、混凝土渗透性、地基渗透性的敏感性分析 ,确定出防渗墙和防渗帷幕的优化方案 ,并对三种库水位工况下的三维渗流场进行了详细的计算分析 ,为工程优化设计提供依据.     

Analytical three-dimensional renormalization for calculating effective permeabilities

The ability to calculate an effective permeability of a heterogeneous reservoir based on knowledge of its small-scale permeability is fundamental to practical numerical reservoir characterization. One elegant technique that forms the basis of this process is renormalization (King, P.R.: Transport Porous Med. 4, 37–58 (1989)). In two dimensions, renormalization can be implemented using a simple analytical formula. In three dimensions, however, no such analytical result exists, and renormalization must be performed using a numerical implementation. In this article, we present a simple analytical approximation to the method of renormalization in three dimensions. A detailed comparison with numerical results demonstrates its accuracy and highlights the significant reduction in computational cost achieved.     

Similarity solutions for buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces

In the present paper similarity solutions for the convective flow induced by buoyancy in a saturated porous medium adjacent to horizontal impermeable surfaces are obtained. The analysis incorporates the variation of permeability from the wall and expressions for boundary layer thickness, local and overall surface heat-flux are obtained. Applications of the results to convective flows in a geothermal reservoir are discussed.     

渗流方程自适应非均匀网格Dagan粗化算法

在粗网格内先统计渗透率在粗网格中的概率分布，利用Dagan渗透率粗化积分方程通过渗透率概率分布计算粗化网格的等效渗透率，并由等效渗透率计算了粗化网格的压强分布，计算压强时还将渗透率自适应网格技术应用于三维渗流方程的网格粗化算法中，在渗透率或孔隙度变化异常区域自动采用精细网格，用直接解法求解渗透率或孔隙度变化异常区域的压强分布。整个求解区采用不均匀网格粗化，在流体流速高的区域采用精细网格。利用本文方法计算了三维渗流方程的压强分布，结果表明这种算法的解在渗透率或孔隙度异常区的压强分布规律非常逼近精细网格的解，在其他区域压强分布规律非常逼近粗化算法的解，计算速度比采用精细网格提高了约100倍。     

低渗岩心驱替过程中的压力分布特征研究

考虑低渗岩心的应力敏感性,通过实验和数值计算,研究了一维水驱油过程中,压力的分布及变化规律,并对其影响因素进行了分析. 结果表明：压力分布由于受到应力敏感性的影响而显示出明显的非线性特征,在前缘面处存在一个明显的分界点;应力敏感性程度、黏度比对压力分布的影响较大.     

Convective Plume Paths in Anisotropic Porous Media

Free convection motions induced by point sources or horizontal line sources of heat are usually assumed to take the form of a vertically orientated plume. In this paper we consider how material anisotropy affects the path of the plume centreline and we show that it is strongly affected by both the anisotropy and the presence of impermeable bounding surfaces. The plume path is a straight line whose angle from the vertical is determined by a balance between the upward buoyancy force, the need for the plume to entrain equal amounts of fluid from the external regions either side of it, and the ease of fluid motion in the direction of the principal axis with the highest permeability.     

Field fluctuations in a heterogeneous elastic material—an information theory approach

Internal stress and strain fields in disordered elastic solids such as multiphase materials or polycrystals are considered. In order to derive a probability distribution for those random internal fields, the information theory entropy is maximized subject to constraints representing the basic equations of elasticity and certain experimental data. Thus one can find the probability distribution which agrees with all known facts but makes no assertions about the internal fields which cannot be supported by the available information. This approach is in accordance with the formal exact solution of the statistical problem if one has complete microstructural information. In case of incomplete microstructural data, useful approximate solutions can easily be obtained. In particular, the following set of data is sufficiently detailed for the prediction of internal field fluctuations: the average strain, the one-point probability density of the random elastic constants, and the effective (overall) elastic constants. Especially the information supplied by the effective elastic constants plays a major role since it reflects the microstructural topology of the heterogeneous material. One obtains Gaussian probability distributions for stress and strain, which are applied to calculate mean values and fluctuations of stresses in a cemented metal carbide and a zinc polycrystal.     

Predicting Vertical Flow Barriers Using Tracer Diffusion in Partially Saturated,Layered Porous Media

Sudden changes in isotopic tracer concentration in pore waters have been interpreted as indicating barriers to vertical advective flow through porous rocks in the subsurface, e.g. step changes in \(^{87}\hbox {Sr}/^{86}\) Sr ratio are often used in the oil and gas industry as a signature of reservoir compartmentalisation. This study shows that this is not necessarily the case. It can take millions of years for such step changes to equilibrate by diffusion if there is no flow resulting from pressure or density gradients even in high permeability, high porosity rocks, particularly if the water saturation is low. Changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers. In contrast changes in sorption without a change in porosity are almost impossible to identify. The time taken for concentration gradients to equilibrate is affected by the layer properties but can be quickly estimated from the harmonic average of the effective diffusion coefficient for each layer and a simple analytical expression for a homogeneous system. This was achieved by performing a sensitivity analysis on different layer properties (porosity contrast, saturation contrast, sorption contrast, thickness ratio) using existing analytical solutions for diffusion in layered systems.     

The use of standard deviation and skewness for estimating apparent permeability in a two-dimensional,heterogeneous medium

In this theoretical study, characteristic or effective permeabilities (referred to as ‘apparent permeability’) of a radial/parallel flow system in a heterogeneous medium are calculated by the Monte Carlo method and the finite element method. The permeability distribution in the radial and parallel flow systems are not the same, as going from Cartesian to cylindrical coordinates changes the probability measure. The Bernoulli trials, the normal distribution or the log-normal distribution, is assumed to be the probability density function of permeability. The results are summarized as follows: (1) when the skewness of the distribution function is equal to zero or nearly equal to zero (that is, when the permeability distribution is regarded to be symmetric), the apparent permeability depends on the standard deviation, but not on the kind of distribution function, (2) when the skewness is not equal to zero, the apparent permeability depends not only on the standard deviation, but also on the skewness, (3) the above facts appeared in the radial and the parallel flow systems.     

Effects of anisotropy on the free convection from a vertical plate embedded in a porous medium

The effects of anisotropy on the steady laminar boundary-layer free convection over a vertical impermeable surface are analysed by using the method of integral relations. If the permeability in the direction orthogonal to the plate is greater than the permeability along the plate, then there is an increase in the temperature field.     

An Experimental Study of CO<Subscript>2</Subscript> Exsolution and Relative Permeability Measurements During CO<Subscript>2</Subscript> Saturated Water Depressurization

Dissolution of CO₂ into brine is an important and favorable trapping mechanism for geologic storage of CO₂. There are scenarios, however, where dissolved CO₂ may migrate out of the storage reservoir. Under these conditions, CO₂ will exsolve from solution during depressurization of the brine, leading to the formation of separate phase CO₂. For example, a CO₂ sequestration system with a brine-permeable caprock may be favored to allow for pressure relief in the sequestration reservoir. In this case, CO₂-rich brine may be transported upwards along a pressure gradient caused by CO₂ injection. Here we conduct an experimental study of CO₂ exsolution to observe the behavior of exsolved gas under a wide range of depressurization. Exsolution experiments in highly permeable Berea sandstones and low permeability Mount Simon sandstones are presented. Using X-ray CT scanning, the evolution of gas phase CO₂ and its spatial distribution is observed. In addition, we measure relative permeability for exsolved CO₂ and water in sandstone rocks based on mass balances and continuous observation of the pressure drop across the core from 12.41 to 2.76 MPa. The results show that the minimum CO₂ saturation at which the exsolved CO₂ phase mobilization occurs is from 11.7 to 15.5%. Exsolved CO₂ is distributed uniformly in homogeneous rock samples with no statistical correlation between porosity and CO₂ saturation observed. No gravitational redistribution of exsolved CO₂ was observed after depressurization, even in the high permeability core. Significant differences exist between the exsolved CO₂ and water relative permeabilities, compared to relative permeabilities derived from steady-state drainage relative permeability measurements in the same cores. Specifically, very low CO₂ and water relative permeabilities are measured in the exsolution experiments, even when the CO₂ saturation is as high as 40%. The large relative permeability reduction in both the water and CO₂ phases is hypothesized to result from the presence of disconnected gas bubbles in this two-phase flow system. This feature is also thought to be favorable for storage security after CO₂ injection.