Factors that influence fluid flow in a single fracture

To investigate the influence of compression, Poisson effect and turbulence on the fluid flow process and the inversion for fracture surface geometries, we simulate two sets of fractures: one with a defined fracture height standard deviation σ constant and a varying autocorrelation length λ and another with a fixed λ and a changing σ. Under compression, the normal stress closes fractures with a large aperture and thus reduces the effective permeability. However, the Poisson effect, which is induced by the compression, has little influence on the fluid flow properties and does not affect the inversion for fracture height standard deviation or the autocorrelation length. When introducing turbulence, we observe a significant difference between the performance of the Navier–Stokes equation and the local cubic law; compared with the Navier–Stokes equation, the local cubic law overestimates the peak value of the breakthrough time curve and effective permeability, thereby underestimating the mean fracture aperture.     

基于各向异性岩石物理的缝隙流体因子AVAZ反演

裂缝型储层表现出较强的各向异性特征.缝隙中充填不同流体时,裂缝储层的地震响应特征也不相同.本文从各向异性岩石物理模型出发,引入可有效识别缝隙流体的指示因子,并研究缝隙充填流体类型、饱和度以及缝隙纵横比与流体因子的相互关系,进而分析不同流体充填时介质的地震响应特征,并基于AVAZ反演方法估测缝隙流体指示因子.首先对缝隙流体因子的敏感性进行了分析,讨论当缝隙充填不同流体时,缝隙流体因子值的变化特征,同时研究了不同流体类型充填时裂缝储层反射系数随方位角和入射角的变化特征.某工区测井数据和复杂裂缝模型应用表明,基于各向异性岩石物理的缝隙流体因子AVAZ反演方法合理、可靠,且具有良好的抗噪性,即当对合成地震记录添加信噪比不小于1/2的随机噪声时,利用AVAZ反演方法估测所得流体因子值与真实值仍然吻合较好.     

Marker-and-cell and Chorin finite difference modeling for fluid flow in a single fracture

It is important to set up a detailed dynamic model of the fluid flow through fractures for understanding many fluid processes in Earth sciences. Numerical simulation is a popular tool for exploring these processes. The objective of this study is to understand fluid flow in fractures. Contrary to the conventional macro-scale modeling approach, micro-scale simulation is carried out. The Navior-Stokes equation solver was developed by a staggered marker-and-cell and the Chorin pressure iterating finite difference approach. We analyze the effects of the Reynolds number and the frequency of pressure fluctuations on flow mainly through visualization. A significant result is that the effect of pressure fluctuation-induced fluid flow can be observed in a broader frequency range. The peak velocity shifts along the spatial axis depending upon the frequency of the pressure fluctuation. An effective frequency band of the pressure fluctuation was identified which dominates dynamic behavior of the flow. Another major finding is that there exits a critical frequency of the pressure fluctuation which controls approximately the flow dynamic behavior. We conclude that it is only possible to estimate the flow behavior from pressure fluctuation, if effective frequency range is properly accounted for.     

Damage evolution and fluid flow in poroelastic rock

We present a formulation for mechanical modeling of the interaction between fracture and fluid flow. Our model combines the classic Biot poroelastic theory and a damage rheology model. The model provides an internally consistent framework for simulating coupled evolution of fractures and fluid flow together with gradual transition from brittle fracture to cataclastic flow in high-porosity rocks. The theoretical analysis, based on thermodynamic principles, leads to a system of coupled kinetic equations for the evolution of damage and porosity. A significant advantage of the model is the ability to reproduce the entire yield curve, including positive and negative slopes, in high-porosity rocks by a unified formulation. A transition from positive to negative values in the yield curve, referred to as a yield cap, is determined by the competition between the two thermodynamic forces associated with damage and porosity evolution. Numerical simulations of triaxial compression tests reproduce the gradual transition from localized brittle failure to distributed cataclastic flow with increasing pressure in high-porosity rocks and fit well experimentally measured yield stress for Berea sandstone samples. We modified a widely used permeability porosity relation by accounting for the effect of damage intensity on the connectivity. The new damage-permeability relation, together with the coupled kinetics of damage and porosity evolution, reproduces a wide range of realistic features of rock behavior. We constrain the model variables by comparisons of the theoretical predictions with laboratory results reporting porosity and permeability variation in rock samples during isotropic and anisotropic loading. The new damage-porosity-permeability relation enables simulation of coupled evolution of fractures and fluid flow and provides a possible explanation for permeability measurements in high-porosity rocks, referred to as the “apparent permeability paradox.” The text was submitted by the authors in English.     

An exact solution for a constant-strength line-sink satisfying the modified Helmholtz equation for groundwater flow

We obtained an exact solution in terms of the discharge potential for a constant-strength line-sink that satisfies the modified Helmholtz equation for groundwater flow, for example for semi-confined flow and transient flow. The solution is obtained by integrating the potential for a point sink (well) along a straight line element. The potential for the point-sink is the modified Bessel function of the second kind and zero order K₀. Since K₀ cannot be integrated directly (in closed form) along a line-element, earlier solutions for a line-sink have been obtained by integrating polynomial approximations to K₀. These approximations, however, are only valid up to a certain distance from the well and consequently impose a limit on the length of the line-sink. In this paper we integrate an exact series representation for K₀ that is valid at any distance from the well, thus allowing integration along line-elements of any length, at least in theory. Numerical difficulties arise when evaluating our expressions at large distances from the line-sink, but these are shown to be of little consequence in practice. We made use of Wirtinger calculus to facilitate integration and also to allow us to arrive at exact expressions for the integrated flux over a poly-line and the total leakage over a domain. These properties are essential when using the solution in the context of the Analytic Element Method (AEM). We demonstrate our solution for the case of semi-confined flow (with leakage) and for the case of transient flow in the context of the Laplace Transform Analytic Element Method (LT-AEM).     

The effect of expansion ratio on the critical Reynolds number in single fracture flow with sudden expansion

Flow in a single fracture (SF) is an important research subject in groundwater hydrology, hydraulic engineering, radioactive nuclear waste repository and geotechnical engineering. An abruptly changing aperture is a unique type of SF. This study discusses the relation between the values of the critical Reynolds number (Re_c) for the onset of symmetry breaking of flow and the expansion ratio (E) of SF, which is defined as the ratio between the outlet (D) and inlet (d) apertures. This study also investigates the effect of inlet aperture d on Re_c for flow in an SF with abruptly changing apertures (SF‐ACA) using the finite volume method. Earlier numerical and experimental results showed that flow is symmetric in respect to the central plane of the SF‐ACA at small Reynolds number (Re) but becomes asymmetric when Re is sufficiently large. Our simulations show that the value of Re_c decreases with the increasing E, and the relationship between the logarithm of Re_c and E can be described accurately using either a quadratic polynomial function or a logarithmic function. However, the relationship of Re_c and d for a given E value is vague, and Re_c becomes even less sensitive to d when E increases. This study also reveals that the hydraulic gradient (J) and flow velocity (v) follow a super‐linear relationship that can be fitted almost perfectly by the Forchheimer equation. The inertial component (J_i) of J increases monotonically with Re, whereas the viscous component (J_v) of J decreases monotonically with Re. The Re value corresponding to equal inertial and viscous components of J (named as the transitional point Re) decreases when E increases, and such a transitional point Re should be closely related to the critical Reynolds number Re_c, although a rigorous theoretical proof is not yet available. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of two-phase fluid flow through compactible reservoirs

Fluid-flow simulators used in the oil industry model the movement of fluids through a porous reservoir rock. These simulators either ignore coupling between the flow and concurring deformation of the solid rock frame or take it into account approximately, in the so-called loose or staggered-in-time mode. In contrast to existing simulators, the one we describe here fully couples two-phase (oil and water) flow to subsurface deformation and simultaneously accounts for all relevant physical phenomena. As such, our flow simulator inherently links time-dependent fluid pressures, saturations, permeabilities and flow velocities to stresses in the whole subsurface. These stresses relate to strains through the non-linear theory of elasticity, allowing us to model time-lapse changes in seismic velocities and anisotropy. The velocity variations manifest themselves in time shifts and reflection amplitudes that are conventionally measured from 4D seismic data. Changes in anisotropy produce time-dependent shear-wave splitting that can be used for monitoring the horizontal stresses.     

The influence of local fluid flow and the microstructure on elastic and anelastic rock properties

The frequency dependent mechanism of local fluid flow was found to be the decisive absorption and dispersion mechanism in fluid containing sandstones. In the ultrasonic frequency range local fluid flow and grain surface effects control the behaviour of highly porous and highly permeable rock if a pore fluid is present. Both mechanisms depend less on macroscopic rock parameters like porosity and permeability than essentially on microscopic parameters like crack size, crack density and grain contact properties. To demonstrate directly the important influence of the microstructure on the rock elastic and anelastic properties the microstructure of a sandstone was artificially changed by thermal cracking. The cracked rock exhibits a clearly changed behaviour at low uniaxial as well as at high hydrostatic pressure despite small changes of porosity and permeability. Fluid effects increase due to cracking. The experimental results are explained by means of a rock, model and local fluid flow. These results emphasize that it is the microstructure which controls the elastic and anelastic rock behaviour, even at high hydrostatic pressure.     

裂缝诱导的TTI介质固液解耦反射系数方程及裂缝和流体参数反演方法

利用地震数据进行裂缝预测及流体识别对于裂缝性储层的勘探开发具有重要意义.针对单组旋转对称的倾斜裂缝诱导的TTI介质, 基于各向异性Gassmann方程, 推导了固液解耦纵波反射系数近似公式, 分析了不同条件下近似公式的精度.根据实际研究工区裂缝储层的特征, 选用高裂缝倾角近似公式, 通过奇异值分解从方位弹性阻抗中估算裂缝密度和裂缝方位, 进而分离和消除各向异性项, 建立流体等效体积模量及孔隙度贝叶斯反演预测方法.模型试算表明, 在信噪比大于2时, 流体等效体积模量、孔隙度和裂缝参数的估计较为可靠, 实际应用也验证了反演方法的有效性.

     

On the modified groundwater flow equation: analytical solution via iteration method

To take into account the variability of the medium through which the groundwater flow takes place, we presented the groundwater flow equation within a confined aquifer with prolate coordinates. The new equation is a perturbed singular equation. The perturbed parameters is introduced and can be used as accurately replicate the variability of the aquifer from one point to another. When the perturbed parameter tends to zero, we recover the Theis equation. We solved analytically and iteratively the new equation. We compared the obtained solution with experimental observed data together with existing solutions. The comparison shows that the modified equation predicts more accurately the physical problem than the existing model. Copyright © 2015 John Wiley & Sons, Ltd.     

On a generalized potential vorticity equation for quasi-geostrophic flow

Summary It is shown that the general nonlinear potential vorticity equation for viscous and conductive fluid in a rotating system can be expressed in terms of the geostrophic stream function for the horizontal velocity alone, provided that the motions are hydrostatic and quasi-geostrophic and the Richardson number is much larger than unity. The form of this equation is identical with that obtained from an asymptotic expansion for a small Rossby number.     

Computed tomography scan imaging of natural open fractures in a porous rock; geometry and fluid flow

Computed tomography scan imaging techniques have been used on core samples to investigate the effect of natural open fractures on reservoir flow in the Snøhvit Gas Condensate Field. Firstly, computed tomography (CT) scanning was used to describe the 3D geometrical properties of the fracture network including orientation and fracture density. Two types of fractures were observed: F1 fractures are short and stylolite related and F2 fractures are longer, cross-cutting the core and without any obvious relationship to stylolites. Secondly, monitoring of single and two phase flow experiments on samples containing these two types of natural open fractures was performed under 10 and 80 bar net confining pressure while using CT scanning. 1-phase miscible flooding experiment shows approximately 3 times higher flooding velocity in an open F2 fracture than in the matrix. 2-phase flooding by gas injection into a 100% water saturated core gave early gas breakthrough due to flow in the fracture and thereafter very little water production. The flow experiments showed that the presence of open fractures has a significant local effect on fluid flow even in a case with relatively high matrix porosity (200–300 mD). The sample containing F1 fractures showed a complex flow pattern influenced both by open fractures and stylolites. The CT scan data enables an exact representation of the fracture network in core scale simulation models and therefore improves the understanding of fracture influence on flow in a fractured porous medium. CT scanning of core samples provides an effective tool for integrating geology and fluid flow properties of a porous fractured medium.     

New methods for measuring the permeability of rock samples for a single-phase fluid

A complex of methodology, instrumentation, and software tools is worked out, which makes it possible to accomplish high-precision measurements of permeability for a single-phase fluid and its anisotropy within a wide range of values both under the normal conditions and high temperature and pressure.The transient method, modified with allowance for the dependence of the properties of a percolating gas on the parameters of its state, enables one to simultaneously determine, from the data of a single experiment, the permeability values for a sample of condensed fluid and the Klinkenberg constant, which characterizes the pore space of rock. This approach made it possible to substantially increase the accuracy of measurements, reducing their labor-intensiveness.The method of varying the flow shape makes it possible to simultaneously determine, in the course of a single experiment, the axial and radial components of the permeability tensor of anisotropic rocks with layered structure. The instrumentation and software implementing these new methodologies are developed and thoroughly tested.     

Effect of NAPL exposure on the wettability and two‐phase flow in a single rock fracture

Surface‐wetting properties are an important cause of changing the groundwater and two‐phase fluid flows. Various factors affecting the surface wettability were investigated in a parallel‐walled glass fracture with non‐aqueous phase liquid (NAPL) (gasoline, diesel, trichloroethylene, and creosote) wetted surfaces. First, the effect of the duration of NAPL exposure on wettability change was considered at pre‐wet fracture surfaces using the various NAPL species, and the result showed that the surface became hydrophobic after the exposure time of NAPL exceeded 2000 min. Second, the initial wetting state of the surface affected the timing when the wettability change begins as well as the extent of the wettability change in an NAPL‐wetted rock fractures. Under the dry condition, the wettability change was completed within a very short time of exposure to NAPL (~5 min), and then it finally reached the intermediate and weakly NAPL wetting (contact angle of 118°). Under the pre‐wet condition, a relatively long time of exposure (~5000 min) was needed to observe the obvious change of the surface wettability, which was changed up to strongly NAPL wetting (contact angle of 142°). Third, the wettability changed by NAPL exposure was stable and maintained for a long time, regardless of water flushing rate and temperature. Finally, the wettability change by the exposure of NAPL on parallel fracture surfaces was evaluated at various groundwater flow velocities. Result showed that groundwater flow velocity has an important impact upon measured contact angle. Although fracture surfaces were exposed to NAPL at the low groundwater flow velocity, the wettability was not changed from hydrophilic to hydrophobic when the contact time between NAPL and mineral surfaces was not sufficient owing to the pulse‐type movement of NAPL. This implies that the variation of exposure pattern due to groundwater flow on the wettability change can be an important factor affecting the wettability change of fracture surface and migration behaviour at natural fractured rock aquifers in case of NAPL spill. Copyright © 2015 John Wiley & Sons, Ltd.     

Some exact solutions for the flow of fluid through tubes with uniformly porous walls

Summary This paper considers an incompressible fluid flowing through a straight, circular tube whose walls are uniformly porous. The flow is steady and one dimensional. The loss of fluid through the wall is proportional to the mean static pressure in the tube. Several formulations of the wall shear stress are considered; these formulations were motivated by the results from Hamel's radial flow problem, boundary layer flows/and boundary layer suction profiles. For each of these formulations exact solutions for the mean axial velocity and the mean static pressure of the fluid are obtained. Sample results are plotted on graphs. For the constant wall shear stress problem, the theoretical solutions compare favorably with some experimental results.Notations A, B, D, E constant parameters - a, b constant parameters - Ai(z), Bi(z) Airy functions - Ai, Bi derivatives of Airy functions - k constant of proportionality betweenV andp - L length of pores - p,p mean static pressure - p ₀ static pressure outside the tube - p ₀ value ofp atx=0 - Q constant exponent - R inside radius of the tube - T wall shear stress - T ₀ shear parameter - t wall thickness - U free stream velocity - ,u mean axial velocity - u ₀ value ofu atx=0 - V,V mean seepage velocity through the wall - v ₀ mean seepage velocity - x,x axial distance along the tube - z transformed axial distance - z ₀ value ofz atx=0 - mean outflow angle through the wall - cos - density of the fluid - wall shear stress - dynamic viscosity of the fluid - over-bar dimensional terms - no bar nondimensional terms The National Center for Atmospheric Research is sponsored by the National Science Foundation     

Cross-section through the frontal Japan Trench subduction zone: Geochemical evidence for fluid flow and fluid–rock interaction from DSDP and ODP pore waters and sediments

Abstract Fluids and sediments from Deep Sea Drilling Project/Ocean Drilling Program Legs (56, 57, 87 and 186) along a transect extending from the subducting plate, across the midslope and upper slope of the Japan Trench forearc were analyzed for B and B isotopes in order to assess their composition and fluid–sediment interaction. At the reference Site 436 on the subducting plate, changes in B contents and B isotopes are controlled by the lithology and diagenesis only. The midslope Sites 440 and 584 showed stronger variations in the B geochemistry, which can be related to diagenesis and tectonic dewatering along faults. The strongest changes in the B geochemistry were observed on the upper slope Sites 1150 and 1151, where profound down‐hole freshening (chlorinities as low as ～310 mmol) coincides with a B enrichment (up to 9.3 × seawater concentration). The B isotope pore fluid profile of Site 1150 displayed a bimodal variation with depth, first increasing to values more positive than seawater, then shifting to lower signatures typical for deep‐seated fluids, whereas Site 1151 showed a constant B decrease with depth. Sites 1150 and 1151 sediments showed B increases with depth to values as high as ～164 p.p.m. and isotopic compositions ranging from ～+4 to ?9‰. A linear decrease in B_solid/B_fluid ratio, suggests that B geochemistry of the upper slope sites is controlled by fluid–rock interaction and deep‐seated fluid flow, whereas constant B_solid/B_fluid ratios were observed at the reference site on the incoming plate. This fluid overprint is probably caused by normal faults in the sediment cover which might be interconnected to deep thrusts in the underlying Cretaceous accreted wedge. This suggests that the erosive Japan Trench margin is characterized by back‐flux of deep‐seated, B‐enriched fluids into the ocean, which is facilitated by extensional normal faulting as a result of tectonic erosion and subsidence.     

Analytical solutions for flow to a well through a fault

We present explicit analytical solutions to problems of steady groundwater flow to a pumping well in an aquifer divided by an infinite, linear fault. The transmissivity of the aquifer is allowed to jump from one side of the fault to the other to model the juxtaposition of host rocks with different hydrologic properties caused by faulting. The fault itself is represented as a thin anisotropic inhomogeneity; this allows the fault to act as a combined conduit–barrier to groundwater flow, as is commonly described in the literature. We show that the properties of the fault may be represented exactly by two lumped parameters—fault resistance and fault conductance—and that the effects of the fault on flow in the adjacent aquifer is independent of the fault width. We consider the limiting cases of a purely leaky and a purely conductive fault where the fault domain may be replaced exactly by internal boundary conditions, and we investigate the effects of fault properties on the flow behavior in the adjacent aquifers. We demonstrate that inferring fault properties based on field observations of head in the aquifer is inherently difficult, even when the fault may be described by one of the two limiting cases. In particular, the effects of a leaky fault and a conductive fault on heads and discharges in the aquifer opposite the fault from the well, are shown to be identical in some cases.     

Reference crop evapotranspiration determined with a modified Makkink equation

A modified version of the Makkink equation is shown to be a suitable alternative for Penman's formula for the determination of the crop reference evapotranspiration E_ref that is used in the so-called crop factor approach. Makkink's equation requires solar radiation and temperature data only. Since 1987 the Royal Netherlands Meteorological Institute has calculated E_ref with the modified Makkink's formula on a routine base. Using solar radiation derived from geostationary satellites, remotely sensed estimates of crop reference evapotranspiration can be obtained. © 1998 John Wiley & Sons, Ltd.     

Characterization of a hydraulically induced crystalline‐rock fracture

Hydraulic fracturing has become an important technique for enhancing the permeability of hydrocarbon source rocks and increasing aquifer transmissivity in many hard rock environments where natural fractures are limited, yet little is known about the nature or behaviour of these hydraulically induced fractures as conduits to flow and transport. We propose that these fractures tend to be smooth based on observed hydraulic and transport behaviour. In this investigation a multi‐faceted approach was used to quantify the properties and characteristics of an isolated hydraulically induced fracture in crystalline rocks. Packers were used to isolate the fracture that is penetrated by two separate observation wells located approximately 33 m apart. A series of aquifer tests and an induced gradient tracer test were performed to better understand the nature of this fracture. Aquifer test results indicate that full recovery is slow because of the overall low permeability of the crystalline rocks. Drawdown tests indicate that the fracture has a transmissivity of 1–2 m²/day and a specific storage on the order of 2–9 × 10^?7/m. Analysis of a potassium–bromide tracer test break through curve shows classic Fickian behaviour with minimal tailing analogous to parallel plate flow. Virtually all of the tracer was recovered, and the breakthrough curve dilution indicates that the swept area is only about 11% of a radial flow field and the estimated aperture is ≤0.5 mm, which implies a narrow linear flow region. These outcomes suggest that transport within these hydraulically induced ‘smooth’ fractures in crystalline rocks is rapid with minimal mixing, small local velocity fluctuations and no apparent diffusion into the host rock or secondary fractures. Copyright © 2016 John Wiley & Sons, Ltd.