Effect of small parameters on the structure of the front formed by unstable viscous-fluid displacement from a Hele-Shaw cell

The process of displacement of a viscous fluid from a Hele-Shaw cell consisting of two plates separated by a small gap is investigated. The front formed when the fluid is displaced from the cell by another, lower-viscosity fluid is unstable. The lower-viscosity fluid breaks through the layer of displaced fluid and forms channels called viscous fingers. As a result, a mixing zone occupied by both displaced and displacing fluids is formed. The structure of the unstable displacement front is investigated when the surface tension forces have no effect on the shape of the fingers. This situation is realized when a water-glycerin mixture is rapidly displaced from the cell by water. Equations taking the inertial and viscous forces acting in the plane of the plates into account are obtained by averaging the Navier-Stokes equations over the cell gap. Using the equations obtained the stability of a plane displacement front traveling in the direction of its normal and the stability of the lateral surfaces of the viscous fingers is investigated when the fluid velocities are parallel to the interface. From the solution for stability of the transverse displacement front it follows that the viscous forces acting in the plane of the plates determine the finger width (when there is no surface tension). Instability also develops in the flow on the longitudinal fluid interface. In this case the destabilizing factor is the inertial forces. Under the action of this instability the fingers, in their turn, lose stability and disintegrate into viscous bubbles.     

Stability of vertical miscible displacements with developing density and viscosity gradients

Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement. If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable. A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.     

A linear stability analysis for immiscible displacements

A linear stability analysis has been performed for an immiscible displacement of a nonwetting phase by a wetting phase in a semi-infinite system of finite width and thickness. It is found that instabilities become a more severe problem as the capillary number based upon the characteristic thickness of the system increases. A displacement can always be stabilized, if the capillary number is sufficiently small. As the aspect ratio (ratio of thickness to width) decreases for a fixed capillary number, the potential for unstable behavior increases. All of this means that a displacement in the laboratory is likely to be more stable than a similar displacement in the field. Since the solution to the base state assumes the analytical solution of Yortsos and Fokas (1983), the effect of the mobility ratio could not be examined.     

Modelling fluid flow in cross-bedded sections

Beadpack experiments and numerical simulations have been carried out to study flow displacements, effluent profiles and streamline patterns for layered systems with flow not parallel to the layers. The effects of layer thickness, permeability contrast, angle of layer to flow direction, mobility ratio and flood rate have been examined. Each of these parameters influence the displacement profiles, and disperse the flood front. Such real effects must be considered when subsuming reservoir heterogeneities in average reservoir parameters in simulation studies, or interpreting core tests.     

High Resolution Imaging of Unstable,Forced Imbibition in Berea Sandstone

An experimental investigation is presented of immiscible, high-mobility ratio forced imbibition in a representative linear homogeneous sandstone. Water floods with mobility ratios (from 1 to 155) at various water injection rates were conducted. Fine-scale (order mm³) in situ water saturation history was collected via X-ray computed tomography (CT). Three-dimensional images were constructed of stable displacement and the initiation and growth of unstable water fingers. Interestingly, viscous fingers do not lead the displacement front by significant distances, counter to experience in miscible systems. In this homogeneous porous medium, both water (displacing phase) injection rate and oil (displaced phase) viscosity have an obvious effect on the stability of the water front. As the oil viscosity and displacement rate increase, the water front becomes less stable. In addition, the so-called shock mobility ratio, as computed from steady-state relative permeability, is found to be predictive regarding displacement front stability. When the shock mobility ratio is greater than 1, the displacement is always unstable. Steady-state relative permeability, however, is found to be a function of viscosity ratio for unstable displacements.     

裂缝气液重力置换的流动实验及仿真

通过可视化模拟实验对真实裂缝下气液重力置换现象进行了模拟,得到了其气液界面现象与常规平板裂缝的差异;通过对压力、置换量等数据的处理证实了气液重力置换的机理;根据实验的液体漏失流量数据提出了钻井过程中漏失、置换、溢流区间;最后根据k-ε双方程模型进行了气液重力置换流动的仿真计算,得到三维裂缝流道中都是钻井液下部侵入,呈现一个近似的直角三角形形状。本文对现场钻井工程方案设计及安全钻井具有指导意义。     

Dynamics of non-Newtonian fluid interfaces in a porous medium: Incompressible fluids

This paper concerns the applications of frontal advance theory to the dynamics of a moving flat interface in a porous medium, when both displacing and displaced fluids are of power law behaviour. The rheological effects of non-Newtonian behaviour of these fluids on the interface position and its velocity are numerically illustrated and discussed with regard to the practical implications in oil displacement mechanisms. The results obtained should be useful in finding an optimal policy of injection in order to control the dynamics of the moving interface in field projects of enhanced oil recovery floods.     

Unstable Displacement of Non-aqueous Phase Liquids with Surfactant and Polymer

In this paper, we study two-phase multicomponent displacement of two immiscible fluids in both homogeneous and heterogeneous porous media. In many applications such as enhanced oil recovery, fluid mixing and spreading can be detrimental to the efficacy of the process. Here, we show that when an initially immobile phase is being displaced by a finite-size slug of solvents (surfactant and polymer), viscous fingering significantly enhances mixing and spreading of solvents. These effects are similar to those caused by medium heterogeneity and lead to poor displacement efficiency. We first quantify the displacement efficiency subject to different mobility ratios, Peclet numbers, and levels of medium heterogeneity. We observe a non-monotonic behavior in displacement efficiency as a function of mobility ratio, indicating that although stable frontal interface is desirable, miscible viscous fingering on the rear interface will eventually disintegrate the solvents slugs and reduce the displacement efficiency. Then, we show that miscible viscous fingering developing on the rear interface of the chemical slug could be greatly suppressed when viscosity contrast is gradually decreased using exponential or linear functions, leading to 10% increase in displacement efficiency while using the same amount of chemicals. To elucidate this low displacement efficiency, we study the evolution of mixing, spreading, and interfacial length and show that while higher viscosity ratios are quite effective in mobilizing the initially immobile phase in 1D displacements, they are in fact detrimental in 2D unstable displacements since they enhance mixing and spreading of solvents.

     

Immiscible Displacement in the Interacting Capillary Bundle Model Part II. Applications of Model and Comparison of Interacting and Non-Interacting Capillary Bundle Models

In the first part of this work (Dong et al., Transport Porous Media, 59, 1–18, 2005), an interacting capillary bundle model was developed for analysing immiscible displacement processes in porous media. In this paper, the second part of the work, the model is applied to analyse the fluid dynamics of immiscible displacements. The analysis includes: (1) free spontaneous imbibition, (2) the effects of injection rate and oil–water viscosity ratio on the displacement interface profile, and (3) the effect of oil–water viscosity ratio on the relative permeability curves. Analysis of a non-interacting tube bundle model is also presented for comparison. Because pressure equilibration between the capillaries is stipulated in the interacting capillary model, it is able to reproduce the behaviour of immiscible displacement observed in porous media which cannot be modelled by using non-interacting tube bundle models.     

Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

We present an analytical formulation useful to interpret the key phenomena involved in non-Newtonian displacement in porous media and an analysis of the results obtained by considering the uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into account compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index n allows transformation of the latter equations via a self-similar variable; further transformation of the equations incorporating the conditions at the interface shows for n<1 the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure distributions; the latter are derived in closed form for any value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative influence of input parameters on model outputs is evaluated by means of associated Sobol indices, calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.     

Effect of vibration on the stability of a plane displacement front in a porous medium

The effect of linearly polarized vibration on the stability of a plane displacement front in a porous medium is studied. The problem of the stability of the motion of a plane displacement front traveling at a constant velocity U under the action of vibration normal to the front is considered. It is shown that under the action of vibration the dynamics of the plane displacement front can be described by the Mathieu equation with a dissipative term. Using the standard averaging method, in the case of high-frequency vibration it is revealed that vibration can only increase the stability of the system. It is found that the vibration stabilizes the plane displacement front with respect to part of the perturbation spectrum.     

Moving line crack accompanied with damage zone subject to remote tensile loading

In the 1920s, a closed-form solution of the moving Griffith crack was first obtained by Yoffe. Based on Yoffe's solution, the Dugdale model for the moving crack case gives a good result. However, the Dugdale model fails when the crack speed is closed to the Rayleigh wave speed because of the discontinuity occurred in the crack opening displacement (COD). The problem is solved in this paper by introducing a restraining stress zone ahead of the crack tip and two velocity functions. The restraining stresses are linearly distributed and related to the velocity of the moving crack. An analytical solution of the problem is obtained by use of the superposition principle and a complex function method. The final result of the COD is continuous while the crack moves at a Rayleigh wave speed. The characteristics of the strain energy density (SED) and numerical results are discussed, and conclusions are given.     

An adaptive artificial viscosity for the displacement shallow water wave equation

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom,the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity(AAV) is proposed and combined with the displacement shallow water wave equation(DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effec...     

New phenomena concerning a screw dislocation interacting with two imperfect interfaces

Dislocation mobility and stability in nanocrystals and electronic materials are influenced by the material composition and interface conditions. Its mobility and stability then affect the mechanical behaviors of the composites. In this paper, we first address, in detail, the problem of a screw dislocation located in an annular coating layer which is imperfectly bonded to the inner circular inhomogeneity and to the outer unbounded matrix. Both the inhomogeneity-coating interface and coating-matrix interface are modeled by a linear spring with vanishing thickness to account for the possible damage occurring on the interface. An analytic solution in series form is derived by means of complex variable method, with all the unknown constants being determined explicitly. The solution is then applied to the study of the dislocation mobility and stability due to its interaction with the two imperfect interfaces. The most interesting finding is that when the middle coating layer is more compliant than both the inner inhomogeneity and the outer unbounded matrix and when the interface rigidity parameters for the two imperfect interfaces are greater than certain values, one stable and two unstable equilibrium positions can exist for the dislocation. Furthermore, under certain conditions an equilibrium position, which can be either stable or unstable (i.e., a saddle point), can exist, which has never been observed in previous studies. Results for a screw dislocation interacting with two parallel straight imperfect interfaces are also presented as the limiting case where the radius of the inner inhomogeneity approaches infinity while the thickness of the coating layer is fixed.     

Experimental displacement patterns in a 2 × 2 quadrant block with permeability and wettability heterogeneities—problems for numerical modelling

The effect of heterogeneities on miscible and immiscible flood displacements in 2D bead packs in quadrant form, 2 × 2 block heterogeneity, with either a permeability or a wettability contrast is the subject of this paper. The physical processes occurring during miscible and immiscible flow and displacement within permeability and wettability quadrant bead pack models have been studied experimentally. This geometry occurs in a number of situations relevant to hydrocarbon production: particularly faults where adjacent rocks have large permeability contrasts with rapid changes, in the laboratory with core butting, in reservoir simulation where grid blocks have different permeability and in reservoirs having near-wellbore damage problems. The model quadrants 1–4, had 1 and 4 and 2 and 3 with identical properties, either in permeability or wettability. Reported are complete unit mobility miscible displacements, then the effects of viscosity differences (mobility modifiers) and finally immiscible displacements on displacement patterns for initial linear injection. The experiments demonstrate that nodal flow occurs for both miscible and immiscible flow, but for immiscible flow there are boundary effects due to capillary pressure differences created by water saturation changes or wettability contrasts which can leave patches of isolated fluid within a quadrant. The displacement patterns for the different models and fluids change significantly with the viscosity and wettability changes, particularly for the immiscible displacements. This is due to the changing capillary pressure between the quadrant blocks as the water saturation change. These are difficult to address in numerical modelling but should be accounted for. Other effects include coupling of all physical processes governing the flow through the node and creations of microzones of trapped residual oil. Our displacement patterns can therefore be a valuable verification benchmark tool for numerical modelling and a calibration data source for those wishing to simulate the effects of capillary pressure under differing wettability conditions and for those investigating upscaling modelling procedures. However, the possible loss of physical reality when averaging must always be considered.     

Gas-gas displacement in a fractured porous reservoir

The results of an experimental investigation of the process of miscible displacement of a gas from a fractured porous reservoir by another gas are presented. It is established that the displacement process is influenced by convective and molecular diffusion. The displacement coefficient depends both on the reservoir properties of the formation and on the technological displacement parameters. The experiments revealed a decrease in the displacement coefficient with increase in the permeability ratio of the joints and the blocks and with decrease in formation pressure. The dependence of the displacement coefficient on the average gas velocity in the reservoir, i.e., the ratio of the gas flow rate to the product of the flow area and the total reservoir porosity coefficient (including the blocks and the joints), is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 74–79, May–June, 1991.     

Modelling the Effect of Surface Tension on Immiscible Displacement in a Saturated Porous Medium

Comments are made about the model employed by Chen and Vafai for forced convection in a porous medium channel, with a surface tension effect at the moving interface between two fluids, when one fluid is displaced by the other. A simple situation is analysed, and the circumstances under which surface tension effects are important in this case are clarified.     

Numerical simulation of the displacement of oil by water

If the mobility of a displacing fluid is greater than the mobility of the displaced fluid, the displacement is unstable (see, for example, [1–3]), and the originally plane displacement front is broken up into irregular tongues. It follows from the linear analysis of stability that initially the amplitude of the perturbation increases exponentially, and according to [1] the extended tongues that are formed move with constant velocity relative to the displaced fluid. The intermediate stages in the development of the instability, like questions relating to a more precise formulation of the problem (which involves giving up the piston displacement approximation) remain unstudied. A natural approach to their study is through numerical simulation, which was realized for the first time in [4, 5]. Some of the results of such an investigation are presented in the present paper. In contrast to [4], the main attention is devoted to the development of regular perturbations. It is shown that for the investigated mobility ratios the development of the perturbations follows the linear theory unexpectedly long, and then arrives at a stationary asymptotic regime. We also investigate the influence of the loss of displacement stability on waterless oil extraction in the case of displacement in homogeneous and inhomogeneous strata.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 58–63, September–October, 1979.We thank L. A. Chudov for advice and discussions.     

自动位移计在树坪滑坡中的应用

用自动位移计对长江干流的树坪滑坡进行了近一年的连续性观测,数据结果显示树坪滑坡体变形的控制因素为降雨和库水波动,滑坡目前变形速率较小,不会失稳。自动位移计通过连续记录数据,对树坪滑坡体的变形特征给予了准确的反映,且数据采集方便,值得在三峡库区边坡监测中推广应用。