Effect of shear-induced contact area and aperture variations on nonlinear flow behaviors in fractal rock fractures

This study experimentally analyzes the nonlinear flow characteristics and channelization of fluid through rough-walled fractures during the shear process using a shear-flow-visualization apparatus.A series of fluid flow and visualization tests is performed on four transparent fracture specimens with various shear displacements of 1 mm,3 mm,5 mm,7 mm and 10 mm under a normal stress of 0.5 MPa.Four granite fractures with different roughnesses are selected and quantified using variogram fractal dimensions.The obtained results show that the critical Reynolds number tends to increase with increasing shear displacement but decrease with increasing roughness of fracture surface.The flow paths are more tortuous at the beginning of shear because of the wide distribution of small contact spots.As the shear displacement continues to increase,preferential flow paths are more distinctly observed due to the decrease in the number of contact spots caused by shear dilation;yet the area of single contacts in-creases.Based on the experimental results,an empirical mathematical equation is proposed to quantify the critical Reynolds number using the contact area ratio and fractal dimension.

Nonlinear Flow Properties of Newtonian Fluids through Rough Crossed Fractures

The nonlinear flow properties of Newtonian fluids through crossed fractures are estimated by considering the influences of length,aperture,and surface roughness of fractures.A total of 252 computational runs are performed by creating 36 computational domains,in which the Navier-Stokes equations are solved.The results show that the nonlinear relationship between flow rate and hydraulic gradient follows Forchheimer’s law–based equation.When the hydraulic gradient is small(i.e.,10^(−6)),the streamlines are parallel to the fracture walls,indicating a linear streamline distribution.When the hydraulic gradient is large(i.e.,10^(0)),the streamlines are disturbed by a certain number of eddies,indicating a nonlinear streamline distribution.The patterns of eddy distributions depend on the length,aperture,and surface roughness of fractures.With the increment of hydraulic gradient from 10^(−6) to 10^(0),the ratio of flow rate to hydraulic gradient holds constants and then decreases slightly and finally decreases robustly.The fluid flow experiences a linear flow regime,a weakly nonlinear regime,and a strongly nonlinear regime,respectively.The critical hydraulic gradient ranges from 3.27×10^(−5) to 5.82×10^(−2) when fracture length=20–100mmandmechanical aperture=1–5mm.The joint roughness coefficient plays a negligible role in the variations in critical hydraulic gradient compared with fracture length and/or mechanical aperture.The critical hydraulic gradient decreases with increasing mechanical aperture,following power-law relationships.The parameters in the functions are associated with fracture length.

The internal circulations on internal mass transfer rate of a single drop in nonlinear uniaxial extensional flow

The internal flow of a droplet in the nonlinear extensional flow field will exhibit more than two internal circulations with the variation of nonlinear intensity(E).In this paper,the effect of positions and sizes of internal circulations on internal mass transfer rate of a single spherical droplet in a nonlinear extensional flow field is studied and compared with that in a linear extensional flow field.The simulation results show that when E≥0,there are two symmetrical internal circulations in the droplet,which is the same with that in a linear extensional flow.The limit value of mass transfer rate Sh is 15,which is equal to that in a linear extensional flow,no matter how large E is.When E≤-3/7,the number of internal flow circulation of a droplet increase to four and the transfer rate Sh increases.When E=-1,the maximum internal transfer rate Sh equals 30 which is twice of that in a linear extensional flow.The generation of new flow circulations in droplets and the circulation positions will enhance mass transfer when E≤-3/7,which provides a new idea for enhancing the internal mass transfer rate of droplets.

One-dimensional large-strain consolidation of soft clay with non-Darcian flow and nonlinear compression and permeability of soil

Geometrical nonlinearity of the soft soil and the deviation of water flow in the soft clay from Darcy's law have been well recognized in practice. However, the theory of consolidation, which can account for both the geometrical nonlinearity and the non-Darcian flow, has not been reported so far. In this contribution, a model for the consolidation of soft clay which can allow for these two factors simultaneously is proposed. Utilizing the finite difference method, the numerical model for this problem is developed. With the numerical model, the effects of the geometrical nonlinearity and the non-Darcian flow on the consolidation of the soft soil are investigated. The results show that when the self-weight stress is calculated by the same method, the rate of the non-Darcian consolidation for the large-strain case is larger than that for the small-strain case, but the difference between them is limited. However, the difference between the consolidation rates caused by the non-Darcian and Darcian flows is significant. Therefore, when the geometrical nonlinearity of the soft clay is considered in calculating the consolidation settlement, due to the complexity of the large-strain assumption, the small-strain assumption can be used to replace it if the self-weight stress for the small-strain assumption is calculated by considering its sedimentation. However, due to the aforementioned large difference between the consolidation rates with consideration of the non-Darcian flow in soft clay or not, it is better to consider the non-Darcian flow law for both the small and large stain assumptions.

An overview on nonlinear porous flow in low permeability porous media

This paper gives an overview on nonlinear porous flow in low permeability porous media, reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows that, deviating from Darcy＇s linear law, the porous flow characteristics obey a nonlinear law in a low-permeability porous medium, and the viscosity of the porous flow fluid and the permeability values of water and oil are not constants. Based on these characters, a new porous flow model, which can better describe low permeability reservoir~ is established. This model can describe various patterns of porous flow, as Darcy＇s linear law does. All the parameters involved in the model, having definite physical meanings, can be obtained directly from the experiments.

Nonlinear flow numerical simulation of low-permeability reservoir

A nonlinear flow reservoir mathematical model was established based on the flow characteristic of low-permeability reservoir.The well-grid equations were deduced and the dimensionless permeability coefficient was introduced to describe the permeability variation of nonlinear flow.The nonlinear flow numerical simulation program was compiled based on black-oil model.A quarter of five-spot well unit was simulated to study the effect of nonlinear flow on the exploitation of low-permeability reservoir.The comprehensive comparison and analysis of the simulation results of Darcy flow,quasi-linear flow and nonlinear flow were provided.The dimensionless permeability coefficient distribution was gained to describe the nonlinear flow degree.The result shows that compared with the results of Darcy flow,when considering nonlinear flow,the oil production is low,and production decline is rapid.The fluid flow in reservoir consumes more driving energy,which reduces the water flooding efficiency.Darcy flow model overstates the reservoir flow capability,and quasi-linear flow model overstates the reservoir flow resistance.The flow ability of the formation near the well and artificial fracture is strong while the flow ability of the formation far away from the main streamline is weak.The nonlinear flow area is much larger than that of quasi-linear flow during the fluid flow in low-permeability reservoir.The water propelling speed of nonlinear flow is greatly slower than that of Darcy flow in the vertical direction of artificial fracture,and the nonlinear flow should be taken into account in the well pattern arrangement of low-permeability reservoir.

Nonlinear fluid flow through three-dimensional rough fracture networks:Insights from 3D-printing,CT-scanning,and high-resolution numerical simulations

Nonlinear flow behavior of fluids through three-dimensional(3D)discrete fracture networks(DFNs)considering effects of fracture number,surface roughness and fracture aperture was experimentally and numerically investigated.Three physical models of DFNs were 3D-printed and then computed tomography(CT)-scanned to obtain the specific geometry of fractures.The validity of numerically simulating the fluid flow through DFNs was verified via comparison with flow tests on the 3D-printed models.A parametric study was then implemented to establish quantitative relations between the coefficients/parameters in Forchheimer’s law and geometrical parameters.The results showed that the 3D-printing technique can well reproduce the geometry of single fractures with less precision when preparing complex fracture networks,numerical modeling precision of which can be improved via CT-scanning as evidenced by the well fitted results between fluid flow tests and numerical simulations using CT-scanned digital models.Streamlines in DFNs become increasingly tortuous as the fracture number and roughness increase,resulting in stronger inertial effects and greater curvatures of hydraulic pressure-low rate relations,which can be well characterized by the Forchheimer’s law.The critical hydraulic gradient for the onset of nonlinear flow decreases with the increasing aperture,fracture number and roughness,following a power function.The increases in fracture aperture and number provide more paths for fluid flow,increasing both the viscous and inertial permeabilities.The value of the inertial permeability is approximately four orders of magnitude greater than the viscous permeability,following a power function with an exponent a of 3,and a proportional coefficient b mathematically correlated with the geometrical parameters.

On Nonlinear Stability Theorems of 3D Quasi-geostrophic Flow

Nonlinear stability criteria for quasi-geostrophic zonally symmetric flow are improved by establishing an optimal Poincard inequality. The inequality is derived by a variational calculation considering the additional invariant of zonal momentum. When applied to the Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode stability criterion provided the channel meridional width is no greater than 0.8605... times its channel length （which is the geophysically relevant case）.

Law of nonlinear flow in saturated clays and radial consolidation

It was derived that micro-scale amount level of average pore radius of clay changed from 0.01 to 0.1 micron by an equivalent concept of flow in porous media. There is good agreement between the derived results and test ones. Results of experiments show that flow in micro-scale pore of saturated clays follows law of nonlinear flow. Theoretical analyses demonstrate that an interaction of solid-liquid interfaces varies inversely with permeability or porous radius. The interaction is an important reason why nonlinear flow in saturated clays occurs. An exact mathematical model was presented for nonlinear flow in micro-scale pore of saturated clays. Dimension and physical meanings of parameters of it are definite. A new law of nonlinear flow in saturated clays was established. It can describe characteristics of flow curve of the whole process of the nonlinear flow from low hydraulic gradient to high one. Darcy law is a special case of the new law. A math- ematical model was presented for consolidation of nonlinear flow in radius direction in saturated clays with constant rate based on the new law of nonlinear flow. Equations of average mass conservation and moving boundary, and formula of excess pore pressure distribution and average degree of consolidation for nonlinear flow in saturated clay were derived by using an idea of viscous boundary layer, a method of steady state in stead of transient state and a method of integral of an equation. Laws of excess pore pressure distribution and changes of average degree of consolidation with time were obtained. Re- sults show that velocity of moving boundary decreases because of the nonlinear flow in saturated clay. The results can provide geology engineering and geotechnical engineering of saturated clay with new scientific bases. Calculations of average degree of consolidation of the Darcy flow are a special case of that of the nonlinear flow.

Nonlinear Stability Analysis of the Zonal Flows at Middle and High Latitudes

An attempt has been made to apply Arnold type nonlinear stability criteria to the diagnostic study of the persistence (stability) or breakdown (instability) of the atmospheric flows. In the case of the blocking high, the cut-off low and the zonal flow, the relationships of the geostrophic stream function versus the potential vorticity of the observed atmosphere are analyzed, which indicates that Arnold second type nonlinear stability theorem is more relevant to the observed atmosphere than the first one. For both the stable and unstable zonal flows, Arnold second type nonlinear stability criteria are applied to the diagnosis. The primary results show that our analyses correspond well to the evolution of the atmospheric motions. The synoptically stable zonal flows satisfy Arnol′d second type nonlinear stability criteria; while the synoptically unstable ones violate the nonlinear stability criteria.

Assessments of the effects of various fracture surface morphology on single fracture flow: A review

Natural rock joint permeability deviates from the classic fluid flow governing equations due to the inher-ent fracture surface roughness(i.e.,contact points,spatial correlation,matching,varying aperture,iso-lated voids,infilling material,tortuosity and channellings)and engineering disturbance such as excavations.To improve the accuracy of fracture permeability evaluation,many efforts have been made in analytical,experimental,and numerical methods.This study reviews the modified mathematical gov-erning equations of fluid flow and classifies them based on different influencing factors,such as friction factor,aperture,tortuosity,inertia,and various in situ stress effects.Various experimental and simulation techniques for the coupled normal-and shear-stress flow problems were assessed,and their advantages and disadvantages were also analysed.Furthermore,different surface roughness descriptions and their impacts on mechanical and hydraulic behaviours were discussed,followed by the potential research directions for fracture flow problems.

Nonlinear Stability of Zonally Symmetric Quasi-geostrophic Flow

By using the conservation laws and the method of variational principle, an improved Arnol′d′s second nonlinear stability theorem for the two-dimensional multilayer quasi-geostrophic model in periodic channel is obtained.

EXACT SOLUTIONS FOR NONLINEAR TRANSIENT FLOW MODEL INCLUDING A QUADRATIC GRADIENT TERM

The models of the nonlinear radial flow for the infinite and finite reservoirs including a quadratic gradient term were presented. The exact solution was given in real space for flow equation including quadratic gradiet term for both constant-rate and constant pressure production cases in an infinite system by using generalized Weber transform. Analytical solutions for flow equation including quadratic gradient term were also obtained by using the Hankel transform for a finite circular reservoir case. Both closed and constant pressure outer boundary conditions are considered. Moreover, both constant rate and constant pressure inner boundary conditions are considered. The difference between the nonlinear pressure solution and linear pressure solution is analyzed. The difference may be reached about 8% in the long time. The effect of the quadratic gradient term in the large time well test is considered.

DEFORMING METRICS WITH POSITIVE CURVATURE BY A FULLY NONLINEAR FLOW

By studying a fully nonlinear flow deforming conformal metrics on a compact and connected manifold, we prove the long time existence and the exponential convergence of the solutions of the flow for any initial metric g0 with the Schouten tensor Ag0 ∈ Γk.

MATHEMATICAL MODEL OF TWO-PHASE FLUID NONLINEAR FLOW IN LOW-PERMEABILITY POROUS MEDIA WITH APPLICATIONS

A mathematical model of two-phase fluid nonlinear flow in the direction of normal of ellipse through low-permeability porous media was established according to a nonlinear flow law expressed in a continuous function with three parameters, a mass conservation law and a concept of turbulent ellipses. A solution to the model was obtained by using a finite difference method and an extrapolation method. Formulas of calculating development index not only before but also after water breaks through an oil well in the condition of two-phase fluid nonlinear flow in the media were derived. An example was discussed. Water saturation distribution was presented. The moving law of drainage front was found. Laws of change of pressure difference with time were recognized. Results show that there is much difference of water saturation distribution between nonlinear flow and linear flow; that drainage front by water moves faster, water breaks through sooner and the index gets worse because of the nonlinear flow; and that dimensionless pressure difference gets larger at the same dimensionless time and difficulty of oil development becomes bigger by the nonlinear flow. Thus, it is necessary that influence of nonlinear flow on development indexes of the oil fields be taken into account. The results provide water-flooding development of the oilfields with scientific basis.

A Nonlinear Sub-grid Scale Model for Compressible Turbulent Flow

The governing equations for large eddy simulation （LES） are obtained by filtering the Navier-Stokes （N-S） equations with standard （non-Favre filtering） spatial filter function. The filtered scale stress due to the standard filtering is then reconstructed by using the Taylor series expansion. The loss of information due to truncating the expansion up to the first derivative term is modeled by a dynamic nonlinear model （DNM）, which is free from any empirical constant and wall damping function. The DNM avoids the singularity of the model and shows good local stability. Unlike the conventional dynamic Smagorinsky model （DSM）, the DNM does not require the plane averaging and reduces the computational cost. The turbulent flow over a double ellipsoid for Reynolds number of 4.25 × 10^6 and Mach number of 8.02 is simulated numerically to validate the proposed approach. The results are compared with experiment data, as well as the data of Reynolds averaged numerical simulation （RANS）.

External mass transfer from/to a single sphere in a nonlinear uniaxial extensional creeping flow

This work systematically simulates the external mass transfer from/to a spherical drop and solid particle suspended in a nonlinear uniaxial extensional creeping flow.The mass transfer problem is governed by three dimensionless parameters:the viscosity ratio(λ),the Peclet number(Pe),and the nonlinear intensity of the flow(E).The existing mass transfer theory,valid for very large Peclet numbers only,is expanded,by numerical simulations,to include a much larger range of Peclet numbers(1≤Pe≤105).The simulation results show that the dimensionless mass transfer rate,expressed as the Sherwood number(5 h),agrees well with the theoretical results at the convection-dominated regime(Pe>103).Only when E>5/4,the simulated Sh for a solid sphere in the nonlinear uniaxial extensional flow is larger than theoretical results because the theory neglects the effect of the vortex formed outside the particle on the rate of mass transfer.Empirical correlations are proposed to predict the influence of the dimensionless governing parameters(λ,Pe,E)on the Sherwood number(Sh).The maximum deviations of all empirical correlations are less than 15%when compared to the numerical simulated results.

Theoretical Computation of Nonlinear System Equations of Heavier Pellets Movements for Two Phase Flow

This paper presents nonlinear ordinary differential equations (ODES) of the heavier pellets movement for two phase flow, which actually represent a system of equations. The usual methods of solution such as Runge -Kutta method and it's datum results are discussed. This paper solves ODES of general form using variable mesh-length, linearizing the nonlinear terms by finite analysis method, fuilding an iteration sequence, and amending the nonlinear terms by iteration . The conditions of convergent operation of iteration solution is checked. The movement orbit and velocity of the pellets are calculated. Analysis of research results and it's application examples are illustrated.

Transfer characteristics of dynamic biochemical signals in non-reversing pulsatile flows in a shallow Y-shaped microfluidic channel: signal filtering and nonlinear amplitude-frequency modulation

The transports of the dynamic biochemical signals in the non-reversing pulsatile flows in the mixing microchannel of a Y-shaped microfluidic device are ana- lyzed. The results show that the mixing micro-channel acts as a low-pass filter, and the biochemical signals are nonlinearly modulated by the pulsatile flows, which depend on the biochemical signal frequency, the flow signal frequency, and the biochemical signal transporting distance. It is concluded that, the transfer characteristics of the dynamic biochemical signals, which are transported in the time-varying flows, should be carefully considered for better loading biochemical signals on the cells cultured on the bottom of the microfluidic channel.

The Effects of Zonal Flow on Nonlinear Rossby Waves

In this paper, we using phase plane method have derived the stability criteria of linear and nonlinear Rossby waves under the conditions of semi-geostrophic approximation and have gotten the solutions and geostrophic vorticity of corresponding solitary Rossby waves. It is pointed out that the wave stability is connected with the distribution of zonal flow and when the zonal flow is different the solitary wave trough or ridge is formed.