Two-dimensional Radial Flow Research Articles

Numerical Simulation of Proppant Transport in Transverse Fractures of Horizontal Wells

Proppant transport and distribution law in hydraulic fractures has important theoretical and field guidance significance for the optimization design of hydraulic fracturing schemes and accurate production prediction. Many studies aim to understand proppant transportation in complex fracture systems. Few studies, however, have addressed the flow path mechanism between the transverse fracture and horizontal well, which is often neglected in practical design. In this paper, a series of mathematical equations, including the rock elastic deformation equation, fracturing fluid continuity equation, fracturing fluid flow equation, and proppant continuity equation for the proppant transport, were established for the transverse fracture of a horizontal well, while the finite element method was used for the solution. Moreover, the two-dimensional radial flow was considered in the proppant transport modeling. The results show that proppant breakage, embedding, and particle migration are harmful to fracture conductivity. The proppant concentration and fracture wall roughness effect can slow down the proppant settling rate, but at the same time, it can also block the horizontal transportation of the proppant and shorten the effective proppant seam length. Increasing the fracturing fluid viscosity and construction displacement, reducing the proppant density and particle size, and adopting appropriate sanding procedures can all lead to better proppant placement and, thus, better fracturing and remodeling results. This paper can serve as a reference for the future study of proppant design for horizontal wells.

Transport Characteristics of Tailing Sand Particles under Slotted Tube Overlapped with Geotextile and Steel Mesh

Compared with a one-dimensional test, a two-dimensional test has the potential to study the transport characteristics of particles in tailings under different filtration materials. Due to the different test conditions and the complex test environment, the existing devices generally have limited measuring range or high accuracy. Thus, it is urgent to develop an advanced device to improve the ability of the transport characteristics of particles. In this study, a two-dimensional radial flow device is designed for analyzing the transport characteristics of particles which combine the water tank with adjustable pressure and the seepage body providing a tailing sand environment. An experimental system is built, and a seepage process is carried out to explore the transport characteristics of particles. The results indicate that when head difference remains stable, the mixture consisting of the tailing sand and water gradually transports to the vicinity of the slotted tube along the diameter direction. With an increase in head difference, the tailing sand particle size in the mixture shows a slow upward trend, which migrates in the tailing sand. And the proportion of tailing sand particles with different sizes varies under different head differences. Separation of the mixture consisting of tailing sand particles and water occurs near infiltration material, while the mixture has different transportation laws under different filtration materials. Under geotextile, most fine tailing sand particles which transport from the edge remain in geotextiles, causing an increase in the proportion of fine particles around the slotted tube. However, most fine particles pass through steel mesh, leading to a decrease in the proportion of fine particles.

An application of spectral linearisation method on the steady two-dimensional radial flow of Au-water and Ag-water nanofluids between two inclined plane walls

The present work purveys the steady two-dimensional viscous incompressible radial flow of Au-water and Ag-water nanofluids between plane walls which either converge or diverge in the presence of MHD effect. A uniform magnetic field is applied. The governing partial differential equations of the present physics and their appropriate boundary conditions are initially cast into the dimensionless forms to reduce into the ordinary differential equation. The resulting equation thus formed is then solved by adopting the spectral linearisation method (SLM) to get the accurate numerical solution. Solution errors and residual norms were analysed to elaborate the convergence and accuracy of the numerical solution. The present results are validated with favourable comparisons to previously published results due to the current investigations' unique cases. Parametric study of the governing parameters, namely the magnetic field strength parameter M, Reynolds number Re, angle of inclination α, and the volume fraction parameter ɸ on the non-dimensional velocity is conducted. The analysis discloses that the profiles of the flow are largely impacted by the physical parameters.

An application of spectral linearisation method on the steady two-dimensional radial flow of Au-water and Ag-water nanofluids between two inclined plane walls

An application of spectral linearisation method on the steady two-dimensional radial flow of Au-water and Ag-water nanofluids between two inclined plane walls

The Slotted Tube Overlay Geotextile in Tailing Pond Radial Seepage Test Model and Analysis of the Permeation Characteristics

Slotted tube drainage system plays an important role in reducing the wetting line of tailing dam. In the past studies, the permeability coefficient of geotextile was measured separately based on one-dimensional test, and the permeability characteristics of the whole system of the slotted tube overlay geotextile were rarely studied. Firstly, a set of radial flow test equipment for infiltration system of the slotted tube overlay geotextile is developed in this study. Then, with the aid of the test device, a two-dimensional radial flow test is conducted to investigate the drainage system of the slotted tube overlay geotextile before and after blocking by tailing sands, with considering different water head difference and the number of geotextile layers. By the two-dimensional radial flow test, it can be concluded that (1) the permeability coefficients of the slotted tube overlay geotextile before and after blocking all have a slight upward trend with the increase of water head difference; (2) with increasing the number of geotextile layers, the permeability coefficients before and after blocking increase first and then decrease, which approaching its minimum value at about two layers; (3) under the seepage pressure, some particles might attach to or stay in geotextile. Therefore, the permeability coefficient of the slotted tube overlay geotextile after blocking is significantly smaller than that before blocking. This study offers the insight to the further research on the permeation characteristics of the slotted tube overlay geotextile system.

Generating one-column grids with fractal flow dimension

The grid generation capability built into the numerical simulator TOUGH for multi-phase fluid and heat flow through geologic media can create one-column grids with linear or radial geometry, corresponding to one-dimensional or two-dimensional radial flow, respectively. The integral-finite-difference-method that TOUGH employs for spatial discretization makes it very simple to generalize the grid-generation algorithm from integer to non-integer (fractal) flow dimension. Here the grid-generation algorithm is generalized to create one-column grids with fractal flow dimension ranging from less than 1 to 3. The fractal grid generation method is verified by comparing numerical simulation results to an analytical solution for a generalized Theis solution for integer and non-integer flow dimensions between 0.4 and 3. It is then applied to examine gas production decline curves from hydraulically fractured shale that is modeled as a fractal-dimensioned fracture network with flow dimensions between 0.25 and 3. Grids with fractal flow dimension are useful for representing flow through fracture networks or highly heterogeneous geologic media with fractal geometry, and may be particularly useful for inverse methods.

An analytical model for flow rectification of a microdiffuser driven by an oscillating source

In this study, we employ the similarity transformation and the AC-circuit analogy to investigate the flow rectification property of a microdiffuser. Driven by an oscillating source, the transient two-dimensional radial flow of viscous fluid in a planar microdiffuser, which consists in a wedge-shaped domain bounded by two circular arcs, is solved. The ratio of flow impedances is determined to evaluate the performance of the flow rectification. By gauging the leverage between viscous and unsteady effects, we discuss the influences of the half angle, the Reynolds number, the excitation frequency, and the ratio of inlet to outlet radii in flow directing properties of the microdiffuser. We find that net flow in the expansion direction only occurs when the product of the half angle and the Reynolds number is small and the Womersley number is low. Moreover, the reversal Womersley number, at which preference of flowing direction switches, decreases with increasing the product of half angle and the Reynolds number or reducing the ratio of inlet to outlet radii. We find that a wider half angle, stronger oscillation, higher frequency, broader inlet, and longer microdiffuser all contribute to the augmentation of the transient inertia and lead to the promotion of converging flow. The specific goal of this study is to provide the means to manipulate the flow rectification effect of a microdiffuser by varying its geometry or the driving conditions.

Velocity Dependent Mobility Calculation Implemented in Finite Element Reservoir Simulator

We implement a novel up-winding scheme for the mobility calculation using the computed velocities in an adaptive finite element (FE) unstructured-mesh reservoir simulator. In the finite-element finite-volume (FEFV) numerical method, the pressure and transport equations are decoupled. The pressure is calculated using finite elements, and the saturation is calculated using finite volumes. Each element is shared between several control volumes -- three for triangles (2D) and four for tetrahedral (3D). Consequently, the saturations used in calculating the mobilities hence updating pressure - are unclear. Some researchers use the average value between the elemental control volumes, or the integration points of the finite elements. For two-dimensional radial flow, this does not produce accurate saturations profiles when compared to the Buckley-Leverett reference solution. In this paper, we present a new formulation to calculate the FE mobility. We use the velocity vector, which is piece-wise constant in first order elements, to find the upstream saturation—where the tail of velocity vector intersects an element. This novel approach produces more accurate saturation profiles than previous methods even with higher order methods. Then, we present some benchmark simulation results where we model vertical spontaneous imbibition driven by capillarity and gravity disequilibrium between a fracture network at the bottom of the simulation domain and the matrix. The results compare favourably with semi-analytical treatments of this problem and experimental measurements. The method presented better models multi-phase displacements in complex reservoirs using FEFV. It can, also, be easily implemented in current FEFV based simulators.

Dynamics of a vortex pair in radial flow

The problem of vortex pair motion in two-dimensional radial flow is solved. Under certain conditions for flow parameters, the vortex pair can reverse its motion within a bounded region. The vortex-pair translational velocity decreases or increases after passing through the source/sink region, depending on whether the flow is diverging or converging, respectively. The rotational motion of a corotating vortex pair in a quiescent environment transforms into motion along a logarithmic spiral in radial flow. The problem may have applications in astrophysics and geophysics.

Development of a Field Design for In Situ Gaseous Treatment of Sediment Based on Laboratory Column Test Data

A testing methodology is presented that supports the development of a field design for in situ gaseous treatment of sediments with diluted hydrogen sulfide. This approach involves the collection of column breakthrough test results at various flow rates, allowing a relationship to be developed between pore velocity of the carrier gas and velocity of the hydrogen sulfide reaction front that permits sizing to the field scale. A regression fit of a set of laboratory column breakthrough test data collected in this study is utilized to illustrate the development of a field design based on a two-dimensional radial flow analytical model. Information regarding treatment time and hydrogen sulfide consumption characteristics associated with in situ gaseous treatment can then be obtained from this model and used as a basis for estimation of treatment schedule and costs. The regression relationship can also be utilized in numerical models in more complex geometries to support the field design of in situ gaseous treatment operations.

Estimation of the sustainable yields of boreholes in fractured rock formations

The simplest way to derive an estimate for the sustainable yield of a borehole is to study the behaviour of drawdowns observed during a hydraulic (also known as a pumping test) of the borehole, through an appropriate conceptual model. The choice of this model is probably the most difficult choice that the analyst of such a hydraulic test has to make, since a wrong model can only lead to the wrong conclusions and failure of the borehole. This paper discusses a semi-analytical and two numerical methods that can be used to simplify the analyses of hydraulic tests in fractured rock formations. The first method, called the Method of Derivative Fitting (MDF), uses a new approach to identify the conceptual model needed in such analyses. This is achieved by characterizing the various flow periods in fractured rock aquifers with numerical approximations of the first logarithmic derivative of the observed drawdown (the derivative of the drawdown with respect to the logarithm of the time). Semi-analytical expressions are used to estimate the influence that boundaries may have on the observed drawdown and the sustainable yield of a borehole — the rate at which a borehole can be pumped without lowering the water level below a prescribed limit. An effort has also been made to quantify errors in the estimates introduced by uncertainties in the parameters, such as the transmissivity and storativity, through a Gaussian error propagation analysis. These approximations and the MDF, called the Flow Characteristics Method (FCM) have been implemented in a user-friendly EXCEL notebook, and used to estimate the sustainable yield of a borehole on the Campus Test Site at the University of the Orange Free State. The first numerical method, a two-dimensional radial flow model, is included here because it allows the user more freedom than the FCM, although it requires more information. One particular advantage of the method is that it allows one to obtain realistic estimates of the storativity and transmissivity of Karoo aquifers in particular, which is required in the estimation of the sustainable yield of a borehole. There is no doubt that a three-dimensional numerical model, the second numerical method discussed here, is the best method with which to analyse a hydraulic test in a fractured aquifer. The method was consequently used to evaluate the accuracy of the implementation of the MDF in the Excel notebook and its application to the borehole on the Campus Test Site. The good agreement between the sustainable yield estimated with the three-dimensional numerical model and the FCM indicates that the FCM can be used with confidence to estimate the sustainable yields of boreholes in fractured media.

Radial flow permeability measurement. Part B: Application

This paper deals with permeability measurement in the context of Resin Transfer Moulding (RTM). A new approach to two-dimensional radial flow permeability measurement with constant inlet pressure is proposed in part A of this paper. In this second part experimental studies are performed to validate the new approach. The new approach is shown to accurately predict the orientation of the tensor axes. It was demonstrated how the convergence chart can be used to check whether the experimental results follow the underlying assumptions of the model. Examples are given of acceptable and unacceptable measurements. Experimental design is introduced to investigate the variation of measured permeability between individual experimental runs. It was concluded that most of the observed variation was the result of flow front measurement sensors used in the experiment. Finally the new approach is compared with current methods. Differences are discussed. It is shown that the new approach extends the capabilities of current methods.

Radial flow permeability measurement. Part A: Theory

This paper deals with permeability measurement in the context of Resin Transfer Moulding (RTM). A new approach to two-dimensional radial flow permeability measurement with constant inlet pressure is proposed. It allows principal permeability to be measured even if the experimental axes are not aligned with the principal direction. This part of the paper looks at the underlying theory of the new approach while part B reports on validation experiments. Formulae are derived which allow to calculate principal permeability and the orientation of the principal axes from flow front measurements in three directions. Transient effects of the developing flow front caused by the circular inlet are discussed and its influence on the measured permeability is illustrated. Numerical studies are performed which show that the shape of the flow front is dependent only on the size of the inlet diameter and the degree of anisotropy. This leads to the development of a formula for estimating the minimum required mould size for permeability measurement.

Theoretical analysis of inertially irrotational and solenoidal flow in two-dimensional radial-flow pump and turbine impellers with equiangular blades

Using the theory of functions of a complex variable, in particular the method of conformal mapping, the irrotational and solenoidal flow in two-dimensional radialflow pump and turbine impellers fitted with equiangular blades is analysed. Exact solutions are given for the fluid velocity along straight radial pump and turbine impeller blades, while for logarithmic spiral pump impeller blades solutions are given which hold asymptotically as (r1/r2)n→0, in whichr1is impeller inner radius,r2is impeller outer radius andnis the number of blades. Both solutions are given in terms of a Fourier series, with the Fourier coefficients being given by the (Gauss) hypergeometric function and the beta function respectively. The solutions are used to derive analytical expressions for a number of parameters which are important for practical design of radial turbomachinery, and which reflect the two-dimensional nature of the flow field. Parameters include rotational slip of the flow leaving radial impellers, conditions to avoid reverse flow between impeller blades, and conditions for shockless flow at impeller entry, with the number of blades and blade curvature as variables. Furthermore, analytical extensions to classical one-dimensional Eulerian-based expressions for developed head of pumps and delivered work of turbines are given.

Shear layers in converging flow of fluid of non-uniform density and viscosity

We present an asymptotic study of steady two-dimensional radial flow between converging plane walls (Jeffery–Hamel flow) when the viscosity μ and density ρ vary with the angular coordinate θ. Two representative situations are considered, the first, being a two-layer system (in which μ and ρ are uniform except for discontinuities at an interface θ = θI), and the other involving a fluid for which μ and ρ vary continuously with θ. The flow is analysed in the asymptotic limit when a parameter c related to the wall pressure gradient is large; this corresponds to converging flow at large Reynolds number. Solutions are derived for the boundary layers at the walls and for the shear layer at the interface; the results are shown to agree well with some exact (numerical) profiles.The solutions obtained are not unique, though for given c they represent the ‘simplest’ type of profile, and the one that seems most likely to be stable. We demonstrate the non-uniqueness by deriving in §3 an alternative solution for the interfacial shear layer. This solution, however, can exist for only restricted ranges of values of the density and viscosity ratios, and involves an outgoing jet, suggesting that it is likely to be unstable.

Heat and mass transfer in a saturated porous wedge with impermeable boundaries

Heat and mass transfer in a two-dimensional radial flow of a viscous fluid through a saturated porous wedge-shaped region with confining walls is studied. Similarity transformations are used for the temperature, velocity and pressure, in order to reduce the describing systems to ordinary differential equations with two-point boundary conditions. Both exact and asymptotic solutions are obtained for the velocity and the temperature. It is found that two distinct solutions (jet flow type and slug flow type) exist for a given set of flow parameters. Specific results are presented for small wedge angles. It is shown that symmetric diverging solutions do not exist above a critical Rayleigh number. An application of the theory to the convection of liquid water in a crude model of a fault zone in a geothermally active area is presented.

A Sensitivity Study Of Cyclic Steam Stimulation Using A 2-D Mathematical Model

Abstract INTERCOMP's steamflood model was used to carry out a sensitivity study of cyclic steam stimulation as applied to the Cold Lake Oil Sands. A 2-D radial system was used and parameters were based on knowledge of the Mannville "C" unit. No history matches were attempted in this study. Variables investigated included steam-treatment size, length of soak, steam injection rate, steam quality, completion interval, effect of natural gas injection and strategy for multiple cycles. It was found that the benefits of a soak period were outweighed by the resulting drop in calendar-day oil rate and that, for the same reason steam should be injected at the greatest feasible rates. Gas injection was shown to be of limited value. The optimum strategy for scheduling cycles involves a continuous increase in treatment size in order to contact unheated bitumen on each cycle. Runs to investigate the effect of completion interval are less easy to interpret, but restricted intervals are beneficial in the short term. Introduction Cyclic steam stimulation has proved to be a technically feasible method of recovering bitumen from the Cold Lake Oil Sands of north-eastern Alberta and several operators are currently conducting field trials of the process. BP's pilot work in the area consisted of two steam-floods carried out between 1966–70 on adjacent legal subdivisions in Oil Sands Lease 64 in the northern part of the Cold Lake deposit as indicated in Figure 1. The second pilot included a few steam stimulation cycles which were, however, insufficient both in size and number to constitute a reasonable test of the applicability of the procedure. This paper reports on a study, using the INTERCOMP steamflood simulator described in Ref. 1, to model cyclic steam stimulation for an oil sands situation. Since many reservoir and fluid parameters were unknown and there was no field data to history-match, the study is regarded as being very preliminary in nature. By using "best-estimate" parameters it was hoped to investigate the problems involved in carrying out such simulations and to perhaps gain some insight into the process mechanisms. In these respects, the study was judged to be quite successful. General levels of performance calculated by the model were reasonable and changes seen in temperature pressure and saturation distributions throughout a cycle are believed to be realistic. A number of runs were carried out, varying certain of the operating parameters, and the resulting changes in performance were analyzed. Representation of the Reservoir-Fluid System The INTERCOMP model has the capability of handling 3-phase flow in a three-dimensional system but, in this study, the grid-block configuration represented two-dimensional radial flow into a single well. As shown in Figure 2, the reservoir was divided into six layers. Parameters for each layer are shown in Table I. Layer 1 has high gas and water saturations and is not typical of the total lease area. A further feature is the presence of a number of hard impermeable layers which are not, however sufficiently continuous to represent a serious impediment to vertical communication.

Solidification of droplets on a cold surface

Title problem has been investigated theoretically and experimentally. In the theory a simple model of two-dimensional radial flow has been used. The degree of flattening ξ m of a droplet depends upon the Weber, Reynolds and Péclet numbers, and upon the freezińg constant U, taken from the solution of a Stefan problem. The agreement of the theory with experiments is not bad if the constant U is taken for the Stefan problem with the isothermal cooling surface.

Effectiveness of Hot-Water Stimulation of Heavy-Oil Formations

Abstract In Venezuela and California, heavy-oil producing well are often stimulated by hot-water injection into the formation. Because of the resulting decrease in oil "Viscosity, wellbore cleanup and other minor thermal effects, the oil production rates usually show an increase over the prestimulation rates. The present investigation involved numerical simulation of the hot-water stimulation process. Heat and mass transport equations for a two-dimensional radial flow system were formulated and solved by two different methods. Tile stability and the sensitivity of the simulator developed were examined. In particular, the variation of the results obtained with changes in thermal conductivity, and also with relative perm abilities, assuming these to be temperature-dependent, were examined. Following this, the simulator was used to investigate the relative benefits and effectiveness of the stimulation of wells in heavy oil formations by hot-water injection. The principal findings of the simulation studies, under a wide variety of conditions, were as follows.For a given heat input and injection rate, the production rate and the oil cut increase with an increase in water temperature.The total oil recovery increases with an increase in the soak time. Although the peak production rate does not change appreciably, the rate decline is slower for the longer soak times, thus showing the importance of conductive heat transfer.More oil is recovered for lower injection rates, all other factors being equal. This again shows the importance of conduction-heating.Hot-water stimulation is more effective for heavier oils. The effectiveness of the stimulation process decreases markedly for oils of moderate viscosities, and it is nil for light oils. The simulator results were relatively insensitive to the thermal conductivity of the formation and the surrounding media, in view of the short injection times involved. Similarly, variations of relative-permeability characteristics with temperature Were found to have a negligible effect, for the data tested. The results of this study indicate that the increases in both rate and recovery obtained from hot-water stimulation are considerably lower than one would expect on the are of some of the reported field data, strongly suggesting that wellbore cleanup is the key factor in the success of a hot-water stimulation program, with oil viscosity reduction playing a secondary role. This is substantiated by the negative results obtained for moderately viscous oils, where the small gain in oil recovery may be offset by the loss of production during the injection period. As a result, it u; concluded that hot-water stimulation should be employed only in viscous oil formations, and its benefits should be carefully examined prior to field implementation. INTRODUCTION A THERMAL STIMULATION that has been employed in some instances, principally in Venezuela and California involves alternate injection of hot water into heavy oil producing formations, much along the lines of cyclic steam stimulation. Such treatment may be more advisable than steam injection if the mechanical condition of the wells is not suited for steam or if the wells are very deep, and high pressures are required, together with relatively low temperatures.

The steady two-dimensional radial flow of viscous fluid between two inclined plane walls

This paper considers the steady two-dimensional radial flow of viscous fluid between plane walls which either converge or diverge. A general solution is obtained in terms of elliptic functions and the various mathematically possible types of flow are discussed.