Proppant Concentration in and Final Shape of Fractures Generated by Viscous Gels

1974 ◽  
Vol 14 (06) ◽  
pp. 531-536 ◽  
Author(s):  
H.R. van Domselaar ◽  
W. Visser
Keyword(s):  
Differential Equations ◽  
Solution Procedure ◽  
Fluid Loss ◽  
Square Root ◽  
Fracturing Fluid ◽  
Breakdown Pressure ◽  
Minimum Width ◽  
Fracture Width ◽  
Final Fracture ◽  
Fracture Shape

Abstract A mathematical procedure is given for calculating proppant concentration and final fracture shape for proppant concentration and final fracture shape for a fracture generated by injection of a viscous gel in which the propping material does not settle. To prevent bridging in the fracture, a decreasing pad prevent bridging in the fracture, a decreasing pad volume is present ahead of the proppant slurry. If combined with a criterion for proppant admittance - expressing the minimum width required for nonbridging particle transport - the developed procedure will result in a realistic design of fracturing treatments. Introduction Hydraulic fracturing is a well known technique for improving the productivity of wells by creating a highly conductive path in the reservoir. This path is made by fracturing the formation through the injection of fluid into a well at pressures above the breakdown pressure. To keep the fracture open after the treatment, propping material is injected with the fracturing fluid. Settling of the proppant can be reduced or even prevented by using viscous oil- and water-based gels as fracturing fluids.From theoretical considerations it follows that the cross-section of a propagating hydraulic fracture is approximately elliptical. The dimensions of the ellipse are determined by the injection rate, the injected volume of fracturing fluid, and fracturing-fluid properties-taking into account the volume of fluid loss to the permeable formation. The fluid loss depends partly on the fluid potential gradients at the fracture walls, resulting in a time-dependent fluid-loss coefficient, which is proportional to the reciprocal of the square root of the exposure time.Since the propping material can cause early screenout, a relatively large sand-free pad of fracturing fluid is injected to initiate the fracture. This pad volume moves ahead of the fluid (gel) containing the propping material. Owing to spurt and filtration losses - highest at the fracture tip but decreasing gradually toward the well - the pad length in the fracture will decrease. The proppant-laden fracturing fluid is also subject to fluid loss, which causes the proppant concentration to increase with distance from the well. The propped fracture width obtained after the fracturing treatment depends on the balance between pad volume and proppant concentration in the fracture. A treatment design should therefore aim at optimization of pad volume, fracturing-fluid volume, and proppant concentration. A design program should deliver practical pumping schedules, which generate fractures of required penetration and conductivity. penetration and conductivity. The present study is a new step toward a more realistic design of fracturing treatments. Differential equations describing the proppant distribution in fractures created by very viscous fluids (no settling) are derived and solved. DEFINITION OF THE MATHEMATICAL MODEL The derivation of the differential equations describing the proppant distribution is based on the following model premises.1. Vertical fractures are of rectilinear shape.2. Two symmetric fracture wings move diametrically from the well.3. Fracture dimensions follow the relations established by Geertsma and de Klerk.4. Gel and proppant move with the same velocity in a piston-like manner.5. Rheological properties of the gel prevent settling of the proppant.6. Fluid-loss is proportional to the square root of the exposure time.7. A decreasing proppant-free pad moves ahead of the proppant suspension.Based on these conditions, a set of differential equations subject to the boundary conditions has been derived (see Appendix A). In Appendix B the applied finite-difference scheme, and in Appendix C the solution procedure are discussed. To describe radial fractures, a simple coordinate transformation has been given in Appendix D. SPEJ P. 531

Fracturing Fluids: Fluid-Loss Measurements Under Dynamic Conditions

1985 ◽  
Vol 25 (05) ◽  
pp. 629-636 ◽  
Author(s):  
L.P. Roodhart
Keyword(s):  
Shear Rate ◽  
Filter Cake ◽  
Fluid Loss ◽  
Square Root ◽  
Fracturing Fluid ◽  
Rock Permeability ◽  
Static Test ◽  
The Third ◽  
Fracturing Fluids ◽  
Third Stage

Abstract When filter-cake-building additives are used in fracturing fluids, the commonly applied static, 30-minute API filtration test is unsatisfactory, because in a dynamic situation (like fracturing) the formation of a thick filter cake will be inhibited by the shearing forces of the fracturing fluid. A dynamic, filter-cake-controlled, leakoff coefficient that is dependent on the shear rate and shear stress at the fracture face is, therefore, introduced. A test apparatus has been constructed in which the fluid leakoff is measured under conditions of temperature, rate of shear, duration of shear, and fluid-flow pattern as encountered under fracturing conditions. The effects of rock permeability, shear rate, and differential pressure on the permeability, shear rate, and differential pressure on the dynamic leakoff coefficient are presented for various, commonly used fracturing-fluid/fluid-loss-additive combinations. Introduction An important parameter in hydraulic fracturing design is the rate at which the fracturing fluid leaks into the formation. This parameter, known as fluid loss, not only determines the development of fracture length and width, but also governs the time required for a fracture to heal after the stimulation treatment has been terminated. The standard leakoff test is a static test, in which the effect of shear rate in the fracture on the viscosity of the fracturing fluid and on the filter-cake buildup is ignored. Dynamic vs. Static Tests The three stages in filter-cake buildup arespurt loss during initiation of the filter cake,buildup of filtercake thickness, during which time leakoff is proportional to the square root of time, andlimitation of filter-cake growth by erosion. In the standard API leakoff test, 1 the fracturing fluid, with or without leakoff additives, is forced through a disk of core material under a pressure differential of 1000 psi [7 MPa), and the flow rate of the filtrate is determined. In such a static test, the third stage-erosion of the filter cake-is absent. In a dynamic situation there is an equilibrium whereby flow along the filter cake limits the filter-cake thickness, and the leakoff rate becomes constant. The duration of each of these stages depends on the type of fluid, the type of additive, the rock permeability, and the test conditions. The differences between dynamic and static filtration tests are shown in Fig. 1, where the cumulative filtrate volume (measured in some experiments with the dynamic fluid-loss apparatus described below) is expressed as a function of time (Fig. la) and as a function of the square root of time (Fig. ]b), The shear rate at the surface of the disk is either static (O s -1 ), or 109 s -1 or 611 s -1. The curves indicate that the dynamic filtration velocities are higher than those measured in a static test and increase rapidly with increasing shear rate. This is in agreement with the observations made by Hall, who used an axially transfixed cylindrical core sample along which fracturing fluid was pumped, while the filtrate was collected from a bore through the center. Fig. la shows how the lines were drawn to fit the data: Vc = Vsp + A t + Bt, .........................(1) where Vc = cumulative volume per unit area, t = filtration time, Vsp= spurt loss, A = static leakoff component, andB = dynamic leakoff component. In static leakoff theory, B =0 and then A =2Cw, twice the static leakoff coefficient.-3 Each of the terms in Eq. 1 represents one of the stages in the leakoff process-spurt loss, buildup of filter cake, and erosion of filter cake. Analysis of the experimental data shows that the spurt loss, Vsp, and the static leakoff component, A, are independent of the shear rate, but the dynamic component, B, varies strongly with the shear rate (see Table 1). This means that, the higher the shear rate, the more the leakoff process is controlled by the third stage. process is controlled by the third stage. One model commonly used is based solely on square-root-of-time behavior with a constant spurt loss. Fig. 1 shows that little accuracy is lost by describing the leakoff with a single square-root-of-time equation: Vc = VsP + m t,...........................(2) where the dynamic leakoff coefficient. Cw = 1/2m, depends heavily on shear. and the spurt loss remains the same as in Eq. 1 and independent of the shear rate Table 2 shows that the error in C, that arises as a result of measuring under static conditions can be more than 100%. SPEJ P. 629

Chemical Evolution of a High- Temperature Fracturing Fluid

1985 ◽  
Vol 25 (05) ◽  
pp. 623-628 ◽  
Author(s):  
C.C. LaGrone ◽  
S.A. Baumgartner ◽  
R.A. Woodroof
Keyword(s):  
Guar Gum ◽  
Chemical Degradation ◽  
Fluid Loss ◽  
High Polymer ◽  
Fracturing Fluid ◽  
Fracture Width ◽  
Fracturing Fluids ◽  
Fluid Properties ◽  
Optimal Ph

Abstract Reservoirs with bottomhole temperatures (BHT's) in excess of 250 deg. F [121 deg. C] and permeabilities of less than 1.0 md are commonly encountered in drilling and completing geothermal and deep gas wells. Successful stimulation of these wells often requires the use of massive hydraulic fracturing (MHF) treatments. Fracturing fluids chosen for these large treatments must possess shear and thermal stability at high BHT'S. The use of conventional fracturing fluids has been limited traditionally to wells with BHT's of 250 deg. F [121 deg. C] or less. Above 250 deg. F [121 deg. C], high polymer concentrations and/or large fluid volumes are required to maintain effective fluid viscosities in the fracture. However, high polymer concentrations lead to high friction pressures, high costs, and high gel residue levels. The large fluid volumes also increase significantly the cost of the treatment. Greater understanding of fracturing fluid properties has led to the development of a crosslinked fracturing fluid designed specifically for wells with BHT's above 250 deg F [121 deg C). The specialized chemistry of this fluid combines a high-ph hydroxypropyl guar gum (HPG) solution with a high-temperature gel stabilizer and a proprietary crosslinker. The fluid remains stable at 250 to proprietary crosslinker. The fluid remains stable at 250 to 350 deg. F [121 to 177 deg. C] for extended periods of time under shear. This paper describes the rheologial evaluations used in the systematic development of this fracturing fluid. In field applications, this fracturing fluid has been used to stimulate successfully wells with BHT's ranging from 250 to 540 deg. F [121 to 282 deg C). Case histories that include pretreatment and posttreatment production data are presented. Introduction Temperatures exceeding 250 deg F [121 deg C) and permeabilities less than 1.0 md are frequently encountered in permeabilities less than 1.0 md are frequently encountered in deep gas and geothermal wells. Successful economic completion of these wells requires that long, conductive fractures with optimal proppant distribution be created. Ultimately, the amount of production from these formations depends on the propped fracture length created. One successful stimulation technique used to create these long fractures is MHF. In these treatments, the fracturing fluids are often exposed to shear in the fracture for prolonged periods of time at high temperatures. Thus the fracturing fluids must exhibit extended shear and thermal stability at the high BHT'S. In addition, the fracturing fluid must not leak off rapidly into the formation, or the fracture may not be extended to the desired length. Many early treatments were limited by fracturing fluids that lost viscosity rapidly at high BHT's because of excessive thermal and/or shear degradation. Creation of a narrow fracture width, excessive fluid loss to the formation, and insufficient proppant transport resulted from the use of these low viscosity fluids. The solution to conventional fracturing fluid deficiencies was to develop a more efficient fracturing fluid (low polymer concentrations) with greater viscosity retention under shear at high temperatures, better fluid-loss control, and lower friction pressures. Generally, the components that make up crosslinked fracturing fluids include a polymer, buffer, gel stabilizer, and crosslinker. Each of these components is critical to the development of the desired fracturing fluid properties. The role of polymers in fracturing fluids is to properties. The role of polymers in fracturing fluids is to provide fracture width, to suspend proppants, to help provide fracture width, to suspend proppants, to help control fluid loss to the formation, and to reduce friction pressure in the tubular goods. Guar gum and cellulosic pressure in the tubular goods. Guar gum and cellulosic derivatives are the most common types of polymers used in fracturing fluids. The cellulosic derivatives are residue-free and thus help minimize fracturing fluid damage to the formation. However, the cellulosic derivatives are difficult to disperse because of their rapid rate of hydration. Guar gum and its derivatives are easily dispersed but produce some residue when broken. Buffers are used in conjunction with polymers so that the optimal pH for polymer hydration can be attained. When the optimal pH is reached, the maximal viscosity yield from the polymer is more likely to be obtained. The most common example of fracturing fluid buffers is a weak-acid/weak-base blend, whose ratios can be adjusted to that the desired ph is reached. However, some of these buffers dissolve slowly, particularly at cooler temperatures. Gel stabilizers are added to polymer solutions to inhibit chemical degradation. Examples of gel stabilizers used in fracturing fluids include methanol and various inorganic sulfur compounds. Other stabilizers are useful in inhibiting the chemical degradation process, but many interfere with the mechanism of crosslinking. The sulfur containing stabilizers possess an advantage over methanol, which is flammable, toxic, and expensive. SPEJ P. 623

scholarly journals Hydraulic Fracturing Behavior in Shale with Water and Supercritical CO2 under Triaxial Compression

Geofluids ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10 ◽  
Author(s):  
Jianming He ◽  
Yixiang Zhang ◽  
Chao Yin ◽  
Xiao Li
Keyword(s):  
Hydraulic Fracturing ◽  
Mechanical Response ◽  
Triaxial Compression ◽  
Injection Rate ◽  
Fracturing Fluid ◽  
Breakdown Pressure ◽  
Fracture Width ◽  
Similar Tendency ◽  
Proper Size ◽  
Shale Reservoir

Comparing to the water fracturing fluid regularly used in the hydraulic fracturing operation, supercritical CO2 (SC-CO2) as a promising nonaqueous fracturing fluid has the great potential for the improvement of production and protection of shale reservoir. This paper presents an experimental study of the mechanical response and fracture propagation of shale fractured using water and SC-CO2 under the different stress status and injection rate. According to the experimental results, SC-CO2 fracturing is more time-consuming due to its compressibility which takes about 20 times more time than hydraulic fracturing using water under the same preset conditions. The breakdown pressure of shale can be affected by not only the anisotropy of itself but also the external factors like injection rate and deviator stress. Similar tendency of the breakdown pressure with the variation of bedding orientation can be observed in both of the fracturing using water and SC-CO2. However, all of the shale specimens fractured using SC-CO2 show smaller breakdown pressure if compared with the shale specimens fractured using water. According to the results of fracture width evolution monitored by circumference during the fracturing, the fracture propping and proper size of the proppant are really important for the hydraulic fracturing.

Optimal Trajectories of Open-Chain Robot Systems: A New Solution Procedure Without Lagrange Multipliers

1998 ◽  
Vol 120 (1) ◽  
pp. 134-136 ◽  
Author(s):  
Sunil K. Agrawal ◽  
Pana Claewplodtook ◽  
Brian C. Fabien
Keyword(s):  
Differential Equations ◽  
Lagrange Multipliers ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Optimal Solution ◽  
Solution Procedure ◽  
Optimal Trajectories ◽  
Point Boundary ◽  
Open Chain ◽  
Robot Systems

For an n d.o.f. robot system, optimal trajectories using Lagrange multipliers are characterized by 4n first-order nonlinear differential equations with 4n boundary conditions at the two end time. Numerical solution of such two-point boundary value problems with shooting techniques is hard since Lagrange multipliers can not be guessed. In this paper, a new procedure is proposed where the dynamic equations are embedded into the cost functional. It is shown that the optimal solution satisfies n fourth-order differential equations. Due to absence of Lagrange multipliers, the two-point boundary-value problem can be solved efficiently and accurately using classical weighted residual methods.

An Improved Method for Measuring Fluid Loss at Simulated Fracture Conditions

1985 ◽  
Vol 25 (04) ◽  
pp. 482-490 ◽  
Author(s):  
Robert Ray McDaniel ◽  
Asoke Kumar Deysarkar ◽  
Michael Joseph Callanan ◽  
Charles A. Kohlhaas
Keyword(s):  
Fluid Flow ◽  
Shear Rate ◽  
Filter Paper ◽  
Filter Cake ◽  
Fluid Loss ◽  
Test Results ◽  
Test Apparatus ◽  
Fracturing Fluid ◽  
The Core ◽  
Static Core

Abstract A test apparatus is designed to carry out dynamic and static fluid-loss tests of fracturing fluids. This test apparatus simulates the pressure difference, temperature, rate of shear, duration of shear, and fluid-flow pattern expected under fracture conditions. For a typical crosslinked fracturing fluid, experimental results indicate that fluid loss values can be a function of temperature, pressure differential, rate of shear, and degree of non-Newtonian behavior of the fracturing fluid. A mathematical development demonstrates that the fracturing-fluid coefficient and filter-cake coefficient can be obtained only if the individual pressure drops can be measured during a typical fluid-loss test. Introduction In a hydraulic fracturing treatment, the development of fracture length and width is strongly dependent on a number of key fluid and formation parameters. One of the most important of these parameters is the rate at which the fracturing fluid leaks, off into the created fracture faces. This parameter, identified as fluid loss, also influences the time required for the fracture to heal after the stimulation treatment has been terminated. This in turn will influence the final distribution of proppant in the fracture and will dictate when the well can be reopened and the cleanup process started. Historically, tests to measure fluid loss have been carried out primarily under what is characterized as static conditions. In such tests, the fracturing fluid is forced through filter paper or through a thin core wafer under a pressure gradient, and the flow rate at the effluent side is determined. Of course, the use of filter paper cannot account for reservoir formation permeability and porosity; therefore, the fluid-loss characteristics derived from such tests should be viewed as only gross approximations. The static core-wafer test on the other hand, reflects to some extent the interaction of the formation and fracturing-fluid properties. However, one important fluid property is altogether ignored in such static core-wafer tests. This is the effect of shear rate in the fracture on the rheology (viscosity) of fracturing fluid and subsequent effects of viscosity on the fluid loss through the formation rock. In the past, several attempts were made to overcome the drawbacks of static core-wafer tests by adopting dynamic fluid-loss tests. Although these dynamic tests were a definite improvement over the static versions, each had drawbacks or limitations that could influence test results. In some of the studies, the shearing area was annular rather than planar as encountered in the fracture. In other cases, the fluid being tested did not experience a representative shear rate for a sufficiently long period of time. An additional problem arose because most studies were performed at moderate differential pressures and temperatures. The final drawback in several of the studies was that the fluid flow and leakoff patterns did not realistically simulate those occurring in the field. In the first part of this paper, we emphasize the design of a dynamic fluid-loss test apparatus that possesses none of these drawbacks. In the second part of the paper, test results with this apparatus are presented for three different fluid systems. These systems areglycerol, a non-wall-building Newtonian fluid,a polymer gel solution that is slightly wall-building and non-Newtonian, anda crosslinked fracturing system that is highly non-Newtonian in nature and possesses the ability to build a wall (filter cake) on the fracture face (see Table 1). The fluids were subjected to both static and dynamic test procedures. In the third part of the paper, results of experiments carried out with crosslinked fracturing fluid for different core lengths, pressure differences, temperatures, and shear rates are compared and the significance of the difference of fluid loss is emphasized. Experimental Equipment and Procedure The major components of the experimental apparatus shown in Fig. 1 are a fluid-loss cell, circulation pump, heat exchanger, system pressurization accumulators, and a fluid-loss recording device. The construction material throughout most of the system is 316 stainless steel. The fluid loss is measured through a cylindrical core sample, 1.5 in. [3.81 cm] in diameter, mounted in the fluid-loss cell. Heat-shrink tubing is fitted around the circumference of the core and a confining pressure is maintained to prevent channeling. Fracturing fluid is circulated through a rectangular channel across one end of the core. SPEJ P. 482^

scholarly journals A new type of experimentally proposed in situ heat/gas clean foam fracturing fluid system

2020 ◽  
Vol 10 (8) ◽  
pp. 3419-3436
Author(s):  
Kuangsheng Zhang ◽  
Zhenfeng Zhao ◽  
Meirong Tang ◽  
Wenbin Chen ◽  
Chengwang Wang ◽  
...  
Keyword(s):  
Heavy Oil ◽  
Pressure Coefficient ◽  
System Optimization ◽  
Inert Gas ◽  
Working Fluid ◽  
Fluid Loss ◽  
Fracturing Fluid ◽  
Fluid System ◽  
New Type

Abstract When cold fluid is injected into low-temperature, low-pressure, low-permeability reservoirs containing wax-bearing heavy oil, cryogenic paraffin deposition and heavy oil condensation will occur, thus damaging the formation. Moreover, the formation pressure coefficient is low and the working fluid flowback efficiency is low, which affects the fracturing stimulation effect. Therefore, an in situ heat/gas clean foam fracturing fluid system is proposed. This system can ensure that conventional fracturing fluid can create fractures and carry proppant in the reservoir, generate heat in situ to avoid cold damage, reduce the viscosity, and improve the fluidity of crude oil. The in situ heat fracturing fluid generates a large amount of inert gas while generating heat, thus forming foam-like fracturing fluid, reducing fluid loss, improving proppant-carrying performance, improving gel-breaking performance, effectively improving crack conductivity, and is clean and environmentally friendly. Based on the improved existing fracturing fluid system, in this paper, a new type of in situ heat fracturing fluid system is proposed, and a system optimization evaluation is conducted through laboratory experiments according to the performance evaluation standard of water-based fracturing fluid. Compared with the traditional in situ heat fracturing fluid system, the fracturing fluid system proposed in this study generates a large amount of inert gas and form foam-like fracturing fluid, reduces fluid loss, enhances the proppant-carrying capacity and gel-breaking performance, improves crack conductivity, the gel without residue and that the gel-breaking liquid is clean and harmless.

scholarly journals Investigating Jeffery-Hamel flow with high magnetic field and nanoparticle by HPM and AGM

2014 ◽  
Vol 4 (4) ◽  
Author(s):  
A. Rostami ◽  
M. Akbari ◽  
D. Ganji ◽  
S. Heydari
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Volume Fraction ◽  
Nonlinear Differential Equations ◽  
Reynolds Numbers ◽  
Solution Procedure ◽  
Algebraic Equations ◽  
Divergent Channel ◽  
Mathematical Operations

AbstractIn this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using two powerful analytical methods, Homotopy Perturbation Method (HPM) and a simple and innovative approach which we have named it Akbari-Ganji’s Method(AGM). Comparisons have been made between HPM, AGM and Numerical Method and the acquired results show that these methods have high accuracy for different values of α, Hartmann numbers, and Reynolds numbers. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.It is necessary to represent some of the advantages of choosing the new method, AGM, for solving nonlinear differential equations as follows: AGM is a very suitable computational process and is applicable for solving various nonlinear differential equations. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration, the solution of the problem can be obtained very simply and easily. It is notable that this solution procedure, AGM, can help students with intermediate mathematical knowledge to solve a broad range of complicated nonlinear differential equations.

Improving the fracturing fluid loss control for multistage fracturing by the precise gel breaking time design

2015 ◽  
Vol 25 ◽  
pp. 367-370 ◽  
Author(s):  
Xin Lin ◽  
Shicheng Zhang ◽  
Qiang Wang ◽  
Yin Feng ◽  
Yuanyuan Shuai
Keyword(s):  
Fluid Loss ◽  
Fracturing Fluid ◽  
Breaking Time ◽  
Loss Control

scholarly journals NEW GEOMETRIC FORMALISM FOR GRAVITY EQUATION IN EMPTY SPACE

2005 ◽  
Vol 14 (06) ◽  
pp. 1009-1022 ◽  
Author(s):  
XIN-BING HUANG
Keyword(s):  
Differential Equations ◽  
Gauge Field ◽  
Empty Space ◽  
Space Time ◽  
Gravity Theory ◽  
Spin Connection ◽  
Square Root ◽  
Field Equations ◽  
Complex Spin ◽  
Set Up

In this paper, a complex daor field which can be regarded as the square root of space–time metric is proposed to represent gravity. The locally complexified geometry is set up, and the complex spin connection constructs a bridge between gravity and SU(1, 3) gauge field. Daor field equations in empty space are acquired, which are one-order differential equations and do not conflict with Einstein's gravity theory.

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