Improvements in Hydraulic Fracture Propagates - Case Study Example

To improve the methods and the change with different factors in the fracture propagates Name Course Instructor Date Table of Contents Table of Contents 2 1.0 Introduction 3 1.2 Objectives of the study 3 1.3 General objective 3 1.4 Specific objectives 4 2.0 Literature review 4 2.2 Operation of the hydraulic fracturing 4 2.3 Fluid Leak off 4 2.4 Linear Elastic Fracture Mechanics 4 2.5 PKN Model 5 2.6 PKN Hydraulic fracture simulation 7 2.6.1 Governing Equations 7 2.6.2 Momentum of the fluid 7 2.7 Local Fluid Mass Balance 8 2.8 Leak Equation 8 2.9 The relationship between pressure and the width 9 3.0 Methodology 10 3.3 Calculated data Table 2.0: Calculation using formula in excel 13 4.0 Analysis Factorial Design 13 4.1 Pareto Chart of the Effects 14 Reference 31 1.0 Introduction When tested in the laboratory, the engineering materials usually fails to reach the theoretical strength. This implies that the material performance in service is different from the expected performance hence components designs normally implores the engineer to help in minimizing failures of the materials (Johnson et al., 2015). There are several factors that influence the performance of different components of materials. Some of the major factors include properties of the material which include load or stress system, maintenance and environment (Lippold 2014). Failures in engineering process normally happens due to design deficiencies resulting from poor selection of materials, defect from manufacturing processes and overloading of materials couples with inadequate maintenance of materials. There is need for engineer to prevent system failures whenever it is anticipated (Boyd 2016). In design, the component structure usually asks to reduce possibilities of failure in the system. The metal failure is somehow complex to study since there are several causes of failures which needs to be taken into consideration (Gonzalez-Chavez et al., 2015). Any structural engineer should be in a position to understand the nature of the system being constructed and be able to understand and relate the material requirement for the system. This way failures will be minimized and in case of any failure earlier detection can be done for necessary measures. In this study, we investigate how to improve the methods and the change with different factors in the fracture propagates. In this way we will be able to minimize failures of materials (Settgast et al., 2017). Different measures will be taken in studying materials failures ti help system designers to reduce failures. 1.2 Objectives of the study This study will have both general and specific objectives 1.3 General objective This research seeks to improve the methods and the change with different factors in the fracture propagates 1.4 Specific objectives 1. To determine the effect of viscosity to the fracture. 2. To determine the effect of rang young modulus and Poisson ratio 3. To determine the effect of flow rate to the fracture 2.0 Literature review 2.1 Hydraulic Fracturing Wan et al. (2014) defines hydraulic fracturing as a simulation treatment which is done on oil and gas well which are situated in a low-permeability layers to help in increasing productivity. Usually a special fracturing fluids are pumped at high pressure to the reservoir well at a given interval to help in the treatment of the open vertical fractures. Normally the wings of the fracture extend away from the wellbore in direction opposite the original stress (Wan et al. 2014). Hydraulic fracture helps in creating high conductivity channel within the wider area of formation and help in bypasses any kind of damage that is present in the near wellbore area. 2.2 Operation of the hydraulic fracturing There are several factors which are involves on how the hydraulic functions. There is both the fluid and the solid mechanics taking place within the operation. 2.3 Fluid Leak off The fluids which are being used in different fracturing activities varies from one fracturing to another (Wan et al. 2014). In open fracture, the fracturing fluid is having direct contact with the fracturing surface, the pressure is also exerted on the surface to push the fracturing fluid into the reservoir well (Maddox 2014). In some cases the fluid may flow into the pore spaces within the rock. 2.4 Linear Elastic Fracture Mechanics The mechanics of fracture deals with the stability of preexisting cracks and their propagation. The foundation of fracture mechanic were laid down by Troiano (2016) when he applied the energy balance theory on the fracture propagation taking into account the total energy consumed in various parts of the fracturing process which is always constant. The Griffiths theory predicts that the critical σc which is applied remotely in the direction which is perpendicular to the length of the fracture. The following equation is used estimate the perpendicular length required to propagate the crack in the fracture:- Where:- E – Is the young modulus of the material being tested? Ƴ- Surface energy per unit area of the Crack a – this is the 0.5 of the crack length 2.5 PKN Model The equation to help in the computation of the fracture length and the width with a fixed height was developed by Perkins and Kern in early 1961. The improvement on the model was later done by Nordgren in 1972. He did this by adding fluid loss to the solution hence this model is commonly known as PKN Model (Wrobel and Mishuris 2013). This models assumes that the toughness of the fracture can be neglected. This is because the required energy for fracture to propagate was significantly less compared with the required fluid level for it to flow along the fracture length and the plane strain behaviour in the vertical direction with constant height of the fracture as it propagate along the horizontal direction (Wrobel and Mishuris 2013). This can be explained using the diagram below:- Figure 1: Schematic diagram for PKN Using the concept of the solid mechanics, if the fracture height hf, is fixed and is smaller than its length, the problem is reduced to two dimensions by the use of the plane strain assumptions (Baez and Kuang 2016). The plane strain in the PKN model is considered in the vertical direction and the response of the rock in each vertical section along the x-direction is usually assumed to be independent on its neighboring vertical plane. Here, the plain strain means that the elastic deformation to open or close, or shear the fracture in the model is concentrated in the vertical planes sections in a more perpendicular direction of the fracture propagation (Baez and Kuang 2016). This fact can only hold if the fracture length is more than its height. On the other hand, if it is monitored on the aspect of the fluid mechanics, the fluid problem found in the PKN model is normally considered as one dimension in an elliptical channel. Here the fluid pressure given by Pf is assumed to be constant in every vertical cross section in a perpendicular to the direction of the propagation of the fluid (Wrobel and Mishuris 2013).. This model help in explaining how the fracturing propagation work both in the solid mechanical aspect and fluid mechanic aspect. 2.6 PKN Hydraulic fracture simulation 2.6.1 Governing Equations PKN model assumes that the height of the fracture H is constant, it has elliptic vertical cross section having maximum width Wm at the center and this propagates X horizontal direction. Provided with the injection rate (Qo) at the wellbore, the characteristics of the fluid, the mechanical characteristics of the rock the magnitude of the minimum in-situ stress (σo) the virgin hole pressure of the formation given by (Po) and the leak-off coefficient (Cl), it is required to establish the history of the pressure in the borehole which is the fracture inlet and also the length and width history of the fracture over a given period (Wrobel and Mishuris 2013). All these statement can be compressed into mathematical equation and it can be explained as below:- 2.6.2 Momentum of the fluid The pressure the fluid fracture is assumed to be uniform across the fracture, varying only in horizontal directional along with fracture propagation (Adachi and Peirce 2013) The equation of the momentum for laminar of the power fluid can be therefore written as follows:- ………………………………………………. (2.1) Where, (i) Pf is the fluid pressure in the fracture (ii) This is the average flow rate per unit height of fracture (iii) This is the average width of the fracture (iv) A is the vertical cross sectional area (v) K is the constitutive constant for power of law fluid (vi) n this is the power law index (Adachi and Peirce 2013) (vii) Ψ This is the Shape factor dependent on the geometry of the fracture cross section adopted by Nolte 1979:- …………………….. (2.2) Where:- Wy is the width of the fracture at the vertical coordinate y For an elliptic cross section, the equation can be written as:- ……………………….. (2.3) The above equation explained the momentum fluid flow equation in the fracture using a given level of pressure. Therefore they are pressure fluid propagation (Baez and Kuang 2016). 2.7 Local Fluid Mass Balance When considering the fluid leak-off, fracture volume change and the fluid injection, the local fracturing fluid mass balance is written as:- Where u is the fluid leak-off velocity which accounts for both sides of the fracture walls 2.8 Leak Equation The leak equation using classical carter leak off theory will be adopted in this case and it is written as:- ………………… (2.5) Where Cl is the leak-off coefficient, t is the time since pumping starts, and is the arrival time of fracture tip at location x. 2.9 The relationship between pressure and the width The width of the fracture W is consisting of two parts: We which is controlled by stress and Wp which is also controlled by net pressure. The Pf is the fluid pressure and σo is the minimum in-situ stress, and Po is the virgin pore pressure. We can now have:- …………………………… (2.6) This is the net stress effect being approximated as pure elastic. Where We is given as ……………………………… (2.7) And Mc is given by Which is the fracture compliance, and v is the Poisson’s ratio, G is the share modulus. Using another method, Mc should be found from an elastic analysis in plane strain. The controlled net pressure poroelastic effect can be now describes as: ……………………………… (2.8) (Baez and Kuang 2016). Where we have:- Is the poroelastic coefficient whose theoretical value ranges from, and f (t*) is an evolutional function which varies between 0 and 1 as t* approaching 0 and respectively. It is used in evaluating the effects of poroelasticity. The symbol t* denotes as a dimensionless fracture surface exposure time since the arrival of the fluid which is defined as:- ………………………… (2.9) Where C is the diffusivity coefficient. For the PKN model, the evolutional function f (t*) is generally dependent on the elastic contrast between permeable and the impermeable layers. In situations of identical elastic property existing in the reservoir and barriers, the following expression f (t*) is derived. ………………………… (2.10) Where: ………………….. (2.11) 3.0 Methodology The section try to achieve objective of this research paper by investigating fracture propagation. Will use excel and Minitab in the calculation of the various values which are necessary to achieve the objective of this study. The data to be used is shown in the table below:- The formula to be used is given by Table 1.0: Factors Effect in the Analysis How to use Minitab:- This paper we will use Minitab 17 to perform analysis. It helps in performing large amounts of statistical calculations with a lot of ease. The software contain several common statistical analysis including but not limited to multiple regression, ANOVA etc. which are already pre-programed in the software. A screen short of the program can be shown below:- Figure 2: How to use Minitab The above data will be placed in the software at the right column before the analysis can be conducted. This software is capable of producing all the graphs and they are in the cross bar and one can chose the type of analysis he or she wishes to perform. Excel Excel is another important data analysis tool which can be used in performing analysis, calculation and even drawing graphs. Excel has already inbuilt formula and when one is using it, he or she must type the formula for the statistical analysis into the cell using acceptable format for the program to perform the required statistical analysis. Excel in most cases are used in conjunction with other programs when doing large amounts of statistical analysis on the input data. It is able to provide an individual with ease calculation but for multiple statistical calculations on a data set it might prove more challenging. For creating graphs for instance, simply use the tab at the top of the screen. Use the insert tab of different graphs you would wish to input. There are varieties of graphs and is upon you to decide which type of the graph you would wish to insert. For both excel and Minitab, when analyzing the data, it is important that the data is formatted in required format and in a way which can be easily understood by the user of the software. The analysis can be presented in different formats depending on which format analyst which to choose. 3.2 Benefits of using software: It is undoubtedly important to use software for meaningful analysis of data. The researcher is not able to explain the findings without using statistical figures to explain the causal effect on the variables in the study. These can only be people by the use of software which will not only make the comparison easy and simple, but also add value to the research result and easy understanding. 3.3 Calculated data Table 2.0: Calculation using formula in excel Time Viscosity Flow rate Modulus Width Poison 60 0.0018 0.003 0.3 15 0.1 118.75 0.0018 0.003 0.31875 15.3125 0.15 177.5 0.0019 0.004 0.3375 15.625 0.16 236.25 0.0019 0.004 0.35625 15.9375 0.17 295 0.0020 0.004 0.375 16.25 0.18 353.75 0.0020 0.005 0.39375 16.5625 0.19 412.5 0.0021 0.005 0.4125 16.875 0.2 471.25 0.0021 0.005 0.43125 17.1875 0.21 530 0.0022 0.005 0.45 17.5 0.22 588.75 0.0022 0.006 0.46875 17.8125 0.23 647.5 0.0022 0.006 0.4875 18.125 0.24 706.25 0.0023 0.006 0.50625 18.4375 0.25 765 0.0023 0.007 0.525 18.75 0.26 823.75 0.0024 0.007 0.54375 19.0625 0.27 882.5 0.0024 0.007 0.5625 19.375 0.28 941.25 0.0025 0.008 0.58125 19.6875 0.29 1000 0.0025 0.008 0.6 20 0.3 4.0 Analysis Factorial Design Factorial design can be described as the factorial experiment having more than one factor in the design and every variable consist of discrete values of levels whose experimental units on all possible. Our final model ended up with three factors, A, C and D, and two of their interactions, AC and AD. This was based on one complete replicate of this design. They are analyzed below:- 4.1 Pareto Chart of the Effects Pareto effect chart are charts histograms but with special design to capture. It arranges elements from the largest to smallest and this can be shown in the diagram below:- Figure 3: Pareto Chart of effects From the chart, time is the highest while viscosity is the least. This is further explanation of the ranges in the table of data analysis. This is also shown in percentage form. Table 3.0: Descriptive Statistics Viscosity N Median Mean Rank Z-Value 0.0018 2 89.375 1.5 -2.24 0.0019 2 206.875 3.5 -1.64 0.002 2 324.375 5.5 -1.04 0.0021 2 441.875 7.5 -0.45 0.0022 3 588.750 10.0 0.38 0.0023 2 735.625 12.5 1.04 0.0024 2 853.125 14.5 1.64 0.0025 2 970.625 16.5 2.24 Overall 17 9.0 The chi-square approximation may not be accurate when some sample sizes are less than 5. 5.2 Normal / Half Plots of the Effects 5.2.1 Normal Plot The plots and its effects are explains per elements as they are produced in the Minitab data analysis tool. Figure 4: Normality time graph The summary data shows that the minimum and maximum is the same mean of 530 and deviation 296.7. The p-value is 0.8774 >0.19 hence accepting null hypothesis and rejecting alternative hypothesis that data follow normal distribution. This is also presented in the graph above. Figure 5: Normality Modulus graph The summary data shows that the minimum and maximum is the same mean of 0.45 and deviation 0.09468. The p-value is =0.8774 >0.19 hence accepting null hypothesis and rejecting alternative hypothesis that data follow normal distribution. This is also presented in the graph above. Figure 6: Normality Time Flow The summary data shows that the minimum and maximum is the same mean of 0.0054706 and deviation 0.0015858. The p-value is =0.37 >0.394 hence accepting null hypothesis and rejecting alternative hypothesis that data follow normal distribution. This is also presented in the graph above. The summary data shows that the minimum and maximum is the same mean of 0.00215 and deviation 0.00022945. The p-value is =0.28 >0.6146 hence accepting null hypothesis and rejecting alternative hypothesis that data follow normal distribution. This is also presented in the graph above. The summary data shows that the minimum and maximum is the same mean of 0.0021765 and deviation 0.05517. The p-value is =0.14 >0.9666 hence accepting null hypothesis and rejecting alternative hypothesis that data follow normal distribution. This is also presented in the graph above. The summary data shows that the minimum and maximum is the same mean of 17.50 and deviation 1.578. The p-value is =0.19 >0.8774 hence accepting null hypothesis and rejecting alternative hypothesis that data follow normal distribution. This is also presented in the graph above. Fluid Viscosity The input data to examine the effect of fluid viscosity is shown in the table below:- Interaction Plot for fracturing width Part (1): The Parallel lines Part (2): Non Parallel lines From the figures and graph above, they show that the fracture width usually increases fluid viscosity and the fracture length decreases with increasing of fluid viscosity. The third graph shows that a higher pumping pressure is normally required when dealing with higher viscosity fracturing fluid under same condition of formation and operation. The results are a true picture of industry practices. Shear modular From the two graphs, they express a shorter and wider fracture can be easily be generated when the formation is soft with low module shear while a longer narrow fracture will be responsible for produced when the formation is hard with high shear modules using the same kind of fracturing fluids under similar operation and condition. Main Effects Plot for Fracturing Width From the results, the main fracturing effect on width is the fluid viscosity and the overall pressure of the fluid itself. Shear effect The above surface and countour graphs above shows how viscosity changes with width and the effect on the fracture. From surface examination, we infer that break strength of the composite is specifically impact by component A/W proportion. The composite made of jute texture fortified epoxy polymer composite were concentrated under different variables, for example, A/W proportion, width and Thickness to know the heap conveying limit and break strength of the composite.Taguchi strategy is utilized to explore that parameter and its level of impact. Surface plot/contour The surface contours of various variables analyzed in the project are shown below. Surface/contour plot of width vs. time Poisson ratio - Surface/contour plot of width vs. time viscosity time flow ratio - Surface/contour plot of width vs. time shear modulus - Surface/contour plot of width vs. time fracture height 2- Summary results of the experiment Seismic speed proportion (compressional and shear wave speed proportion,) and Poisson's proportion were acquired from compressional and shear waves utilizing the seismic refraction estimations for surface soils and shallow dregs. The right around 1.5 proportion saw in the above areas indicate the way that the proportion of incompressibility is practically solidarity. Negative Poisson's proportions were gotten, coming about because of proportions not exactly 3- Comparison between surface plot and contour plot The surface plot gives clear picture of the behaviour of the Poisson's proportion and other variables compared with contour plots. Reference Adachi, J.I. and Peirce, A.P., 2013, May. 5153-NEAR-TIP ASYMPTOTIC ANALYSIS OF A PKN FLUID-DRIVEN FRACTURE WITH NON-LOCAL ELASTICITY EQUATION. In ICF11, Italy 2005. Baez, J. and Kuang, Y., 2016. Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy. Applied Sciences, 6(11), p.352. Boyd, G.M. ed., 2016. Brittle fracture in steel structures. Elsevier. Gonzalez-Chavez, M., Dahi Taleghani, A. and Olson, J.E., 2015, February. A cohesive model for modeling hydraulic fractures in naturally fractured formations. In SPE Hydraulic Fracturing Technology Conference. Society of Petroleum Engineers. Johnson Jr, R.L., Abul Khair, H.F., Jeffrey, R.G., Meyer, J.J., Stark, C. and Tauchnitz, J., 2015. Improving fracture initiation and potential impact on fracture coverage by implementing optimal well planning and drilling methods for typical stress conditions in the Cooper Basin, central Australia. In The APPEA Conference Proceedings, 55, extended abstract. Lippold, J.C., 2014. Welding metallurgy and weldability. John Wiley & Sons. Maddox, S.J., 2014. Fatigue strength of welded structures. Woodhead publishing. Settgast, R.R., Fu, P., Walsh, S.D., White, J.A., Annavarapu, C. and Ryerson, F.J., 2017. A fully coupled method for massively parallel simulation of hydraulically driven fractures in 3‐dimensions. International Journal for Numerical and Analytical Methods in Geomechanics, 41(5), pp.627-653. Troiano, A.R., 2016. The role of hydrogen and other interstitials in the mechanical behavior of metals. Metallography, Microstructure, and Analysis, 5(6), pp.557-569. Wan, Y.J., Tang, L.C., Gong, L.X., Yan, D., Li, Y.B., Wu, L.B., Jiang, J.X. and Lai, G.Q., 2014. Grafting of epoxy chains onto graphene oxide for epoxy composites with improved mechanical and thermal properties. Carbon, 69, pp.467-480. Wrobel, M. and Mishuris, G., 2013. Efficient pseudo-spectral solvers for the PKN model of hydrofracturing. International Journal of Fracture, 184(1-2), pp.151-170. Zhang, H., Liu, Y., Kuwata, M., Bilotti, E. and Peijs, T., 2015. Improved fracture toughness and integrated damage sensing capability by spray coated CNTs on carbon fibre prepreg. Composites Part A: Applied Science and Manufacturing, 70, pp.102-110. Read More

2.3 Fluid Leak off The fluids which are being used in different fracturing activities varies from one fracturing to another (Wan et al. 2014). In open fracture, the fracturing fluid is having direct contact with the fracturing surface, the pressure is also exerted on the surface to push the fracturing fluid into the reservoir well (Maddox 2014). In some cases the fluid may flow into the pore spaces within the rock. 2.4 Linear Elastic Fracture Mechanics The mechanics of fracture deals with the stability of preexisting cracks and their propagation.

The foundation of fracture mechanic were laid down by Troiano (2016) when he applied the energy balance theory on the fracture propagation taking into account the total energy consumed in various parts of the fracturing process which is always constant. The Griffiths theory predicts that the critical σc which is applied remotely in the direction which is perpendicular to the length of the fracture. The following equation is used estimate the perpendicular length required to propagate the crack in the fracture:- Where:- E – Is the young modulus of the material being tested?

Ƴ- Surface energy per unit area of the Crack a – this is the 0.5 of the crack length 2.5 PKN Model The equation to help in the computation of the fracture length and the width with a fixed height was developed by Perkins and Kern in early 1961. The improvement on the model was later done by Nordgren in 1972. He did this by adding fluid loss to the solution hence this model is commonly known as PKN Model (Wrobel and Mishuris 2013). This models assumes that the toughness of the fracture can be neglected.

This is because the required energy for fracture to propagate was significantly less compared with the required fluid level for it to flow along the fracture length and the plane strain behaviour in the vertical direction with constant height of the fracture as it propagate along the horizontal direction (Wrobel and Mishuris 2013). This can be explained using the diagram below:- Figure 1: Schematic diagram for PKN Using the concept of the solid mechanics, if the fracture height hf, is fixed and is smaller than its length, the problem is reduced to two dimensions by the use of the plane strain assumptions (Baez and Kuang 2016).

The plane strain in the PKN model is considered in the vertical direction and the response of the rock in each vertical section along the x-direction is usually assumed to be independent on its neighboring vertical plane. Here, the plain strain means that the elastic deformation to open or close, or shear the fracture in the model is concentrated in the vertical planes sections in a more perpendicular direction of the fracture propagation (Baez and Kuang 2016). This fact can only hold if the fracture length is more than its height.

On the other hand, if it is monitored on the aspect of the fluid mechanics, the fluid problem found in the PKN model is normally considered as one dimension in an elliptical channel. Here the fluid pressure given by Pf is assumed to be constant in every vertical cross section in a perpendicular to the direction of the propagation of the fluid (Wrobel and Mishuris 2013).. This model help in explaining how the fracturing propagation work both in the solid mechanical aspect and fluid mechanic aspect. 2.6 PKN Hydraulic fracture simulation 2.6.1 Governing Equations PKN model assumes that the height of the fracture H is constant, it has elliptic vertical cross section having maximum width Wm at the center and this propagates X horizontal direction.

Provided with the injection rate (Qo) at the wellbore, the characteristics of the fluid, the mechanical characteristics of the rock the magnitude of the minimum in-situ stress (σo) the virgin hole pressure of the formation given by (Po) and the leak-off coefficient (Cl), it is required to establish the history of the pressure in the borehole which is the fracture inlet and also the length and width history of the fracture over a given period (Wrobel and Mishuris 2013).

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