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## Material Information- Title:
- Equilibrium dry beach widths for arbitrary nourishment sediment size distributions and arbitrary native profiles
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- Equilibrium dry beach widths for arbitrary nourishment sediment size distributions and arbitrary native profiles
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- Chial, Loraine K.
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- Gainesville, Fla.
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- Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
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UFL/COEL-2002/010
EQUILIBRIUM DRY BEACH WIDTHS FOR ARBITRARY NOURISHMENT SEDIMENT SIZE DISTRIBUTIONS AND ARBITRARY NATIVE PROFILES by Loraine K. Chial Thesis 2002 EQUILIB3RIUJM DRY BEACH WIDTHS FOR ARBITRARY NOURISHMENT SEDIMENT SIZE DISTRIBUTIONS AND ARBITRARY NATIVE PROFILES By LORAINE K. CHIAL A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2002 ACKNOWLEDGMENTS I would like to express my sincere gratitude to Dr. Robert G. Dean, my graduate advisor, for his wisdom, insight, and dedication to his work and his students. I would also like to thank Dr. Daniel M. Hanes and Dr. Ashish J. Mehta for serving on my supervisory committee. I am grateful to all the faculty and staff of Coastal Engineering for their knowledge and support throughout the last two years. I would like to thank my parents for always giving me the liberty of making my own choices and supporting me through my decisions. I thank my sisters for their inspiration, encouragement, and "aiyaaaaaa's." Special thanks go to the "Coastal Fraternity," especially to Freebird, Answer-man, Brain, Dan the Man, Big Kat, Magic Wanz, Jengle, The Spy, and "J"(rd), for the long days, the good times, and for trying to get me "acclimatized." I am very thankful to all my friends for their support and the great moments we shared. I am grateful to the University of Florida and the Department of Civil and Coastal Engineering for awarding me an Alumni Fellowship and thereby granting me the liberty of pursuing my research interests independently. TABLE OF CONTENTS page ACKNOWLEDGMENTS ......................................................................1 LIST OF TABLES ...............................................................................v LIST OF FIGURES ............................................................................. vi ABSTRACT.................................................................................... iX 1 INTRODUCTION...........................................................................1. Problem Description .........................................................................1 Previous Work in Determining Additional Dry Beach Width ............................ 2 Scope of This Thesis .......................................................................... 3 2 BACKGROUND .............................................................................S The Phi Scale for Characterizing Sediment Sizes........................................... 5 Previous Method for Representing Native Beach Profiles................................. 5 Previous Methods for Representing Fill Sediment Sizes................................... 7 Granulometric Basis ...................................................................... 7 Equilibrium Beach Profile Method...................................................... 9 Single sediment size ................................................................ 11 Non-uniform sediment size across profile........................................ 14 Two-parameter representation (two sediment sizes) ........................... 15 3 METHODOLOGY ........................................................................... 17 Representing the Native Profile with a Least Squares Fit ................................ 17 Accounting for Arbitrary Fill Sediment Sand Size Distributions ....................... 18 Description of Assumptions and Programs................................................. 19 Fill Sediment Finer than the Native (ffiner.m)........................................ 26 Fill Sediment Coarser than the Native (fcoarser.m).................................. 28 4 RESULTS AND DISCUSSION............................................................. 32 Comparison with Previous Methods........................................................ 32 Delray Beach Nourishment Project: A Case Study Application ....................... 44 Further Improvements........................................................................ 49 5 SUMMARY AND) CONCLUSIONS ............................ I.......................... 51 APPENDICES A PROGRAM LISTING FOR FFINER.M ................................................... 54 B PROGRAM LISTING FOR FCOARSER.M.............................................. 71 LIST OF REFERENCES ....................................................................... 94 BIOGRAPHICAL SKETCH ................................................................... 96 LIST OF TABLES Table page 2-1. A values (rn"3) for given sediment diameters d (mm).................................... 7 3-1. Comparison of earlier two-size representation to present four-size method......... 25 LIST OF FIGURES Figure pa4e 2-1. Profile scale parameter, A for varying sediment diameter, D (from Dean and Dalrymple 2002) .......................................................................... 6 2-2. Overfill factor based on Dean's (1974) method (from Dean and Dalrymple 2002). ..8 2-3. Sand transport resulting from nourishment project equilibration (a) Plan view, and (b) elevation view (from Dean and Dalrymple 2002)................................ 10 2-4. Three types of equilibrium profiles for nourishment projects (from Dean 1991). (a) Intersecting profile AF > AN; (b) non-intersecting profile; (c) submerged profile AF and V for h /B=2.0 (from Dean 199 1) ................................................ 13 2-6. Variation of additional dry beach with nourishment volume density for three different fill sediment sizes. h. = 6 mn, B =2.0 m. (from Dean and Dalrymple 2002)...................................................................................... 14 2-7. Comparison of the effect of representing the nourishment sediment with one vs. two sediment sizes. h. =6.1 mn, B=1.5 mn (from Dean 2001) ........................... 16 3-1. Example of a least squares fit of the 2/3 power rule to the profile at the Army Corps of Engineers Field Research Facility in Duck, NC (measured profile modified from Lee and Birkemeier 1993)........................................................ 18 3-2. Definition of input variables .............................................................. 19 3-3. Example of dividing the grain size distribution into two portions at 4,.. In this example, the mean size of the fill sediment is finer than the native. DN 0.20 mm, DF =0.14 mm, qF0F .5 ........................................................... 21 3-4. Example of dividing the fill sediment grain size distribution into four bins. In this example, the fill sediment is finer than the native and Bin 1 has the same mean size as the native distribution. Bins 2, 3 and 4 have equal cumulative frequencies of occurrence. DN = 0.20 mm, 4- = 0. 14 mm, qF = 0.5............................. 21 3-5. Sediment size bins are placed with decreasing sediment size in the offshore direction. DN = 0.20 mm, DF = 0.14 mm, aF = 0.5, B = 2 m, slope_nat = 1/30, slope_place = 1/20, h, = 6 m, V= 300 m3/m (Vact = 188 m3/m and Vun.mixed= 112 m 3/m ) ...................................................................................................................... 23 3-6. Final EBP and Ayo obtained when the depth farthest offshore is equal to h.. DN = 0.20 mm, DF = 0.14 mm, 'F = 0.5, B = 2 m, slope fnat = 1/30, slope place = 1/20, h.=6m, V=300m3/m, Ay,= 26.3 m ............................................................ 24 3-7. Procedure for establishing the active volume from a trial EBP for cases with Dp < DN. DN = 0.20 mm, DF = 0.14 mm, O'F = 0.5, B = 2 m, slopenat = 1/30, slope-place = 1/20, h. = 6 m, V= 300 m3/m, Vact = 188 m3/m, A = 0.100 m13 (coarsest bin) ................... ........................... 27 3-8. EBP showing decreasing trend in grain size offshore .......................................... 28 3-9. Exam ple of trial Y ............................................................................................... 30 3-10. Res_VAcT is calculated once the correct Y, value for a given X% is found. DN 0.20 mm, DF = 0.275 mm, crF = 0.5, V= 300 m3/m, B = 2 m, h. = 6 m, slopenat = 1/30, slope..lace = 1/20 ............................................................... 31 4-1. Comparison of predicted additional dry beach width for different methods of representing the nourishment sediment ............................................................ 33 4-2. Nourishment sediment size distribution for the case of DF > DN .............................. 35 4-3. Four sediment size bins used byfcoarser.m (DF > DN) ....................................... 36 4-4. Nourishment sediment size distribution for the case of DF < DN ......................... 37 4-5. Four sediment size bins used byffiner.m (DF < DN) ............................................ 38 4-6. Effect of the sorting of nourishment sediments on the additional dry beach width (D F > D N) ............................................................................................................... 39 4-7. Effect of the sorting of nourishment sediments on the additional dry beach width (D F < D N) .............................................................................................................. 40 4-8. Variation of the additional dry beach width with DF when DN is constant ...... 42 4-9. Example of an arbitrary nourishment sediment size distribution and its log-normal approximation. (a) Frequency.distribution; (b) cummulative frequency ..... 43 4-10. Average profiles for the 1992 Delray Beach nourishment project ..................... 46 4-11. Borrow sand sediment size distribution (Delray Beach 1992 nourishment) ..... 47 4-12. Ffiner.m results for the 1992 Delray Beach nourishment project (V=327 m3/m)...47 4-13. Ffiner.m results for the 1992 Delray Beach nourishment project (V=1 87 m3/m)...48 4-14. Measured cross-shore sediment size distribution for Delray Beach (January 1997).48 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EQUILIB3RITUM DRY BEACH WIDTHS FOR ARBITRARY NOURISHMENT SEDIMENT SIZE DISTRIBUTIONS AM) ARBITRARY NATIVE PROFILES By Loraine K. Chial December 2002 Chair: Robert G. Dean Department: Civil and Coastal Engineering Due to the increasing popularity of beach nourishment as the preferred solution to many beach erosion problems, a need has arisen to improve current methods used to predict the performance of these projects. The additional dry beach width resulting from the equilibration of beach nourishment projects is of particular interest. For this purpose, two Matlab programs were developed to determine the additional dry beach width for a nourishment project based on a more realistic representation of the fill than provided by earlier methods. The procedures presented allow representation of the nourishment sediment by either a log-normal sediment size distribution or by a user-defined arbitrary distribution. The proposed methods then account for the fill sediment by using four representative size bins. It is assumed that finer sediments will be moved offshore while coarser sediments will remain closer inshore as the profile is reworked towards equilibrium. With this method, a portion of the fill placed is left "undisturbed" while the remainder becomes the sorted "active volume." In addition, it is assumed the native beach can be represented by either a single sediment size or a measured arbitrary native profile. One program,fcoarser.m, is applicable when the fill sediment is coarser than the native sediment. Another program,ffiner.rm, is applicable when the fill sediment is finer than the native. The additional dry beach widths predicted with these programs are compared to the predictions obtained with earlier methods. It was determined that although the predictions provided byfcoarser.m are qualitatively significant, they might not be as quantitatively reliable. However, since most nourishment projects are carried out with fill sediment finer than the native,ffiner.m is applicable in most cases. The 1992 Delray Beach nourishment project is presented as a case study application. Recommendations for further improvement of the methodology presented are included. CHAPTER 1 INTRODUCTION Problem Description As erosion plagues beaches worldwide, engineers have strived to devise methods to counteract these recessive trends. Lately, the tendency in coastal engineering practice in the United States and in many other countries has been to avoid the construction of "hard" structures (groins, seawalls, breakwaters, etc.) for shore protection purposes. Instead, beach nourishment has become a more attractive solution because it directly addresses the cause of erosion: a shortage of sediment in the system. It is also a preferred method because it is more natural, it provides a recreational beach while serving as storm protection, and results in environmental benefits to the area. Although beach nourishment projects have been widely constructed throughout the world, not all aspects of the evolution of such projects are clearly understood. Of prime importance to design engineers is the ability to predict the performance of such projects, including the cross-shore and longshore (planform) evolution. This objective is made complex by the unpredictability of storms affecting the project and any variability in the placement geometry and sediment characteristics. One particular issue of interest to design engineers is the additional dry beach width that results when the project equilibrates in the cross-shore direction. This is the dry beach width associated with the equilibrium beach profile (EBP) that occurs when destructive and constructive forces are balanced. Equilibrium beach profiles are known to share certain characteristics: (1) they are generally concave upwards, (2) coarser sediment particles result in steeper slopes than finer particles, (3) the beach face is roughly planar, and (4) steep storm waves tend to transport sand seaward into bars and result in milder slopes (Dean 1991). Many different equations have been proposed to describe equilibrium beach profiles. The power law form of the EBP equation developed by Bruun (1954) and Dean (1977) has been widely used to describe this shape. The equation is h=Ay 2/3 [1-1) where h is the water depth at an offshore distance y from the shoreline, and A is the profile scale parameter, which is a function of the energy dissipation and indirectly the grain size. Other equations used to describe equilibrium beach profiles have been proposed by Larson (198 8), Bodge (1992), Komar and McDougal (1994) and Dean (Dean and Dalrymple 2002). Previous Work in Determining Additional Dry Beach Width Based on the 2/3 power law form of the EBP equation, numerous attempts have been made to determine the additional dry beach width, Ay, that results from the placement of a given nourishment volume density, V, per unit beach length. Based on Equation 1-1, Dean (199 1) developed equations to determine Ay, assuming the native and nourishment sediments were each represented by a single sediment size, which was assumed to be uniform across the profile. However, recognizing that most beaches exhibit a decreasing sediment size in the offshore direction, and that nourishment sediments from a borrow pit include a range of particle sizes, modifications to Equation 1-1 were incorporated to account for these variations (Dean 2001). One alternative allows the sediment size to'be represented as a piecewise continuous distribution across the profile. Another method assumes the A values vary linearly between points where the sediment size is known (Dean and Charles 1994). The effect of representing the nourishment sediment with a range of sizes can have a significant effect on the resulting additional dry beach width. Dean (2000) incorporated this possibility into the calculation of Ay, This newer method assumes a log-normal distribution of nourishment sediment sizes, and considers a cross-shore distribution of the fill sediment. The nourishment sediment size distribution is divided into two portions: one having the same mean size as the native sediment, and the other the mean size of the remaining nourishment sediment distribution. The coarser of these portions is placed closer inshore, while the finer portion is placed farther offshore. The 2/3 power law is applied to each portion. This method permits representation of the nourishment sediment by two sizes. Scope of This Thesis The purpose of this thesis is to investigate a more realistic representation of the native profile and the nourishment sediment distribution. This will hopefully lead to a more accurate prediction of the additional dry beach width resulting from a given nourishment volume density. The method is no longer limited by the description of the native profile with a single sediment size nor by representing the fill sediment as a single size or with a normal distribution. For this purpose, two computer programs were developed to determine Ay,, for nourishment projects with arbitrary native profiles and arbitrary nourishment sediment size distributions. One program is applicable to cases where the nourishment sediment is finer than the native, while the other one is applicable when the nourishment sediment is coarser than the natiave. Chapter 2 presents necessary background theory and more detailed descriptions of previous methods used to represent native profiles and nourishment sediment sizes. The alternative approaches and assumptions used in this study are included in Chapter 3. This chapter also contains descriptions of the Matlab programs developed. In Chapter 4, comparisons are made to results obtained from earlier methods and to a monitored beach nourishment project. Also, a number of recommendations that have surfaced from this study are included. Chapter 5 provides a summary, conclusions and recommendations for future work. Finally, the Appendices contain the Matlab program listings and sample input and output files. CHAPTER 2 BACKGROUND The Phi Scale for Characterizing Sediment Sizes The phi scale is widely used in geology as a means to quantify sediment sizes. It was first introduced by Krumbein (1936), who based it on powers of two. The phi (0) size is given by where D is the sediment particle diameter in mm. This scale is very useful because the size distribution of many sediments can be approximated by a normal distribution when the sediment size is given in phi units. This log-normal probability density function is described by (0-jj)2 1 2= ;[222' where g and a the mean size and standard deviation, respectively, are given in phi units. Previous Method for Representing Native Beach Profiles As mentioned earlier, one of the most popular representations for equilibrium beach profiles is the 2/3 power law presented as Equation 1 -1 (h =Ay"3). This formulation was first proposed by Bruun (1954), who determined it empirically by examining profiles from Denmark and Mission Bay, CA. Dean (1977) found supporting evidence for this based on a study of some 502 profiles extending from the, eastern end of Long Island to the Texas/Mexico border. In addition, Dean (1977) determined that the 2/3 power law was consistent with uniform wave energy dissipation per unit water volume in the breaking zone. Based on linear shallow water wave theory, it can be shown that A is a function of wave energy dissipation (Dean 1977), but it can also be associated with sediment particle size. Moore (1982) examined numerous laboratory and field profiles and determined the relationship depicted by the curve in Figure 2-1, between A and the effective diameter of sediment particles, D. For ease of use, Table 2-1 presents A values for 0.01 mm increments of sediment diameters ranging from 0.100 to 1.09 mm. It should be noted that A has units of length to the one-third power. Sediment Fall Velocity, w (cm/s) 0 ,01 0 1 1.0 10.0 100.0 1.0 Suggested EmpiricalA{mS) Reltionship A vs. 0 ~From Hughes'From Individual Field Field Results A 0067 W4 Profiles Where a Rang of Sand Sizes was Given 0,10 --"\_ __ Laboratory Results 0.001 0.01 0,1 1.0 10.0 100.0 Sediment Size, D (mm) Figure 2-1. Profile scale parameter, A for varying sediment diameter, D (from Dean and Dalrymple 2002). In earlier work by Dean (1991 and 2000), the additional dry beach width resulting from a beach nourishment project is determined assuming that the native profile is represented by a single sediment size. The shape of the native profile is given by Equation 1-1. According to this formulation, the shape of the profile is a function of the profile scale parameter, A, only. Therefore, given a single representative sediment diameter, the entire native profile is defined. This is not very realistic, for it is known that in nature the sediment size varies across the profile. Table 2-1. A values (m13) for given sediment diameters d (mm). d (mm) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.063 0.0672 0.0714 0.0756 0,0798 0.084 0.0872 0.0904 0.0936 J.(096 0).2 0.100 (1.10)3 0.106 0.109 0.112 0.115 0.H17 0,19 0.121 0.123 o.3 0.125 0.127 0.129 0.131 0.133 )0.135 0.137 0.139 0.141 0.143 0.4 0.145 0.1466 0.1.482 11498 0.514 0.153 0.1546 0.1562 0.1578 0.1594 0.5 0_16t 0.1622 0.1 (3 4 01646 0.1658 0,167 0.1682 0,1694 0. 17M1 0.1718 0,6 0.173 0.1742 0.1754 0,1766 0,1778 0.179 0.1802 0.1814 0.1826 0.1838 0.7 0.185 0.1859 0..1868 0,1877 0.1886 0.1895 0.1904 0.11t3 0.1922 ().1931 0.8 0.194 01948 1.1956 0.1964 0.1972 0.198 0.1988 0.1996 0.2004 0.2012 0.9 0.202 0.2028 0.2036 0.2044 0.2052 0.206 0.2068 0.2076 0.2084 0.2092 1.0 0.210 0,2108 (1.2116 0.2124 0.2132 0.2140 0.2148 0-2156 0.2164 0.21.72 (from Dean and Dalrymple 2002) Previous Methods for Representing Fill Sediment Sizes Through experience, monitoring and research, design methods for beach nourishment projects have improved with time. Earlier methods were based only on the comparison of native and nourishment sediment sizes, i.e. a granulometric basis. More recently, the concept of equilibrium beach profiles has been incorporated into design as a means of accounting for the profile response in general and equilibrated dry beach width in particular which is of greatest interest to the stakeholders. Granulometric Basis Krumbein and James (1965) proposed a method which assumed that both native and nourishment sediment sizes were represented by log-normal distributions (Equation 2-2). They considered that the portion of the fill with the same distribution as the native sediment would be "compatible". In this manner, they discounted both the portions of the nourishment distribution which were finer and coarser than the native. They then defined the "overfill" factor as the number of units of nourishment material required to provide one equivalent unit of native sediment. Since it did not seem reasonable to discount the coarser portion of the fill sediments, Dean (1974) modified the Krumbein and James method, discounting only the finer portion of the fill sediment by requiring that the mean of the remaining portion of the distribution be the same as the mean of the native sediment distribution. This resulted in a reduction of the overfill factor calculated. The overfill factor, K, determined with this method is shown in Figure 2-2. Here, "F" represents the nourishment sediment while "N" represents the native sediment. 2.0 - Note: and ar are tNe ~Moan and Standard DevOion Measured .0 ",, * 0.1 012 0.4 0 6 0810 2.0 3.0 1.0 0- 8 1.0 Figure 2-2. Overfill factor based on Dean's (1974) method (from Dean and Dalymple 2002). The method adopted by the U.S. Army Corps of Engineers Shore Protection Manual (1984) was the procedure developed by James (1974). This method assumes that once in place, the finer portion of the fill sediment will be winnowed out until the sediment size distribution remaining is the same as the native sediment distribution. In addition to the overfill factor (RA), James defined a renourishment factor (Ri). The renourishment factor, is the ratio of the frequency of renourishment for a project constructed with a particular borrow material to the frequency of renourishment if the project were constructed with fill sand with the same distribution as the native. The new Coastal Engineering Manual (U.S. Army Corps of Engineers 2002) presents the overfill factor for use only in cases where the native sand size is significantly different from the fill sand. However, it no longer recommends the use of the renourishment factor for beach nourishment design. Equilibrium Beach Profile Method Defining the compatibility of a given nourishment material through granulometric considerations has the disadvantage of not representing the forces that shape beach profiles and not focusing on the equilibrated dry beach width which is of greatest concern to the stakeholders. The following methods are based on the concept of equilibrium beach profiles and are founded on the 2/3 power law postulated by Bruun (1954) and Dean (1977), h=Ay213. According to this method, once a beach nourishment project is in place it will be modified by waves toward an equilibrium shape. As shown in Figure 2-3, the project evolves in the cross-shore and alongshore directions. Planform equilibration occurs as alongshore "spreading out" losses. This occurs because the project attempts to approach the planform existing without beach nourishment. In cross-shore equilibration, the project will tend to attain an equilibrium profile that depends on the volume of sand 10 placed ( ', the berm height (B), the native and fill sediment sizes (DN and DF, respectively), and the depth of closure (h,). The depth of closure is that depth to which a profile will equilibrate. As defined by Hallerneier (1978), it is the "limit of intense bed activity." Hallermeier determined the closure depth to be h. = 2.28H- 68.5 H,' CgrT2 [2-3] where g is gravity, He is the effective significant wave height which is exceeded only twelve hours per year, and T is the corresponding wave period. He can be approximated from the annual mean significant wave height, H, and the corresponding wave height standard deviation, TH, He = H+5.6cr- [2-4] -Original Shoreline --. Spreading Our Lossos Sand moves Offshore to J Equilibrate Profile - Nournshed Shoreline 'Spreading Ou" Losses (a) Beach Width (Coarse Sand) Initial Placed Profile / IEquitibrated Pmrie (Co/~ ,Sand) Original Profile Eqilibrated Profile (FPe Figure 2-3. Sand transport resulting from nourishment project equilibration (a) Plan view, and (b) elevation view (from Dean and Dalrymple 2002). The Coastal Engineering Manual (U.S. Army Corps of Engineers 2002).stipulates that present guidance from the National Research Council supports the use of equilibrium beach profile concepts for beach nourishment project design. Different methods of determining the evolution of a beach nourishment project have been developed based on equilibrium beach profile concepts. A number of these, methods are discussed below. Single sediment size Dean (199 1) developed a method to determine the additional dry beach width, Ay, resulting from the equilibration of a nourishment project. Assuming that the native and fill sand can each be represented by a single sediment size, DN and DE, respectively, he determined that three different types of profiles could occur: intersecting, nonintersecting and submerged, as depicted in Figure 2-4. Dean (1991) developed equations t o calculate AYo, as a function of the placement volume density ( V, volume per unit length), berm height (B), fill and native sediment scale parameters (AF and AN, respectively), closure depth (h.), and width of the beach profile out to the closure depth on the native profile (W.), in terms of the following relationship of non-dimensional parameters, Ay,=f( V AF B.)[25 W. BW.'A,'I4 [-5 where W.and AF and AN are functions of the sediment sizes DF and D, respectively. Graphical representations of Dean's (1991) equations have been developed as design aides. Figure 2-5 shows an example for the ease of h*/B = 2.0. This is the method presently used by the Army Corps of Engineers as a measure of compatibility (U.S.A.C.E. 1994). The Coastal Engineering Manual provides the equations relevant to this method but does not recommend their use for final fill volume calculations (U.S. Army Corps of Engineers 2002). Figure 2-4. Three types of equilibrium profiles for nourishment projects (from Dean 1991). (a) Intersecting profile AF > AN; (b) non-intersecting profile; (c) submerged profile AF < AN. 13 10.0 ...... ..... ..... ............ .. V02 0.0 0.005 0.001 i .. V......002 0 1.0 2.0 2.8 Figure 2-5. Variation of non-dimensional additional dry beach width Ayo/W. as a function of A' and V' for h./B=2.0 (from Dean 1991). Figure 2-6 shows examples of the variation of Ayo with Vwhen the nourishment sediment is represented by a single size. Included are the cases of fill sediment coarser than, compatible with, and finer than the native for Dv = 0.2 mm. 14 -Range for Beach S Nourishment Project ARIAN=1.2 ./ 180 DF=0 275mmn D4=0.2mm- -" 160 (DFrDN-0.2mm) , 140 Intersectin No ng se D 3.-0.1 N.mm , S120 --.". U " 10 00 o s ars 40 20 P 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Volume Added (m31/m) Figure 2-6. Variation of additional dry beach with nourishment volume density for three different fill sediment sizes. h. = 6 m, B = 2.0 m. (from Dean and Dalrymple 2002). Non-uniform sediment size across profile In reality, the sediment obtained from a borrow pit contains a range of sizes. Also, the fill sediment tends to be reworked by waves such that the coarser particles are located inshore and the finer sediments farther offshore. A modification of Equation 1-1 was developed assuming the sediment size to be piecewise continuous across the profile (Dean 2001). Based on D = D., S ,J Y7 and assuming uniform wave energy dissipation for each segment, it was found that h(y)= [h(, / + A "2(y y)]2, y. This is equivalent to having a separate 2/3 power equilibrium equation for each segment (y, h(y)= h,3/2+ [(A. +m(y-Y,))]52} m n A [2-9] YI -Y. Two-parameter representation (two sediment sizes) The following method for calculating Ayo considers two sediment sizes to represent the nourishment sediment (Dean 2000). It is assumed that the fill sediment can be represented by a log-normal distribution given two parameters: the mean (uF) and the standard deviation (OF). The native sediment is represented by a single sediment size, PN. The fill sediment is then divided into two portions, one having the same mean size as the native distribution, the other the mean size of the remaining fill sediment distribution. The coarser portion (uFf) is placed farther inshore while the finer portion (JF2) is placed seaward of it. It is also assumed that all the fill sediment is sorted during the equilibration process, i.e. none is left unmixed. This method always results in nonintersecting profiles and less extreme results than those obtained from the use of a single nourishment sediment size (Dean 2000). An example of the effect of representing the fill sediment by two sizes is shown in Figure 2-7. In this example, DN= 0.2 mm, B = 2 m, and h. = 6 m. Two cases are considered: 1) nourishment sediment coarser than the native (DF= 0.275 mm, UF= 0.5), and 2) nourishment sediment finer than the native (DF = 0.140 mm, 0F = 0.5). The darker lines represent the cases where the nourishment sand is represented by a single sediment size (i.e. TF = 0), as discussed previously. The lighter lines represent the results obtained with the two-size representation of the fill sediment (i.e. UF # 0). As shown, the effects of the two-size representation on Ayo are considerable. In the case of the fill sediment being coarser than the native, the effect of a > 0 is to decrease the resulting Ay, and for the case of fill finer than the native, the effect is to increase Ayo. Overall, a range of sediment sizes moderates the associated additional shoreline changes. 500- 300 !___ C,, 1 200.......... .. O .,.' ... 0 ....... . ~A 0 100 200 300 Volume Density (yd3/ft) Figure 2-7. Comparison of the effect of representing the nourishment sediment with one vs. two sediment sizes. h. = 6.1 m, B=l.5 m (from Dean 2001). D 0.2 mm N 00 ..... ..... ...... ................ (V ................ CHAPTER 3 METHODOLOGY Representing the Native Profile with a Least Squares Fit The methods presented previously assumed that the native beach could be described with a single sediment size. The present method allows the native beach to be represented by either a single sediment size or an arbitrary (measured) native profile. In the first case, the A value corresponding to the given native sediment size, DN, is used to calculate the shape of the native profile according to the 2/3 power form of the EBP equation. In the second case, the programs permit the user to specify an arbitrary native profile by providing values of offshore distance and their corresponding depths. The profile is not required to be monotonic, hence the use of barred profiles is acceptable. If an arbitrary profile is defined, the programs carry out a least squares fit of the specified profile to the shape described by the 2/3 power EBP equation. From this, a representative value of A is determined for later use as a descriptor of the native profile shape. This scale parameter is calculated from AN- 4/3[31 where y, is the offshore distance (in) and hr, is the corresponding water depth measured (in). This procedure results in a smoothed profile which is more convenient for use in later calculations and is probably a more realistic representation of the actual profile shape than that based on a single sediment size. Figure 3-1 shows an example of this procedure. From the calculated sediment scale parameter, a native sediment size, DN, is obtained from Table 2-1. 0 Measured Profile -Least Squares Fit 2 A 010 mIr3 3 g4 05 6 7 8 0 100 200 300 400 500 600 700 800 Distance Offshore (m) Figure 3-1. Example of a least squares fit of the 2/3 power rule to the profile at the Army Corps of Engineers Field Research Facility in Duck, NC (measured profile modified from Lee and Birkemeier 1993). Since the 2/3 power form of the EBP equation does not describe the beach profile above the SWL, a separate method was adopted to describe this portion of the native profile. It was assumed that this portion of the profile is linear and is uniquely defined by two parameters: the berm height and the natural beach face slope. Accounting for Arbitrary Fill Sediment Sand Size Distributions The methodology presented in this study permits a more realistic description of the nourishment sediment than previous methods. The user is not restricted to defining a single sediment size to describe the entire fill. Instead, the programs allow the nourishment sand size to be represented in two ways: 1) a log-normal distribution defined by the mean, DF (mm), and the standard deviation, CTF (phi units); and 2) an arbitrary distribution defined by sediment sizes (phi units) and their corresponding cumulative frequency of occurrence (proportion finer than). Description of Assumptions and Programs As mentioned previously, two MATLAB programs were developed for this study. One is used for the case of fill sediment coarser than the native (fcoarser.m); the other for the case of fill sediment finer than the native (ffiner.m). The input required by the programs includes the native sediment size (mm) or an arbitrary native profile (m); the nourishment sediment distribution (log-normal or arbitrary); the berm height, B (m); closure depth, h. (m); natural beach face slope, slope._nat (m/m); placement volume density, V (m3/m); and placement slope, slopeplace (m/m). The placement slope is assumed to be steeper than the natural beach face slope. Figure 3-2 illustrates the input variables required by the programs. .1 I i ', \ j ..Native 1 oe t ~~p-~c j 2t 0 100 200 300 400 500 Distance Offshore (m) Figure 3-2. Definition of input variables. From the given nourishment sediment size distribution, the programs determine the cumulative frequency distribution (proportion finer than). The distribution is then divided in two portions. One portion is required to have the same mean size as the native distribution; the other portion contains the remainder of the fill distribution. The limit between these two portions, 0., is shown in Figure 3-3 and is determined by iteration. The latter of the two portions is then subdivided into three bins with equal probability of occurrence. This process results in the fill sediment being represented by four grain size bins, each with a corresponding fraction of the fill sediment volume. Figure 3-4 presents an example of dividing the grain size distribution into four bins. The centroid of each bin, as present in the given fill sediment size distribution, is chosen as the "representative" size for the bin, Jo f (b)do 0,ep =0 [3-2] Ob ff ()dob . where 0,ep is the representative size for a bin and 0a and ob are the limits of the bin. For the case of a log-normal distribution of sediment sizes, Equation 3-2 becomes F2~ e 4TT 01-V = r ek vJ eF( / / [3-3] Ierf O jU erA,-U (" aF1- "( 'CF-2 ) where erf(z) is the error function of argument z given by erf(z) 2 [34 0 2.5 3 Send Size (.) Figure 3-3. Example of dividing the grain size distribution into two portions at 4.. In this example, the mean size of the fill-sediment is finer than the native. DN = 0.20 mim, DF= 0.14 mm, T-F = 0.5. Cumulative Nourishment Sediment Size Distribution (Log-Normal) .8 .6 .2 .....i ,... ,.. .. ..... ...... ...... ...... 1 1.5 2 2.5 3 3.5 4 4.5 Sediment Size (*) Nourishment Sediment Size Distribution (Log-Normal) 6n 4 2 Bin Bin Bin Bin 132i3 4i 1 1.5 2 2.5 3 Sediment Size (0) 3.5 4 4.5 Figure 3-4. Example of dividing the fill sediment grain size distribution into four bins. In this example, the fill sediment is finer than the native and Bin 1 has the same mean size as the native distribution. Bins 2, 3 and 4 have equal cumulative frequencies of occurrence. DN= 0.20 mm, DF = 0.14 mm, (7F = 0.5. It can be shown that by initially dividing the fill sediment distribution in two portions as described above, it is guaranteed that a solution can be attained such that the volume of fill sand moved offshore will extend exactly to the closure depth. Next, the given volume of nourishment sediment is placed on the beach at the specified slope. It is assumed that a percentage of the placed volume remains in-place, undisturbed and un-mixed, while the rest of the placed volume, known as the active volume, is reworked, sorted, and moved offshore as the profile equilibrates. This partitioning of the placement volume into active and un-mixed portions is illustrated in Figure 3-5. The previously determined size bins are then successively placed offshore from the un-mixed volume according to Equation 2-7, k(y)= [h(y. Y /2 + A,,3/2 (Y Y" ] 213, "< Y < Y"J[3-5] The programs ensure that the volume of each grain size placed corresponds to the availability of this size bin according to the given fill sediment distribution. In addition, they guarantee that the total volume of sand moved offshore is equal to the active volume. The sediment size bins are placed such that the size decreases offshore, as shown in Figure 3-5. This trend in grain size variation has been observed commonly in nature. An example of such sediment sorting was noted following the fifth Hunting Island beach nourishment project (Kana and Mohan 1998). Here it was observed that the proportion of coarse material decreased in the offshore direction, and there was a shift towards finer material offshore. The requirement of accounting for sorting processes in profile equilibration responds to suggestions by Medina et at. (1994), who studied the variation of sediment size and beach profile as a function of time and space. Results from this study indicate that erosional processes on a beach do not occur as mass movements of sediment. Instead, each grain size responds independently to hydrodynamic forces, requiring that sediment sorting processes be taken into account to realistically represent nature. -2 -, '------ ... % ... Native \ \ \ I SWL Placement -1 Active Volume ESP J 2 Un-mixed Volume "% A=0.100m/3 A=0.080ml/3 A=0.058m 5 6 7 0 100 200 300 400 500 Distance Offshore (in) Figure 3-5. Sediment size bins are placed with decreasing sediment size in the offshore direction. DN = 0.20 mm, DF = 0. 14 mm, qF = 0.5, B = 2 m, slope_nat =1/30, slopejplace =1/20, h. = 6 m, V= 300 m3/ (a, = 188 M3 /m and Vun..mixed =112 m 3/M). Since the equations for EBP's do not describe the profile shape above the SWL, the shape of this portion of the profile had to be assumed. Above the SWL, the equilibrium beach profiles were extended as straight lines with slopes equal to those of the native beach face up to the pre-project berm height. Based on these assumptions, an iterative process leads to a solution for the EBP. Iterations are stopped when the resulting equilibrium beach profile offshore depth limit (after the entire active volume is placed offshore) is within 0.01 mn of the specified closure depth. The additional dry beach width, Ayo, corresponding to this final profile is given as the solution for the specified placement volume. Figure 3-6 shows an example of the final profile resulting from this procedure. AY0 Native -2 .. .. .. % S W L --- Placement - EBP 0 -1 :S% CL 3- h=h. 4 5 6 7 8 0 100 200 300 400 500 Distance Offshore (m) Figure 3-6. Final EBP and Ayo obtained when the depth farthest offshore is equal to h.. DN = 0.20 mm, DF = 0.14 mm, 0rF = 0.5, B = 2 m, slope_nat = 1/30, slope_place = 1/20, h. = 6 m, V= 300 m3/m, dyo= 26.3 m. As described above, the programs assume that cross-shore sediment transport is dominant over longshore transport, thus considering the time scales required for crossshore profile equilibration to be shorter than those required for planform equilibration. Therefore, longshore transport is disregarded in the procedures described above. In addition, longshore sediment transport is known to vary along the length of a nourishment project, being more pronounced at the ends of project areas, where a greater loss of sediment is expected. Therefore, the results obtained from these programs might also be interpreted as more applicable to the central region of the project area, where the effects of longshore transport are less pronounced (Work and Rogers 1998). The output of the programs includes plots of the sediment size frequency distribution, the cumulative frequency distribution (proportion finer than) and the final equilibrium beach profile. The resulting value of the additional dry beach width (Ay,) and the percentage of the nourishment volume which becomes the active volume (PercentVA cT) are also provided. Table 3-1 summarizes the most significant similarities and differences between the earlier method of representing the nourishment sediment with two sediment sizes and the current four-bin method. Table 3-1. Comparison of earlier two-size representation to present four-size method. Similarities One fill sediment size bin has the same mean size as the native sediment Cross-shore sediment size variation (decreasing size in offshore direction) :~~WjDifferences . Two-bin Four-bin 2 representative fill sediment sizes 4 representative fill sediment sizes Log-normal fill sediment size distribution Log-normal or arbitrary size distribution Single native sediment size Single native size or arbitrary native profile Entire volume is sorted Only active volume is sorted and moved Vertical beach face Sloping beach face The details of the procedures used in arriving at the final EBP for the case of fill sediment coarser and finer than the native are different. Following is a discussion of some relevant issues for each case. Fill Sediment Finer than the Native (ffiner.m) For the case of nourishment sediment finer than the native and assuming a log-normal distribution of fill sand, the value of can be obtained iteratively from O"F e(0._ F)'/2 2 YN := PF + [. [3-6] 1 [1 +erf7;*LJ- F] where /F and #N (phi units) are the mean sediment size of the fill and native sediment size distribution, respectively, and 0F (phi units) is the standard deviation of the fill distribution. The procedure for arriving at the final EBP requires guessing at values of Ayo and determining the corresponding EBP. As noted previously, the iterative process is stopped when the offshore limit of the EBP is the same depth as h. ( 0.01 m). For a given value of dyo the entire fill volume is initially placed in a trial EBP according to the A value of the coarsest nourishment sediment size bin. This trial EBP for Ayo is used along with the previously determined fill placement profile to determine the active volume, VACT. Figure 3-7 shows an example of this trial equilibrium profile and the associated VAcT. This calculation assumes that once the profile equilibrates, the un-mixed portion of the placed volume will take the shape dictated by the coarsest grain size bin. Once VAcTis determined, this volume of fill is placed offshore from the unmixed portion of the fill. The four size bins are placed one after the other beginning with the coarsest bin, which is placed closest to shore. The program ensures that the volume of each sediment size placed corresponds to the percentage that is available according to the segmented sediment size distribution. Once the entire active volume is placed in the offshore section of the EBP, the final offshore depth (h Jinal) is determined and compared with h. to determine if the current value for Ayo is correct. Figure 3-8 shows an example of the placement of the different size bins and the resulting EBP. Appendix A contains the program listing forffiner.m, a sample input file, and the corresponding output. -2 --- %-- Native -SWL I I -- Placement - \\V Trial EBP ACT 2 --. "A" from coarsest bin 7 -.. 0 100 200 300 400 500 Distance Offshore(m) Figure 3-7. Procedure for establishing the active volume from a trial EBP for cases with DF A=0.100m1/3 '%,% A=0.080ml/3 5 A- 0.058 1 6 7I I I 0 100 200 300 400 500 Distance (m) Figure 3-8. EBP showing decreasing trend in grain size offshore. Fill Sediment Coarser than the Native (fcoarser.m) For the case of nourishment sediment coarser than the native and assuming a lognormal distribution of fill sand, the value of 0* can be obtained iteratively from PM[1 lerf('-/PF ] [3-7] The procedure used in the case of fill sediment coarser than the native is more complicated than that for fill sediment finer than the native. Instead of placing the sediment size bins from onshore to offshore, this procedure places the offshore bins first. Due to this complication, the active volume becomes an added unknown that must be solved for iteratively. The program commences by assuming a value for X%, the percentage of the nourishment volume that is active, i.e. X% = AC ) X 100%. For each value ofXo (and consequentially, VACT), an iterative procedure is used to determine the distance offshore that h, should be placed such that once the fill sediment bins are placed inshore from this point, the total volume placed as active volume is equal to the assumed VAcT ( 1%). This offshore distance, Yo, is shown in Figure 3-9. Once the correct value of Y, for the given value ofXAiD is found, the actual active volume that results from the inshore portion of this EBP is determined. This resulting active volume, ResVAcr, is shown in Figure 3-10. If the difference between Res_VAcT and VACT is less than 1%, this value of X% and its corresponding Y, and EBP are used to obtain the value of Ayo. If the difference between ResVAcT and VACT is greater than 1%, a new guess of X% is assumed and the procedure is repeated. In placing the grain size bins across the profile, it is assumed that the profile follows the shape dictated by Equation 3-2. However, if the profile slope exceeds the specified natural beach face slope, from that point inshore, the profile is replaced by a straight line with the slope of the natural beach face. This is done to simplify complications that arise from geometry issues and to eliminate (to some extent) the unrealistic infinite slope that results at the shoreline when using the 2/3 power rule equation for EBPs. If the equilibrium profile intersects the placement profile underwater, the un-mixed part of the fill volume is assumed to be shaped according to the A value corresponding to the mean of the fill sediment distribution, DF. The mean size, DF was selected rather than the representative size of the coarsest bin (as done inffiner.m) because for the ease of DF > DN, the coarsest bin represents a small percentage of the total placement volume. If would be unrealistic to assume that the entire un-mixed volume would take on this shape. Appendix B contains the program listing forfcoarser.m, a sample input file, and the corresponding output. -2 Native - SWL --- Placement -1 ~ -EBP C y I h. 2 ....,, a."k. 100 150 200 250 300 350 400 Distance Offshore (m) Figure 3-9. Example of trial Y. -50 0 50 100 150 200 250 300 350 400 450 Distance Offshore (m) Figure 3-10. Res VAcT is calculated once the correct Yo value for a given X% is found. DN= 0.20 mm, DF = 0.275 mm, orF = 0.5, V= 300 m3/m, B = 2 m, h = 6 m, slope_nat = 1/30, slope_ place = 1/20. CHAPTER 4 RESULTS AND DISCUSSION Comparison with Previous Methods The effect of representing the nourishment sediment with four sediment size bins was compared to results obtained with earlier methods. For this comparison, it was assumed that the fill sediment could be represented with a normal distribution of sediment sizes. The results obtained by representing the fill sediment by a single size and with a twoparameter normal distribution (two size bins), are compared with the results obtained applyingfcoarser.m and ffiner.m (four size bins). For each method, the cases of fill sediment coarser and finer than the native are presented. Also included are the results obtained when assuming a single fill sediment size compatible with the native. The effect of a varying placement volume density on the additional dry beach width is depicted for all the above cases in Figure 4-1. Throughout the example, the parameters used included: native sediment size, DN =0.20 mm; berm height, B = 2 mn; and closure depth, hlz= 6 m. For the cases with a normal distribution of fill sediment sizes, the standard deviation was taken as qF = 0.5. For cases with fill sediment coarser than the native, DF = 0.275 mm; whereas for cases with fill sediment finer than the native, DF 0. 14 mm. Additional input for the more recent programs included a natural beach face slope, slope _nat = 1/30 and a fill placement slope, slope place = 1/20. In Figure 4-1, the bold lines represent the cases where the nourishment sediment is represented by a single sediment size. The dash-dotted lines are the two-parameter representation assuming a log-normal distribution of nourishment sediment sizes and two size bins. The lines with markers represent the solutions obtained usingfcoarser.m andffiner.m (both representing the nourishment sediments with log-normal distributions and four size bins). Comparison of Dry Beach Width for Varying Values of Volume Placed F=0.5 0 100 200 300 400 500 Placement Volume Density (m3Im) 600 700 Figure 4-1. Comparison of predicted additional dry beach width for different methods of representing the nourishment sediment. In studying the results obtained withffiner.m andfcoarser.m and comparing them with results from previous methods, it was determined thatfcoarser.m might not accurately represent some scenarios. Although the qualitative results illustrate expected trends, quantitative comparison with results from previous methods should be performed with much caution and good judgement. This is especially the ease for conditions when the placement volume is small and when DF is only slightly larger than DN. Further explanation of these circumstance and their effect on the results is given below. Althoughfcoarser.m must be used with caution, its results are still significant in studying the effects and complications of describing an equilibrium profile with a variety of sediment sizes for the case of DF > DN. Although the case of DF > DN is relevant, most beach nourishment projects constructed make use of fill sand finer than the native (DF < DN) due to restrictions imposed by the availability and feasibility of obtaining coarser sand. Therefore, ffiner.m should be applicable to most nourishment projects. Figure 4-2 presents the nourishment sediment size distribution for the case of nourishment sediment coarser than the native (example in Figure 4-1). Figure 4-3 presents the four size bins corresponding to this scenario. As shown in Figure 4-1, the Adyo values predicted byfcoarser.m are always greater than those predicted by the two-bin method. This can be attributed to the newly introduced concept of "active volume". The two-bin method assumes that the entire fill volume placed on the beach will be reworked and sorted such that the portion with the coarser size is placed inshore and the portion with the finer size is placed offshore from it. Therefore, unless oF is large, the finer portion does not remain on the beach as part of the beach berm, since this upper portion of the profile is preferentially composed of the coarser size. However, using the concept of an "active volume",fcoarser.m assumes that a portion of the fines remains inshore as part of the un-mixed volume and contributing to the additional dry beach width as part of the beach berm. Therefore, unlike the two-bin method, withfcoarser.m, not all the fines are moved offshore. This results in a smaller percentage of the fine sand being moved offshore. In addition, the coarse bin used by the two-bin method is subdivided into three bins when usingfcoarser.m. As a result, fcoarser.m uses some bins that are coarser than the one used by the two-bin method. The coarser bins result in steeper slopes and wider beaches. As observed in Figure 4-1, the Jyo values obtained withfcoarser.m are initially smaller than those obtained by assuming a single nourishment sediment size. This is expected sincefcoarser.m is based on a log-normal distribution of sediment sizes. Therefore, it accounts for a range of sizes including some that are finer than DF. These finer sediments are sorted out and moved offshore, where their effect on the dry beach width is not as pronounced. As Vincreases, the Ayo values obtained withfcoarser.m surpass those predicted for a single nourishment sediment size. This can be attributed to the four size bins thatfcoarser.m uses to represent the fill sediment. The coarser bins will have a representative sediment size that is larger than the single size used by the single grain size assumption. The steeper equilibrium slopes resulting from these coarser sediments yield a wider beach. This effect is larger for increasing values of V. Figure 4-2. Nourishment sediment size distribution for the case of DF > DN. 0. 0 WO. . -0. 0. 0. Cummulative Nourishment Sediment Size Distribution (Log-Normal) I I I i .8 ............... .4 .2 ,0 0. 1.5 2 2.5 .6 .4 .2 0 0.5 1 1.5 2 2.5 Sediment Size () Nourishment Sediment Size Distribution (Log-Normal) 3 3.5 1.5 2 Sediment Size (#) Figure 4-3. Four sediment size bins used byfcoarser.m (DF > DN). Figure 4-4 presents the nourishment sediment size distribution for the case of nourishment sediment finer than the native (example in Figure 4-1). Figure 4-5 presents the four size bins corresponding to this scenario. As is observed in Figure 4-1, for the case of nourishment sediment finer than the native, the Ayo values predicted byjfiner.m are always larger than those predicted with the two-bin and the single sediment size assumptions. Dividing the fill sediment distribution into four bins with one having the same mean as the native guarantees that a finite Ay value will always be obtained with ffiner.m. This is not the case when a single sediment size is used to represent the fill. As seen in Figure 4-1, for that case, there is a threshold value of Vbelow which no additional dry beach width will result. The procedure used byffinerm divides the sediment distribution into four bins guaranteeing that one of the bins will have a mean size greater than DF. This results in greater Ayo values than obtained when assuming the entire fill is of size DF. The A,,y values predicted byffiner.m are also expected to be greater than those obtained with the two-bin method. This can be attributed to the "active volume" concept explained above. In addition, sinceffiner.m subdivides the finer bin used by the two-bin method into three smaller bins, one or two of these will be guaranteed to be coarser than the original bin, resulting in a wider beach. 1 1,5 2 2.5 3 Sand Size (0) 3.5 4 4.5 Figure 4-4. Nourishment sediment size distribution for the case of DF < DN. 0 0* I" 0. 4) .2 - 0. E E S0. Cummulative Nourishment Sediment Size Distribution (Log-Normal) .8 .6...... . .4 .2 1 1.5 2 2.5 3 3.5 4 4.5 Sediment Size (40 Nourishment Sediment Size Distnbution (Log-Normal) Sediment Size () Figure 4-5. Four sediment size bins used byffiner.m (DF < DN). The effect of varying the sorting of the nourishment sediment size distribution was studied forffiner.m andfcoarser.m. For DF > DN, it was determined that increasing o'F decreases the predicted dry beach width, as shown in Figure 4-6. As 0F increases, the proportion of the fill sediment size distribution having the same mean as the native increases. Consequently, the proportion of the fill sediment size distribution corresponding to the coarser bins decreases. The total percentage of fines in the fill increases, leading to a decreasing additional dry beach width. Although the mean sizes of the coarser bins increase with increasing ry, these bins become very small, which decreases their effect. In addition, they become so steep that they are quickly replaced with the critical natural slope of the beach, slope nat. For the case of DF 7 shows that the trend is the opposite. As cF increases, the proportion of the fill sediment distribution having the same mean size as the native increases. Since this corresponds to the coarsest sediment size bin, it results in an increase in the additional dry beach width. Although the mean sizes of the three finer bins decrease, tending to decrease the resulting additional dry beach width, this does not occur because the effect of these bins is diminished since the proportion of the fill sediment corresponding to these sediment sizes decreases with increasing qF. 80 . 70 60I 50 U 'U M 40 30 20 10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Sorting of Nourishment Sediment, OF (0) Figure 4-6. Effect of the sorting of nourishment sediments on the additional dry beach width (DF > DN). _N = 0.20 mm DF = 0.275 mm B=2m h.=6m V = 300 m3/m slope_nat = 1/30 slope-place = 1/20 40 35 m 20 0 0 5 0 0.0 0.5 1.0 1.5 2.0 Sorting of Nourishment Sediment, rF (0) Figure 4-7. Effect of the sorting of nourishment sediments on the additional dry beach width (DF < DN). Figure 4-8 shows the effect of varying DF while holding Dv constant. The results labeled as biclassf correspond to those obtained using the Fortran program for the twobin method. Figure 4-8 includes Ayo predictions corresponding to DF > DN and DF < DN for the two-bin and four-bin methods. The star plotted for DF = DN corresponds to the assumption of a single size compatible sediment. As mentioned previously, the dry beach widths predicted by the four-bin method are greater than those predicted by the two-bin method. For the case of DF < Div, as the value of DF approaches DN, the additional dry beach width predicted byffiner.m approaches that predicted by the two-bin method (biclass.fi. This occurs because the procedure for dividing the sediment distribution into four bins assigns an increasing percentage to the bin with the same mean as the native as DF approaches DN. Therefore, the resulting segmented distribution closely approximates that used by the two-bin method. In addition, as DF approaches DN, the additional dry beach width predicted by both methods approaches the value obtained from the compatible sediment assumption. This occurs because the resulting distributions allocate most of the volume to the bin with mean DN, leaving very little to the coarser bins. Therefore, they essentially represent the fill sediment as a single size with mean DN. Accordingly, it is expected that as DF approaches DV, the predictions provided byfcoarser.m and biclassf for DF > DN should converge to the result predicted by the compatible sediment assumption. As seen in Figure 4-8, this is the case for biclass.f. As illustrated, for the case ofDF > DN, the two-bin assumption requires that a threshold value for DF be surpassed before an increase in the additional dry beach width is obtained. When DF is close to DN, the sediment volume allocated to the coarser bin is so small that it all resides within the berm. The total additional dry beach is then determined by the finer bin (DN) until DF is large enough that the corresponding coarse volume extends beyond the berm and thus is active. The results portrayed in Figure 4-8 indicate that forfcoarser.m a "jump" occurs near DF= DN, indicating that the results obtained from this program might not be as reliable. This 5 m jump near DNv implies that the program may be over-predicting the additional dry beach width. A closer look at the program's assumptions and routines indicates that the problem may reside in the geometrical complications that stem from the many different configurations the equilibrium profile may take close to its intersection with the placement profile, especially when the sediment volume corresponding to the coarser bins is small. It is recommended that further refinement in this area of the program be achieved. However, this may also identify a shortcoming in the basic assumptions offcoarser.m. If-the suggested modifications are included and it is found that their effect is very significant, it may be that the assumptions allot excessive importance to very small volumes of sediment. It is questionable whether such small volumes of coarse sediment should have such a great effect on the equilibrium profile and the corresponding additional dry beach width. 80 DN = 0.20 mm 70 OF = 0.5 B=2m hb. =6 m 60 slopenat = 1/30 slope place = 50 w 40 30 *0 3 S20 10 0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Mean Nourishment Sediment Size, DF (mm) Figure 4-8. Variation of the additional dry beach width with DF when Dj is constant. It may be concluded from Figure 4-8 that the estimates obtained with ffiner.m and fcoarser.m may only be an improvement over those predicted by the two-bin method when the native sediment size is sufficiently different from the mean nourishment sediment size. Figure 4-9 presents an example of a skewed nourishment sediment size distribution that would previously have been required to be approximated by a log-normal distribution. The programsffiner.m andfcoarser.m allow the actual size distribution to -4-- ffiner.m --I- biclass.f (DF -X-biclass.f (DF>DN) be used in determining the additional dry beach width resulting from nourishing a beach with this fill sand. Nourishment Sand Size Distribution (Arbitrary) Sand Size (4,) Cumulative Distribution (Finer Than) -1 0 1 2 3 4 5 6 Sand Size (4) Figure 4-9. Example of an arbitrary nourishment sediment size distribution and its lognormal approximation. (a) Frequency.distribution; (b) cummulative frequency. Delray Beach Nourishment Project: A Case Study Application Between November and December 1992, Delray Beach was nourished with approximately 914,800 cubic meters of sediment obtained from offshore borrow pits. The project was constructed over 3.14 km and represented the third periodic renourishment of the beach since its initial nourishment in 1973. Figure 4-10 presents the average profiles representing the project in the pre-nourishment (October 1992), postnourishment (December 1992) and 48-month post-placement (January 1997) conditions. These were obtained by averaging the profiles corresponding to the middle half of the project area for each survey. The common point for each set of profiles averaged was taken to be the waterline. Figure 4-11 shows the borrow sediment size distribution with a mean of 0.27 mm and a sorting of 0.74 0. The parameters for the project were used as input forffiner.m. The closure depth was taken as 7.3 m, the berm height as 2.6 m, and the equivalent A as 0.137 m It was determined from the averaged profiles that the natural beach face slope was approximately 0.151 m/m while the placement slope was milder at approximately 0.0491 in/m. Becauseffiner.m requires the placement slope to be steeper than the natural slope, the estimated slopes for Delray Beach were unsuitable for use with the program. Program runs were carried out assuming that the estimated natural slope, placement slope, and the average of both was applicable. It was found that the variation in the predicted additional dry beach was within 0.5 m. The resulting predicted equilibrium beach profile for the case of a placement volume of 327 m3/m (estimated from the averaged 1992 profiles) is shown in Figure 4-12. According to the average profiles presented in Figure 4-10, the additional dry beach width for January 1997 is 21.0 m. Ffiner.m predicts an equilibrium dry beach width of 29.0 m; whereas biclassfpredicts a value of 24.8 m. Thus, both programs overestimate the additional dry beach width. One shortcoming of both programs is that they do not take into account longshore spreading losses that occur as sediment from the project area is transported into the neighboring beaches. This would result in a decreased additional dry beach width. In addition, Figure 4-10 shows a large bar in the January 1997 profile. This bar is the result of the offshore sediment transport caused by Hurricane Gordon in November 1994. As is seen, the profile may still be recovering from this storm and the bar may still be moving onshore. This accumulation of sand in a bar results in a decreased dry beach width. Since the profile may not be in equilibrium due to Hurricane Gordon, it is difficult to compare the predictions offfiner.m and biclassfwith the January 1997 data. An additional run offfiner.m was carried out assuming that volume was conserved (i.e. the placement volume corresponds to the volume of fill sand remaining in January 1997, as determined from the average profiles). Figure 4-13 shows the resulting equilibrium beach profile this case ( V=1 87 m3/m). The additional dry beach width predicted by ffiner.m was 16.2 m, while that predicted by biclass.fwas 14.0 m, and as noted earlier, the measured width was 21.0 m. If the fill sediment were assumed to be represented by a single grain size (the mean of the borrow sand size distribution, DF = 0.27 mm), the corresponding A value would be 0.119 m1/3. A least squares fit of the 2/3 power form of the EBP equation to the average January 1997 profile yields an A of 0.153 m1/3. Therefore, representing the fill sediment by a single size would result in a milder profile and a narrower beach width than was measured in January 1997. This indicates that representing the fill sediment with a distribution of sediment sizes might be more realistic than assuming the entire fill is composed of a single sediment size. Figure 4-14 presents the average profile surveyed in January 1997 and the mean sediment sizes of samples collected along a survey line within the middle half of the project. .. Pre-nourishment (Oct 92) Post-nourishment (Dec 92) 48-month post-placement (Jan 1997) t SWL It t.\ i I -200 -100 0 100 200 300 400 500 600 Distance Offshore (m) Figure 4-10. Average profiles for the 1992 Delray Beach nourishment project. 0.6- ; 0.6 -distribution 0.5 0.4 u- 0.3 0,2 0.1 0' -3 -2 -1 0 1 2 3 4 5 Sand Size (#) Figure 4-11. Borrow sand sediment size distribution (Delray Beach 1992 nourishment). bWL . Pre-nourishment (Oct 92) % .Post-nourishment (Dec 92) 48-month post-construction (Jan 97) -2.5- 1. ffiner.m prediction (V=327 m3/m) 0 7.5 -, 10 -1 00 0 100 200 300 400 500 Distance Offshore (m) Figure 4-12. Ffiner.m results for the 1992 Delray Beach nourishment project (V=327 m /m). Distance Offshore (m) Fi ure 4-13. m /m). Ffiner.m results for the 1992 Delray Beach nourishment project (V=187 a) Profile -5 10 5 -100 0 100 200 300 400 500 600 Distance Offshore (m) 10.6 N 0.4 af) 0.2 200 300 Distance Offshore (m) Figure 4-14. Measured cross-shore sediment size distribution for Delray Beach (January 1997). As is seen from the significant effect of Hurricane Gordon on the beach profile, it is obvious that a beach profile may not be solely determined by sediment size. Other factors, such as wave height may play an important role in determining a beach's equilibrium profile. As an attempt to study this effect on Delray Beach, a comparison of the A values before and after the storm was carried out for the average profiles outside the nourishment area. As determined from the WIS hindeast study for this area, the average significant wave height during Hurricane Gordon was 2.5 m for a nearby offshore station. This represents double the WIS estimated annual average of 1.25 m. A least squares fit of the 2/3 power form of the EBIP equation (allowing the origin of the profile to float) was used for the average profiles outside the nourishment area for the pre-storm (December 1993) and post-storm (December 1994 and December 1995) conditions. The pre-storm A value was 0.15 rn"3 while the post-storm values where 0. 145 rn"3 and 0. 14 rn/3 for 1993 and 1994, respectively. The pre-storm profile became milder after the storm. This may be attributed to the effect of the large wave heights associated with Hurricane Gordon. The increased wave energy experienced during the storm also resulted in an increase in the depth of closure for the area. The pre-storm closure depth was estimated at 5.5 m; whereas the after-storm value was 7.2 m. Further Improvements The complications encountered in this study for the case of DF > DN are helpful in identifying possible courses of action to be taken in future attempts to describe these profiles. One possibility is that representing the nourishment distribution with four bin sizes might be unrealistic, since the coarsest bin sediment sizes might become disproportionally large as DF approaches L)N, giving a seemingly great importance to a small volume of sediment. The sediment distribution might be represented with two bins, as is done by biclass.f. An improvement to this would be to include the concept of the active volume. Further studies should be carried out to determine the dependence of A on factors other than the sediment size. As suggested above, one possible parameter would be the wave height. In addition, it is recommended that monitoring procedures for beach nourishment projects include additional sand sampling throughout the project lifetime. This would aide in the understanding of the cross-shore variation of sediment sizes with equilibration and might improve the assumptions made by programs such asffcoarser.m and ffiner.m. CHAPTER 5 SUMMARY AND CONCLUSIONS Two Matlab programs were developed to determine the additional dry beach width resulting from a beach nourishment project carried out with sediments characterized by a range of sediment sizes. One program is applicable for the case of fill sediment coarser than the native (fcoarser.m); the other one is applicable for the case of fill sediment finer than the native (ffiner.m). Current methodology to predict additional dry beach widths is based on assuming that the nourishment and native sediments can each be represented by a single sediment size. However, it is well know that in nature, beaches and sand available from borrow pits exhibit a distribution of grain sizes. The programs developed in this study allow the nourishment sediment to be represented more realistically by either a log-normal distribution with a specified mean and standard deviation, or by a user specified arbitrary sediment size distribution. In addition, the programs allow the native profile to be described by either a single sediment size or by an arbitrary native profile. The methodology applied assumes that in the time scales applicable to cross-shore equilibration, longshore spreading losses are negligible. The nourishment sediment size distribution is divided into four bins. One bin is required to have the same mean size as the native beach sediment. The remainder is subdivided into three bins with equal cummulative frequencies of occurrence. The methodology applied assumes that as a project equilibrates, a portion of the fill sand will remain undisturbed and un-mixed in the onshore portion of the profile, while another portion, the "active volume", will be sorted and moved offshore. The active volume is assumed to equilibrate with the finer sediment being located farthest offshore, consistent with observations of natural profiles. A comparison of the predictions obtained with this four-bin representation of the fill sediment to the results obtained from previous methods indicates that due to the effect of the active volume, fcoarser.m andffiner.m estimate greater dry beach widths than previous methods for most cases. The effect of the different assumptions offcoarser.m andffiner.m is most noticeable when DF is sufficiently dissimilar to DN. As expected for DF very similar to DN, the results fromffiner.m converge to the predicted dry beach width obtained assuming a single size fill compatible with the native. Howeverfcoarser.m exhibits a noticeable "jump" in this case. This and other unexpected results indicate that the procedure applied tofcoarser.m might not be applicable to cases when DF is very similar to DN. In such cases, the volumes allocated to the coarse grain size bins are very small, leading to geometrical complications. In addition, results indicate that a disproportionate significance may be placed on very small volumes of coarse sediment. Although the quantitative results provided byfcoarser.m might not be as reliable because of these issues, the qualitative information supplied by the program is still valuable. Since most nourishment projects are carried out with fill sediment finer than the native, most scenarios can be covered withffiner.m. The 1992 Delray Beach nourishment project was used as a case study. The additional dry beach width predicted withffiner.m was larger than the measured value for the 48month post-construction survey. The discrepancy can be attributed to longshore spreading losses and the effects of Hurricane Gordon in 1994. The storm conditions resulted in movement of sediment to a large offshore bar and a milder beach profile. A number of improvements forfcoarser.m are recommended in this study. An attempt might be made to address the geometrical complications encountered. Another option would be to revert back to a two-bin representation of the fill sediment for cases of nourishment sediment coarser than the native. The concept of an "active volume" could then be incorporated in this two-bin representation. In addition, it is recommended thatffiner.m andfcoarser.m be modified to account for longshore spreading losses and the possible loss of fines offshore from the closure depth. Further improvement might result from the inclusion of the effect of wave height on the beach profile shape. In addition, a detailed study of the dependence of the profile scale paramater, A, on wave height and other factors should be carried out. APPENDIX A PROGRAM LISTING FOR FFINER.M % ffiner.m (MAIN PROGRAM) % DF clear all warning off % DEFINE GLOBAL VARIABLES global miuF miuN sigmaF V B slopenat slope_place ANdes A-rep dyl yeqb delta_Vact Vl Dy yI n hint yint V global Vact FI numbins hI h eqb Percent Vact dyo global num den bin farb phiarb % LOAD INPUT DATA inputdata numbins=4; % TABLE OF A VALUES d tab=0.1:0.01:1.09; d_tab=(round(dtab.*100))./100; A_tab=[0.063 0.0672 0.0714 0.0756 0.0798 0.084 0.0872 0.0904 0.0936 0.0968 0.1 0.103 0.106... 0.109 0.112 0.115 0.117 0.119 0.121 0.123 0.125 0.127 0.129 0.131 0.133 0.135 0.137... 0.139 0.141 0.143 0.145 0.1466 0.1482 0.1498 0.1514 0.153 0.1546 0.1562 0.1578 0.1594... 0.161 0.1622 0.1634 0.1646 0.1658 0.167 0.1682 0.1694 0.1706 0.1718 0.173 0.1742 0.1754... 0.1766 0.1778 0.179 0.1802 0.1814 0.1826 0.1838 0.185 0.1859 0.1868 0.1877 0.1886... 0.1895 0.1904 0.1913 0.1922 0.1931 0.194 0.1948 0.1956 0.1964 0.1972 0.198 0.1988... 0.1996 0.2004 0.2012 0.202 0.2028 0.2036 0.2044 0.2052 0.206 0.2068 0.2076 0.2084... 0.2092 0.210 0.2108 0.2116 0.2124 0.2132 0.2140 0.2148 0.2156 0.2164 0.2172]; % IF GIVEN DN DETERMINE THE CORRESPONDING SEDIMENT SCALE PARAMETER if nat==1 miuN=-(log(DN)./log(2)); DN=(round(DN.*100))./100; k=find(dtab==DN); ANdes=Atab(k); end % IF GIVEN AN ARBITRARY NATIVE PROFILE DETERMINE THE LEAST SQUARES FIT OF 2/3 POWER RULE if nat==2 AN des=sum(h arb.*y arb.^(2/3))./sum(yarb.^(4/3)); q=find(Atab q2=ql+l; DN=dtab(ql)+(dtab(q2)-d tab(ql))./(Atab(q2)-A tab(ql)).*(ANdesA_tab(ql)); miuN=-(log(DN)./log(2)); end % IF GIVEN A LOG-NORMAL SEDIMENT SIZE DISTRIBUTION: if fill==1 % Determine CDF miuF=-(log(DF)./log(2)); phi=miuF-4.*sigmaF:(miuF+4.*sigmaFmiuF+4.*sigmaF)./30:miuF+4.*sigmaF; f=l./sigmaF./sqrt(2.*pi).*exp(-(phi-miuF) .A2./2./sigmaF.^2); F=1-0.5.*(l+erf((phi-miuF)./sqrt(2)./sigmaF)); % Determine phi star guess phistar=(miuN+miuF)./2; phi star=fzero('eqtnl', guess phistar,optimset('disp','off')); % Plot nourishment sediment size distribution along with phi_star figure plot(phi,f) hold on plot([miuF miuF],[0 max(f)]) plot([miuN miuN],[0 max(f)],'r') plot( Ephistar phi star], [0 max(f)] 'g') xlabel('Sand Size (phi)'); ylabel( 'Frequency'); legend('distribution','miu F','miu N', 'phi *') title('Nourishment Sand Size Distribution (Log-Normal)') axis( [miuF-4*sigmaF miuF+4*sigmaF 0 max(f)]) % Plot cummulative nourishment sediment size distribution along with phi_star figure plot(phi,F) hold on plot([miuF miuF], [0 1]) plot([miuN miuN],[0 1],'r') plot([phi_star phi_star] [0 1], 'g') xlabel ('Sand Size (phi)'); ylabel ( 'Cummulative Frequency'); legend( 'cummulative distribution', 'miu F', 'miu N', 'phi_*'); title('Cummulative Nourishment Sand Size Distribution (LogNormal)'); axis([miuF-4*sigmaF miuF+4*sigmaF 0 1]) % Determine limits of bins (for numbins) Fstar=1-0.5.*(l+erf((phi_star-miuF)./sqrt (2)./sigmaF)); deltaF=Fstar./(numbins-1); FI (1)=1; FI (2)=Fstar; for i=3:numbins FI(i)=FI(i-1) -deltaF; end FI(numbins+l) =0.0; phi_FI=sqrt(2).*sigmaF.*erfinv(1-2.*FI)+miuF; % Determine phi_rep for bins for i=l:numbins ul=(phi_FI(i)-miuF)./sigmaF./sqrt(2); u2=(phi_FI(i+l)-miuF)./sigmaF./sqrt(2); phi rep(i)=sqrt(2./pi).*sigmaF.*(exp(-ul.A2)-exp(u2.^2))./(erf(u2)-erf(ul))+miuF; d_rep(i)=2.^(-phi rep(i)); drep(i)=(round(drep(i).*100))./100; if d_rep(i)>=0.l j=find(dtab==d rep(i)); A_rep(i)=A tab(j); elseif d_rep(i)<0.1 A_rep(i)=0.4.*d_rep(i).^(0.8); end end end % IF GIVEN AN ARBITRARY CUMMULATIVE FILL SEDIMENT DISTRIBUTION: if fill==3 Dbins=length(Darb)-1; phi_arb=-log2 (Darb); for q=1:Dbins farb(q)=Farb(q)-F arb(q+1); end % Find phi 84 and phi 16 corresponding to 84% and 16% cummulative % distribution clear q ql q2 q=find(Farb==0.84); if isempty(q)==0 D_84=D arb(q); phi 84=-log2(D_84); else ql=find(F_arb>0.84); ql=max(ql); q2=ql+1; D_84=Darb(q2)+(0.84-F arb(q2)).*(Darb(ql)D_arb(q2))./(F arb(ql)-F arb(q2)); phi 84=-log2(D_84); end clear q ql q2 q=find(F arb==0.16); if isempty(q)==0 D 16=D arb(q); phi 16=-log2(D_16); else ql=find(Farb>0.16); ql=max(ql); q2=ql+l; D_16=D arb(q2)+(0.16-Farb(q2)).*(Darb(ql)D_arb(q2))./(Farb(ql)-Farb(q2)); phi_16=-log2(D_16); end phi mean=(phi_84+phi_16) ./2; phi_sorting=(phi_16-phi_84)./2; DF_mean=2. A (-phimean) DN if DF mean>=DN sprintf('Warning: DF>=DN, THIS PROGRAM IS NOT APPLICABLE!') pause end % Determine phi star centroid=phi_arb(I); bin=0; while centroid num=O; den=0; for i=l:bin num=num+f_arb(i).*(phi arb(i+l)phiarb(i)).*(phi arb(i)+phi arb(i+l))./2; den=den+f-arb(i).*(phiarb(i+1)-phiarb(i)); end centroid=num./den; end if centroid==miuN phi_star=phi_arb(i+l); else guessphi_star=phiarb(i); num=num-f arb(i). *(phiarb(i+l) phi arb(i)).*(phi_arb(i)+phiarb(i+l))./2; den=den-farb(i).*(phiarb(i+1) -phi arb(i)); phiarb(i) phi arb(i+1) phistar=fzero( 'eqtnl0',guessphi_star,optimset ( 'disp', 'off')) if phi star>phi_arb(length(phi arb)) I phi_star end end % Plot sediment size distribution area=0; for i=l:Dbins area=area+farb(i).*(phi arb(i+ 1) -phi_arb(i)); end figure plot([phimean phi_mean],[0 max(f_arb)./area+.05],':b'); hold on plot([miuN miuN],(O[0 max(farb)./area+.05]j,':r'); plot([phi_star phi star],[0 max(farb)./area+.051,':g'); for i=l:Dbins plot([phiarb(i) phi arb(i)], [0 f_arb(i)./area]); plot([phiarb(i) phi arb(i+l)],[f _arb(i)./area farb(i)./area]); plot([phi_arb(i+l) phi arb(i+l)],[0 farb(i)./area]); end plot([phi arb(i+l) phi arb(i+l)], f_arb(i)./area 0]); xlabel( 'Sand Size (phi) '); ylabel( 'Frequency'); legend('miu F', 'miu N', 'phi *', 'distribution'); title('Nourishment Sand Size Distribution (Arbitrary)'); % Plot cummulative frequency sediment size distribution (finer than) figure plot(phi arb,Farb,'.-') hold on plot ( [phi mean phimean], [0 11) plot([miuN miuN],[0 1],'r') plot([phi star phi star],[0 1],'g') xlabel('Sand Size (phi)'); ylabel('Cummulative Frequency (Finer than) '); legend( 'cummulative distribution', 'miu F', 'miu N', 'phi_*'); title('Cummulative Nourishment Sand Size Distribution (Arbitrary)'); % Determine limits of bins clear q ql q2 q=find(phi arb q2=ql+l; Fstar=(F arb(ql)-F arb(q2)).*(phi star-phi_arb(q2))./(phiarb(ql)phi_arb(q2))+F_arb(q2); delta F=F star./(numbins-l); FI(1)=l; FI(2)=F_star; for i=3:numbins FI(i)=FI(i-1)-delta_F; end FI(numbins+l)=0.0; phi_FI(1)=phiarb(1); phi_FI(2)=phi_star; for i=3:numbins clear q ql q2 q=find(F_arb>FI(i)); ql=max(q); q2=ql+l; phi_FI(i)=(FI(i)-F_arb(q2)).*(phiarb(ql)phi_arb(q2))./(Farb(ql)-F arb(q2))+phiarb(q2); end phi FI(numbins+l)=max(phiarb); clear q ql q2 % Determine phi_rep for bins phi_rep (1) =miuN; for i=2:numbins clear q phi hold F hold sum t q=find((phi arb>phi FI(i))&(phi arb Fhold=1FI(i)Farb(q) FI(i+l)]; sum=0; for t=l:length(phi hold)-1; sum=sum+(Fhold(t+l)Fhold(t)).*(phi hold(t)+phi hold(t+1))./2; end phi rep(i)=sum./(Fhold(length(Fhold))-Fhold(1)); end clear q % Determine Arep from phi_rep d_rep=2.A (-phi_rep); drep=(round(drep.*100))./100; for i=l1:numbins if drep(i)>=0.l j=find(dtab==d rep(i)); A rep(i)=A tab(j); elseif drep(i)<0.l Arep(i)=0.4.*drep(i).^ (0.8); end end end % DETERMINE PLACEMENT PROFILE guess yI=200; yI=fzero('eqtn2',guess yI,optimset(,disp','off')); dyI--yI+B./slope_nat-B./slopeplace-ANdes.*yI.A(2./3)./slope_place; hI=AN des.*yI.^(2./3); % ITERATE TO DETERMINE EQUILIBRIUM BEACH PROFILE max_dyo=dyl-B./slope_nat+B./slope_place; q=1; trial_dyo(q)=maxdyo-5; dyo=trialdyo(q); trial_hfinal(q)=findfinalh(dyo); while trial_hfinal(q) trial dyo(q)=trial_dyo(q-l) -5; dyo=trial_dyo(q); trialhfinal(q)=findfinalh(dyo); end next_dyo=trial dyo(q)-(trial hfinal(q)-hstar).*(trialdyo(q-l)trial dyo(q))./(trial_hfinal(q-l)-trialhfinal(q)); dyo=next_dyo; next_hfinal=findfinalh(dyo); diff=abs(nexthfinal-hstar); while diff>0.01 if next hfinal>hstar dyoright=trial_dyo(q-l); hfinal_right=trial_hfinal(q-1); dyoleft=nextdyo; hfinal left=next hfinal; elseif nexthfinal hfinal_right=nexthfinal; dyo_left=trial_dyo(q); hfinalleft=trialhfinal(q); end next dyo=dyo_left-(hfinal left-hstar).*(dyo_rightdyo left)./(hfinal right-hfinal_left); dyo=nextdyo; nexthfinal=findfinalh(dyo); diff=abs(next_hfinal-hstar); end % DETERMINE NATIVE PROFILE ynat=0o: (y eqb(numbins+l))./85:y_eqb(numbins+l); hnat=AN_des.*ynat.^(2/3); h_nat=[-B hnat]; y_nat=[-B/slope_nat ynat]; % PLOT ARBITRARY NATIVE PROFILE ALONG WITH THE LEAST SQUARES FIT PROFILE if nat==2 figure y_nat2=0:max(y arb) ./85:max(yarb); h nat2=AN des.*ynat2.A^(2/3); hnat2= [-B hnat2]; ynat2=[-B/slope_nat ynat2l; plot(y-nat2,h nat2,'.-b') set(gca, 'ydir', 'reverse') hold on plot(yarb,h_arb, '+-r') plot([min(y_nat) max(ynat)],[o0 0],'k') xlabel ('Distance from Shoreline (m)');ylabel('Depth (m)'); legend( 'Smoothened Native Profile','Original Arbitrary Profile', 'SWL'); title('Comparison of Arbitrary and Smoothened Native Profiles') end % PLOT NATIVE, PLACEMENT AND EQUILIBRIUM PROFILES % PLOT NATIVE PROFILE figure plot(y_nat,hnat,'-b') set(gca, 'ydir', 'reverse') % PLOT WATER LINE hold on plot(Emin(ynat) max(ynat)], 0 0] 'k') % PLOT PLACEMENT PROFILE yplace=lmin(y_nat) min(ynat+dyl) yI]; hplace=[min(h_nat) min(h_nat) hI]; plot(yplace,h_place,'-r'); % PLOT RESULTING EBP hold on for n=1:numbins clear yplot hplot yplot=yeqb(n):(y_eqb(n+l)-y eqb(n))./40:y_eqb(n+1); hplot=(h-eqb(n).^(3./2)+Arep(n) .A(3./2).*(yplot-y_eqb(n))).^(2./3); plot(yplot,hplot, '-m') end plot(tynat(1)+dyo dyo], [-B 01],'-m'); % PLOT LIMITS OF EACH SEDIMENT SIZE BIN for n=2:numbins+l h corr(n)=AN des.*(yeqb(n)) A(2./3); plot([yeqb(n) yeqb(n)], [heqb(n) h corr(n)], 'm') end legend('Native','SWL','Placement','EBP'); title('Beach Profiles') xlabel('Distance (m)'); ylabel('Depth (m)'); axis( [ynat(l) yeqb(length(y_eqb ))+5 -B-0.25 AN_des*(max(yeqb))^(2/3)+0.25]) % PRINT FINAL ANSWERS TO SCREEN sprint f ('******************************FINALANSWE*******************************,) dyo PercentVact %Findfinalh.m (function that loops for dyo) function [h_final] = findhfinalh(dyo) global miuF miuN sigmaF V B slope_nat slope_place ANdes A-rep dyl dyo yeqb delta_Vact Vl Dy yI n hint yint V2 global Vact FI numbins hI h eqb PercentVact sprintf('------------------------------ NEW-GUESS----------------------dyo % DETERMINE (yint, hint) AND Vact FOR THIS GUESS OF dyo guessyint=yI; yint=fzero('eqtn3',guess yint,optimset('disp','off')); hint=A rep(1).*(Arep(1).*(yint-dyo) .^(2/3)./slope_place+dylB./slopenat+B./slopeplace-dyo).^(2/3); Vactl=-0.5.*B.A2./slopenat+B.*(dyl-dyo)+0.5.*B.^2./slope_place; Vact2=3./5.*A_rep(1).*(yint-dyo).^(5./3)-O0.5.*hint.^A2./slope place; Vact=Vactl+Vact2 Percent Vact=Vact./V.*100 % DETERMINE VOLUME OF FILL SEDIMENT CORRESPONDING TO EACH GRAIN SIZE BIN for i=l:numbins delta Vact(i)=(FI(i)-PI(i+l)).*Vact; end % DISTRIBUTE Vact ALONG THE OFFSHORE PORTION OF THE PROFILE yeqb(1)=dyo; h_eqb(1)=0; yeff(1)=(hint./A_rep(1)). ^(3./2); Dy(1)=yeff(1)-yint; Vl=0.5.*hI.^2./slope_place-0.5.*hint.^2./slope_place3./5.*Arep(1).*((Dy(1)+yI) ^(5./3)-(Dy(1)+yint) ^(5./3)); if Vl hend=(hint^(3/2)+A-rep(1).^(3/2)*(yI-yint)).^(2/3); guessy2=yI+100; solve_y=fzero('eqtn4',guess y2,optimset('disp','off')); while isnan(solve_y)==l guessy2=guess y2+50; solvey=fzero('eqtn4',guess y2,optimset('disp','off')); end yeqb(2) =solvey; if isnan(y_eqb(2))==1 sprintf('ERROR: NO CONVERGENCE!!!! NEED TO INCREASE INITIAL GUESS FOR EQTN4. Hit any key to continue') pause end % heqb(2)=Arep(l).*(y_eqb(2)+Dy(1)).^ (2/3); h_eqb(2)=(h eqb(1)^(3/2)+Arep(l) .A (3/2).*(y_eqb(2)y_eqb(1))) .^(2/3); % yplot=yint:(y_eqb(2)-yint)./10:yeqb(2); % hplot=(hint^'(3/2)+Arep(1) .^(3/2) .*(yplot-yint)).^(2/3); % plot(yplot,hplot,'-m') for n=2:numbins yeff(n)=(h eqb(n)./A rep(n)).^(3./2); Dy(n)=yeff(n)-y_eqb(n); guessy=y eqb(n); solvey=fzero( 'eqtn5',guess y,optimset( 'disp', 'off')); yeqb(n+1) =solvey; heqb(n+l)=(h eqb(n) .A(3/2)+A rep(n) .^(3/2).*(y eqb(n+l)y_eqb(nf)) .^(2/3); % clear yplot hplot % yplot=y_eqb(n):(y_eqb(n+l)-yeqb(n))./10:yeqb(n+1); % hplot=(heqb(n). ^ (3./2)+A_rep(n). A (3./2).*(yplotyeqb(n))) ^(2./3); % hplotcheck=Arep(n).*(yplot+Dy(n)) .^(2./3); % plot(yplot,hplot,'-m') end h_final=max(h eqb) end if Vl>delta Vact(1) sprintf('Vl > delta_Vact(l) ') guess_y=yI; ystop=fzero('eqtn6',guess y,optimset( 'disp','off')); h_stop=hint+slopeplace.*(y_stop-yint); y_eqb(2)=y_stop; h_eqb(2)=(hint. (3./2)+A rep(1).^(3./2).*(yeqb(2)-yint)).^(2./3); %heqb(2)=A rep(l).*(y_eqb(2)+Dy(1)).^(2/3); % clear yplot hplot % yplot=yint:(yeqb(2)-yint)./10:y_eqb(2); % hplot=(hint^A(3/2)+Arep(1).^(3/2).*(yplot-yint)).^(2/3); % plot(yplot,hplot, '-m') yeff(2)=(h_eqb(2)./A_rep(2)).^A(3./2); Dy(2)=yeff(2)-y_eqb(2); V2=0.5.*hI. ^2./slope_place-0.5.*h stop.^2./slope_place3./5.*Arep(2).*((Dy(2)+yI).^(5./3)-(Dy(2)+yeqb(2)).^(5./3)); if V2>=delta Vact(2) sprintf('WARNING!!!!! V2>delta_Vact(2) But, it is OK if dyo converges. Hit any key to continue'); end guess y=yI; solve_y=fzero('eqtn7',guess y,optimset('disp','off')); y eqb(3)=solve y; h_eqb(3)=(heqb(2)^(3/2)+Arep(2).^(3/2).*(yeqb(3)yeqb(2))) .(2/3); % n=2; % clear yplot hplot % yplot=y_eqb(n):(yeqb(n+l)-y_eqb(n))./10:y_eqb(n+l); % hplot=(h eqb(n).^(3./2)+A rep(n).A^(3./2).*(yplotyeqb(n))) .A(2./3); % hplotcheck=A rep(n).*(yplot+Dy(n)) .A^(2./3); % plot(yplot,hplot, '-m') for n=3:numbins yeff(n)=(h eqb(n)./Arep(n)).^(3./2); Dy(n)=yeff (n)-y_eqb(n); guessy=y_eqb(n); solve y=fzero ( 'eqtn5',guess y,optimset( 'disp', 'off')); yeqb(n+l)=solvey; h eqb(n+l)=(h_eqb(n) .^(3/2)+Arep(n).A(3/2).*(y_eqb(n+l)yeqb(n) ) ) .^ (2/3); % clear yplot hplot % yplot=y eqb(n):(y_eqb(n+l)-y eqb(n))./10:y_eqb(n+l); % hplot=(heqb(n) .(3./2)+A rep(n). ^(3./2).*(yploty-eqb(n))) ^(2./3); % hplotcheck=A rep(n).*(yplot+Dy(n)) .A(2./3); % plot(yplot,hplot, '-m') end hfinal=max(heqb) end % Smaller functions used by ffiner.m and/or fcoarser.m function eql=eqtnl(phi_star) global miuF miuN sigmaF eql=miuF-sqrt(2/pi)*sigmaF*exp(- ((phi_starmiuF)/sigmaF/sqrt(2)) ^2) / (1+erf((phi star-miuF)/sigmaF/sqrt(2)) ) -miuN; function eq2=eqtn2(yI) global V B slopenat slope_place ANdes eq2=-yI+V/B-0.5*B/slope nat+O.5*B/slopeplace3/5/B*ANdes*yI^(5/3)+O.S*AN desA 2*yIA (4/3)/B/slope_place+ANdes*yI^(2/ 3)/slopeplace; function eq3=eqtn3(yint) global Arep slopeplace dyl B slopenat dyo cl=A rep(1)./slope_place; c2=dyl-B./slopenat+B./slopeplace; eq3=-yint+cl.*(yint-dyo).A(2/3)+c2; function eq4=eqtn4(solve_y) global ANdes deltaVact V1 A-rep Dy yI eq4=delta Vact(1)-Vi-3/5.*ANdes.*(solve_y.^(5/3)yI.A (5/3)T+3/5.*A_rep(1).*((Dy(1)+solvey).^(5/3)-(Dy(1)+yI).A(5/3)) function eq5=eqtn5(solve_y) global deltaVact n AN des y_eqb Arep Dy eq5=deltaVact(n)-3/5.*ANdes.*(solve_y.A(5/3)yeqb(n).^(5/3))+3/5.*Arep(n).*((Dy(n)+solve_y) .^ (5/3)(Dy(n)+y_eqb(n)) .^ (5/3)); function eq6=eqtn6(ystop) global delta Vact slopeplace Arep Dy hint yint eq6=deltaVact(1)-O.5./slope-place.*((hint+slopeplace.*(ystopyint)).^2-hint.^2)+3./5.*Arep(1).*((Dy(1)+y_stop).A(5./3)(Dy(1)+yint).^(5./3)); function eq7=eqtn7(solve_y) global ANdes delta Vact V2 Arep Dy yI eq7=deltaVact(2)-V2-3/5.*ANdes.*(solvey.^A(5/3)yI.A^(5/3)+3/5.*A rep(2).*((Dy(2)+solvey).^(5/3)-(Dy(2)+yI).^(5/3)); function eq8=eqtn9(solution) global AN des y n Arep Dy deltaVact eq8=3./5.*AN des.*(y(n+1).^(5./3)-solution).^(5./3))3./5.*Arep(n).*((y(n+l)+Dy(n)).^(S./3)-(solution+Dy(n)).^(5./3))deltaVact(n); function eq9=eqtn9(yint) global dyl B slopenat slopeplace A rep Dy eq9=dyl B./slopenat+B./slope_place+A rep(l) ./slopeplace.*(yint+Dy(l)) ^(2./3) -yint; function eqlO=eqtnl0 (phi_star) global num den bin miuN f arb phi arb eql0=(num+farb(bin).*(phistarphi arb(bin)).*(phi_star+phi arb(bin))./2) ./(den+farb(bin).*(phi_starphi arb (bin)))-miuN; function eqll=eqtnll (phi star) global num den bin miuN farb phiarb eqll=(num+f_arb(bin).*(phiarb(bin+l)phi star) (phi_star+phi arb(bin+l))./2) ./(den+f_arb(bin). *(phi_arb(bin +1)-phi_star))-miuN; function eq50=eqtn5O(Yo_max) global Vact yI hstar ANdes eqS0=Vact-yI.*hstar-Yo_max.*(AN_des.*Yomax. A(2./3)-hstaryI.*ANdes./Yomax. ^(1./3)); function eq51l=eqtn51 (solution) global ANdes y n Arep Dy delta Vact eqSl=3./5.*AN des.*(y(n+l). A(5./3)-solutin.^(5./3))3./5.*Arep(n).*((y(n+l)+Dy(n)).^(5./3)-(solution+Dy(n)). (5./3)) delta Vact(n); function eq52=eqtn52(yint) global dyl B slope nat slope place Arep Dy eq52=slopeplace.*(yint-dyI+B./slopenat-B./slope_place)A_rep(1).*(yint+Dy(1)).A(2./3); function eq53=eqtn53(ynew) global Vadd slope_place hplcl dyl B slope nat A-rep isnany Dy y eq53=-Vadd+0.5./slope_place. (hplcl. 2-(slope_place. (ynewdyl+B./slope_nat-B./slopeplace)) 2) 3./5.*A_rep(isnany+l).*((Dy(isnany+l)+y(isnany+l)).A(5./3)(Dy(isnany+l)+ynew).A(5./3)); function eq54=eqtn54(ynew) global Vadd2 slopeplace h eqbwat dyl B slope_nat y_eqbwat eq54=-Vadd2+0.5./slopeplace.*(h eqbwat.A2-(slope_place.*(ynewdyl+B./slopenat-B./slopeplace)) .^2)+0.5*slopenat. (yeqbwatynew).A^2; function eq55=eqtn55 (phi star) global miuF sigmaF miuN eq55=miuF+sqrt(2./pi).*sigmaF.*exp( ( (phi starmiuF)./sigmaF./sqrt(2)) .A2)./(l-erf((phistar-miuF)./sigmaF./sqrt(2)))miuN; function eq56=eqtn56(yI) global V B slopenat slope_place AN_des eq56=-yI+V/B-0.5*B/slope nat+0.5*B/slopeplace3/5/B*ANdes*yI^(5/3)+O.5*ANdes^2*yIA(4/3)/B/slopeplace+ANdes*yI^(2/ 3)/slopeplace; function eq57=eqtn57(ycom) global slope_place dyl B slopenat A-rep Dy eq57=slope_place.*(ycom-dyI+B./slopenat-B./slopeplace)A_rep(l).*(ycom+Dy(1)).^(2./3); function eq8l=eqtn8l(ystop) global slopenat Arep n Dy eq81=-slope_nat+2./3.*Arep(n)./(ystop+Dy(n)).^(1./3); function areaeqb=Inteqb(a,c) global slope_nat hstop ystop areaeqb=0.5.*slope nat.*(c.^2-a.^2)+(hstop-slope_nat.*ystop).*(c-a); function areaplc=Intplc(a,c) global slopeplace dyl B slopenat areaplc=0.5.*slope_place.*(c.^2-a.^2)+(-slope_place.*(dyIB./slope nat+B./slope_place)).*(c-a); % inputdata.m (to input data)) % TO FACILITATE ENTRY OF INPUT DATA. VARIABLES. % CHOOSE ONE OPTION TO REPRESENT THE SEDIMENT. % VARIABLES REQUIRED FOR ALL CASES V=300 B=2 hstar=6 slopenat=l./30 slopeplace=l./20 % REPRESENTING THE NATIVE PROFILE. % PICK ONE OPTION, COMMENT OUT THE % Option 1: DN=0.20 nat=1 % Option 2: %y_arb=[]; %harb=[]; %nat=2 FILL IN VALUES OF REQUIRED NATIVE PROFILE AND THE FILL OTHER ONE. Single sediment size, D(mm) Arbitrary native profile % REPRESENTING THE FILL SEDIMENT SIZE DISTRIBUTION % PICK ONE OPTION, COMMENT OUT THE OTHERS. % Option 1: DF=0.14 sigmaF=0.5 fill=l % Option 3: distribution %D_arb=[]; %F arb=[]; %fill=3 Log-normal distribution Arbitrary fill sediment cummulative size 70 Sample Output ans = ******************************FINAL-ANSWER******************** dyo = 26.2952 Percent Vact = 62.5670 APPENDIX B PROGRAM LISTING FOR FCOARSER.M % fcoarser.m (MAIN PROGRAM) % DF > DN close all clear all warning off % DEFINE GLOBAL VARIBLES global numbins hstar hI y h incomplete straightline nstop ystop hstop yB perdif Res Vact caseid global miuF miuN sigmaF V B slope_nat slope place AN_des Vact yI hstar y n Arep Dy deltaVact hstop ystop dyl global Yo Yo_min Yomax dyo y_nat hnat AF global num den bin farb phi arb test test test2 k % LOAD INPUT DATA inputdata numbins=4; % TABLE OF A(mA1/3) VALUES FOR A GIVEN D(mm) dtab=0.1:0.01:1.09; dtab=(round(dtab.*100))./100; Atab=[0.063 0.0672 0.0714 0.0756 0.0798 0.084 0.0872 0.0904 0.0936 0.0968 0.1 0.103 0.106... 0.109 0.112 0.115 0.117 0.119 0.121 0.123 0.125 0.127 0.129 0.131 0.133 0.135 0.137... 0.139 0.141 0.143 0.145 0.1466 0.1482 0.1498 0.1514 0.153 0.1546 0.1562 0.1578 0.1594... 0.161 0.1622 0.1634 0.1646 0.1658 0.167 0.1682 0.1694 0.1706 0.1718 0.173 0.1742 0.1754... 0.1766 0.1778 0.179 0.1802 0.1814 0.1826 0.1838 0.185 0.1859 0.1868 0.1877 0.1886... 0.1895 0.1904 0.1913 0.1922 0.1931 0.194 0.1948 0.1956 0.1964 0.1972 0.198 0.1988... 0.1996 0.2004 0.2012 0.202 0.2028 0.2036 0.2044 0.2052 0.206 0.2068 0.2076 0.2084... 0.2092 0.210 0.2108 0.2116 0.2124 0.2132 0.2140 0.2148 0.2156 0.2164 0.2172]; % IF GIVEN DN DETERMINE THE CORRESPONDING SEDIMENT SCALE PARAMETER if nat==l miuN=- (log (DN)./l1og(2)); DN=(round(DN.*100))./100; k=find(d_tab==DN); ANdes=Atab(k); end % IF GIVEN AN ARBITRARY NATIVE PROFILE DETERMINE THE LEAST SQUARES FIT OF 2/3 POWER RULE if nat==2 ANdes=sum(h arb.*yarb.^(2/3))./sum(y_arb.A(4/3)); q=find(Atab q2=ql+1; DN=d tab(ql)+(d_tab(q2)-d tab(ql))./(Atab(q2)-A tab(ql)).*(AN desA_tab(ql)); miuN=-(log(DN)./log(2)); end % IF GIVEN A LOG-NORMAL SEDIMENT SIZE DISTRIBUTION: if fill==l % Determine nourishment sediment size distribution miuF=-(log(DF)./log(2)); AF=A tab(find(d tab==(round(DF.*100)./100))); phi=miuF-4.*sigmaF:(miuF+4.*sigmaFmiuF+4.*sigmaF)./30:miuF+4.*sigmaF; f=1./sigmaF./sqrt(2.*pi).*exp(-(phi-miuF) .A^2./2./sigmaF.A^2); F=1-0.5.*(l+erf((phi-miuF)./sqrt(2)./sigmaF)); % Find proportion of nourishment sand that has the same mean size as the native guess-phistar=(miuN+miuF)./2; phistar=fzero( 'eqtn55ss',guessphistar,optimset( 'disp', 'off')); F_star=1-0.5.*(l+erf((phi_star-miuF)./sqrt(2)./sigmaF)); % Plot nourishment sediment size distribution figure plot(phi,f) hold on plot((miuF miuF],[0 max(f)]) plot([miuN miuN],[o0 max(f)],'r') plot( [phi star phi star],[0 max(f)], 'g') xlabel('Sand Size (phi)'); ylabel('Frequency'); legend('distribution','miuF','miuN', 'phi_*') title('Nourishment Sand Size Distribution (Log-Normal)') axis ([miuF-4*sigmaF miuF+4*sigmaF 0 max(f)]) % Plot cummulative nourishment sediment size distribution figure plot(phi,F) hold on plot([miuF miuF],[0 11]) plot([miuN miuN],[0 11,'r') plot([phi star phi_star],[0 11], 'g') xlabel( 'Sand Size (phi) '); ylabel('Cummulative Frequency'); legend('cummulative distribution', 'miu F', 'miu N', 'phi *'); title('Cummulative Nourishment Sand Size Distribution (LogNormal) '); axis([miuF-4*sigmaF miuF+4*sigmaF 0 11) % Divide the rest of the distribution into (numbins-l) bins delta_F=(l-Fstar)./(numbins-1); delta_per=delta_F; for i=l:numbins-2 delta_per= [deltaper deltaF]; end deltaper=1delta-per Fstar]; check=sum(delta per); % Determine the limits of each grain size bin FI(1)=l.0; for i=2:numbins-1 FI(i)=FI(i-l) -deltaF; end FI(numbins)=Fstar; FI(numbins+l) =0.0; phi FI=sqrt(2).*sigmaF.*erfinv(1-2.*FI)+miuF; % Determine phi rep for each bin for i=l:numbins ul=(phi_FI(i)-miuF)./sigmaF./sqrt(2); u2=(phi_FI(i+l)-miuF)./sigmaF./sqrt(2); phi_rep(i)=sqrt(2./pi).*sigmaF.*(exp(-ul.^2)-exp(u2.^2))./(erf(u2)-erf(ul))+miuF; d-rep(i)=2.A^(-phi_rep(i)); if drep(i)>=0.1 & d_rep(i)<=l.09 drep(i)=(round(d_rep(i).*100))./100; j=find(dtab==d rep(i)); A_rep(i) =A tab(j); elseif drep(i)<0.1 A_rep(i)=0.4.*d_rep(i).^(0.8); elseif drep(i)>l.09 & d rep(i)<=10 Arep(i)=0.21.*d_rep(i)^0.A.311; elseif drep(i)>10 & d_rep(i)<=100 Arep(i)=0.257.*d_rep(i).^0.224; elseif d-rep(i)>100 & drep(i)<=520 Arep(i)=0.288.*drep(i).A^0.199; end end end % IF GIVEN AN ARBITRARY CUMMULATIVE FILL SEDIMENT DISTRIBUTION: if fill==3 Dbins=length(Darb)-1; phi_arb=-log2(D_arb); for q=l:Dbins farb(q)=Farb(q)-F_arb(q+l); end % Find phi_84 and phi_16 corresponding to 84% and 16% cummulative % distribution clear q ql q2 q=find(F arb==0.84); if isempty(q)==0 D 84=D arb(q); phi 84=-log2(D_84); else q1=find(F_arb>0.84); ql=max(ql); q2=ql+l; D_84=Darb(q2)+(0.84-F_arb(q2)).*(D_arb(ql)D_arb(q2))./(F arb(ql)-Farb(q2)); phi_84=-log2(D_84); end clear q q1 q2 q=find(F arb==O.16); if isempty(q)==0 D_16=D_arb(q); phi 16=-log2(D_16); else ql=find(Farb>0.16); ql=max(ql); q2=ql+l; D_16=Darb(q2)+(0.16-Farb(q2)) D_arb(q2))./(F arb(ql)-Farb(q2)); phi_16=-log2(D_16); end phi mean=(phi_84+phi_16)./2; phi_sorting=(phi_16-phi_84)./2; DFmean=2.^(-phimean) DN . (Darb(ql)- if DF mean<=DN sprintf=('Warning: DF<=DN, THIS PROGRAM IS NOT APPLICABLE') pause end AF=Atab(find(d tab==(round(DF_mean.*100)./100))); % Determine phi_star centroid=phi_arb(Dbins+l1); bin=Dbins+1; while centroid>miuN bin=bin-1; num=0; den=0; for i=Dbins: -1 :bin num=num+farb(i).*(phiarb(i+l)phiarb(i)).*(phi_arb(i)+phi_arb(i+l1))./2; den=den+f_arb(i).*(phi arb(i+l)-phiarb(i)); end centroid=num./den; end if centroid==miuN phi_star=phi_arb(i); else num=num-f arb(i).*(phi_arb(i+l)phi_arb(i)).*(phi_arb(i)+phiarb(i+l))./2; den=den-farb(i).*(phiarb(i+l)-phi arb(i)); guess_phi_star=phiarb(i+l); phistar=fzero( 'eqtnll',guess_phi_star,optimset( 'disp', 'off')); if phi star>phi_arb(length(phi arb)) Iphi_star end % Plot sediment size distribution area=0; for i=l:Dbins area=area+farb(i).*(phiarb(i+l)-phi arb(i)); end figure plot([phimean phimean],[0 max(f_arb)./area+.05],':b'); hold on plot([miuN miuN],[0 max(f arb)./area+.05],':r'); plot([phi_star phi_star],(0 max(farb)./area+.05],':g'); for i=l:Dbins plot([phi_arb(i) phiarb(i)],[0 f_arb(i)./area]); plot(tphi_arb(i) phiarb(i+1)], [farb(i)./area f_arb(i)./area]); plot([phi_arb(i+l) phi_arb(i+1)], [0 farb(i)./areal); end plot([phi_arb(i+l) phi_arb(i+l)], 1f_arb(i)./area 0]); xlabel('Sand Size (phi)'); ylabel('Frequency'); legend('miu F','miu N', 'phi_*','distribution'); title('Nourishment Sand Size Distribution (Arbitrary)'); k Plot cummulative frequency sediment size distribution (finer than) figure plot(phi_arb,Farb,'.-') hold on plot([phi mean phimean], [0 1]) plot([miuN miuN],[0 1],'r') plot([phi_star phi_star],[0 l],'g') xlabel ('Sand Size (phi)'); ylabel('Cummulative Frequency (Finer than) '); legend( 'cummulative distribution', 'miu F', 'miu N', 'phi_*'); title( 'Cummulative Nourishment Sand Size Distribution (Arbitrary)'); % Determine limits of bins clear q q q2 q=find(phi_arb q2=ql+l; F_star=(F arb(ql)-Farb(q2)).*(phi_star-phi_arb(q2))./(phi arb(ql)phi arb(q2))+Farb(q2); deltaF=(1-Fstar)./(numbins-l); delta per=deltaF; for i=1:numbins-2 deltaper=[deltaper delta_F]; end delta per=[deltaper F_star]; check=sum(deltaper); FI (1)=l; for i=2:numbins-i FI(i)=FI(i-1)-delta F; end FI(numbins)=Fstar; FI(numbins+l)=0.0; phi_FI(1)=phi arb(l); for i=2:numbins-1 clear q ql q2 q=find(F_arb>FI(i)); ql=max(q); q2=ql+l; phiFI(i)=(FI(i)-Farb(q2)).*(phiarb(ql)phiarb(q2))./(F_arb(ql)-F arb(q2) )+phi arb(q2); end phi_FI(numbins)=phi_star; phi_FI(numbins+l)=max(phi_arb); clear q ql q2 W Determine phi_rep for bins for i=l1:numbins-1 clear q phi hold F hold sum t q=find((phi arb>phi_FI (i))&(phi arb Fhold=[FI(i)Farb(q) FI(i+l)]; sum=O; for t=1:length(phi hold)-1; sum=sum+(Fhold(t+1)Fhold(t)).*(phi hold(t)+phihold(t+1))./2; end phi_rep(i)=sum./(Fhold(length(Fhold))-Fhold(l)); end clear q phi rep(numbins)=miuN; % Determine A-rep from phi_rep drep=2.^(-phi rep); for i=l1:numbins if drep(i)>=0.1 & d_rep(i)<=l.09 d rep(i)=(round(d_rep(i).*100))./100; clear j j=find(dtab==d-rep(i)); A_rep(i)=A_tab(j); elseif drep(i)<0.1 Arep(i)=0.4.*d rep(i).^(0.8); elseif drep(i)>1.09 & d rep(i)<=10 A_rep(i)=0.21.*drep(i).^0.311; elseif d rep(i)>10 & drep(i)<=100 A_rep(i)=0.257.*drep(i).^0.224; elseif d rep(i)>100 & d rep(i)<=520 A rep(i)=0.288.*d rep(i).^0.199; end end end % DETERMINE dyl FOR THE GIVEN PLACEMENT VOLUME, V guessyI=100; yI=fzero( 'eqtn56',guess_yI,optimset( 'disp', 'off')); while isnan(yI) ==1 guessyl=guessy+50; yI=fzero('eqtn56',guessyl,optimset('disp', 'off')); end dyl=yI+B./slope nat-B./slopeplace-ANdes.*yI. (2./3)./slopeplace; hl=ANdes.*yI.^(2./3); % LOOP TO FIND X% p=1; trialX (p)=50; X=trial X(p); trial Vact(p)=trial_X (p)./100.*V; Vact=trialVact(p); deltaVact=delta-per.*trialVact(p); Yomin=(hstar./AN_des).A(3./2); guess=Yomin; Yomax=fzero('eqtn50', guess, optimset('disp', 'off')); trialResVact(p)=loopfcoarserl (X); difVact(p)=(trialResVact(p)-trialVact(p))./trialVact(p).*100; trialYo(p)=Yo; while dif_Vact(p)<0 p=p+l; trial X(p)=trialX (p-1) -10; X=trialX (p); trial Vact(p)=trialX(p)./100.*V; Vact=trial Vact(p); deltaVact=deltajper.*trial Vact(p); Yomin=(hstar./ANdes).^(3./2); guess=Yo_min; Yo max=fzero('eqtn5o0', guess, optimset('disp','off')); trial Res Vact(p)=loopfcoarserl (X); difVact(p)=(trialResVact(p)-trial Vact(p))./trialVact(p).*l00; trial Yo(p)=Yo; end upperlim X=trialX(p-l); upperlimDif=difVact(p-1); lowerlim X=trial X(p); lowerlimDif=dif_Vact(p); while abs(difVact(p))>1 p=p+l; trial X(p) =lowerlimX-lowerlim Dif. ((upperlim XlowerlimX) ./(upperlim Dif-lowerlim Dif)); X=trial_X(p); trial_Vact(p)=trialX (p)./100.*V; Vact=trialVact(p); delta_Vact=deltaper.*trial Vact(p); Yo_min=(hstar./ANdes).^(3./2); guess=Yo_min; Yomax=fzero( 'eqtn50', guess, optimset( 'disp', 'off')); trial_Res Vact(p)=loopfcoarserl(X); dif_Vact(p)=(trial_Res Vact(p)-trialVact(p))./trialVact(p).*100; trial Yo(p)=Yo; if dif_Vact(p)<0 upperlim X=trial_X(p); upperlimDif=difVact(p); else lowerlimX=trialX(p); lowerlimDif=difVact(p); end end % PRINT FINAL ANSWERS TO SCREEN sprintf( '******************************FINALANSWE******************************',) dyo PercentVact=X caseid % loopfcoarserl.m (function that loops for X%) % For the given guess of X%, loop for the correct value of Yo. % Determine the resulting active volume Res Vact for these values % of X% and Yo. function [ResVact]=loopfcoarserl(X) global numbins hstar hI y h incomplete straight line nstop ystop hstop yB perdif Res Vact caseid global miuF miuN sigmaF V B slope nat slope place ANdes Vact yI hstar y n A-rep Dy deltaVact hstop ystop dyl global Yo Yomin Yo max dyo y_nat h_nat AF test testl test2 k test= []; testl=[]; test2=[]; q=l; trial Yo(q)=Yo_max; Yo=trial Yo(q); trial change(q)=loopfcoarser2(Yo); trialperdif(q)=perdif; trial_case=caseid; while trial_change(q)<0 q=q+l; trial Yo(q)=trialYo(q-l) -5; if trial_Yo(q) end Yo=trial_Yo(q); trial_change(q)=loopfcoarser2(Yo); trialperdif(q)=perdif; trialcase=char(trial case,caseid); end upperlim=trial_Yo(q-l); lowerlim=trial_Yo(q); r=l; new_Yo(r)=0.5.*(upperlim+lowerlim); Yo=newYo(r); newchange(r)=loopfcoarser2(Yo); new perdif(r)=perdif; newcase=caseid; while new_change(r)-=0 if new_change(r)>0 lowerlim=new Yo(r); end if new_change(r)<0 upperlim=newYo(r); end r=r+l; new_Yo(r)=0.5.*(upperlim+lowerlim); 80 Yo=new Yo(r); newchange (r) =loopfcoarser2 (Yo); new_perdif (r) =perdif ; new case=char(newcase,caseid); end %test testl %test2 %keyboard % loopfcoarser2.m (function that loops for Yo) % For the given guess of Yo for the current value of X% (i.e. Vact), % determine if the volume that fits offshore up to (Yo,h*) is equal to Vact (+/- 1%). % This is used to determine if the guess of Yo should be increased (change=5), % decreased (change=-5), or if it is the correct value for the current X% (change=0). function [change] = loopfcoarser2(Yo) global numbins hstar hI y h incomplete straight_line nstop ystop hstop yB perdif ResVact caseid global miuF miuN sigmaF V B slopenat slopeplace ANdes Vact yI hstar y n Arep Dy delta_Vact hstop ystop dyl global Yo Yomin Yomax dyo y_nat hnat AF test testl test2 k yeqbwat=0; Yo; perdif=NaN; y=[]; h=[]; y(numbins+l) =Yo; h(numbins+l)=hstar; incomplete=0; straight line=0; nstop=0; ystop=1[]; h_stop=l; yB=[]; % Place Vact starting from offshore at Yo and coming inshore using eqtn 2-7. % If the profile intersects the SWL or becomes steeper than slope nat, % it is replaced with a straight line with slope=slope_nat. for n=numbins:-l:l Dy(n)=(h(n+l)./Arep(n)).^(3./2)-y(n+l); guess=y(n+l); solution=fzero('eqtn51',guess,optimset('disp','off')); y(n)=solution; h(n)=A rep(n).*(y(n)+Dy(n)).^(2./3); % If the equilibrium profile shape intersects the SWL in this bin, % find the point at which this occurs. % If this is the case, incomplete=l. if isnan(solution)==l incomplete=l; isnany=n; nstop=isnany; ystop=(2./3.*Arep(n)./slope_nat).^3-Dy(n); if ystop>y(n+l) ystop=y(n+l); end hstop=A rep(n).*(ystop+Dy(n)) .^(2./3); yB=ystop-l./slope_nat.*(B+hstop); %y=y(l:nstop-l); break end % If the equilibrium profile slope >= slopenat in this bin, % find the point at which this occurs. % If this is the case, straight_line=l. endslope=2./3.*A_rep(n)./(y(n)+Dy(n)).^(l./3); if endslope>=slope_nat straight line=1; nstop=n; ystop=(2./3.*Arep(n)./slopenat) .^3-Dy(n); if ystop>y(n+l) ystop=y(n+l); end hstop=A rep(n).*(ystop+Dy(n)) ^(2./3); yB=ystop-I./slope_nat. (B+hstop); %y=y(1:nstop-1); break end end % Determine the native profile ynat=0:Yo./75:Yo; h_nat=ANdes.*y nat.^A(2./3); hnat=[-B hnat]; ynat= [-B/slope nat ynat]; % Plot the native profile figure(3) clf plot(y_nat,hnat) set(gca,'ydir','reverse','-b') hold on % Plot SWL plot([min(y_nat) max(y_nat)],[0 O],'k') % Plot placement profile y_place=[min(y_nat) min(y_nat+dyl) yI]; hplace=[min(hnat) min(hnat) hI]; plot(y_place,h place, '-r'); % Plot trial equilibrium profile for the current X% and Yo guess % substituting with straight lines where required from above steps if straight_line==l sprintf('Very steep slope replaced with slope nat'); for n=nstop:numbins clear yplot hplot yplot=y(n): (y(n+l)-y(n))./20:y(n+l); hplot=(h(n).^(3./2)+Arep(n).^(3./2).*(yplot-y(n))).^(2./3); plot(yplot,hplot, '-gT') end plot(y,h, .g'); elseif incomplete==l sprintf('Not all bins fit underwater'); for n=l1:numbins clear yplot hplot yplot=y(n):(y(n+l)-y(n))./20:y(n+l); hplot=(h(n).^(3./2)+A rep(n).^(3./2).*(yplot-y(n))).^(2./3); plot(yplot,hplot,'-m') end clear yplot hplot yplot=-Dy(isnany):(y(isnany+l)+Dy(isnany))./15:y(isnany+l); hplot=Arep(isnany) .*(Dy(isnany)+yplot). A (2./3); plot(yplot,hplot, ':m') plot(y,h, '.m'); else sprintf('All bins fit with their usual slope'); for n=l1:numbins clear yplot hplot yplot=y(n):(y(n+l)-y(n))./20:y(n+l); hplot=(h(n).^(3./2)+A rep(n).^(3./2).*(yplot-y(n))).^(2./3); plot(yplot,hplot, '-y') end plot(y,h, '.y'); end if incomplete==0 & straight line==0 if y(1)>yl sprintf('Case 3a: Decrease Yo'); caseid='3a'; change= -5; elseif y(1) if y(1) caseid='3c'; change=+5; elseif y(1)>yplcyl if y(2) %sprintf('Case 3b with y(2) Vadd=3./5.*AN des.*(yl.^(5./3)-y(1).^(5./3))-Intplc(y(1),yI); ystop=(2./3.*A_rep(1)./slopenat).^3-Dy(1); hstop=Arep(1).*(ystop+Dy(1)).A^(2./3); guessycom=y(l); ycom=fzero ('eqtn57' ,guess ycom,optimset( 'disp','off')); hcom=Arep(l).*(ycom+Dy(1)).^(2./3); if ycom>ystop sprintf('Case 3b.l'); caseid='3b.1'; Vol=Intplc(ycom,y(1))-3./5.*Arep(1).*((y(1)+Dy(1)).^(5./3)(ycom+Dy(1)).^(5./3)); yx=ycom; hx=hcom; DyF=(hx./AF).^ (3./2)-yx; y_eqbwat= -DyF; clear yplot hplot yplot=yx:(y(1)-yx)./10:y(1); hplot=A rep(1) .*(yplot+Dy(1)). ^(2./3); plot(yplot,hplot,'-c') clear yplot hplot yplot=y_eqbwat: (yx-y_eqbwat)./20:yx; hplot=AF.*(yplot+DyF) .^ (2./3); plot(yplot,hplot, '-c') clear yplot hplot plot([ynat(l) ynat(l)+yeqbwat y_eqbwat],[-B -B 0],'-c'); elseif ycom caseid=' 3b.2'; yx=(hstop-slope_nat.*ystop+slope place. (dyIB./slopenat+B./slope_place))./(slopeplace-slopenat); hx=slopenat.*yx+hstop-slopenat.*ystop; Vl=Intplc(ystop,y(1))-3./5.*A rep(1).*((y(1)+Dy(1)).^(5./3)(ystop+Dy(1)) .^(5./3)); V2=Intplc(yx,ystop) -Inteqb(yx,ystop); Vol=Vl+V2; DyF=(hx./AF).^(3./2)-yx; yeqbwat=-DyF; clear yplot hplot yplot=ystop:(y(1)-ystop)./10:y(1); hplot=A rep(l) .*(yplot+Dy(l)).^ A(2./3); plot(yplot,hplot, '-c') plot([yx ystop],[hx hstop],'c'); clear yplot hplot yplot=yeqbwat:(yx-y eqbwat)./20:yx; hplot=AF.*(yplot+DyF) ^(2./3); plot(yplot,hplot,'c') plot ([ynat(il) ynat(l)+y_eqbwat y_eqbwat],[-B -B 0], '-c'); end sum remain=Vadd; difference=Vol-sum_remain; perdif=difference. / sum_remain.*100; if abs(perdif)>1 if Vol>sum remain sprintf('Vol>sum remain: change=-5; end if Vol decrease Yo'); increase Yo'); if abs(perdif)<1 sprintf('This is the value of Yo for X%'); change=0; dyo=yeqbwat; y_plcwat=dyI-B./slope_nat+B./slopeplace; ResVact=3./5.*AF.*(yx+DyF).^(5./3)Intplc(yplcwat,yx)+0.5.*B.A2./slopenat+B.*(dyI-y_eqbwatB./slope nat)+0.5.*B.^2./slope_place; if abs((ResVact-Vact)./Vact.*100)>l if ResVact>Vact sprintf('Res_Vact>Vact: Need to increase X'); end if ResVact Need to decrease X'); end end if incomplete==l I straight_line==l yx=(hstop-slope nat.*ystop+slope_place.*(dylB./slope_nat+B./slopeplace))./(slope_place-slopenat); hx=slope_nat.*yx+hstop-slope_nat.*ystop; yp=hstop./slopeplace+dyl-B./slopenat+B./slopeplace; if yB>dyl-B./slope_nat caseid='la'; sprintf('Case la: Decrease Yo'); change=-5; elseif yB sprintf('Case 1c: Increase Yo'); change=5; elseif yB %Vcrit=B.^2./slopenat-0.5.*B.^2./slope_nat0.5.*B. ^2./slope_place; %if Vcrit % caseid='1b'; % change=5; % %keyboard %else % sprintf('WARNING Case lb!!!') % %keyboard yplcwat=dylI-B./slope_nat+B./slope_place; sumremain=sum(deltaVact(1:nstop)); if ystop>yI if ystop==y(nstop+1) sprintf('Case lb.la'); caseid='lb.la'; Vl=3./5.*AN des.*(y(nstop+l).^(5./3)-yI.^(5./3))Inteqb(yI,y(nstop+l)); V2=Intplc(yx,yI)-Inteqb(yx,yI); Vol=Vl+V2; elseif ystop-=y(nstop+l) sprintf('Case lb.lb'); caseid='lb.1b'; Vl=3./5.*AN des.*(y(nstop+l).A(5./3)-ystop.^(5./3))3./5.*Arep(nstop).*((Dy(nstop)+y(nstop+l)).^(5./3)(Dy(nstop)+ystop). ^A(5./3)); V2=3./5.*ANdes.*(ystop.A(5./3)-yI.A(5./3))Inteqb(yI,ystop); V3=Intplc(yx,yI)-Inteqb(yx,yI); Vol=Vl+V2+V3; end elseif ystop sprintf('Case lb.2a'); caseid='lb.2a'; Vadd=3./5.*AN des.*(yI.A(5./3)-ystop.^(5./3))Intplc(ystop,yl); sumremain=sumremain+Vadd; Vl=Intplc(yx,ystop)-Inteqb(yx,ystop); Vol=Vl; elseif ystop-=y(nstop+l) & y(nstop+l)>yI sprintf('Case lb.2b'); caseid='lb.2b'; Vl=3./5.*AN_des.*(y(nstop+l).^(5./3)-yI.A(5./3))3./5.*Arep(nstop).*((Dy(nstop)+y(nstop+l)).A(5./3)(Dy(nstop)+yI).^(5./3)); V2=Intplc(ystop,yI)3./5.*Arep(nstop).*((Dy(nstop)+yI).A(5./3)-(Dy(nstop)+ystop).^(5./3)); V3=Intplc(yx,ystop)-Inteqb(yx,ystop); Vol=Vl+V2+V3; elseif ystop-=y(nstop+l) & y(nstop+l)<=yI sprintf('Case lb.2c'); caseid='lb.2c'; Vadd=3./5.*ANdes.*(yI.A^(5./3)-y(nstop+l). (5./3))Intplc(y(nstop+l),yl); sumremain=sumremain+Vadd; Vl=Intplc(ystop,y(nstop+l))3./5.*Arep(nstop).*((Dy(nstop)+y(nstop+l)).A(5./3)(Dy(nstop)+ystop).^(5./3)); V2=Intplc(yx,ystop)-Inteqb(yx,ystop); Vol=Vl+V2; end end difference=Vol-sum_remain; perdif=difference./sum_remain.*100; if yx>=y_plcwat DyF=(hx./AF). ^(3./2)-yx; y_eqbwat = DyF; plot([yx ystop],[hx hstopl,'c'); clear yplot hplot yplot=y_eqbwat:(yx-yeqbwat)./20:yx; hplot=AF.*(yplot+DyF) .^(2./3); plot(yplot,hplot, 'c') plot([ynat(l) ynat(l)+y eqbwat yeqbwat],[-B -B 01,'-c'); else clear yplot hplot yplot=[ynat(l) yB ystop]; hplot=[-B -B hstop]; plot(yplot,hplot, '-+c') if abs(perdif)>l if Vol>sumremain sprintf('Vol>sumremain: change=-5; end if Vol end end decrease Yo'); increase Yo'); if abs(perdif) change=0; if yx>=yplcwat DyF=(hx./AF).A(3./2)-yx; y_eqbwat=-DyF; ResVact=3./5.*AF.*(yx+DyF).A(5./3)Intplc(yplcwat,yx)+0.5.*B.^2./slopenat+B.*(dyI-yeqbwatB./slope nat)+0.5.*B.^2./slopeplace; else yeqbwat=ystop-hstop./slopenat; ResVact=0.5.*(B+hx).*(dyI-y eqbwat); end %if abs((Res Vact-Vact)./Vact.*100)<=l dyo=y eqbwat; %end if abs((Res_Vact-Vact)./Vact.*100)>l if Res_Vact>Vact sprintf('ResVact>Vact: Need to increase X'); end end end if ResVact Need to decrease X'); %end end axis([ynat(1) max(ynat)+5 -B-0.25 max(hnat)+0.25]) 88 legend('Native','SWL','Placement', 'EBP') %sprintf ('%%%%%%%%%%%%%%%%%%%%%%%%% ) %caseid %Vact./V.*100 %change %perdif %keyboard test= [test Yo]; testl= testl ' caseid]; test2= [test2 y-eqbwat]; k=0.5.*B.^2./slopenat+B.*(dyl-yeqbwatB./slopenat)+0.5.*B.^2./slope_place; % Smaller functions used by ffiner.m and/or fcoarser.m function eql=eqtnl(phi_star) global miuF miuN sigmaF eql=miuF-sqrt(2/pi)*sigmaF*exp(-((phi_starmiuF)/sigmaF/sqrt(2))^2)/(l+erf((phi star-miuF)/sigmaF/sqrt(2)))-miuN; function eq2=eqtn2(yI) global V B slope nat slopeplace AN des eq2=-yI+V/B-0.5*B/slopenat+0.5*B/slope place3/5/B*AN_des*yI^(5/3)+O.5*AN_des^2*yI^(4/3)/B/slope_place+AN des*yI^(2/ 3)/slopeplace; function eq3=eqtn3(yint) global Arep slopeplace dyl B slopenat dyo cl=A rep(1)./slope_place; c2=dyl-B./slopenat+B./slopeplace; eq3=-yint+cl.*(yint-dyo).^(2/3)+c2; function eq4=eqtn4(solvey) global ANdes delta Vact Vl A rep Dy yI eq4=delta_Vact(1)-V1-3/5.*ANdes.*(solvey.^(5/3)yI.^(5/3))+3/5.*Arep(1).*((Dy(1)+solve_y).A(5/3)-(Dy(1)+yI).^(5/3)); function eq5=eqtn5(solvey) global deltaVact n AN des y_eqb Arep Dy eq5=deltaVact(n)-3/5.*ANdes.*(solve_y.^(5/3)y eqb(n).^ (5/3) ) +3/5.*Arep (n) .*((Dy(n)+solve_y)^ (5/3)(Dy(n)+y_eqb(n)).^(5/3)); function eq6=eqtn6(ystop) global delta Vact slopeplace Arep Dy hint yint eq6=delta Vact(1)-O.5./slopeplace.*((hint+slope place.*(y_stopyint)).^2-hint.^2)+3./5.*A_rep(1).*((Dy(1)+y stop).^(S./3)(Dy(1)+yint).^(5./3)); function eq7=eqtn7(solvey) global ANdes delta Vact V2 A-rep Dy yI eq7=deltaVact(2)-V2-3/5.*AN des.*(solvey.^(5/3)yI.A (5/3))+3/5.*Arep(2).*((Dy(2)+solvey) .^ (5/3)-(Dy(2)+yI) .^ (5/3)); function eq8=eqtn8(solution) global AN_des y n Arep Dy deltaVact eq8=3./5.*ANdes.*(y(n+l).A(5./3)-solution) .^A(5./3))3./5.*A rep(n).*((y(n+l)+Dy(n)).^(5./3)-(solution+Dy(n)).^(5./3))delta Vact(n); function eq9=eqtn9(yint) |