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Porewater pressure response of an artificial island to wave loading Yogendrakumar, Muthucumarasamy 1983

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POREWATER PRESSURE RESPONSE OF AN ARTIFICIAL ISLAND TO WAVE LOADING by MUTHUCUMARASAMY YOGENDRAKUMAR B.Sc(Eng.), University Of Peradeniya, S r i Lanka, 1 9 8 0 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING DEPARTMENT We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1 9 8 3 © MUTHUCUMARASAMY YOGENDRAKUMAR, 1 9 8 3 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: January 20, 1984 D e d i c a t e d Appah a n d Ammah ABSTRACT Plans for the future development of hydrocarbon reserves in the Western Canadian A r c t i c are based on the use of Caisson retained and Tanker islands as platforms for exploration d r i l l i n g and future production. At present, the design of these islands are based on current geotechnical engineering design procedures. As exploration progresses towards deeper waters, the need for secure designs i s indeed necessary. To be able to achieve t h i s , one requires more sophisticated a n a l y t i c a l procedures with the a b i l i t y to quantify the probable response of these islands to environmental loadings. The chief environmental loads are due to ice, wave and earthquake. A computer based method of analysis i s presented for determining the porewater pressure response of these islands to wave loading. The method considers both d i s s i p a t i o n and generation e f f e c t s during wave loading. It also takes into account o„f the effect of increasing porewater pressure on s o i l properties. The computer program was used to analyse three d i f f e r e n t a r t i f i c i a l islands subjected to d i f f e r e n t patterns of storm waves, each of duration 6 hours. The results of the analyses are presented and discussed. iv TABLE OF CONTENTS Page No. DEDICATION i i ABSTRACT i i i TABLE OF CONTENTS iv LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENTS x i i i CHAPTER 1 INTRODUCTION 1 1.1 Conventional A r t i f i c i a l Islands 1 1.2 Caisson Retained Islands 3 1.3 Scope 5 1.4 Thesis Outline 7 CHAPTER 2 GENERAL ASPECTS OF WAVE INDUCED RESIDUAL POREWATER PRESSURES 9 2.1 Introduct ion 9 2.2 Mechanism For Porewater Pressure Generation 9 2.3 Dissipation E f f e c t s On Wave Induced Residual Porewater Pressure 10 2.4 Wave Induced I n s t a b i l i t y 11 2.5 Review Of An a l y t i c a l Methods 15 2.5.1 Seed and Rahman Method 15 2.5.2 Siddharthan and Finn Method 17 CHAPTER 3 GENERAL THEORY 19 3.1 Assumptions and Idealizations 19 3.1.1 Storm Waves 19 3.1.2 S o i l P r o f i l e and Ocean Floor 22 3.2 Derivation Of Governing Equation 22 3.3 Estimation Of Rate Of Porewater Pressure Generation 23 3.4 Solution Technique 26 3.5 Variation In Volume Compressibility 29 3.6 S o i l Moduli Variation 29 3.6.1 Modification Of Bulk Modulus 31 3.6.2 Modification Of Shear Modulus 32 3.7 Establishing Equivalent Uniform Storm 34 3.8 Linear Wave Theory 35 CHAPTER 4 FINITE ELEMENT FORMULATION OF THE PROPOSED METHOD 38 4.1 Introduction 38 4.2 Formulation of F i n i t e Element Equations 40 4.2.1 Interpolation Function 41 4.2.2 Element Matrix Equation 42 4.2.3 Global Matrix Equation 47 CHAPTER 5 ISLAND GEOMETRIES AND SOIL PROPERTIES FOR WAVE ANALYSES 50 5.1 Island Configuration 50 5.2 Specified Storm Waves 55 v i 5.3 S o i l Properties 56 5.3.1 Basic S o i l Properties 56 5.3.2 Derived S o i l Properties 56 5.3.3 Selection Of I n i t i a l Volume Compressibility 59 5.4 Liquefaction Strength Curve 66 CHAPTER 6 WAVE INDUCED RESIDUAL POREWATER PRESSURE ANALYSIS 70 6.1 General 70 6.2 Response of Islands on Sand Foundation 70 6.3 Wave Induced Porewater Pressure Response Of Island 1 to 6m 6 hour Storm 71 6.3.1 Wave Induced Porewater Pressure Response Of Island 1 to 4m 6 hour Storm 85 6.3.2 Comparison Of Performance to the Two Different Storms 95 6.4 Wave Induced Porewater Pressure Response Of Island 2 99 6.5 Wave Induced Porewater Pressure Response Of Island 3 112 6.6 Summary and Comparison Of Results of Analyses on Sand Foundation 122 v i i CHAPTER 7 EFFECT OF ROCKFILL COVER ON WAVE INDUCED POREWATER PRESSURES 128 7.1 Introduction 128 7.2 Effect Of Cover on Porewater Pressure Response Of Island 1 to 6m 6 hour Storm 128 7.2.1 Effect Of Permeability of Cover Material 141 7.3 Effect Of Cover on Porewater Pressure Response Of Island 1 to 4m 6 hour Storm 143 7.4 Effect Of Cover on Porewater Pressure Response Of Island 2 148 7.5 Effe c t Of Cover on Porewater Pressure Response Of Island 3 158 CHAPTER 8 EFFECT OF FOUNDATION CONDITIONS ON POREWATER PRESSURES 167 8.1 Response Of Island 1 167 CHAPTER 9 SUMMARY AND CONCLUSIONS 172 REFERENCES 176 v i i i LIST OF TABLES No. Page No. 5.1 Details Of Islands 51 5.2 Specified Storms Of the Islands 57 5.3 S o i l Properties Selected For Wave Analyses 58 5.4 Compressibilities of Cohesionless Material in Given Stress Range For Relative Densities 65 6.1 Maximum Porewater Pressure Response At Sections Of Island 1 At the End of the 6m 6 hour Storm 81 6.2 Maximum Porewater Pressure Response At Sections Of Island 1 At the End of the 4m 6 hour Storm 91 6.3 Comparison of Porewater Pressure Response to Two Different Storms 96 6.4 Maximum Porewater Pressure Response At Sections Of Island 2 At the End of the 9m 6 hour Storm 109 6.5 Maximum Porewater Pressure Response At Sections Of Island 3 At the End of the 12m 6 hour Storm 120 6.6 Drainage Characteristics Requirement to Limit Porewater Pressure Ratio to Specified Levels For Different Storms Of Duration 6 Hours. 124 6.7 Predicted Maximum Depth Of Liquefaction At C r i t i c a l Locations For Different Storms Of Duration 6 Hours and Different Permeabilities For Islands On Sand Foundation. 125 ix 7.1 Effect Of 1m Coarse Cover On Maximum Porewater Pressure Response At Sections Of Island 1 At the End Of 6m 6 hour Storm 138 7.2 Effe c t Of 1m Coarse Cover On Maximum Porewater Pressure Response At Sections Of Island 1 At the End Of 4m 6 hour Storm 147 7.3 Effe c t Of 1m Coarse Cover On Maximum Porewater Pressure Response At Sections Of Island 2 At the End Of 9m 6 hour Storm 157 7.4 Effe c t Of 1m Coarse Cover On Maximum Porewater Pressure Response At Sections Of Island 3 At the End Of 12m 6 hour Storm 166 X LIST OF FIGURES No. Page No. 1.1 Caisson Retained Island 4 3.1 Rate of Porewater Pressure Generation 25 3.2 Basic Equation and Solution Domain 27 3.3 Liquefaction Strength Curve 28 3.4 Variation of Volume Compressibility With Porewater Pressure Ratio 30 3.5 Wave Pressure and Definitions of Terms - Linear Wave Theory 37 5.1 Sections of Island 1 For Wave Induced Residual Porewater Pressure Analyses 52 5.2 Sections of Island 2-For Wave Induced Residual Porewater Pressure Analyses .53 5.3 Sections of Island 3 For Wave Induced Residual Porewater Pressure Analyses 54 5.4 Oedometer Test Results For a Libyan Sand 61 5.5 Effect Of Density On Compressibility At Low Excess Porewater Pressure 63 5.6 Volumetric Strain Vs Root V e r t i c a l E f f e c t i v e Stress 67 5.7 Liquefaction Strength Curve Of Sand 69 6.1 Section-AA;J Residual Porewater Pressure 72 to j Response At the end Of to 6.7 Section-GG^/ 6m 6 hour Storm 78 xi 6.8 Section-AA; Shear Stress Ratio D i s t r i b u t i o n At the Start of the 6m 6 hour Storm 80 6.9 Maximum Porewater Pressure Response Of Island 1 At the End of the 6m 6 hour Storm 83 6.10 Maximum Porewater Pressure Response Of Island 1 At the End of the 6m 6 hour Storm 84 6.11 Porewater Pressure Response At the End of 6m 6 hour Storm For Different Permeabilities 86 6.12 Section-AA;} Residual Porewater Pressure 87 6.13 Section-CC;V Response At the end Of 88 6.14 Section-DD;J 4m 6 hour Storm 89 6.15 Maximum Porewater Pressure Response Of Island 1 At the End of the 4m 6 hour Storm 92 6.16 Maximum Porewater Pressure Response Of Island 1 At the End of the 4m 6 hour Storm 93 6.17 Porewater Pressure Response At the End of 4m 6 hour Storm For Different Permeabilities 94 6.18 Time History Of Porewater Pressure At 3m Below Island Surface At Section CC 98 6.19 Section-PP;^ Residual Porewater Pressure 100 to V Response At the end Of to 6.25 Section-W ; J 9m 6 hour Storm 106 6.26 Section-QQ; Flow Through Interface At the End of 9m 6 hour Storm 107 6.27 Maximum Porewater Pressure Response Of Island 2 At the End of the 9m 6 hour Storm 110 xi i 6.28 Section-HH;! Residual Porewater Pressure 113 to ^ Response At the end Of to 6.34 Section-NN;J 12m 6 hour Storm 119 6.35 Maximum Porewater Pressure Response Of Island 3 At the End of the 12m 6 hour Storm 121 7.1 Section-AA;1 Effect Of Cover On Porewater 129 to V Pressure Response At the end Of to 7.7 Section-GG;' 6m 6 hour Storm 135 7.8 Effect of Cover on Time History Of Pore Pressure At 3m Below Origi n a l Island Surface At Section CC 140 7.9 Section-AA; Effect of Cover Permeability on Porewater Pressure Response At the end of 6m 6 hour Storm 142 7.10 Section-AA;1) Eff e c t Of Cover On Porewater 144 7.11 Section-CC ;V Pressure Response At the end Of to 7.12 Section-DD ;J 4m 6 hour Storm 146 7.13 Section-PP;-) Effect Of Cover On Porewater 149 to 7 Pressure Response At the end Of to 7.19 Section-W ; J 9m 6 hour Storm 155 7.20 Section-HHn Eff e c t Of Cover On Porewater 159 to S Pressure Response At the end Of to 7.26 Section-NN;J 12m 6 hour Storm 165 8.1 Section-AA; Residual Porewater Pressure Pesponse At the end Of 6m 6 hour Storm 168 8.2 Section-CC; Residual Porewater Pressure Pesponse At the end Of 6m 6 hour Storm 169 8.3 Section-AA; Shear Stress Ratio D i s t r i b u t i o n 171 ACKNOWLEDGEMENTS I would especially l i k e to thank Professor W.D.Liam Finn, my major advisor, for his encouraging suggestions and support shown throughout t h i s research. I am indebted to Dr. M.de St.Q.Isaacson for his comments and discussion which have enhanced the quality of thi s study. I am especially grateful to my wife, Uma, without whose love, patience and understanding I would never have completed this graduate studies. 1 CHAPTER 1 INTRODUCTION 1 .1 Conventional A r t i f i c i a l Islands Dramatic advances have taken place in offshore d r i l l i n g for both exploration and production of hydrocarbons ever since the f i r s t commitment of the o i l industry to offshore works. Numerous innovative offshore d r i l l i n g methods have been proposed to suit offshore environments generally considered to be ho s t i l e and remote. These include a r t i f i c i a l islands, concrete gravity structures,submersible concrete gravity structures such as the 'Monopad' and the 'Cone'(Stenning et al,1979),bottom-founded mobile rigs and several other types of floa t i n g r i g s . Of these innovative methods, a r t i f i c i a l d r i l l i n g islands are the popular mode in offshore d r i l l i n g in the Mackenzie delta area and the southern Beaufort sea in the Western Canadian A r c t i c . A r t i f i c i a l islands are man-made islands and serve as platforms for exploration d r i l l i n g . The conventional a r t i f i c i a l islands can be divided into d i f f e r e n t main groups depending on the construction techniques. (1) Islands,known as ice islands, b u i l t during winter by trucking on-land gravels and dumping them on the sea bed after removing the ice by cutting i t into blocks. Slope protection is provided after completion of the island. Adequate free board is 2 also provided so that these islands could be used during summer. This type of island i s suitable for water depths less than 2 to 3 metres. (2) Islands b u i l t within an underwater retaining wall consisting of sandbags. The f i l l material required for construction of the island i s hauled in by barges from an offshore borrow p i t . Slope protection above the water l e v e l i s usually provided by additional sandbags. (3) Islands constructed as hydraulic f i l l s with material excavated by suction dredges from art offshore and/or onshore borrow p i t and pumped as a slurr y through a f l o a t i n g pipeline d i r e c t l y onto the island. Slope protection i s provided by a s a c r i f i c i a l beach surrounding the island. This type of island is suitable for intermediate water depths. The technical f e a s i b i l i t y of conventional a r t i f i c i a l islands, p a r t i c u l a r l y in the offshore environment of Beaufort sea, i s influenced to a great extent by the following factors. F i r s t l y , suitable f i l l i n g material must be available in abundance close to the island location. Secondly,enough construction power and equipment must be available on s i t e to haul f i l l i n g material from borrow p i t s and to complete the construction of the island within the limited time available for construction during the summer season. Thirdly,a reasonable construction season must be available so that d r i l l i n g equipment can be moved onto the island in time. 3 The cost for island construction increases substantially in deeper water and at locations where suitable f i l l i n g material cannot be found l o c a l l y . 1 . 2 Caisson Retained Islands The scarcity of suitable f i l l i n g material for island construction, the increased cost involved in transporting suitable f i l l material to the s i t e , coupled with the experience and confidence gained through the performance of existing conventional a r t i f i c i a l islands have given r i s e to the concept of caisson retained a r t i f i c i a l islands. These are islands b u i l t by b a l l a s t i n g reuseable concrete caissons onto a previously b u i l t berm and b a c k f i l l i n g the i n t e r i o r by sand and gravel. The concrete caissons form the geometry of the island and are connected at the corners by-steel doors to retain the f i l l . The maximum set down depth of a set of caissons is fixed,generally around 6 to 9 metres and in the case of deeper water, the underwater berm would be constructed to within the maximum set down depth of the water surface. Once exploration i s complete, the caissons would be floated onto a new location as a ring. Figure 1 . 1 shows schematically a t y p i c a l caisson retained island. The caisson retained islands, also known as CRI have the advantage that they require much less quantity of f i l l material than conventional a r t i f i c i a l islands. Further,these are not subject to s i g n i f i c a n t erosion during or after construction. It also offers the advantage that i t can be constructed more 4 Figure 1.1; Caisson Retained Island (After De Jong and Bruce, 19 5 speedily. The above factors provide the at t r a c t i o n for i t s use in the Beaufort sea in the Western Canadian A r c t i c , where the construction period is extremely limited and uncertain. The p o s s i b i l i t y that CRI could be converted into a production island with appropriate modifications and approvals and also that i t could provide o i l storage within caissons are added advantages of the caisson retained islands. 1.3 Scope The a r t i f i c i a l d r i l l i n g islands b u i l t to date in Beaufort sea are for the purpose of gas and o i l explorations and therefore, they are, at t h i s stage, temporary in character. Further, a l l of them have been constructed in shallow waters within the landfast ice zone, except for a few more recent islands that have been constructed in intermediate water depths on the shear zone which separates the landfast ice from the floes of f i r s t year and multi-year pack ice. These islands are proven to be resistant to wave and ice attacks. However, once gas and o i l explorations progress towards deeper waters, a r t i f i c i a l d r i l l i n g islands w i l l become exposed to harsher offshore environments. They w i l l have interactions with much more mobile ice packs than encountered before. Also they w i l l be exposed to open-water fetches of up to several thousands kilometres depending on wind dir e c t i o n and ice conditions. Therefore, deep water islands have to be designed on the basis of revised design procedures, generally trending towards greater stringency, to ensure their long term success. 6 One of the more interesting and perhaps, more important aspect that has to be given consideration in designing a r t i f i c i a l islands in both deep and intermediate water depths is the wave induced porewater pressure during a storm and i t s implications for the s t a b i l i t y of the island. It has been realised that the magnitude of wave induced porewater pressure at any location in berm and seafloor depends not only on the intensity of the storm but also on the contemporaneous rates of generation and d i s s i p a t i o n of porewater pressure, which in turn depends on the liquefaction, the-drainage and compressibility c h a r a c t e r i s t i c s of s o i l deposits. In practice, s u i t a b i l i t y of f i l l material for berms i s based on the c r i t e r i a drawn mainly from past experience. Generally sand and/or gravel with an average grain size of 150 microns or greater and with less than 10% s i l t are accepted to be most suitable for f i l l material. However, scarcity of such clean sand and gravel in the Beaufort sea area, the economic imperatives coupled with an extremely limited construction period make i t almost impossible to have a good quality control on the material dredged for berm construction that would ce r t a i n l y meet the accepted standard for the f i l l . Therefore, when less permeable f i l l i s used, i t i s possible that during a storm, the porewater pressure may build up substantially, perhaps even to liquefaction l e v e l s , causing great concern for the s t a b i l i t y of the islands. It is also possible that residual porewater pressures after a storm can cause substantial reduction in s t a b i l i t y of the island. In order to handle these 7 conditions, a proper understanding of wave induced porewater pressures during and after a storm i s e s s e n t i a l . A part of the study ca r r i e d out in this thesis i s directed towards finding answers to such potential problems mentioned above. B a s i c a l l y , various analyses were conducted to establish the l e v e l of porewater pressures induced at selected sections of a t y p i c a l a r t i f i c i a l d r i l l i n g island during a moderate storm. Variation in berm configuration, variation in s o i l strata comprising the seafloor s o i l p r o f i l e and their drainage and compressibility c h a r a c t e r i s t i c s were also considered in the analyses to determine their significance on the induced porewater pressure. A l l wave induced porewater pressure analyses were conducted using the computer program STABW3, which was developed by Yogendrakumar, Siddharthan and Finn. It is a modified version of STABW (Siddharthan and Finn, 1979;1982). Some important elements of the program STABW3 are presented in Chapter 3 and 4. 1.4 Thesis Outline Chapter 2 discusses extensively the important aspects of wave induced residual porewater pressure analysis which include the mechanisms of porewater pressure generation and dis s i p a t i o n during wave loading. It also contains a brief review of existing a n a l y t i c a l methods for the determination of wave induced porewater pressures. Chapter 3 deals with the general theory of wave induced residual porewater pressure analysis. The assumptions of the 8 theory are examined and the procedures for incorporating the modifications in s o i l properties caused by increasing porewater pressures are discussed. The motivation for the development of STABW3 program and the formulation of the f i n i t e element equations involved are presented in Chapter 4. The selection of s o i l parameters and other relevant data required for wave induced residual porewater pressure analysis are presented in Chapter 5. Chapter 6 discusses the results of the wave induced residual porewater pressure analysis for di f f e r e n t drainage c h a r a c t e r i s t i c s . The effect of r o c k f i l l cover and foundation conditions are presented in Chapter 7 and 8 respectively. The summary and main conclusions based on the results of the analyses are presented in Chapter 9 . 9 CHAPTER 2 GENERAL ASPECTS OF WAVE INDUCED RESIDUAL POREWATER PRESSURES 2.1 Introduction The wave induced porewater pressure response is a result of a complex interaction between waves and seafloor. However, with certain assumptions and id e a l i z a t i o n s with respect to storm c h a r a c t e r i s t i c s and seafloor c h a r a c t e r i s t i c s , i t i s possible to devise a simple a n a l y t i c a l tool to evaluate wave induced porewater pressures with an accuracy generally acceptable for engineering purposes. Such assumptions and ideal i z a t i o n s are often extensive and are discussed in the next Chapter along with the development of theory. 2.2 Mechanism For Porewater Pressure Generation The mechanism that i s responsible for the generation of residual porewater pressure under the action of waves i s well understood. The waves, as they pass by, create dynamic wave pressure on the seafloor. There are numerous wave theories available to compute the amplitude of the pressure wave, each of which has i t s own assumptions and a p p l i c a b i l i t y based primarily on wave c h a r a c t e r i s t i c s and water depth. Most researchers determine pressure wave amplitude using linear wave theory which assumes the seafloor to be r i g i d and impermeable. Some aspects of the linear wave theory are presented in Section 3 . 8 . This moving harmonic pressure wave on the seafloor creates shear stresses, c y c l i c in nature, the magnitude of which depend 10 on the material properties of the underlying s o i l forming the s o i l p r o f i l e . These c y c l i c shear stresses generate porewater pressures in the underlying s o i l due to the creation of volumetric s t r a i n potential (Martin, Finn and Seed, 1975). 2.3 Dissipation Effects On Wave Induced Residual  Porewater Pressure Unlike earthquakes, storms last much longer, often several hours. Therefore, unlike in earthquake analyses, the common assumption that an undrained condition prevails cannot be adopted in wave induced residual porewater pressure analyses. The analysis assuming undrained conditions w i l l lead to higher porewater pressure response than w i l l actually occur and as a result undue conservatism in design w i l l result from using this approach. To avoid t h i s , i t i s necessary to take into account both d i s s i p a t i o n as well as the generation of porewater pressure. The net porewater pressure response w i l l be the resultant of the two opposing processes mentioned above. Diffusion within and drainage out of the free draining boundary, constitute the di s s i p a t i o n e f f e c t s . These may be substantial in those s o i l s in which drainage can take place e a s i l y . Seed and Rahman (1977) have i l l u s t r a t e d the significance of incorporating d i s s i p a t i o n effects on the wave induced residual porewater pressure response. In general, the changes in porewater pressure response produced by incorporating d i s s i p a t i o n e f f e c t s , w i l l depend primarily on (1) the geometric d e t a i l of the s o i l p r o f i l e , 11 (2) the compressibility and permeability c h a r a c t e r i s t i c s of s o i l layers forming the s o i l p r o f i l e . 2.4 Wave Induced I n s t a b i l i t y The a r t i f i c i a l d r i l l i n g islands b u i l t to date are temporary in character with design l i v e s of a few years at most. The designs of these islands are based only on f a i r l y simple geotechnical engineering design assessments. As exploration progresses towards deeper waters, the need for more secure designs for both temporary and production islands becomes necessary. To be able to achieve t h i s , one requires more sophisticated geotechnical engineering a n a l y t i c a l procedures with the a b i l i t y to quantify the probable response of the islands to environmental loadings. The chief environmental loads are due to ice, waves and earthquakes. The rel a t i v e importance of the ris k from each of thi s sources depends on the location of the island. A brief review of the i n s t a b i l i t y a r i s i n g from wave loading and the methods available in current engineering practice to handle wave related i n s t a b i l i t y are presented herein. The kinds of i n s t a b i l i t y that arise from wave loading f a l l into two main categories. The f i r s t one i s due to the instantaneous stress f i e l d generated by a passing wave. If the intensity of the passing wave i s strong enough, then the eff e c t i v e stresses associated with the wave loading vi o l a t e the Mohr-Coulomb f a i l u r e c r i t e r i o n and consequently f a i l i n g or yiel d i n g w i l l occur in the seafloor or island slope. 1 2 The other kind of wave induced i n s t a b i l i t y arises from the cumulative e f f e c t s of waves which create residual porewater pressures. The consequences of the wave induced porewater pressure are of two types. The f i r s t type i s liquefaction related. If the induced porewater pressures at t a i n a value equal to the i n i t i a l e f f e c t i v e overburden pressure, then s o i l w i l l loose a l l shear strength temporarily. Depending on the conditions such as density of s o i l , s t r a t i f i c a t i o n , slope of the ground surface and nature of supported structures, t h i s temporary loss of strength may result in serious engineering problems. The most common form of these problems, as far as a r t i f i c i a l islands are concerned, are sand b o i l s , excessive subsidence, s l i d e s and. foundation f a i l u r e . The second type of consequence a r i s i n g from wave induced porewater pressures i s somewhat less dramatic but s t i l l of major concern. Even i f the wave induced porewater pressures do not reach l e v e l of lique f a c t i o n , they reduce the i n s i t u e f f e c t i v e stresses and shear resistance of the s o i l so that i t becomes more susceptible to large scale deformation under a passing large wave or gravity stresses. Another possible consequence of residual porewater pressure i s the potential for settlement. The wave induced residual porewater pressure w i l l eventually dissipate, at rates dictated by the drainage c h a r a c t e r i s t i c s of the s o i l . This dis s i p a t i o n w i l l be accompanied by a decrease in volume of the voids which may be reflected in corresponding settlements at the surface. The amount of settlement w i l l depend on the l e v e l 13 of induced porewater pressure, the extent of affected zone and nature of overburden material. At present, the analysis of instantaneous wave induced porewater pressure i s best investigated througth the general computer program STAB-MAX (Siddharthan et al,l979). It i s a coupled e f f e c t i v e stress analysis taking into account the coupling of the sand skeleton and pore water in r e s i s t i n g the waves. The study of the response of seabed to wave loading by Yamamoto (1978) and Madsen(l978) provided the base for STAB-MAX. Yamamoto, in his study, assumed hydraulic isotropy and deposits of f i n i t e and i n f i n i t e depth. On the other hand, Madsen assumed deposits of i n f i n i t e depth but included hydraulic anisotropy. The computer program STAB-MAX is thus a generalisation of the Yamamoto-Madsen solutions to layered s o i l s with hydraulic anisotropy and deposits of f i n i t e depth. A limited f i e l d v e r i f i c a t i o n of the c a p a b i l i t y of STAB-MAX has been reported by Finn et a l (1982). The computer programs available at present for predicting residual porewater pressure and estimating liquefaction potential under wave loading are OCEAN1 (Seed et a l , 1977) and STABW (Siddharthan et al,l979). In thi s thesis, another computer program STABW3 i s developed. This p a r t i c u l a r program is an extended version of STABW. A brief review of the analyses incorporated in these programs is presented in Section 2.5. The analyses incorporated in the STAB-MAX, STABW, and STABW3 are a l l based on the assumption of l e v e l seafloor. Their application to gentle slopes may be acceptable for p r a c t i c a l 14 purposes. But as the slope get steeper the prediction of porewater pressure based on these programs becomes increasingly conservative. The main sources that are responsible for the conservative predictions of porewater pressures are: (i) greater drainage from a sloping seafloor than from a l e v e l one ( i i ) the presence of s t a t i c shear stresses in a sloping seafloor which tend to retard the rate of porewater pressure generation. The s t a b i l i t y of a sloping seafloor may be evaluated by l i m i t i n g equilibrium methods of analysis. Henkel (1970) was the f i r s t to provide an a n a l y t i c a l framework for the s t a b i l i t y of sloping seafloor under a wave loading. His method i s a t o t a l stress s t a t i c method. The method considers the l i m i t i n g equilibrium state of a c i r c u l a r s l i p surface for undrained conditions, taking into account wave pressures on the seafloor, gravity loads and the undrained strength of the s o i l . The main objection for t h i s method i s that i t does not include the true c y c l i c nature of the wave loading and the porewater pressure associated with i t which are so v i t a l for the s t a b i l i t y of seafloor slopes. Finn and Lee (1979) proposed an e f f e c t i v e stress s t a b i l i t y analysis applicable to steeper slopes under wave loading. The method is a modification of Sarma's (1973) method of s l i c e s to include the wave pressures generated by the waves. The method considers an acting force system on the s l i d i n g mass consisting of gravity loads, wave pressure on seafloor, and instantaneous 15 and residual porewater pressures acting on the f a i l u r e surface of the s l i d i n g mass. The main a t t r a c t i o n of th i s method l i e s on the fact that i t recognises the true c y c l i c nature of the wave loading and take into account of porewater pressures associated with the wave loading. 2.5 Review Of An a l y t i c a l Methods 2.5.1 Seed And Rahman Method Seed and Rahman (1977) were the f i r s t to propose an a n a l y t i c a l procedure for evaluation of wave induced residual porewater pressure that takes into account both generation and dis s i p a t i o n e f f e c t s . The procedure i s quite similar to that developed for evaluating l i q u e f a c t i o n potential under earthquake loading (Seed, et a l 1971) except for the manner of determining the induced c y c l i c shear stresses. Their method of analysis contains two separate phases. In the f i r s t phase, the wave induced shear stresses are computed using the computer program STR1. The program evaluates the shear stresses using the theory of e l a s t i c i t y , for the di f f e r e n t wave components constituting the spec i f i e d design storm. The shear stresses computed at the top of the s o i l deposit for each wave component are then used to establish the equivalent uniform storm using procedures proposed by Seed et a l (1975). This enables the complex wave storm loading to be represented by an equivalent uniform wave storm loading consisting of an equivalent number of uniform cycles of a spe c i f i e d shear stress r a t i o . 1 6 In the second phase, the wave induced residual porewater pressures are computed through the f i n i t e element computer program OCEAN 1. The c y c l i c shear stresses induced by the established equivalent wave loading are used in thi s program to estimate the residual porewater pressures. Some aspects of the theory involved, p a r t i c u l a r l y the ones which are common to both the Seed-Rahman method of analysis and the method of analysis to be developed in this thesis are presented in Chapter 3. The computation of c y c l i c shear stresses i s accomplished by a f i n i t e element analysis of an idealized two dimensional problem, requiring two e l a s t i c constants, for convenience, chosen to be the shear modulus and bulk modulus. The shear and bulk moduli are functions of mean normal e f f e c t i v e stresses and as the porewater pressure increases, the mean normal e f f e c t i v e stress decreases, resulting in the degradation of shear and bulk moduli. This, in turn, affects the computed shear stresses. Therefore, in general, i t i s important to include degradation of shear and bulk moduli in computation of c y c l i c shear stresses and thereby to obtain reasonable estimates of the rate of porewater pressure generation. Although the Seed-Rahman method of analysis takes into account of the variation in volume compressibility due to the eff e c t of increasing porewater pressure, i t never considers the degradation of the shear and bulk moduli in the computation of c y c l i c shear stresses. It i s a known fact that in the case of deep uniform deposit, the maximum c y c l i c shear stress induced by wave loading i s independent of the e l a s t i c constants. In 17 these cases, the degradation of moduli in the computation of shear stresses are not necessary. However, in the case of f i n i t e and non-uniform deposits, considered to be the general case, the shear stresses depend on the e l a s t i c constants and i t is essential to modify the s o i l properties for the effect of increasing porewater pressure. In order to handle the most general case of non-uniform deposits a method of analysis which considers the degradation of s o i l properties along with variation of volume compressibility for the ef f e c t of increasing porewater pressure is needed. Such a method of analysis was f i r s t proposed by Siddharthan and Finn (1979) and i t i s outlined b r i e f l y in the next section. 2.5.2 Siddharthan and Finn Method The method of analysis proposed by Siddharthan and Finn (1979, 1982) is b a s i c a l l y a generalization of the Seed and Rahman method. In this method of analysis, the stress analysis phase is combined with the residual porewater pressure analysis phase into a single f i n i t e element computer program STABW. In th i s way, i t is possible to modify e l a s t i c constants repeatedly to be comparable with the current value of porewater pressure and to re-evaluate c y c l i c shear stresses and thereby the rate of porewater pressure generation. The program has the option of carrying out analysis with or without s o i l property modifications for the effect of increasing porewater pressure. Apart from t h i s improvement, the other main difference in this approach from the Seed-Rahman approach i s the manner by 18 which the equivalent uniform storm i s established. Instead of the procedure adopted by Seed and Rahman which uses a simple weighting curve to determine equivalence, the more general procedure proposed by Lee and Chan (1972) i s used. The d e t a i l s of the procedure are presented in Section 3.7. 19 CHAPTER 3 GENERAL THEORY 3.1 Assumptions and Idealizations Most methods of analysis require that the problem under consideration be idealized in some way so that a convenient model may be formulated. The wave induced residual porewater pressure analysis to be developed in thi s thesis i s no exception to t h i s . The assumptions and i d e a l i z a t i o n s implied in defining storm c h a r a c t e r i s t i c s , ocean and s o i l p r o f i l e s are described in this section. The assumptions involved in other elements of the analysis, for example, in the development of governing equations and in the computation of wave induced shear stresses, are presented in sections where they are developed. 3.1.1 Storm Waves The offshore wave environment i s a random process dependent on wind speed, water depth, mudline c h a r a c t e r i s t i c s and various other factors. However, in practice, i t i s customary to define the sea state at any time by two important variables, namely, wave heights and periods existing at that time. The common parameters that characterizes the sea state in the s t a t i s t i c a l sense are the s i g n i f i c a n t wave height H s and the s i g n i f i c a n t wave period T§. The s i g n i f i c a n t wave height is a n a l y t i c a l l y defined as the average height of the highest t h i r d of the waves and the s i g n i f i c a n t wave period is the mean period 20 of the waves chosen for the determination of the s i g n i f i c a n t wave height. The s i g n i f i c a n t wave height and s i g n i f i c a n t wave period can be estimated by wave hindcasting techniques which involve the application of wind data. These are determined d i r e c t l y in terms of wind speed U, fetch F and duration t over which the wind acts. More dire c t information on wave c h a r a c t e r i s t i c s can be determined from a continuous record of surface elevation obtained from a wave recorder. The important parameters required to define the ov e r a l l wave c h a r a c t e r i s t i c s are: (i) the zero-crossing period,T z, defined as the average period between sucessive zero up-crossing, ( i i ) the crest period,T c , defined as the average period between sucessive crests-, ( i i i ) the v e r t i c a l distance from the lowest trough to the highest c r e s t . In t h i s thesis, the storm waves are described in terms of si g n i f i c a n t wave height and s i g n i f i c a n t wave period. The two approaches widely used for analysis involving wave loading are the discrete wave method of analysis and the spectral method of analysis. In thi s thesis, the discrete method of analysis i s used. This approach makes no attempt to model wave loading process as a random excitation but assumes that the process can be s p l i t into discrete waves each of which has a specified period associated with i t . The d i s t r i b u t i o n of wave heights in a wave storm is assumed to be given by a Rayleigh density function and i t i s often specified in terms of 21 s i g n i f i c a n t wave height in the form, p(H) = 1 - exp{-2(H/H s) 2} (3-1) where, H = wave height Hg= s i g n i f i c a n t wave height p(H) = prob a b i l i t y density function For a given s i g n i f i c a n t wave height, the pro b a b i l i t y of occurence of a wave of height H, occurring between H, and H 2, where H,< H < H 2, is given by, P{H) = p(H,) - p(H 2) (3-2) The pro b a b i l i t y of occurence P(H) given by equation (3-2) is assumed to be associated with a wave of height (H, + H 2)/2. The maximum wave height in the wave height d i s t r i b u t i o n i s assumed to be the breaking height associated with a s t i l l water depth d and for shallow water cases, i t can be calculated using following equation suggested by McCowan (Sarpakaya et a l , 1981), H m = 0.78 d (3-3) For analysis, waves of height greater than H m are assumed to be waves of height of H^ ,. In other words, the t o t a l number of waves of height greater than H,^  in the d i s t r i b u t i o n are added to the number associated with the wave of height H w. The Rayleigh d i s t r i b u t i o n enables the storm to be represented by many waves, each of them d i f f e r i n g in c h a r a c t e r i s t i c s . It i s assumed that these waves have ch a r a c t e r i s t i c s in accordance with linear wave theory, which describes the wave by i t s period, wave height and water depth. 22 Some aspects of the linear wave theory are presented in Section 3.8. The waves are assumed to travel predominantly in one di r e c t i o n , that i s , the eff e c t of d i r e c t i o n a l randomness i s assumed to be ne g l i g i b l e . Also, the shoaling e f f e c t s , the wave scouring effects and the d i f f r a c t i o n effects in modifying the responses are not taken into account. 3.1.2 S o i l P r o f i l e and Ocean Floor The entire s o i l p r o f i l e i s assumed to comprise of horizontally layered s o i l s , each of them are of i n f i n i t e l a t e r a l extent. The properties of s o i l deposits are assumed to vary only in the v e r t i c a l d i r e c t i o n and each deposit is divided into layers each with uniform properties. The ocean floor is assumed to be p a r a l l e l to s t i l l water l e v e l . 3.2 Derivation Of Governing Equation As discussed previously the governing equation of wave induced porewater pressure response in an offshore environment should incorporate the eff e c t of both di s s i p a t i o n and generation. In developing the governing equation, i t is assumed that Darcy's flow is v a l i d . Hence from the one dimensional continuity equation in z d i r e c t i o n , 5/6z { k z/?f w • 5u/5z } = 5e/6t (3-4) where, u = excess porewater pressure kz= c o e f f i c i e n t of permeability in z ( v e r t i c a l ) d i r e c t i o n T w= unit weight of water 23 e = volumetric s t r a i n , reduction considered to be po s i t i v e . Consider an element of s o i l with excess porewater pressure u. Suppose i t undergoes a change of Au in excess porewater pressure during an interval of time At, then during that time i n t e r v a l , i t w i l l be subjected to a certain number of cycles of c y c l i c shear stress, which in turn, w i l l cause an increase in porewater pressure given by (6u 3/6t).At, where (6u g/6t) i s the rate of porewater pressure generation. If the change in bulk stress i s neglected then the volume change be of the element in that i n t e r v a l of time i s given by, Ae = mv( AU - 6ug/6t .At) (3-5) where, my = c o e f f i c i e n t of volume compressibility. Now as At—*0, be/bt = mv ( 5u/6t - 6u g/5t ) (3-6) From equations (3-4) and (3-6), b/bz { ^Aui • bu/bz } my ( 5u/5t - 5u g/6t ) (3-7) Equation (3-7) is the governing equation for porewater pressure response to storm waves. It has been used previously by Finn et al (1976) for the analysis of seismically induced porewater pressures. 3.3 Estimation Of Rate Of Porewater Pressure Generation The rate of porewater pressure generation required in equation (3-7) can be determined by the procedure proposed by Seed and Rahman (1977). The basic steps involved are given 24 herein. The rate of porewater pressure increase can be written in the form, 5u 9/6t = 5ug/6N . 5N/6t (3-8) where N i s the number of stress cycles during the storm. The values of 5ug/6N can be obtained from undrained te s t s . However, for p r a c t i c a l purposes, the relationship between u g and N can be expressed in terms of number of cycles N L required for i n i t i a l l iquefaction in the following form, us/ov'0 = 2/TT arc Sin ( N/Nu )' / 2 0 (3-9) where, a^Q = i n i t i a l v e r t i c a l e f f e c t i v e stress 8 = an empirical constant The relationship in equation (3-9) i s given in Figure 3.1 for d i f f e r e n t values of 6. The value of 8 = 0.7 i s t y p i c a l for clean medium sands. D i f f e r e n t i a t i o n of equation (3-9) with respect to N and si m p l i f i c a t i o n y i e l d s , 6u3/6N = ay'0 /( 07rN,_ ) . l / f ( r u ) (3-10) where, f ( r u ) = S i n ^ 9 " ' ) (0.57rru) . Cos(0.5jrr u) r u = porewater pressure r a t i o , u/o^ Also, 5N/6t = N e q/T p (3-11) where Neq = equivalent number of uniform stress cycles corresponding to the established equivalent uniform storm with duration Tp. Therefore, from equations (3-8), (3-10) and (3-11) Figure 3 . 1 ; Rate of Porewater Pressure Generation 26 6u g/6t = o^/(07rTp) . (N eq/N u) . 1/f ( r u ) (3-12) The rate of porewater pressure generation 5ug/5t, at any time, can be calculated from equation (3-12) knowing the value of porewater pressure at that time. 3.4 Solution Technique With the rate of porewater pressure generation given by equation (3-12), i t i s now possible to solve equation (3-7) for the domain and boundary shown in Figure 3.2. The formulation of the proposed f i n i t e element method is outlined in d e t a i l in Section 4.2. To compute the rate of porewater pressure generation from equation (3-12) at any location, one needs to know N L, the number of cycles to cause i n i t i a l wave induced li q u e f a c t i o n . Nj_ can be conveniently computed from a li q u e f a c t i o n strength curve such as the one shown in Figure 3.3. The shear stress r a t i o induced by the equivalent storm at the location of interest can be used to est a b l i s h the appropriate N L values. To establish a liquefaction strength curve, c y c l i c loading tests, usually the c y c l i c simple shear test or the c y c l i c t r i a x i a l test, can be performed on representative undisturbed samples. In the case of tests performed in the t r i a x i a l apparatus, a correction factor has to be applied to the results to account for the two dimensional plane s t r a i n condition of ocean wave loading. A correction factor between 0.60 to 0.70 i s considered reasonable. However, in the case of tests conducted in simple shear apparatus, the correction factor is not necessary, as i t 3 . 2 ; Basic Equation and Solut ion Domai Figure 3 .3 ; L iquefact ion Strength Curve (After Seed and Rahman, 1977) oo 29 provides the closest representation of f i e l d conditions (Seed, 1979). 3.5 Variation In Volume Compressibility The volume compressibility of s o i l increases with increase in porewater pressure. The volume compressibility can be computed using the following equation proposed by Martin (1 976) . mv/mvo = e* /( 1 + y + 0.5y2) (3 -13) where, B y = A.r a A = 5(1.5 - D r) -2T>t-B = 3 x 2 D(- = r e l a t i v e density r u= porewater pressure r a t i o mv= volume compressibility at porewater pressure r a t i o , u/a; 0 mvo= i n i t i a l volume compressibility at zero porewater pressure r a t i o Results from equation (3 -13) for D r = 50% is given in Figure 3 .3 . 3 . 6 S o i l Moduli Variation As discussed e a r l i e r in Section 2.5.1, the e l a s t i c constants used in the c y c l i c shear stress analysis have to be modified for the effect of porewater presssure. In the present 3 0 31 wave induecd residual porewater pressure analysis, the s o i l moduli are modified in the manner described below. 3.6.1 Modification Of Bulk Modulus A comprehensive study by Duncan et a l (1978) reveals that bulk modulus depends on minor p r i n c i p a l e f f e c t i v e stress and the variation can be approximated by an equation of the form, Bm = K b P q ( a ' 3 / P 0 )"* (3-14) where B = bulk modulus m Kfc,= bulk modulus constant m = bulk modulus exponent PQ = atmospheric pressure, expressed in the same units as a' 3 and B^ . a' 3 = average e f f e c t i v e minor p r i n c i p a l stress, assumed to be Kj, o^0 With porewater pressure u, the minor p r i n c i p a l e f f e c t i v e stress is given by a' 3 = ( K oo V 0 - u ) (3-15) Therefore, from equations (3-1.4) and (3-15), the compatible bulk modulus B^j. for the current l e v e l of porewater pressure is given by Bmk/Bmo = < (*o°vo " u ) / ( K 0 a V Q )}"» (3-16) where B m o is the i n i t i a l bulk modulus at zero porewater pressure and K 0 i s the c o e f f i c i e n t of earth pressure at rest. Equation (3-16) represents the modification of bulk 32 modulus for the effect of porewater pressure adopted in thi s thesi s. 3.6.2 Modification Of Shear Modulus Seed and Idriss (1970) developed a relat i o n s h i p for the determination of maximum shear modulus G mo X (at shear strains less than 10~^%) in the form Gmax = 1ooo k 2 m f l x (o'n j'Z (3-17) where, = mean normal e f f e c t i v e stress in psf k2jyioix = parameter which depends on s o i l type and r e l a t i v e density Dr It i s also suggested that k2max r o r sands (Byrne,1981) is as follows; k2max = (15 + 0.6.Dr) (3-18) For gravels and s i l t s , k 2 i s given as k2»viax = ( 1 5 + ° - 6 5 D r ) F (3-19) The parameter F depends on s o i l type and t y p i c a l values of F are, F = 2.0 for gravels F = 0.6 for s i l t s The empirical equation for the determination of values of maximum shear modulus, proposed by Hardin and Drnevich (1972) is of the form G m c* = 320.8 {(2.973 - e)*/(1 + e)} (OCR)* (o'm /Pa fz (3-20) where, 33 e = void r a t i o OCR = overconsolidation r a t i o fe = parameter that depends on the p l a s t i c i t y index of the s o i l . The equations (3-17) and (3-20) imply that Grnax depends on a^, and i s proportional to io'^ )'^. This allows a modification of shear modulus for the effect of porewater pressure in the form, Gmfc - Gm 0 l°'m/°m >"2 <3-2D where, Gm{. = compatible shear modulus for the current l e v e l of porewater pressure u, G ^ Q = i n i t i a l value of shear modulus at zero porewater pressure, a'mt = mean normal e f f e c t i v e stress at current.level of porewater pressure u, a' m o = mean normal e f f e c t i v e stress at zero porewater pressure. and o'm^ can be calculated using the following equations; °*0 = (1 + 2 K 0)/3 a'Vo (3-22) a' w t = (1 + 2 K a)/3 ( a; o - u ) (3-23) where a y 0 = i n i t i a l v e r t i c a l e f f e c t i v e stress. In t h i s thesis, the modification of shear modulus for the effect of porewater pressure, is taken into account in the form given by equation (3-21). 34 3.7 Establishing Equivalent Uniform Storm As pointed out e a r l i e r in Section 2.5.2, the Lee and Chan (1972). The very f i r s t step i s to select a reference wave in the wave height d i s t r i b u t i o n resulting from Rayleigh d i s t r i b u t i o n (see Section 3.1.1). The maximum wave or a wave of height close to the maximum wave height i s often chosen as the reference wave. Now, the shear stress r a t i o  T/°^0 t a t the top of the deposit ( i e , z—»-0) i s calculated for each of the wave components in the storm and also for the selected reference wave. Using these r/a^0 values at z=0, for each of the waves involved, the number of cycles to cause i n i t i a l l iquefaction (N L) i s computed from an appropriate liquefaction strength curve, such as the one shown in the Figure 3.3. The equivalent number of cycles, N^q, for the selected reference wave can be calculated from the equation, equivalent storm i s established using the method proposed by Nl (3-24) where, ^Leq = number of cycles required to cause i n i t i a l l i q u e f a c t i o n obtained from appropriate liquefaction strength curve, corresponding to the shear stress r a t i o T/O\ vo at z=0 for the selected reference wave. M L^ = number of cycles required to cause i n i t i a l l i q u e f a c t i o n obtained from liquefaction strength curve, corresponding to the shear stress r a t i o 35 r/o^ 0 at z = 0 for the i wave component. N£ = number of waves of the i*** wave component in the wave storm. Hvo = t o t a l number of wave components representing the wave storm. 3.8 Linear Wave Theory Linear wave theory has been used in this thesis for the purposes l i s t e d below: (1) To describe each of the wave components in the storm. (2) To compute the pressure wave loading on the seafloor required for the cal c u l a t i o n of shear stresses due to each of the wave components. The theory assumes that the seafloor to be r i g i d and impermeable. According to the theory, the equation of the wave p r o f i l e of a wave of height H and period T i s given by: Y s = H/2 Cos { 2TT (x/L - t/T) } (3-25) and the wave length L can be obtained from the following i m p l i c i t equation: L = (p.5gT 2A) tanh (27rd/L) (3-26) where, d = s t i l l water depth g = acceleration due to gravity x = space coordinate in horizontal d i r e c t i o n t = time coordinate The pressure wave loading Ap, imparted on the seafloor by 36 the wave i s given by: Ap = p o Cos { 2TT ( X / L - t/T) } (3-27) where, Po = 0.5tffcH / { Cosh (2ird/L) } tfft = density of sea water. The d e f i n i t i o n of terms and other elements of linear wave theory are shown in Figure 3 . 5 . 37 Ocean F loor Figure 3 . 5 ; Wave Pressure and D e f i n i t i o n s of Terms - L inear Wave Theory. 38 . CHAPTER 4 FINITE ELEMENT FORMULATION OF THE PROPOSED METHOD 4 .1 Introduct ion The method of analysis developed in this thesis for the evaluation of wave induced residual porewater pressures is an extended version of Siddharthan-Finn method with an apparent difference in the degree of the polynomial used in the interpolation for the porewater pressure f i e l d . The f i n i t e element computer program STABW3 uses a complete cubic polynomial interpolation function for the porewater pressure f i e l d , whereas STABW uses a linear interpolation function. The motivation for using a higher degree, polynomial in the interpolation i s for the reason stated below. It has been observed that when a f i n i t e s o i l deposit is analysed for the wave induced residual porewater pressures, a l l existing a n a l y t i c a l methods, which were b r i e f l y reviewed in the previous chapter, indicate higher residual porewater pressures at lower elevations for cases with s o i l having higher permeabilities than for cases with lower permeabilities, while a l l other potential variables remain the same. It i s believed that t h i s phenomenon is due to the increased downward flow of water associated with cases where the s o i l has a higher c o e f f i c i e n t of consolidation, that i s , higher permeability. In order to examine and v e r i f y the above phenomenon, i t is neccessary to know the time history of flow through nodal points. This cannot be achieved through the existing methods 39 reviewed in Chapter 2 because of the fact that they use a linear interpolation function for the porewater pressure f i e l d . One requires a higher degree polynomial to include flow {in the form k z(du/dz)} as- a nodal variable. A complete cubic interpolation function is chosen in the STABW3 f i n i t e element formulation. This requires two nodal variables per node to uniquely define the porewater pressure f i e l d . The porewater pressure u at the node and the flow through the nodal point in the form k z(du/dz), are selected as the required nodal variables. The formulation allows the determination of the time history of residual porewater pressure response at any depth within the domain and also the flow through the interface at nodal points. Since a higher degree p o l y R o m i a l is used in the interpolation, i t i s apparent that higher accuracy and faster convergence may be achieved for the solution. Though most of the important aspects are common for STABW and STABW3, the noticeable difference occurs in the formulation of the f i n i t e element equations as a result of differences in interpolation function. The other main difference in STABW3 f i n i t e element formulation comes from the fact that the terms in functional J {see equations (4-12) to (4-15)} are accounted for in a di f f e r e n t manner than in STABW f i n i t e element formulation. The f i n i t e element formulation of STABW3 is given in the next section and the important aspects involved in the development of this method of analysis are already outlined in Chapter 3. 40 4 . 2 Formulation Of F i n i t e Element Equations The basic equation (see sect ion 3.2) governing the res idual porewater pressure response i s , 3 f^z 3u. ,9u ^ Ug s 3z V 1? = m v ("3T — 5 T ) ( 4 - i ) w At any instant of time, the r ight hand side of equation (4-1) may be considered to be a function of z only . Hence equation (4-1) reduces to , 3 3u. . >. - z (-37) = Q(z> ( 4-2) w The funct ional J for a d i f f e r e n t i a l equation of the form, as in equation (4-2) i s , (|^) 2 + 2 Q(z)u]dz Y dZ 0 Expanding the above, D . z ,3u,. ,D J = ( H ) U ] 0 " 0 w with boundary condi t ions , u = 0 at z = 0 and i H . = 0 at z = D, 3z the boundary term in the funct ional vanishes. Hence, D J - - j 17* ( I I ) 2 + 2 Q ( 2 ) U ] D Z ( 4 - 3 ) 0 w Suppose the s o i l deposit i s considered as an assemblage of f i n i t e number of elements, then J t ° t a l = elements J e l e m e n t (4-4) 41 4.2.1 Interpolation Function In order to evaluate the functional, an interpolation function for u must be selected. Let us choose a cubic interpolation function for u. This would be more than enough to s a t i s f y the completeness c r i t e r i o n . Now, 2 3 u «= a1 + a 2 n + a 3 n + a^n (4-5) where, n = l o c a l coordinate system, and a l t o aA = c o e f f i c i e n t s which need to be evaluated. Each elememt has two nodes1, therefore, two nodal variables per node i s required to uniquely define u. Let us choose u and q as two nodal variables, where (—) q = k M z From equation (4-5) |H = a 2 + 2a 3n + 3a An 2 Then, q = kz (a 2 + 2a 3n + 3 a ^ p 2 ) (4-6) Now, consider the i t h element with nodes i and i+1 and thickness dj . • r A t n - 0 , u - u . and q - q ± ; n A t n - d ± , u •= u 1 + 1 and q - q ± + 1 -42 U s i n g t h e s e i n e q u a t i o n s ( 4 - 5 ) and ( 4 - 6 ) t o g e t , u i " a l (4-7) M i z 2 (4-8) u i + l " a l + a 2 d i + a 3 d i 2 + a A d i 3 (4-9) 4i+l k (a 0 +2a.d, + 3a.d/) z 2 3 i 4 I (4-10) S o l v i n g f o r a 1 to a^ f r o m e q u a t i o n s (4-7) t o (4-10) and bac k s u b s t i t u t i n g i n t o e q u a t i o n (4-5) w i l l y i e l d , N u. — — l I N.u. J - l J J where (4-11 ) e u. = — i > < "i+1 q i + l and u s i n g x = n/d. (1 - 3 X 2 + 2 X 3 ) (d./k )(1 - 2 X 2 + X 3 ) i z 2 3 ( 3 A Z - 2 X J ) N, = (d./k ) ( - X 2 + X 3 ) A i z 4.2.2 E l e m e n t M a t r i x E q u a t i o n Consider J e l e m e n t from equation (4 -3) , - J element 0 d. I-5- ( | V + 2 Q(n)u]dn w k - „ . 3u r 2 , OUv Z . , dU g. , . [ 7 ~ W + 2 - - d T ) u ] d T 1 w where e e e h + I 2 - X 3 ( 4 - 1 2 ) 0 d z /duN2 , — (-rr) dn Y 9n w 2 n uC-l^dn V • dt and i . 3u 2 m u(—&)dr •V o l Now c o n s i d e r I, 0 1 z ,011*2 , — (—) dn Y dn w k d , . , — & d A Y o A W From e q u a t i o n ( 4 - 1 1 ) , u = N.u., j = 1,4, , s u b s t i t u t i n g t h i s , 1 k d. z 1 Y " k " k w N'.u. N'u. dX, 1=1,4 and j = l,4 3 1 , 3u, 2 — • d J N'.N.' U , dX Y „ i J k j 44 [Se] '{uJ} (4-13) where [Sg] - 4 x 4 symmetric matrix with the general term given by, k 2 — • o \ N ' N : dX w or '11 12 z 5 Y d . w 1 '12 = S 21 1 _1_ 5 ' r w '13 '31 -S 11 '14 '41 '12 S r% r\ n r '22 4 . 1 15 k Y z w '23 " S32 = " S 21 '24 = S 42 1 d i 15 k Y z w '33 '34 = S 11 -S 12 '44 = S 22 Now c o n s i d e r i„ e , 2 a , 0 1 2 m u(-|7)dn V d t 2 m d., N,u. N.U dX v i k k j 2 mvd. u(|H)d.X 31, 3u, 2 m d, N, N. U dX v i k j 45 - H>E] <V (4-14) where [D g] - 4 x 4 symmetric m a t r i x w i t h the g e n e r a l term g i v e n by, 2 m d . N, N. dX v i k j or '11 35 v i '24 "42 70 m v 7 ^ D 1 2 D 11 v i 21 105 k '33 "11 '13 D31 = 35 V i D14 = D41 13 v i 210 k '34 "43 D44 = D22 D,„ = -D 12 m d . 2 v i '22 105 '23 = D32 = "D14 Now c o n s i d e r i3 1 du 2 m u (-?-*) dn V Q t 0 L e t us assume a s i m p l e l i n e a r v a r i a t i o n f o r (3u /3t) g 3u 3u 3u as, 3t (1-X) ( J f ) i + (-^) 3t 46 To accomadate this into the f i n i t e element formulation, l e t (3u g/3t) as 3u = ( M l M2 M3.MA) J 3u 3u ^3t ;i+l where M1 = (1-X) M2 = MA = 0 M3 = X Using this technique, 3u 2 m d, N,u, M. {T-*}dX v i k k j 3t 31. 3u, 3u 2 m d. N.M.{-r-*}dX v i k j 3t 3u [Re] U ^ ) e ) ( 4 - 1 5 where [R ] = 4 x 4 symmetric matrix with the general term 6 1 given by, or 2 m d. N,M, dX V I K j R31 = 10 V i 11 10 v l -R„= R41 1 m d / "15 - R , m d . 1 v i 4 * ~ 21 10 k 47 A l l o t h e r t e r m s a r e z e r o . 4.2.3 G l o b a l M a t r i x E q u a t i o n s Now, 3J , ' 3I. C 31* 31* element _ 1 + 2 3 d{u±e] 3{ U i 6} 3{ U i e} 3{ u e) From e q u a t i o n s ( 4 - 1 3 ) , ( 4 - 1 4 ) a n d ( 4 - 1 5 ) 3J , 3u e l e men t - [se]{u.e} + [De]{ue] + [Re]{(a^)e> 3{u. } l U s i n g v a r i a t i o n a l p r i n c i p l e s , 3J 3{u} = 0 T h a t i s , I "element = {0} elements 3{u. } I 3u I [S e]{u. e} + lD e]{U e) + [R £]{(—£) } = {0} elements Summing up w o u l d y i e l d t h e g l o b a l m a t r i x e q u a t i o n a s a 3u [S](u> + [D]{f} + [ R ] ^ } = {0} { 4 _ 1 6 ) The g l o b a l m a t r i c e s [D] and \ R ] a r e f u n c t i o n s o f c o m p r e s s i b i l i t y m v a n d , h e n c e , v a r y w i t h p o r e w a t e r p r e s s u r e r a t i o . The g l o b a l m a t r i x [s] i s c o n s t a n t f o r a g i v e n p r o b l e m . The m a t r i x e q u a t i o n ( 4 - 1 6 ) c a n be t r e a t e d a s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n and be i n t e g r a t e d o v e r t h e t i m e i n t e r v a l 48 t , t + At t o g e t 9u t S ] 6 { u t + A t } + a{u t)]At + [ D ] U u t + A t } - {ut}] - [R]{-^*)At = {0} ( 4 - 1 7 ) where a+B = 1 and s u b s c r i p t s t and t + A t c o r r e s p o n d t o v a l u e s a t time t and t + A t r e s p e c t i v e l y . M a t r i c e s [5] a n d \R] a r e c o n s t r u c t e d u s i n g a v e r a g e v a l u e s o f v a r i a b l e s b e t w e e n t i m e t and t + A t . R v a l u e s g r e a t e r than o r e q u a l t o 0.5 c o r r e s p o n d s t o d i f f e r e n t a p p r o x i m a t i o n s . H owever, i n t h i s p r o g r a m a v a l u e o f £ = 0.5 i s u s e d . E q u a t i o n ( 4 - 1 7 ) c a n be g r o u p e d t o g e t h e r t o f o r m , [AQKu t + A t) - {BQ1}+(BQ2} u _ i e ) where [AQ] = [S]6At = [D] {BQ1} = (-[S]cxAt + [5]){u t) 3u {BQ2} =. [R]{(^r £ ) ) • At W i t h s p e c i f i c a t i o n o f b o u n d a r y c o n d i t i o n s , a p p r o p r i a t e c o l u m n s and rows o f [A Q] a r e s t r u c k o u t t o f o r m a n e t g l o b a l m a t r i x [ A Q * ] . So a r e c o r r e s p o n d i n g rows o f { B Q 1 } and { B Q 2 } t o form net v e c t o r s {B Q 1 *} and { B Q 2 * } . H e n c e , [ A Q * ] { u t + A t } = { B Q 1 * } + { B Q 2 * } = { BQ* } ( 4 - 1 9 ) The p r o g r a m h a s t h e o p t i o n o f c o m p r e s s i b i l i t y , e i t h e r v a r y i n g o r r e m a i n s c o n s t a n t . I n t h e e v e n t o f c o n s t a n t 49 c o m p r e s s i b i l i t y , equation (4-19) i s solved ins tant l y for every time step. However,, in the event of varying compress ib i l i t y , equation (4-19) i s solved i t e r a t i v e l y . Each time var iable matrices [ D ] and £R ] are ca lcu la ted using the best current estimate of nodal v a r i a b l e s . The i t e r a t i v e procedure i s repeated u n t i l a s p e c i f i e d accuracy or a s p e c i f i e d maximum number of i t e r a t i o n s i s obtained, whichever occurs f i r s t . 50 CHAPTER 5 ISLAND GEOMETRIES AND SOIL PROPERTIES FOR WAVE ANALYSES 5.1 Island Configuration Wave induced residual porewater pressure analyses using STABW3 were conducted for three d i f f e r e n t islands at water depths 12m, 21m and 31m respectively. The other d e t a i l of the islands are presented in Table 5.1. The berm configuration and the v e r t i c a l sections selected for the analyses are shown in Figure 5.1 to Figure 5.3. Since only v e r t i c a l sections are considered in the analyses, the slope of the berm does not d i r e c t l y a f f e c t the method of analysis except for defining variations in water depth. However, the slopes play a major role in the structural s t a b i l i t y of the islands and also in containing the flow of l i q u e f i e d s o i l in cases of l i q u e f a c t i o n . To be able to analyse an island for wave induced porewater pressure, the v e r t i c a l sections considered has to be of i n f i n i t e l a t e r a l extent and moreover the top p r o f i l e has to be p a r a l l e l to s t i l l water surface. But in r e a l i t y , the sections, for example AA to GG of island 1, are not of i n f i n i t e l a t e r a l extent and also they are not p a r a l l e l to s t i l l water surface. The actual l a t e r a l length and the top p r o f i l e of those sections depend on the shape and slope of the berm. In the analyses conducted in t h i s thesis, a l l such sections are assumed to be of i n f i n i t e l a t e r a l extent and the top p r o f i l e to be p a r a l l e l to the s t i l l water surface. Is land No. Bern, height(m) S t i l l Water depth(m) Set down depth(m) 1 6.0 12.0 6.0 2 15.0 21.0 6.0 3 25.0 31.0 6.0 Table S'i ; D e t a i l s Of Islands 52 ! 30 m * * Sand F E D c 8 A F i g u r e 5 - 1 ; S e c t i o n s o f I s l a n d 1 f o r Wave I n d u c e d R e s i d u a l P o r e w a t e r P r e s s u r e A n a l y s i s . 5 3 S W L Sa Seafloor ' • - — ' nd Bei •m SI ope 1 9tn :5 II w 8 - ^ ^ p 6m ism Seaf loor Sand 50 m V "77—J T i ' r u T 5 R Q p " Figure 5>2; Sect ions of Is land 2 for Wave Induced Residual Porewater Pressure Ana lys i s . — ¥ Sanr, Seafloor 1 - — J Berm srn Slope 1:5 « J | L 13 rr) k 15 m 'L———— I7rrj M i l — 21 m 6rrj 25rrj 50 m Seaf loor Sand N M L. K 3T 1 H Figure 5 - 3 ; Sections of Island 3 for Wave Induced Residual Porewater Pressure Ana lys is . 55 The above assumption i s j u s t i f i a b l e in the cases of berms of very gentle slopes. For berms of sharp slopes, i t i s believed that the above assumption would lead to conservative estimates of wave induced residual porewater pressures because of the effect of s t a t i c shear stresses. The presence of s t a t i c shear stresses i s to retard the rate of porewater pressure generation and hence the resulting porewater pressure response with the above assumptions would be higher than as i t actually would be. The conservative nature is also due to fact that drainage would be faster in a slope than in horizontal ground. The e f f e c t s of s t a t i c shear stresses and the slope of berms can be taken into account by various ways as b r i e f l y described below. 1. By using a porewater pressure generation model that takes into account of the influence of s t a t i c shear stresses in the development of porewater pressure during c y c l i c loading. An example would be the model proposed by Finn et a l (1978). 2. By using a modified equivalent permeability to cater for the increase in the drainage resulting from sloping ground. 3. By using a modified strength curve to cater for preshearing and preconsolidation. However, these types of refinements are rarely required because of the uncertainity associated with the information gathered from an offshore s i t e . 5.2 Specified Storm Waves Storm waves are described by three parameters. They 56 are the s i g n i f i c a n t wave height, the s i g n i f i c a n t wave period and the duration. Each of these parameters depends on the location of the s i t e and several other factors related to the s i t e . For the purpose of the analyses, storms are spec i f i e d for each of the island and the d e t a i l s are given in Table 5.2. The number of cycles in each case, i s given by (6 x 3600)/8, which is equal to 2700. 5. 3 S o i l Properties 5.3.1 Basic S o i l Properties The basic s o i l properties such as densities, r e l a t i v e density, void r a t i o , s p e c i f i c gravity of s o l i d etc used in the analyses are shown in Table 5.3. 5.3.2 Derived S o i l Properties I n i t i a l Shear Modulus For sands and gravels; The i n i t i a l value of shear modulus for sand and gravel were computed using the equations (4-17), (4-18) and (4-19). For sand of D t - = 50%, from equation (4-18), K 2 max = 4 5. For gravel and r o c k f i l l of D = 50%, from equation (4-19), k 2Max = 9 5 • required for shear modulus G m < J X c a l c u l a t i o n in equation (4-17) was calculated using the following equation, = (1 + 2K 0)/3 . o v o (5-1) Is land No. (m) (sec) (hrs) 4:01 1 1 • 8.0 6.0 6 .0 . 2 9.0 8.0 6.0 3 12.0 8.0 6.0 Table 5-2 ; Spec i f ied Storms O f the Islands 58 Property S o i l Type Sand Gravel Clay Tota l Unit Weight(kN/m3) 19.0 19.5 18.0 Sub. Unit Weight(kN/m3,) 9.0 9.4 8.0 S p e c i f i c Gravi ty 2.65 2.67 2.67 Void Ratio 0.85 0.65 0.90 Relat ive Density (%) 50.0 50. -Angle of Internal F r i c t i o n (deg) 33. 37. 22. I n i t i a l Compress ib i l i t y (mc/kN) 3x 1 0 ' £ 1 .9x10"* 1 0 ^ V e r t i c a l permeabi l i ty (cm/sec) 1 o" 3 -1 0~ 4 10.0 10"' Empi r ica l Constant 0.70 0.10 0.10 Bulk Modulus exponent 0.50 0.50 0.0 Poisson Ratio 0.35 0.25 0.45 Table 5 - 3 ; S o i l Propert ies Selected For Wave Analyses 59 K0 was calculated using the equation, K 0 = 1 - sin <j>' (5-2) where </>' = angle of internal f r i c t i o n . For clays; The i n i t i a l shear modulus G m Q x for clays was computed from the following equation, where S u = undrained strength of clay. I n i t i a l Bulk Modulus The bulk modulus, B, for sand, gravel and clay were computed using the e l a s t i c relationship, where v = i n i t i a l Poisson r a t i o , G = shear modulus. 5.3.3 Selection Of I n i t i a l Volume Compressibility A close examination of the governing equation for the wave induced porewater pressure in- Section 3.2 reveals that the parameter that plays the most c r u c i a l role in determining the levels of porewater pressure that may develop in the berm i s the c o e f f i c i e n t of consolidation defined as k^/my*^ . For th i s reason, both k z and mv are equally important and should be determined experimentally in order to obtain the most r e a l i s t i c estimates of wave induced porewater pressures. Unfortunately, no experimental data on compressibility are available on G m v l x = 1000 S u (5-3) B/G = 2 ( l + v ) / 3 ( 1 - 2 * ) (5-4) 6 0 potential sand f i l l . A brief review on selected compressibility data on sand is presented herein. The compressibility of sand is usually determined in an oedometer test. The sand i s confined in a s t i f f s tainless steel ring and v e r t i c a l settlements under increasing v e r t i c a l e f f e c t i v e stress are recorded. A t y p i c a l oedometer test result i s shown in Figure 5 . 4 , where the volumetric s t r a i n , e v%, i s plotted against v e r t i c a l e f f e c t i v e stress, Oy The c o e f f i c i e n t of volume compressibility m^  i s defined as, mv = de v/do^ ( 5 - 5 ) where de v = change in volumetric st r a i n corresponding to a small change in e f f e c t i v e v e r t i c a l stress, do^ . Therefore, the slope of the experimental curve plotted in the form shown in Figure 5 . 4 , is the c o e f f i c i e n t of volume compressibi1ity. Two d i f f e r e n t phases of loading can be i d e n t i f i e d in Figure 5 . 4 . The f i r s t one corresponds to v i r g i n loading where the e f f e c t i v e stress i s always increased. The other phase corresponds to rebounding where the e f f e c t i v e stress i s reduced. It i s noticeable that the compressibi1ies (slope of the curve) under these two loading phases are quite d i f f e r e n t and the compressibility being higher under v i r g i n loading. It is also seen from the Figure 5 . 4 that during unloading the rebound compressibility increases and further the amount of increase depends on the l e v e l of unloading. During wave loading, residual porewater pressures are 61 8 I \ 1 s N L s \ s. — 0 10 20 30 40 Vtrttul Urns, t. Of/cm') Figure 5 . A ; Oedometer Test Results For a Libyan Sand ( A f t e r Lambe and Whitman, 1969) 6 2 generated. As a result of t h i s , the e f f e c t i v e stress regime i s changed. That i s , the i n i t i a l e f f e c t i v e stress, a^0 , i s reduced by the increase in porewater pressure to result in a new current e f f e c t i v e stress of (o v o-u) while ay 0 i t s e l f remain constant. Therefore, in effect the sand i s rebounding during c y c l i c loading which means appropriate mv values have to be obtained from rebound portions of experimental curves. The rebound compressibility values have to be also adjusted depending on the l e v e l of residual porewater pressures. The data on compressibility of sand due to rebounding is very limited. The major contribution in this area i s the experimental data by Lee and Albaisa (1974). Based on their comprehensive study, Seed et a l (1976) proposed variations of rebound compressibility with increasing porewater pressure at constant t o t a l stress for sands at various r e l a t i v e densities. The important conclusion emerged from the study of Seed et a l (1976) regarding the v a r i a t i o n of compressibility r a t i o , expressed as the r a t i o of current compressibility to the compressibility at low excess porewater pressure, i s that for values of porewater pressure r a t i o upto 60%, neither the grain size nor the r e l a t i v e density have a marked influence on the compressibility r a t i o s . The rebound compressibility of sandy s o i l s can be determined from Figure 5.5. For the analysis conducted in t h i s thesis, the selected rebound compressibility of sand f i l l (D r = 50%) i s 3.0 x 10~5 m2/kN (0.15 x 10"5 f t 2 / l b ) , which agrees quite well with the corresponding value from the Figure 5.5. 0 20 40 60 80 100 Relative Density, D r - % Figure 5 - 5 ; E f fec t Of Density On Compress ib i l i t y At Low Excess Porewater Pressure. (After Martin and Seed, 1978) 64 Rebound compressibility of sand can also be computed from rebound modulus, E,. , given by da^ /de v . The rebound compressibility i s thus the reciprocal of the rebound modulus. Based on the experimental study, Martin et a l (1976) developed an expression for Er at any current e f f e c t i v e stress , in terms of i n i t i a l e f f e c t i v e v e r t i c a l stress , as E = U' v )'" m/mk 2(a v o )r)~™ (5-6) With appropriate values of m, n and k 2 for the sand, E and thereby rebound compressibility can be computed. Rebound compressibility can also be determined from the compressibility under v i r g i n loading. The factor by which the compressibility value for v i r g i n loading be divided in order to obtain rebound compressibility i s often recommended to be atleast 2. Table 5.4 presents the compressibility data quoted by Lambe and Whitman (1979) for v i r g i n loading for dif f e r e n t s o i l s under two dif f e r e n t stress ranges. In the low stress range, the rebound compressibility computed from above data i s about 1.0 x 10~5 m2/kN for dense sand and 3.6 x 10~5 m2/kN for loose sand. Therefore, for medium dense sand, rebound compressibility in the range 1.0 x 10~5 m2/kN and 3.6 x 10~5 m2/kN may be expected. It i s observed that the selected rebound compressibility of 3 x 10"5 m2/kN for the medium dense sand f i l l in the analyses reported in th i s thesis f a l l s within t h i s range. The oedometer measurements of compressibility are often found to be unreliable because of the errors involved in the standard oedometer equipment. The primary sources of errors are -S 2 Virgin Compressibility my (10 ) m /kN Soil Relative For For Density 62-103 kN/m2 200 - 510 kN/m2 Uniform Gravel 0 3.30 1 .67 1 mm< D < 5 mm 100 0.85 0.56 Well Graded Sand 0 7.24 3.92 0.02mm< D < 1 mm 100 1 .93 0.82 Uniform Fine Sand 0 6 .81 2.84 0.07mm< D <0.3mm 100 1 .96 0.83 Uniform S i l t 0 35.71 5 .81 0.02mm< D <0.07mm 100 2.64 1 .32 Table S-4-i Compressibilities of Cohesionless Material in Given Stress Range For Relative Densities 0% and 100%. (After Lambe and Whitman, 1979) Note; For Rebound Compressibilities, the above values have to be divided by atleast 2. 66 due to; 1. Compressibility of the oedometer system. This i s found to be comparable with that of sand and thus very d i f f i c u l t to correct, 2. Side f r i c t i o n , 3. High void spaces at the contact of the consolidation ring, 4. Improper contacts with the top and bottom porous stones, 5. Inaccurate r e l a t i v e density measurements resulting from small specimen si z e . Because of these errors, the compressibility of sand i s often overestimated. For this reason, Cornforth (1974) studied the compressibility of sand in the t r i a x i a l apparatus. The tests were conducted on Brasted sand under K 0 conditions. He observed that the consolidation curves were parabolic and there is a linear r e l a t i o n between volumetric s t r a i n and root v e r t i c a l e f f e c t i v e stress as, X is dependent on the dry r e l a t i v e density, RDD, of the sand. The results from his study are presented in Figure 5.6. For Dr = 50%, X = 0.026 From equation (5-7), = x U v ) V z (5-7) mv = de v/da^ = 0.005 X (o' v ) (5-8) Therefore, for D r = 50%, m = 0.0001 3 u \ , ) f ' / 2 (5-9) V For mean e f f e c t i v e v e r t i c a l stress of 10 kN/m2, mv = 4 x IO"5 m 2/kN 6? 3 a 2 " u -U to -t-> (1) e £ 4 Root V e r t i c a l E f f e c t i v e Stress ; J«T • k /^ni* 10 20 30 40 50 60 Condition coefficient of Linearity x e 70 / Dense o - o i 9 50/. Med. Dense 0-026 A 30/ Loose 0*040 • 10/ Very LooAt o. o l^ Figure 5 - 6 ; Volumetric S t ra in Vs Root V e r t i c a l E f f e c t i v e Stress (After Cornfor th , 1974) 68 Since this value i s for v i r g i n loading, i t should be divided by a factor of 2. Therefore, the rebound compressibility i s , mv = 2.0 x 10~5 m 2AN Hence, for the stress range of interest, the selected value of rebound compressibility compares well with the value calculated above. 5.4 Liquefaction Strength Curve The liquefaction strength curve for sand of = 50% used in the analyses is given in Figure 5.7. The curve was deduced from the strength curve for sand of D f = 54% presented in Figure 3.3, by reducing the c y c l i c shear stress r a t i o by a factor 50/54. Number Of Cycles To L iquefac t ion , NL 5 - 7 ; L iquefact ion Strength Curve of 70 CHAPTER 6 WAVE INDUCED RESIDUAL POREWATER PRESSURE ANALYSIS 6.1 General The wave induced residual porewater pressure analyses reported herein were conducted using computer program STABW3. In a l l cases, modifications of s o i l properties for the effect of porewater pressure was taken into account in the manner discussed in Section 3.6. The importance of incorporating s o i l property modification for the effe c t of increasing porewater pressure has been discussed already in Section 2.5. The porewater pressure responses established in the analyses are a l l free f i e l d responses. The d i s t o r t i o n of porewater pressure response due to the presence of any structures were not considered in the analyses. Hence, considerable caution should be exercised in interpreting the responses in the v i c i n i t y of any structures placed on the berm. 6.2 Response of Islands on Sand Foundation The f i r s t series of analyses on islands 1, 2 and 3 s i t t i n g on a sand foundation susceptible to liqu e f a c t i o n were conducted for di f f e r e n t drainage c h a r a c t e r i s t i c s of the sand f i l l . The i n i t i a l compressibility of the sand f i l l for thi s particular series of analyses i s taken as 3.0 x 10~5 m2/kN. The other properties of the sand f i l l and seafloor sand are given 71 in Table 5.3. The ef f e c t of diss i p a t i o n on the porewater pressure response i s controlled by the value kz/mv Tf^ . Having selected the same compressibility value for a l l the analyses, i t is now possible to compare the effect of drainage c h a r a c t e r i s t i c s , that i s , the effect of variation in k z on the porewater pressure response. 6.3 Wave Induced Porewater Pressure Response Of Island 1  to 6m, 6 hour Storm Figure 6.1 to 6.7 show the residual porewater pressures induced at the end of a 6 hour storm with s i g n i f i c a n t wave height of 6m at selected sections AA to GG of island 1 for the d i f f e r e n t permeability values of k z = 10"3 cm/s and k z = 10"" cm/s. For the permeability value of 10"3 cm/s, the results indicate that there i s no liquefaction at any of the sections considered in the analyses in constrast to results with k z 10"" cm/s, where liqu e f a c t i o n occurs to substantial depth at a l l sections. This c l e a r l y shows the significance of drainage on the wave induced porewater pressure response. For the case with k z = 10"3 cm/s, the i n i t i a l drainage i s much greater than in the case with k z = 10"" cm/s, and as a result the porewater pressures developed in the former case are much lower. It i s interesting to note, at thi s point, that the analyses assuming undrained conditions would have predicted liquefaction to depths as much as 9 to 10 m in a l l the sections of the island. 72 Porewater pressure r a t i o , 26 h 32 r 36 UJ 1 I I I I Figure 6 . 1 ; Sect ion-AA; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. Figure 6 . 2 ; Sect ion-BB; Residual Porewater Pressure Respon At the end of 6m 6 hour Storm. 74 36' ' ' ' ' 1 Figure 6 . 3 ; Sect ion -CC; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 75 6.4; Sect ion-DD; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 7 6 Porewater pressure r a t i o , u / ^ S 2 h 361 1 1 1 — 1 1 Figure 6.5; Sec t ion -EE ; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 77 2&Y Sfcl 1 1 1 — - r J Figure 6 . 6 ; S e c t i o n - F F ; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 78 Porewater pressure rat io , u /*vo 3 2 h 561 1 1 1 1 -I Figure 6 . 7 ; Sect ion-GG; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 79 The residual porewater pressure d i s t r i b u t i o n s presented in Figures 6.1 to 6.7 show the same trend; the maximum porewater pressure r a t i o , u/a^ 0 , occurs very near the top and decays rather steadily as the depth increases. In the case with k z = 10"4 cm/s, the decay starts to occur beneath the zone of li q u e f a c t i o n . This decaying trend i s similar to the ty p i c a l shear stress r a t i o d i s t r i b u t i o n , such as the one shown in Figure 6.8. This i s as expected because as the shear stress r a t i o decreases, the number of cycles required to cause i n i t i a l l i q u e faction, N L, increases. Now from equation (3-12), the porewater pressure generation i s inversely proportional to N L. Hence, the porewater pressure generated would be higher at the top and decrease as the depth increases. -It i s also seen that as the depth of water increases, for example,,6 m at section AA to 12 m at secton GG, the porewater pressure response increases up to a certain c r i t i c a l location and then starts to decrease. The above trend i s apparent for both values of k z. For analyses with k z = 10"3 cm/s, the maximum porewater pressure r a t i o developed at sections AA to GG are such that i t increases from 47% at section AA to 66% at section CC and decreases to 22% at section GG. On the other hand, for analyses with kz = 10"" cm/s, the maximum depth of liquefaction increases from 6.5m at section AA to 9m at section CC and then decreases to 6m at section GG. Table 6.1 shows the maximum porewater pressure response in terms of porewater pressure r a t i o , u/a v o , for the case with kj. = 10"3 cm/s and in terms of depth of liquefaction for the case 80 Shear stress r a t i o . ' V o ^ 361—1 1 _i_ , , I igure 6.8; Section-AA; Shear Stress Ratio D i s t r i b u t i o n At the start of the Storm. 81 Maximum pwp Response Section Water (d/H s) pwp Ratio L iquefact ion depth(m) (%) depth(m) k = 1 0"*5cm/s k = 1 0~*cm/s AA 6 1 .00 47 6.5 BB 7 1.17 • 62 8.0 CC 8 1 .33 67 9.0 DD 9 1 .50 65 8.5 EE 10 1 .67 43 7.0 FF 1 1 1 .83 29 6.5 GG 12 2.00 22 6.0 Table 6 - 1 ; Maximum Porewater Pressure Response At Sections of Is land 1 At the End of the 6m 6 hour Storm 82 with kz = 10~" cm/s at the selected sections of the island 1. The above results are presented in Figure 6.9 and 6.10 respectively. From these figures, i t can be concluded that the effect of the storm is severe at one par t i c u l a r section and the location of thi s c r i t i c a l section in terms of i t s water depth is given approximately by 1.4 to 1.5 times the s i g n i f i c a n t wave height of the storm, regardless of the k 2 values. Results from Figures 6.1 to 6.7 also indicate that for a par t i c u l a r location, there exists a c r i t i c a l value of permeability that would prevent any liquefaction at that location during storm a c t i v i t y . If the permeability i s greater than t h i s c r i t i c a l value then there would not be any liquefaction and i f the permeability is less than t h i s c r i t i c a l value then li q u e f a c t i o n would occur at thi s p a r t i c u l a r section. The level of porewater pressure r a t i o induced or the extent of the liquefaction zone depends on by how much the permeability is greater or lesser than the c r i t i c a l value. Having recognised that section CC is the closest to the c r i t i c a l location, where the eff e c t of the 6m, 6 hour storm i s f e l t severely, analyses were conducted at section CC, to determine the c r i t i c a l value of permeability that would prevent liquefaction or in other words, that would l i m i t the porewater pressure r a t i o to within 95 to 100%. This c r i t i c a l value of permeability would serve as the minimum permeability required to prevent liquefaction within the entire island for the spe c i f i e d 6m, 6 hour storm. Figure 6.9; Maximum Porewater Pressure Response of Island 1 At the end of 6m 6 hour Storm. 84 Maximum Depth of l i q u e f a c t i o n (m) -2o cn •rt <D > ro cn -rt c n a cu T 3 -P ro » O o ro Figure 6.10; Maximum Porewater Pressure Response of Island 1 At the end of 6m 6 hour Storm. 85 Figure 6.11 shows the residual porewater pressure d i s t r i b u t i o n at section CC for the di f f e r e n t values of k z between 10"3 and 10"" cm/s with the i n i t i a l compressibility in each of these cases kept as 3.0 x 10"5 m2/kN. It is seen from the figure that the c r i t i c a l value of permeability that i s required to l i m i t porewater pressure development below 95 to 100% of the i n i t i a l e f f e c t i v e stress for the specified 6m, 6 hour storm i s around 8.0 x 10"" cm/s. It i s interesting to note that the permeability required to l i m i t porewater pressure ra t i o to 65% or less within the entire island for the spec i f i e d storm is 10"3 cm/s. It is often convenient to interpret these results in terms of c o e f f i c i e n t of consolidation. The i n i t i a l c o e f f i c i e n t of consolidation, Cy 0 , required to meet the above mentioned c r i t e r i a are 2.7 x 10"2 m2/s and 3.4 x 10"2 m2/s respectively. 6.3.1 Wave Induced Porewater Pressure Response of Island 1 to  4m, 6 hour Storm Figures 6.12 to 6.14 show the residual porewater pressure induced at the end of the 4m, 6 hour storm at section AA, CC and DD of island 1 for di f f e r e n t k z values of 10"3 and 10" tt cm/s. It i s evident from these figures that the porewater pressure response show the same kind of steady decaying trend as the e a r l i e r response with the 6m storm. The apparent difference being that the porewater pressure response shows a steady decrease as the water depth increases from 6m at section 86 Figure 6.11; Porewater Pressure Response At the end of 6m 6 hour Storm for D i f fe ren t P e r m e a b i l i t i e s . 87 Porewater pressure ratio,M/*v« 32r 36U 1 1 1 . I Figure 6 . 1 2 ; Sect ion-AA; Residual Porewater Pressure Response At the end of 4m 6 hour Storm. 88 24H 28 H 32 36l 1 1 i , I Figure 6.13; Sect ion -CC; Residual Porewater Pressure Response At the end of 4m 6 hour Storm. 89 24 28 3 2 3 $ L 1 1 . . | Figure 6 . 1 4 ; Section-DD; Residual Porewater Pressure Response At the end of 4m 6 hour Storm. 90 AA to 9m at section DD. For analyses with k z = 10"3 cm/s, the maximum porewater pressure r a t i o developed at section AA i s 19% and at section DD is 10%. Further, for the case with k z = 10"" cm/s the results indicate that the depth of liquefaction at section AA is 6.5m and i t reduces to 2m at section DD. The maximum porewater pressure response for the two cases of k 2 values are presented in Table 6.2 and the results are p l o t t e d in Figures 6.15 and 6.16. Both figures show that the maximum porewater pressure occurs at section AA, where the water depth i s 1.5 times s i g n i f i c a n t wave height. Section AA could not be guaranteed d i r e c t l y as the c r i t i c a l location for the 4m storm since there can be a c r i t i c a l location at water depths less than 6m. However, as far as the island is concerned, the minimum water depth to island surface is 6m and accordingly section AA can be treated as the c r i t i c a l location for the 4m storm. Figure 6.16 also indicates that during the 4m storm, the depth of water beyound which liquefaction would not occur for the k z value of 10"" cm/s is approximately given by 2.42 times the s i g n i f i c a n t wave height. Additional analyses were conducted at section AA, the c r i t i c a l section for the 4m, 6 hour storm for d i f f e r e n t permeability values between 10"3 and 10"" cm/s to determine the c r i t i c a l permeability value that would prevent li q u e f a c t i o n within the entire island during the storm a c t i v i t y . The results of these analyses are presented in Figure 6.17. It i s evident 91 Section Water depth(m) (d/Hf) Maximum pwp Response pwp Ratio (%) k = 1 0"Scm/s L iquefact ion depth(m) k = 1 6"*cm/s AA 6 1 .50 19 6.5 CC 8 2.00 13 5.0 DD 9 2.25 10 2.0 Table 6 -2; Maximum Porewater Pressure Response At Sections of Island 1 At the End of the 4m 6 hour Storm 92 Maximum Porewater pressure rat io ,M/^ Figure 6 .15; Maximum Porewater Pressure Response of Island 1 At the end of 4m 6 hour Storm. 93 Figure 6 .16; Maximum Porewater Pressure Response of Is land 1 At the end of 4m 6 hour Storm. 94 32 3 6 Porewater pressure ratio,^/GJO O.l 0.4 o«6 o-8 ««0 5 - k*= 2 . 5x 1 0~* cm/s 6 - k r= 10"^ cm/s sect ion AA * Figure 6 . 1 7 ; Porewater Pressure Response At the end of 4m 6 hour Storm for D i f fe rent Permeab i l i t i es . 95 that the c r i t i c a l value of permeability that l i m i t s the porewater pressure r a t i o to within 95 to 100% i s 2.5 x 10"" cm/s while a value' of 3.5 x'10"" cm/s i s s u f f i c i e n t to l i m i t porewater pressure r a t i o to 65% or less. The corresponding values of i n i t i a l c o e f f i c i e n t of consolidation are 8.5 x 10"3 m2/s and 1.19 x 10"2 m2/s respectively. 6.3.2 Comparison of Performance to the Two Different Storms Table 6.3 provides the opportunity to compare the effect of storm c h a r a c t e r i s t i c s on the induced porewater pressure response at the selected sections AA, CC and DD. The results c l e a r l y indicate that the more severe the intensity of the storm, the higher the maximum porewater pressure response w i l l be. For example, the maximum porewater pressure r a t i o developed for the case k z = 10"3 cm/s is 19% (at section AA) due to 4m storm, while i t is 67% (at section CC) due to 6m storm. The ef f e c t of storm intensity i s also c l e a r l y seen in the permeability requirement for the c r i t e r i a discussed e a r l i e r . The permeability required to l i m i t porewater pressure r a t i o to within 95 to 100% i s 2.5 x 10"" cm/s and 8.0 x 10"" cm/s for the 4m and 6m storm respectively. This indicates that the requirement i s more stringent in the case of severe storms. The same trend i s apparent in the requirement to l i m i t porewater pressure r a t i o to 65% or l e s s . To understand the eff e c t of the storm intensity more 96 Section Water depth (m) pwp r a t i o (%) kz = 1 0 3cm/s L iquefact ion depth(m) k z = 1 0""cm/s Hs=4m H,=6m H a = 4m • * Hs = 6m' AA 6 19 47 6.5 6.5 CC 8 13 67 5.0 9.0 DD 9 10 65 2.0 8.5 Table 6-3; Comparison of Maximum Porewater Pressure Response to Two D i f fe ren t Storms 97 c l e a r l y , i t is. perhaps important to compare the time history of the porewater response during the storms. Such a comparison at a pa r t i c u l a r depth 3m below the island top surface at section CC i s highlighted in Figure 6.18. As can be seen from the figure, the residual porewater pressure builds up steadily in the case with k z = 10~3 cm/s u n t i l the end of the storm and dissipates f a i r l y rapidly after the storm a c t i v i t y . In contrast to t h i s , the porewater pressure in the case with k z = 10"" cm/s builds up f a i r l y rapidly, attains liquefaction level and dissipates rather slowly after the storm. The constrasting behavior can be attributed to the differences in drainage c h a r a c r t e r i s t i c s . For both values of k z, the rate of residual porewater pressure build up i s found to be higher during the 6m storm than the 4m storm. It is also noticeable that for the case with k z = 10"" cm/s, liquefaction l e v e l i s attained within f i r s t half an hour of the 6m storm, whereas for the 4m storm, 3 hours is required to attain liquefaction l e v e l . The di s s i p a t i o n after the end of storms i s faster for 4m storm than the 6m storm. The difference in the rate of residual porewater pressure build up can be attributed to the difference in N c <j, the equivalent number of cycles of the reference wave, since a l l other potential variables remain the same i n i t i a l l y for the two storms. The major contributing factor to the residual porewater pressure i s the porewater pressure generated due to the action of c y c l i c shear stresses. It can be seen from equation (3-12) that ug is d i r e c t l y proportional to (Necj/NL) for a given T — 1 » i 1 ! — i 1 r End of storm Time (hours) Figure € . 1 8 ; Time History Of Porewater Pressure At 3m Below Island Surface At Section CC. CO 99 porewater pressure r a t i o . Since the difference in N u is marginal at the start of the storms, the dominating parameter becomes Ne<j. The values of Ngq for the 4m and 6m storm at section CC are 100 and 559 respectively. Thus, the resulting net porewater pressure response during the storm a c t i v i t y i s higher for the 6m storm than the 4m storm. 6.4 Wave Induced Porewater Pressure Response of Island 2 The porewater pressure response at selected sections PP to W of island 2 at the end of the 6 hour storm with s i g n i f i c a n t wave height of 9m for the two di f f e r e n t k z values of 10"3 and 10"" cm/s are shown in Figures 6.19 to 6.25. The results indicate that liquefaction occurs at a l l sections for the case with 10"" cm/s and at a l l sections, except PP, for the case with k z f 10"3 cm/s. It is also apparent that the zone of liquefaction in the case with kx - 10"" cm/s i s deeper than with k z = 10"3 cm/s. However, analyses with k 2 = 10"3 cm/s indicate higher porewater pressures at depths well below the zone of liquefaction than with k z = 10"ft cm/s. This kind of behaviour can be attributed to the dominant influence of the d i f f u s i o n of porewater pressure within the p r o f i l e . To i l l u s t r a t e the above argument, the flow through the nodal point, in the form k z.(du/dz), at the part i c u l a r section QQ predicted by STABW3 analysis i s presented in Figure 6.26 for the two cases of k z value. It appears that at the end of the storm, the effect of the top drainage boundary is f e l t only up Porewater pressure r a t i o , " / ^ o 0-2. o-4 o-6 o-S 1 0 40 • SO 60 Figure 6 . 1 9 ; Sect ion -PP; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. Porewater pressure ratio,U/<*vo io V TOY 4-1 40 5 0 I-6 0 r Figure 6 . 2 0 ; Section-QQ; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. 10 20 £, 3 0 a D AO no 60 k2 = 1 Cf3 cm/s k r = 1 0"4 cm/s Figure 6 . 2 1 ; Sect ion-RR; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. Porewater pressure ratio, u/0vo 6oh Figure 6 . 2 2 ; Sect ion -SS ; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. 104 Porewater pressure ratio,u/j^, ° 0-3 Q.4 0-6 0-6 1-0 Figure € . 2 3 ; Sect ion -TT ; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. 105 Porewater pressure ratio,M/fer^, O 0-2 O.J. 0.6 o-8 ».0 Figure 6 .24 ; Sect ion-UU; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. Porewater pressure ratio,ty<rj 0 O Q-2 Q>A O-fc «-8 M? Figure 6 . 2 5 ; S e c t i o n - W ; Residual Porewater Pressure Response At the end of 9m 6 hour Storm. Figure 6 . 2 6 ; Sect ion-QQ; Flow Through Interface At the end of 9m 6 hour Storm. 1 0 8 to a certain depth, 10 to 11 m and beyound th i s depth there is substantial downward flow. It i s also seen that in the case with k z = 10"3 cm/s, the flow through the top drainage boundary is much higher, as much as 10 times, than in the case with k,. = 10"" cm/s. However, in both cases, liquefaction occurs presumably because of the higher rate of porewater pressure generation in the top few metres. The depth of liquefaction i s shallower in the case with k z = 10"3 cm/s because of higher drainage through the top boundary. At lower elevations, the difference in the rate of porewater pressure generation for the two values of k z are marginal because of the fact that N L remains the same as a result of very low r / a ^ values. Hence, the only factor that could influence the porewater pressure response, especially under the circumstance that the eff e c t of the top drainage boundary is not f e l t e f f e c t i v e l y at depths, i s the difference in d i f f u s i o n of porewater pressure within the p r o f i l e . Figure 6.26 c l e a r l y indicates that the downward flow at lower elevations is higher in the case with k z = 10"3 cm/s than with k z = 10"" cm/s. This i s because of the higher C v value associated with i t . This increased downward d i f f u s i o n and high porewater pressures at the top few metres make the porewater pressure higher at lower elevations with k z = 10"3 cm/s. The maximum porewater pressure response at sections of island 2 at the end of the spec i f i e d storm is summarized in Table 6.4 for the two values of k z considered. The results are plotted in Figure 6.27. The porewater pressure response shows 109 Sect ion Water depth (m) (d/H s) Maximum pwp Response L iquefact ion depth (m) k z= 1 0~3 cm/s kz = 1 0" + cm/s PP 6 0.67 0.0 7 .0 ' QQ 8 0.89 6.0 9.0 RR 10 1.11 8.0 11.5 SS 12 1 .33 10.5 12.5 TT 14 1 .55 11.0 1 3 .5 UU 16 1 .78 9.0 12.0 W 18 2.00 2.0 8.5 Table 6 » 4 ; Maximum Porewater Pressure Response At Sections of Island 2 At the End of the 9m 6 hour Storm 1 10 Figure 6.27; Maximum Porewater Pressure Response of Island 2 At the end of 9m 6 hour Storm. 111 the same kind of trend as before; that i s , the response for both cases increases with depth of water u n t i l a c r i t i c a l depth is reached and then decreases beyound that depth. The c r t i c i c a l location in this case i s around section TT and the c r i t i c a l water depth in terms of the s i g n i f i c a n t wave height of the storm is approximately given by 1.50 H s. It i s also found that, within the l i m i t s of data, the location is unique and i s not dependent on the drainage c h a r a c t e r i s t i c s of the berm material. This agrees very well with the results obtained in the analyses involving island 1. It can be inferred from Figure 6.27 that the depth of water beyound which liquefaction would not occur for the specified storm, depends on the drainage c h a r a c t e r i s t i c s of the berm mate-rial. For the case with k z = 10"3 cm/s, the depth in terms of s i g n i f i c a n t wave height i s 2.10 H s. It would appear that in the case of kz= 10"' cm/s, thi s depth i s increased considerably, as high as to 2.55 H s. This would mean that liquefaction is possible even at the bottom most section of island 2, as the maximum depth of island 2 is 2.34 . Additional analyses conducted at section TT of island 2, the closest section to c r i t i c a l location for the 9m, 6 hour storm, reveal that the permeabilities required to l i m i t porewater pressure r a t i o to 95 to 100% and 65% or les s , are 2 x 10"3 cm/s and 3 x 10"3 cm/s respectively. 1 12 6.5 Wave Induced Porewater Pressure Response of Island 3 The porewater pressure response at selected sections HH to NN of island 3 at the end of the specified 12m, 6 hour storm for the two diff e r e n t k z values of 10"3 cm/s and 10"4 cm/s are presented in Figure 6.28 to 6.34. It i s observed that li q u e f a c t i o n occurs at a l l sections in the case with k z = 10"4 cm/s and only at sections II to MM in the case with k 2 = 10"3 cm/s. The zone of liq u e f a c t i o n predicted for k z = 10"3 cm/s i s deeper than for k e = 10"3 cm/s. However, as in the case of analyses involving island 2, higher porewater pressures are predicted at depths below the zone of liquefaction for the case with k z = 10"3 cm/s than for k r = 10"" cm/s. The explanation for t h i s behaviour has been presented in section 6.4. The maximum porewater pressure response at sections of island 3 at the end of the spec i f i e d storm for both values of k z are given in Table 6.5 and the results are plotted in Figure 6.35. From this figure, i t is apparent that the effect of the storm is f e l t severely at a c r i t i c a l location, where the water depth in terms of the s i g n i f i c a n t wave height i s approximately given by 1.50 H s, regardless of the k z value. These results agree with the similar results obtained from analyses of islands 1 and 2. Figure 6.35 suggests that the depth of water beyound which liquefaction would not occur for the k z values of 10"3 cm/s and 10"4 cm/s are 2.20 H s and 2.50 H 9 respectively. Porewater pressure ratio,U/crjo so • 60-Figure 6 . 2 8 ; Sect ion-HH; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure ra t io ,U/o^ 0;4 0 - 6 Q.A I O Figure 6 . 2 9 ; S e c t i o n - I l ; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. 10 20 4 j 30 a Q Porewater pressure ratio,U /cr^J ' 0 2 0.4 o-6 o-a 1.1 • • 1 k z= I0" 4cm/s k x= I0" 5cm/s 4r0 SO 60 Figure 6 .30 ; S e c t i o n - J J ; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure ratio,u/o^0' O 0-2 0 4 O - t O-fi i-O I 1 1 1 1 r • O h Figure 6 . 3 1 ; Sect ion-KK; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure r a t i o , U/CJ^ ' ° °:2 °-6 o-fl 10 kz= 10"^ cm/s k2= 10" acm/s Figure 6 .32 ; S e c t i o n - L L ; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure ratio,M/Ov© o 02 0-4 0.6 o-a 1.0 Figure 6 . 3 3 ; Section-MM; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure r a t i o , U/a^'0 O 0-2 0 - 4 0-6 o-S IO kz= 10~3 cm/s k r = 1u~4cm/s Figure 6.34; Sect ion-NN; Residual Porewater Pressure Response At the end of 12m 6 hour Storm. 120 Section Water depth (m) (d/H s) Maximum pwp Response L iquefact ion depth (m) k x = 1 0" 3 cm/s k z = 1 0" 4cm/s HH 6 0.50 0.0 5.0 II 10 0.83 6.0 9.0 JJ 14 1.17 9.0 13.0 KK 16 1 .33 12.5 15.0 LL 18 1 .50 13.0 16.0 MM 22 1.83 9.0 12.0 NN 26 2.17 0.0 6.5 Table 6 - 5 ; Maximum Porewater Pressure Response At Sections of Island 3 At the End of the 12m 6 hour Storm 121 Maximum Depth of l i que fac t ion (m) 4. S 12 16 2 0 Figure 6 . 3 5 ; Maximum Porewater Pressure Response of Island 3 At the end of 12m 6 hour Storm. 1 22 Additional analyses were conducted at section LL to establish the value of k 2 required to prevent liquefaction within the island and also to l i m i t porewater pressure r a t i o to 65% or l e s s . The respective k 2 values are found to be 2.5 x 10"3 cm/s and 4.0 x 10"3 cm/s. 6.6 Summary and Comparison of Results of Analyses  On Sand Foundation Based on the results of the analyses involving three islands on sand foundations, the following conclusions and comments are made. The porewater pressure response during and after the storm strongly depends on the storm c h a r a c t e r i s t i c s , the drainage and compressibility c h a r a c t e r i s t i c s of the berm material and the s t i l l water depth at sections of interest. Further, at a pa r t i c u l a r section, the porewater pressure response at a location depends on the depth of that location from the top island surface. As the depth increases, the porewater pressure r a t i o developed at any instant of time shows a steady decay similar to the d i s t r i b u t i o n of wave induced c y c l i c stress r a t i o , r / a v o . The effect of a storm i s f e l t most strongly at a s p e c i f i c location, regardless of the drainage and compressibility . The water depth to th i s location i s given by 1.50 times the s i g n i f i c a n t wave height. This shows that within the range of data investigated, this c r i t i c a l location i s unique for a sp e c i f i e d storm and for a severe storm the c r i t i c a l location i s 123 deeper than for a mild storm. For a given storm, the porewater pressure response increases with depth u n t i l the c r i t i c a l water depth and decreases beyound the c r i t i c a l water depth. The permeabilities for a given i n i t i a l compressibility or the i n i t i a l c o - e f f i c i e n t of consolidation, Cyo , required to l i m i t the maximum porewater pressure r a t i o to a certain s p e c i f i e d l e v e l depends on the s i g n i f i c a n t wave height and duration of the storm. Table 6.6 shows the permeabilities and the corrresponding i n i t i a l values of the c o - e f f i c i e n t of consolidation, required for d i f f e r e n t storms of duration 6 hours to l i m i t maximum porewater pressure r a t i o within the islands analysed to just liquefaction (95 - 100%) and to 65%. It i s evident from these results that the requirements become tougher as the storm becomes more severe. As stated e a r l i e r , the factor that governs the rate of diss i p a t i o n and thereby the net porewater pressure response is kz/mvo or, the i n i t i a l c o - e f f i c i e n t of consolidation, C v o , defined as k /mVo Thus, i f analyses were to be carried out with combinations of k z and m V o values such that the r a t i o k z / m vo remains the same in each case, then the resulting porewater pressure response to a specified storm would be id e n t i c a l in each case. This p r i n c i p l e can be applied to a l l analyses presented so far. The predicted maximum depth of liqu e f a c t i o n , that i s , the depth of liquefaction at the c r i t i c a l location of each island for the spec i f i e d storms of duration 6 hours, for the two values of k z are presented in Table 6.7, along with an estimate 124 I sland No. Storm H s (m) 95-100% (u/<rvy L imit 65% (U/< r ,^) L imit k t (cm/s) Cve(m*/s) k z (cm/s) C V 0(m2/s) 1 4 2.5x1 0"4 0.85x1 0~ z 3.5x1 0~4 1 .20x1 0 T £ 1 6 8.0x1 0~4 2.70x1 0"-2 1.0x1 0"? 3.40X10" z 2 9 2.0x10"* 6.80x1 0" 2 3.0X10" 3 1 .02x10"' 3 12 2 . 5 X 1 0 " 3 8.50x1 0" 2 4.0x10"'' 1 .36x10"' Table 6*6; Drainage C h a r a c t e r i s t i c s Requirement To Limit Porewater Pressure Ratio To Spec i f ied Levels For D i f fe rent Strorms of Duration 6 Hours. 125 Maximum Depth of L iquefact ion (m) Island No. Storm Hs(m) STAB-W3 Analysi s 'Undrained' Analys i s K z = 10"3 cm/s 1^  = 1 0'4 cm/s 1 4 0.0 6 . 5 7.5 1 6 0.0 9.0 10.0 2 9 11.0 13.5 13.5 3 i I 1 2 13.5 16.0 i 6.0 Table 6-7; Predicted Maximum Depth of L iquefact ion At C r i t i c a l Locat ions D i f fe rent Storms and Permeabi l i t ies For Islands On Sand Foundations. 126 of depth of liquefaction from analyses assuming undrained conditions. It i s interesting to note that for mild storm and higher k z values, the error involved in estimating maximum depth of liqu e f a c t i o n assuming undrained conditions i s high. It is quite possible in some instances that 'undrained' analyses may predict depth of liquefaction to be several metres, whereas analyses incorporating d i s s i p a t i o n e f f e c t s , such as STABW and STABW3 analyses, would predict no liquefaction at a l l . The c l a s s i c example for such a case i s the analysis involving 4m, 6 hour storm. It i s c l e a r l y seen from the results in Table 6.7, that 'undrained' analysis predicted liquefaction at c r i t i c a l section to be as much as to 7m, but STABW3 analyses predicted only 6m for the case with k z = 10"" cm/s and no liquefaction for the case with k 2 = 10"3 cm/s. The above example c l e a r l y shows the significance of incorporating d i s s i p a t i o n effects to avoid unduly conservative estimates of depth of liqu e f a c t i o n during wave loading. The results also suggest that in cases of severe storms and lower k z values, such as k z < 10"" cm/s, the islands are p r a c t i c a l l y undrained in the analyses and the estimates based on 'undrained' analyses and STABW3 analyses are the same. However, for locations other than the c r i t i c a l location, even for low permeabilities, there may be s i g n i f i c a n t errors involved, i f the depth of liquefaction i s predicted assuming undrained conditions. If the wave pattern and the drainage c h a r a c t e r i s t i c s are such that liquefaction occurs at the c r i t i c a l location, then 127 the depth of water beyound which li q u e f a c t i o n would not occur, i s dependent on the drainage c h a r a c t e r i s t i c s and the wave pattern i t s e l f . The results with k z = 10"4 cm/s indicate that the best estimate of the above depth for the 6 hour storm, in terms of the s i g n i f i c a n t wave height i s 2.50 H s. The best estimate of the above depth for the case with k z = 10"3 cm/s i s 2.20 H$. However, these estimates are based on such a limited number of analyses that they should be viewed with caution. F i n a l l y , the results of analyses conducted on islands on sand foundations, indicate l i q u e f a c t i o n or high levels of porewater pressures are possible for drainage c h a r a c t e r i s t i c s of sand generally encountered in practice during moderate wave conditions of p r a c t i c a l l y feasible duration. Thus, i t i s important to resort to some kind of remedial measures that would bring down the wave induced porewater pressures to acceptable l e v e l s . One of the popular and e f f i c i e n t means of suppressing liquefaction potential i s to provide coarse, r e l a t i v e l y free draining material on top of the island surface, where the island i s susceptible to l i q u e f a c t i o n . The eff e c t of such measures i s examined in the next chapter. 1 28 CHAPTER 7 EFFECT OF ROCKFILL COVER ON WAVE INDUCED POREWATER PRESSURES 7. 1 Introduction One of the p r a c t i c a l remedial measures to bring down the wave induced porewater pressure to acceptable levels is to provide coarse cover on top of the island surface. This chapter examines the effect of r o c k f i l l in reducing the wave induced porewater pressures. The choose of r o c k f i l l as a suitable cover has the advantages that i t not only reduces the wave induced porewater pressures s i g n i f i c a n t l y but also stays intact without being eroded heavily. For analyses involving cover, the permeability and the thickness of the r o c k f i l l cover are taken as 10 cm/s and 1m respectively. Within the limited range of data investigated, the results from the previous chapter suggest that in cases of possible l i q u e f a c t i o n , the cover has to be extended to water depths beyond 2.50 H s in the case with k z = 10"" cm/s and 2.20 H s in the case with k z = 10"3 cm/s. 7.2 Effect of Cover on Porewater Pressure Response of  Island 1 To 6m, 6 hour Storm Figures 7.1 to 7.7 show the effect of 1m cover of a coarse material on the poreawter pressure response at sections from AA to GG of island 1 at the end of 6m, 6 hour storm for the two di f f e r e n t permeabilities of the sand f i l l . It i s 129 Porewater pressure r a t i o , 4 ^ v o 9 q-2. Q-4* Q-6 o-B i.o Figure 7 - 1 ; Sect ion-AA; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 130 Porewater pressure ratio,iVo*vo 3 6 l " ' L Figure 7>2 ; Sect ion -BB; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 131 Porewater pressure r a t i o , u / * * » o 0 2 o-4- 0^ 6 o_-8 LP 36I 1 I i 1 1 Figure 7 - 5 ; Sect ion -CC; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 1 32 Porewater pressure r a t i o , u/0vo 0-2. 0-4 Ob O- f l I O Figure ?>A; Sect ion-DD; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 1 33 Porewater pressure ra t io^/^o o o-z o-i 0 6 o-8 up 36' I 1 1 L Figure 7- 5 ; Sec t ion -EE ; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 134 Figure 7-. 6 ; S e c t i o n - F F ; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 135 Porewater pressure r a t i o , u / ^ o O O-Z 0 « 4 0 * 6 O'A i<0 3 2 3 6 * 1- no coverl . , . - 3 , 2- 1m cover) k * = 1 0 c m / s 3- no cover . . 4 - 1m coverJ k*= 1 0 c m / s Figure 7>7 ; Sect ion-GG; E f fec t of Cover on Porewater Pressure Response At the end of 6m 6 hour Storm. 1 36 evident from these figures that the effect of the coarse cover is to reduce the porewater pressure response s i g n i f i c a n t l y and in some cases to negligible l e v e l s . Unlike the porewater pressure response without the cover, the response with cover shows that the maximum porewater pressure r a t i o occurs not at the top, but at some depth from the top of the o r i g i n a l island surface and thereafter shows a steady decay. The reduction in porewater pressures at the top few metres i s apparently due to the influence of the free draining non-liquefiable cover at top. It i s also seen that the porewater pressure response at the end of the storm is much higher for the case with kz = 10"* cm/s than with k z = 10"3 cm/s. This i s as expected because for the same depth of cover , the porewater pressure response with' k r = 10"3 cm/s has to be less because of the more pervious nature of the s o i l . This also makes i t clear that the resulting response for the same depth of cover, depends on the drainage c h a r a c t e r i s t i c s of the sand f i l l ; the greater the permeability of sand f i l l , the smaller the maximum porewater pressure response w i l l be. At section AA, for the case with k z = 10"* cm/s, analysis shows that with 1m coarse cover the maximum porewater pressure r a t i o developed at the end of the storm i s 51%. In contrast,analysis shows that without 1m coarse cover there would be liquefaction up to a depth 6.5m. In the case with kz = 10"3 cm/s, analyses with and without the cover show that the maximum porewater pressure r a t i o developed at the end of the 1 37 storm are 11% and 47% respectively. Section CC i s of interest, because i t is closest to the c r i t i c a l section for the 6m, 6 hour storm. The results of the analysis with 1m cover for the case with k 2 = 10"' cm/s indicate that liquefaction i s limited to a zone of 1m extent between the depth 2m and 3m, while without cover, s o i l would liquefy up to a depth of 9m. In the case with k z = 10"3 cm/s, the maximum porewater pressure r a t i o predicted from analyses with and without cover are 14% and 67% respectively. Similar reductions in porewater pressure response are apparent at a l l other sections of island 1. The quantitative comparison of the maximum porewater pressure response at the end of the 6m, 6 hour storm from analyses with and without the 1m top coarse cover for the two values of kx are summarized in Table 7.1. In the event of liquefaction, that i s , porewater pressure r a t i o of 100%, the figures in brackets indicate the predicted depth of liq u e f a c t i o n . Results in Table 7.1 show that the suppression in the porewater pressure response i s very s i g n i f i c a n t and for the range of permeability above 10"3 cm/s, 1m coarse cover i s s u f f i c i e n t to bring down the porewater pressures to negligible l e v e l s . In di r e c t contrast to t h i s , in the case of ^=10"' cm/s, the 1m top coarse cover i s i n s u f f i c i e n t to suppress liquefaction at section BB to EE. In these cases, the thickness of the cover has to be increased s u f f i c i e n t l y to bring down the wave induced porewater pressure to levels considered to be 138 Maximum Porewater Pressure Rat io , U/^c , (%) Sect ion k z = I0~3cm/s Ic^  = 1 CP* cm/s No Cover 1m Cover No Cover 1m Cover AA 47 11 100(6.5) 51 BB 62 12 100(8.0) 100(1.0) CC 67 14 100(9.0) 100(2.0) DD 65 * 14 100(8.5) 100(2.0) EE 43 • 13 10-0(7.0) 100(1.5) FF 29 9 100(6.5) 50 GG 22 8 100(6.0) 40 Table E f fec t Of Cover On Maximum Porewater Pressure Response At Sect ions Of Is land 1 At the End Of 6m 6 hour Storm. Note; F igures in brackets ind icate the extent of the zone of l i q u e f a c t i o n in metres. 1 39 saf e. The reduction in porewater pressure response in the case of analyses with coarse cover, may be attributed to the following reasons; F i r s t l y , the presence of pervious and non-1iquefiable cover reduces the build up of the porewater pressure because of the easy drainage at the top. Secondly, the cover reduces the water depth and consequently the wave composition of the storm changes due to the breaking of higher waves. The changes in the wave composition a l t e r s the shear stress d i s t r i b u t i o n with depth and in turn, the i n i t i a l values of N u required for the estimation of porewater pressure generation, as in equation (3-12), increase. This increase would result in a slower rate of porewater pressure generation. The reduced rate of porewater pressure generation coupled with the increased di s s i p a t i o n effects give r i s e to the reduced residual porewater pressure responses. Figure 7.8 c l e a r l y i l l u s t r a t e s the effect of coarse cover on the rate of residual porewater pressure build up during the 6m, 6 hour storm as compared to the rate of bu i l d up without the cover at a depth 3m below the o r i n i g a l top surface at section CC. It can be seen from Figure 7.8 that in both cases of k z values, the rate of porewater pressure b u i l d up during the storm a c t i v i t y i s reduced considerably in the cases with the coarse cover on top. The reduction is much more apparent in the case with k z = 10"3 cm/s than with k z = 10"" cm/s. O I 2 S 4 - 5 6 7 6 S Time (hours) Figure 7 - « 3 ; E f fec t of Cover on Time History Of Pore Pressure At 3m Below Or ig ina l Island Surface At Sect ion CC. o 141 7.2.1 Effect Of Permeability Of Cover Material Figure 7.9 shows the influence of the permeability of the cover material on the induced porewater pressure response at section AA of island 1 at the end of the 6m, 6 hour storm. It is c l e a r l y seen that there is no difference in the response when the permeability of the cover is increased from 10 cm/s to 100 cm/s. Hence i t appears that the use of highly pervious materials as cover does not seem to produce any further reductions in the porewater pressure response and the e f f i c i e n t way to reduce porewater pressures to desired levels i s to resort to the use of cover with an increased thickness. 142 Porewater pressure r a t i o , u / ^ e O o.Z C4. 0 < 6 0 ' < 9 t>o T k r = 10" 5 cm/s k c= 10 to 100 cm/s 2 - / k z= 10"* cm/s L l£= 10 to 100 cm/s Figure "7-.9 ; Sect ion-AA; E f f e c t of Cover Permeabi l i ty on Porewater Pressure Response At the End of the 6m 6 hour Storm. 143 7.3 Effect of Cover on Porewater Pressure Response of  Island 1 to 4m, 6 hour Storm Figures 7.10 to 7.12 show the effect of 1m top coarse cover on the porewater pressure response at sections AA, CC and DD of island 1 at the end of 4m, 6 hour storm. The results show that the predicted reductions in porewater pressure response with 1m coarse cover are very s i g n i f i c a n t . For example, at section AA, the c r i t i c a l section for the 4m, 6 hour storm, for kz = 10"" cm/s, the results from the analysis with the cover shows that maximum porewater pressure r a t i o developed at the end of the storm is 36%, while analysis without cover shows that there would be liquefaction to a depth of 6.5m. At the other sections, the resulting porewater pressure response with the '1m cover are ne g l i g i b l e for both the cases of permeability. Table 7.2 compares maximum porewater pressure response with and without the 1m of coarse cover at the end of the 4m, 6 hour storm at sections AA, CC and DD of island 1. The results suggest that the 1m of coarse cover is s u f f i c i e n t to bring down porewater pressures to safe levels for the range of permeability greater than 10"* cm/s during the 4m, 6 hour storm. In the case of kj, = 10"3 cm/s, the need for the coarse cover protection to reduce porewater pressures is not necessary since the developed porewater pressure without the presence of cover are unlikely to exceed 20% during the 4m storm. But i f the permeability of the sand f i l l i s around 10"* cm/s, i t i s essential to have a top coarse cover protection and 1 4 4 Porewater pressure ratio,u/ovo Figure 7 . i 0 ; Sect ion-AA; E f f e c t of Cover on Porewater Pressure Response At the End of the 4m 6 hour Storm. 145 Porewater pressure rat io ,U/a^ O o>2 o«4 0-6 0-8 J-O Figure 7 • U; Sect ion -CC; E f f e c t of Cover on Porewater Pressure Response At the End of the 4m 6 hour Storm. 146 Figure Section-DD; E f fec t of Cover on Porewater Pressure Response At the End of the 4m 6 hour Storm. 147 Maximum Porewater Pressure Rat io , o/o-^ t (%) Sect ion k z= 10"S cm/s = IO - 4 cm/s No Cover 1m Cover No Cover 1m Cover AA 19 10 100(6.5) 35 CC 13 5 100(5.0) 24 DD 10 4 100(2.0) 18 Table 7>2 ; E f fec t Of Cover On Maximum Porewater Pressure Response At Sections Of Is land 1 At the End Of 4m 6 hour Storm. Note; F igures in brackets ind icate the extent of the zone of l i q u e f a c t i o n in metres. 1 48 a cover of thickness 1m is s u f f i c i e n t to reduce porewater pressures to acceptable l i m i t s . As mentioned e a r l i e r , the cover has to be extended beyond water depth 1Om (as given by 2.50 times H s) to eliminate the p o s s i b i l i t y of liq u e f a c t i o n during the 4m storm. 7.4. Effect of Cover on Porewater Pressure  Response of Island 2 The effect of the 1m thick coarse cover on the porewater pressure response at sections PP to W of island 2 at the end of the spe c i f i e d storm are shown in Figures 7.13 to 7.19. These figures indicate that there is considerable reduction in the porewater pressure response as a result of the top coarse cover and also the response has the same trend as seen before in the analyses of island 1. In the case with k z = 10~3 cm/s, the results indicate that the effect of 1m cover is to reduce the porewater pressure response to just below liqu e f a c t i o n levels at sections RR, SS and TT and to very neg l i g i b l e levels at other sections. But in the case with k z = 10"" cm/s, the effect of cover i s to reduce the thickness of the zone of liquefaction s i g n i f i c a n t l y . In these cases, the liquefaction is limited to a l o c a l i z e d zone of a few metres in extent. For example, at section TT, the thickness of the zone of liquefaction is reduced from 13.5m to 7m and the l o c a l i z e d zone of liquefaction extends from depth 2m to depth 9m from the o r i g i n a l sand berm top surface. 149 Porewater pressure ratio,u/^7c' o oa. 0.4- e -6 o>8 uo * • • Figure 7 - i 3 ; Sect ion -PP; E f fec t of Cover on Porewater Pressure Response At the end of 9m 6 hour Storm. 150 Porewater pressure r a t i o , u / « ^ o ' Figure 7«14- ; Sect ion-QQ; E f fec t of Cover on Porewater Pressure Response At the end of 9m 6 hour Storm. 151 Porewater pressure r a t i o , 0_ o-2 o « 4 c - 6 o-8 i-O Figure 7<15; Sect ion-RR; E f fect of Cover on Porewater Pressure Response At the end of 9m 6 hour Storm. Porewater pressure r a t i o , u / ° v c O Q'2 0.4. o-6 ©•* \-o * ' t * Figure 7 -16 ; Sec t ion -SS ; E f fec t of Cover Porewater Pressure Response the end of 9m 6 hour Storm. 153 Porewater pressure r a t i o , U A ^ » Figure Sect ion -TT ; E f fec t of Cover on Porewater Pressure Response At the end of 9m 6 hour Storm. 154 Porewater pressure r a t i o , u / « w O o-2 o-4 o«6 o>8 i.o 60 1 -no coverl y-^ 2- lm coverj z 3- no coverl ^ 4- 1m coverj z 1 0~* cm/s 1 OP4 - cm/s Figure 7-*l<8; Sect ion-UU; E f fec t of Cover on Porewater Pressure Response At the end of 9m 6 hour Storm. Porewater pressure ratio,U/o^ 0 0 -3 Q -4 Q - 6 o - f i 1-0 • • • Figure S e c t i o n - W ; E f fec t of Cover Porewater Pressure Response the end of 9m 6 hour Storm. 156 The comparison of maximum porewater pressure response with and without the 1m thick coarse at sections PP to W of island 2 at the end of the storm i s presented in Table 7.3. It i s evident from the results that, as in the case without cover, the maximum porewater pressure r a t i o in the case with cover occurs around section TT. Results also indicate that the cover of 1m thickness i s not s u f f i c i e n t to suppress liquefaction when the permeability i s 10"" cm/s. An increase in thickness of cover is required to prevent liquefaction in these cases. However, in the case of k z = 10~3 cm/s, a cover thickness of 1m is s u f f i c i e n t only beyond water depth of 14m and an increased cover would be necessary up to water depth 14m to bring down porewater pressures to safer levels during the spec i f i e d storm a c t i v i t y . 157 Maximum Porewater Pressure Rat io , U / ^ , (%) Section k x = 10"3 cm/s k,= I0" 4cm/s No Cover 1m Cover No Cover 1m Cover PP 70 11 100(7.0) 100(2.0) QQ 100(6.0) 41 100(9.0) 100(2.5) RR 100(8.0) 90 100(11.5) 100(4.5) SS 100(10.5) 92 100(12.5) 100(6.0) TT 100(11.0) 95 100(13.5) 100(7.0) " UU 100(9.0) 27 100(12.0) 100(4.5) W 100(2.0) 16 100(8.5) 100(3.0) Table 7*3 ; E f fec t Of Cover On Maximum Porewater Pressure Response At Sections Of Island 2 At the End Of 9m 6 hour Storm. Note; F igures in brackets ind icate the extent of the zone of l i q u e f a c t i o n in metres. 1 58 7.5 Effect of Cover on Porewater Pressure  Response of Island 3 Figures 7.20 to 7.26 show the effect of 1m coarse cover on the porewater pressure response at sections HH to NN of island 3 at the end of the sp e c i f i e d storm for the two diffe r e n t permeabilities. Even though s i g n i f i c a n t changes in the porewater pressure response is apparent, the results indicate that liquefaction s t i l l occurs to considerable depths at sections II to MM for the case when k z = 10"" cm/s. Liquefaction or high levels of porewater pressure are indicated for the case with k 2 = 10"3 cm/s at sections JJ to LL. Table 7.4 shows the maximum porewater pressure response at sections of island 3 at the end of the storm with and without 1m of coarse cover. It appears that 1m of coarse cover is not s u f f i c i e n t to suppress liquefaction during the specified storm for both permeabilities considered. However, in the case of k z 10~3 cm/s, 1m cover i s s u f f i c i e n t beyond water depth 18m (section LL) although increased cover i s required for water depths above 18m. In the case of k z = 10"" cm/s, 1m cover i s s u f f i c i e n t only beyond water depth 22 m and for shallower depths the thickness of cover has to be increased. Porewater pressure ra t io , u /0vc o o.a. e - 4 o -6 o-s i.o Figure 7 - 2 0 ; Sect ion-HH; E f fec t of Cover on Porewater Pressure Response At the end of 12m 6 hour Storm. 160 Porewater pressure r a t i o , u/tf*.' o p-2 0 4 . Q-6 6-8 1 0 Figure 7 . 2 i ; S e c t i o n - I I ; E f fec t of Cover on Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure r a t i o , Wo^'c O O-Z O-A- 0'6 o*fi l-O Figure 7-22', S e c t i o n - J J ; E f fec t of Cover Porewater Pressure Response the end of 12m 6 hour Storm. Porewater pressure r a t i o , u / ° v c O o-2 0-4- 0-6 O-S l-O Figure 1.23; Sect ion-KK; E f fec t of Cover on Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure r a t i o , u/trvl O o-2 O'd- 0'6 1.0 Figure 1'2A; Sec t ion -LL ; E f fec t of Cover on Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure r a t i o , u / ^ O A O b 60 1- no cover? 1 0 - 3 c m / s 2- 1m coverj z 3- no cover? k 1 0 - 4 c m / s . 4- lm coverJ * * • • * Figure 7 -25; Section-MM; E f fec t of Cover on Porewater Pressure Response At the end of 12m 6 hour Storm. Porewater pressure r a t i o , O o-2 OA- 0 6 o-8 « o Figure 7-2^; Sect ion-NN; E f fec t of Cover on Porewater Pressure Response At the end of 12m 6 hour Storm. 166 Maximum Porewater Pressure Rat io ,^/a^ Q , (%) Sect ion k z = 1 Cf a cm/s k x= 10" 4 cm/s No Cover 1m Cover No Cover 1m Cover HH 80 12 100(5.0) 100(1.0) II 100(6.0) 90 100(9.0) 100(4.0) JJ 100(9.0) 98 100(13.0) 100(6.0) KK 100(12.5) 100(1.0) 100(15.0) 100(7.0) LL 100(13.0) 100(2.5) 100(16.0) 100(7.5) MM 100(9.0) 22 100(12.0) 100(5.0) NN 18 . 5 100(6.5) 18 Table 7-4-j E f fec t Of Cover On Maximum Porewater Pressure Response At Sect ions Of Island 3 At the End Of 12m 6 hour Storm. Note; F igures in brackets ind icate the extent of the zone of l i q u e f a c t i o n in metres. 167 CHAPTER 8 EFFECT OF FOUNDATION CONDITIONS ON POREWATER PRESSURES 8.1 Response of Island 1 The porewater pressure response at the end of the 6m, 6 hour storm at sections AA and CC of island 1 s i t t i n g on a clay foundation is presented in Figure 8.1 and 8.2 for the case of k z = 10"3 cm/s. These figures show that the porewater pressure response is very much dependent on the undrained shear strength Sg of clay immediately below the sand f i l l . It is also apparent that the harder the clay, greater the porewater pressure response w i l l be. The l i m i t i n g magnitude of the porewater pressures i s the one corresponding to the r i g i d base. For instance, at section- AA, the maximum porewater pressure r a t i o developed at the end of the storm i s 66% and 42% for Su values of 50 kPa and 30 kPa respectively. On the other hand, at section CC, the corresponding maximum induced porewater pressure ratios are 45% and 24% respectively. The l i m i t i n g wave induced porewater pressures at section AA and CC are 95% and 50% respectively. The reason for the porewater pressure response being d i r e c t l y dependent on the undrained shear strength i s due to the fact that in current engineering practice the shear modulus is related to the undrained shear strength on a one-to -one basis (Seed et al,l970). Any increase in undrained shear strength of the clay increases the shear modulus and hence a l t e r s the shear stress d i s t r i b u t i o n with depth in the sand 168 Porewater pressure ratio,u/<*7o 7L 1 1 1 1 I Figure <3 * I ; Sect ion-AA; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 169 Porewater pressure ratio,u/a^> 51 1 1 1 1 1 Figure 8-2. ; Sect ion -CC; Residual Porewater Pressure Response At the end of 6m 6 hour Storm. 170 f i l l . Figure 8.3 c l e a r l y demonstrates the resu l t i n g increase in the shear stress r a t i o d i s t r i b u t i o n in the sand f i l l at section AA at the start of the storm, as a result of the increase in undrained shear strength from 30 kPa to 50 kPa. The consequence of the increase in shear stress d i s t r i b u t i o n as in the above case, w i l l be reflected not only on the i n i t i a l values of N u but also in the calculation of the number of waves of the equivalent storm Ncq in such a manner that N L values become smaller and Ne<j value becomes larger. For example, at section AA, the N L value corresponding to the f i r s t layer of the sand f i l l for the cases of S u being 30 kPa and 50 kPa i s 10.77 and 10.0 respectively. The - corresponding values of Neej are 1076 and 1086 respectively. Now, under these conditions, the factor (Ne*j/NL) which governs the rate of porewater pressure generation at any instant of time, increases with increasing S u and so does the rate of porewater pressure generation (see equation 3 -12). This increased rate of porewater pressure generation at the start of the storm results in greater porewater pressure response for the case of higher S u value. 171 Shear s t ress rat io^e/^v* 0>08 Q.IO Q.I2 o-l-i O.lfc Q . l8 71 i t i i Figure c?'3; Sect ion-AA; Shear Stress Rat io D i s t r i b u t i o n At the s tar t of 6m 6 hour Storm. 1 72 CHAPTER 9 SUMMARY AND CONCLUSIONS A simple method of analysis for the determination of wave induced porewater pressures is presented. The method considers both di s s i p a t i o n and generation effects during wave loading. It also considers the e f f e c t of increasing porewater pressures on s o i l properties, namely shear modulus, bulk modulus and volume compressibi1iy. The method was incorporated into a f i n i t e element computer program STABW3. The program uses a cubic polynomial interpolation function for the porewater pressure f i e l d . The computer program was used to analyse three d i f f e r e n t a r t i f i c i a l islands b u i l t up to a set down depth of 6m in water depths 12m, 21m and 31m respectively. The islands were subjected to d i f f e r e n t patterns of storm waves each of 6 hours duration. The porewater pressures induced in each of the islands by the storm waves were computed for d i f f e r e n t drainage c h a r a c t e r i s t i c s of the berm material. The effect of incorporating a non l i q u e f i a b l e coarse cover on top of the island surface on the induced porewater pressure response was also examined. A brief examination of the effect of foundation conditions on the induced porewater pressure response was also reported. In t h i s study, the r e l a t i v e density of the berm material has been assumed to be 50%. This means that the sand has a 1 73 r e l a t i v e l y low resistance to liquefaction which tends to dramatise the effect of wave action. Current construction practices tend to give r e l a t i v e density of the range from 50% to 70%. At the higher r e l a t i v e densities, the zone of liq u e f a c t i o n w i l l be greatly reduced and phenomenically the same kind of behavoir w i l l be obtained. Hence, the numerial values cannot be considered to be generally applicable. The following general conclusions can be drawn from the results of the analyses. These were based on such a limited number of analyses and therefore they have to viewed with caut ion. 1. For homogeneous island berms on sand • foundations, the effect of the waves i s f e l t strongly and severely at a particular location. The water depth, D c, to t h i s c r i t i c a l location i s primarily dependent on the s i g n i f i c a n t wave height, H s, of the storm and i s given approximately by Dc = 1.50 H s regardless of the drainage c h a r a c t e r i s t i c s of the berm material. 2. For islands on sand foundations, the water depth beyound which liq u e f a c t i o n would not occur during a storm i s dependent on wave parameters and the drainage c h a r a c t e r i s t i c s of the berm material. For a storm of 6 hours duration and for i n i t i a l volume compressibility of 3 x 10"5 m2/kN, the water depth, in terms of H s, beyound which liquefaction would not occur, is 174 given approximately by 2.20 H s and 2.50 H s for k 2 values of 10"3 cm/s and 10"" cm/s respectively. 3. For islands on sand foundations, the drainage c h a r a c t e r i s t i c s of the berm material required to l i m i t the porewater pressure response below liquefaction l e v e l s , i s dependent on the wave c h a r a c t e r i s t i c s ; the requirements becomes more stringent the more severe the storms. 4. The effect of non l i q u e f i a b l e r e l a t i v e l y free draining coarse cover material placed on top of the berm slope i s to reduce the porewater pressure response during wave loading. The reduction in porewater pressure response for a given permeability of the berm material i s dependent on the cover thickness provided. Moreover, the increase in the permeability of the cover material does not seem to produce further s i g n i f i c a n t reduction in the porewater pressure response. Hence, in order to suppress porewater pressure response to the desired l e v e l s , i t is more e f f e c t i v e to increase the thickness of the coarse cover rather than to resort to the use of much more pervious material as cover. 5. The suitable thickness of cover required to suppress liquefaction at a pa r t i c u l a r section of interest during a storm depends on the drainage c h a r a c t e r i s t i c s of the berm material and the wave parameters. The cover thickness required to completely suppress li q u e f a c t i o n for given drainage 175 c h a r a c t e r i s t i c s o f t h e berm m a t e r i a l , i s g r e a t e r f o r a s e v e r e s t o r m t h a n f o r a m i l d e r s t o r m o f t h e same d u r a t i o n ; A g a i n f o r a g i v e n s t o r m , i t i s h i g h e r f o r l e s s p e r v i o u s berm m a t e r i a l t h a n f o r more p e r v i o u s m a t e r i a l . 6. F o r i s l a n d s on c l a y f o u n d a t i o n s , t h e p o r e w a t e r p r e s s u r e r e s p o n s e d u r i n g t h e wave l o a d i n g i s d e p e n d e n t on t h e u n d r a i n e d s h e a r s t r e n g t h o f t h e c l a y i m m e d i a t e l y b e l o w t h e s a n d berm. The h a r d e r t h e c l a y f o u n d a t i o n , t h e h i g h e r t h e p o r e w a t e r p r e s s u r e r e s p o n s e w i l l b e , up t o t h e l i m i t i n g m a g n i t u d e c o r r e s p o n d i n g t o a r i g i d b a s e . 176 REFERENCES 1. Bercha, F.G and Stenning, D.G (1979),"Arctic Offshore Deepwater Ice-Stucture Interactions", Proceedings, Eleventh Annual Offshore Technology Conference, Houston, Texas, Paper No. 3632, Vol.4, pp. 2377-2386. 2. Biot, M.A (1941),"General Theory Of Three Dimensional Consolidation", Journal Of Applied Physics, Vol.12, February, pp. 155-164. 3. 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Finn, W.D.Liam, I a i , S and Ishihara, K (1982), "Performance of A r t i f i c i a l Offshore Islands Under Wave and Earthquake Loading; F i e l d Data Analyses", Proceedings, Fourteenth Annual Offshore Technology Conference, Houston, Texas, Paper No.4220, Vol.1, pp. 661-671. 12. Finn, W.D.Liam and Lee, M.K.W (1979), "Seafloor S t a b i l i t y Under Seismic and Wave Loading", Proceedings, S o i l Dynamics in the Marine Environment, ASCE National Convention and Exposition, Boston, Mass., A p r i l 2-6, Preprint 3604, pp 1-25. 13. Finn, W.D.Liam, Lee, K.W and Martin, G.R (1977), "An Ef f e c t i v e Stress Model for Liquefaction", Journal of Geotechnical Engineering D i v i s i o n , ASCE, Vol.103, N 0 . G T 6 14. Finn, W.D.Liam, Martin, G.R and Lee, M.K.W (1978), "Comparison of Dynamic Analyses for Saturated Sands", Proceedings, ASCE Earthquake Engineering and S o i l Dynamics Conference and Exhibit, Pasadena, C a l i f o r n i a , June. 15. 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A (1974), "Earthquake Induced Settlements in Saturated Sands", Journal of Geotechnical Engineering Div i s i o n , Vol.100, No.GT4, A p r i l , pp. 387-406. 26. Lee, K.L and Chan, K (1972), "Number of Equivalent S i g n i f i c a n t Cycles in Strong Motion Earthquakes", Proceedings, International Conference on Microzonation, Seattle, Vol.11, pp. 609-627. 27. Lee, Michael.K.W and Finn, W.D.Liam (1975), "DESRA-1; Program for Dynamic E f f e c t i v e Stress Response Analysis of S o i l Deposits Including Liquefaction Evaluation", S o i l Mechanics Series No.36, Department of C i v i l Engineering, University of B r i t i s h Columbia, Vancouver, B.C. 28. 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Siddharthan, R and Finn, W.D.Liam (1979,1982),"STABW; Analysis of I n s t a b i l i t y Induced in Seafloor Sands by Cumulative Effects of Storm Waves", Unpublished. 46. Sleath, J.F.A (1970), "Wave Induced Pressures in Beds of Sand", Journal of Hydraulic D i v i s i o n , ASCE, Vol.96, No. HY2, February, pp.367-378. 47. Stenning, D.G and Schumann, C.G (1979), "Arctic Production Monocone", Proceedings, Eleventh Annual Offshore Technology Conference, Houston, Texas, Paper No. 3630, Vol.4, pp. 2357-2366. 48. Yamamoto, T (1978), "Sea Bed I n s t a b i l i t y From Waves", Proceedings, Tenth Annual Offshore Technology Conference, Houston, Texas, Paper No. 3262, Vol.3, pp. 1819-1824. 

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