218 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Analytical Analysis of Two - Dimensional Consolidation of Soil Ohioze Osemobora*, Ebuka Nwankwob, Solomon Iyekec, Adedayo Aladenikad a,b,c,dDepartment of Civil Engineering, University of Benin, Benin-City, Nigeria aEmail: fredosemobor@yahoo.com bEmail: nwankwoebuka@yahoo.co.uk cEmail: solo.iyeke@gmail.com dEmail: admofocon2002@yahoo.com Abstract Consolidation is the gradual reduction in volume of a saturated soil due to drainage of some of the pore water, the process continuing until the excess pore water pressure set up by an increase in total stress has completely dissipated; the most common case is that of one dimensional consolidation. In reality, the Terzaghi’s 1- dimensional consolidation theory has been found to be highly conservative and at best only an estimation of the actual consolidation. Accurately predicting consolidation in soil has led to the development of 2-dimensional consolidation solutions. In this paper analytical solution (using the separation of variables method) have been provided for 2-dimensional consolidation equation. Keywords: Consolidation; Two-dimensional; Pore water pressure; Analytical method, Soil. 1. Introduction In many real problems of soil mechanics, the conditions are basically two-dimensional as in the case of consolidation of hydraulically deposited soils. Terzaghi in 1925 proposed the first theory to consider the rate of consolidation for saturated cohesive soils. The theory was based on certain assumptions including that the flow of water during consolidation is only in the vertical direction. Some of the assumptions made by Terzaghi are not fully satisfied in actual field problems. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 219 The results obtained from the use of his theory in solving practical problems are at best approximations. According to [1], one of the major limitations of the Terzaghi’s theory is that in the field, the consolidation is usually not one dimensional. Several researchers [2-7] have made advances towards Terzaghi’s theory of consolidation into two dimensions. Also, in the work of Razouki and his colleagues [8], design charts for the rate of settlement of embankments on soft soils on the basis of 2D consolidation during construction and post construction periods were developed. They also concluded that the post construction settlement becomes insignificant for relatively thick deposit with permeability in the horizontal direction much higher than that in the vertical direction. In this research, an analytical solution was developed herein to provide acceptable solutions for a two dimensional consolidation problem. 2. One Dimensional Analytical Solution to Consolidation Equation The solution of the basic differential equation of one-dimensional equation can be obtained using Fourier series [1]. Let us express the hydrostatic excess pressure 𝑒𝑒� as; 𝑒𝑒� = 𝑓𝑓1(𝑧𝑧) βˆ— 𝑓𝑓2(𝑑𝑑) (1) Where; 𝑓𝑓1(𝑧𝑧) & 𝑓𝑓2(𝑑𝑑) indicate some function of z and t, respectively. Substituting the above value of 𝑒𝑒 into the consolidation equation shown in equation (1) 𝐢𝐢𝑣𝑣 �𝑓𝑓2(𝑑𝑑) πœ•πœ•2 πœ•πœ•π‘§π‘§2 [𝑓𝑓1(𝑧𝑧)]οΏ½ = 𝑓𝑓1(𝑧𝑧) πœ•πœ•[𝑓𝑓2(𝑑𝑑)] πœ•πœ•π‘‘π‘‘ Or πœ•πœ•2 πœ•πœ•π‘§π‘§2 [𝑓𝑓1(𝑧𝑧)] 𝑓𝑓1(𝑧𝑧) = πœ•πœ• πœ•πœ•π‘‘π‘‘ [𝑓𝑓2(𝑑𝑑)] 𝐢𝐢𝑣𝑣𝑓𝑓2(𝑑𝑑) The left-hand side of the above equation is a function of z only and the right-hand side is a function of t only. In order words, if the left-hand side is equal to some constant (say, -A2) when t is taken as a variable and the right- hand side is equal to the same constant when z is considered as a variable. Thus, πœ•πœ•2 πœ•πœ•π‘§π‘§2 [𝑓𝑓1(𝑧𝑧)] = βˆ’π΄π΄2𝑓𝑓1(𝑧𝑧) And American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 220 πœ•πœ•2 πœ•πœ•π‘§π‘§2 [𝑓𝑓2(𝑑𝑑)] = βˆ’π΄π΄2𝐢𝐢𝑣𝑣𝑓𝑓2(𝑑𝑑) Equation (1) has the solution given by 𝑓𝑓1(𝑧𝑧) = 𝐢𝐢1𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝑧𝑧 + 𝐢𝐢2 sin𝐴𝐴𝑧𝑧 (2) where: 𝐢𝐢1 and 𝐢𝐢2 are constants of integration e is the base of the hyperbolic or Napierian logarithm. Substituting the above equations into eqn. (2), 𝑒𝑒� = [𝐢𝐢1𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝑧𝑧 + 𝐢𝐢2 sin𝐴𝐴𝑧𝑧]𝐢𝐢3π‘’π‘’βˆ’π΄π΄ Ξ›2𝐢𝐢𝑣𝑣𝑑𝑑 𝑒𝑒� = [𝐢𝐢4𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝑧𝑧 + 𝐢𝐢5 sin𝐴𝐴𝑧𝑧]π‘’π‘’βˆ’π΄π΄Ξ›2𝐢𝐢𝑣𝑣𝑑𝑑 (3) Where: 𝐢𝐢4 and 𝐢𝐢5 are other constants, such that 𝐢𝐢5 = 𝐢𝐢1𝐢𝐢3 and 𝐢𝐢5 = 𝐢𝐢2𝐢𝐢3 The constants 𝐢𝐢4 and 𝐢𝐢5 can be determined from the boundary conditions: (i) t = 0, 𝑒𝑒� = 𝑒𝑒�𝑖𝑖, for any value of z where 𝑒𝑒�𝑖𝑖 is initial hydrostatic pressure (ii) 𝑑𝑑 = ∞ 𝑒𝑒� = 0, for any value of z (iii) 𝑧𝑧 = 0 𝑒𝑒� = 0, for any value of t (iv) 𝑧𝑧 = 𝐻𝐻(= 2𝑑𝑑), 𝑒𝑒� = 0, for any value of t For the boundary condition (iii) Equation (3), gives 𝐢𝐢4= 0. Therefore Equation. (3) becomes 𝑒𝑒� = 𝐢𝐢5 sin(𝐴𝐴𝑧𝑧)π‘’π‘’βˆ’π΄π΄Ξ›2𝐢𝐢𝑣𝑣𝑑𝑑 For boundary condition iv, 𝑒𝑒� = 0 at 𝑧𝑧 = 𝐻𝐻 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 221 Therefore, 𝐢𝐢5 sin(𝐴𝐴𝑧𝑧)π‘’π‘’βˆ’π΄π΄Ξ›2𝐢𝐢𝑣𝑣𝑑𝑑 = 0 The above equation is satisfied if 𝐴𝐴𝐻𝐻 = 𝑛𝑛𝑛𝑛, Where: 𝑛𝑛 is any integer. The equation can be written in the following form: 𝑒𝑒 = 𝐡𝐡1 sin οΏ½ 𝑛𝑛𝑛𝑛 𝐻𝐻 οΏ½ π‘’π‘’βˆ’(𝑝𝑝𝑖𝑖Λ2/𝐻𝐻Λ2)𝐢𝐢𝑣𝑣𝑑𝑑 + 𝐡𝐡2 sin οΏ½ 2𝑛𝑛𝑧𝑧 𝐻𝐻 οΏ½ π‘’π‘’βˆ’(4𝑝𝑝𝑖𝑖Λ2/𝐻𝐻Λ2)𝐢𝐢𝑣𝑣𝑑𝑑 + … + 𝐡𝐡𝑛𝑛 sin οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝐻𝐻 οΏ½ π‘’π‘’βˆ’π‘›π‘›Ξ›2βˆ—piΞ›2/𝐻𝐻Λ2)𝐢𝐢𝑣𝑣𝑑𝑑 + … .. Or 𝑒𝑒 = οΏ½ 𝐡𝐡𝑛𝑛 sin οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝐻𝐻 οΏ½ π‘’π‘’βˆ’π‘›π‘›Ξ›2βˆ—piΞ›2/𝐻𝐻Λ2)𝐢𝐢𝑣𝑣𝑑𝑑 (4) 𝑛𝑛=∞ 𝑛𝑛=1 Where; 𝐡𝐡1,𝐡𝐡2 , … …𝐡𝐡𝑛𝑛 are constants. 3. Two Dimensional Analytical Solution to Consolidation Equation As stated by [1], the governing equation for two dimensional consolidations is given below; π‘šπ‘šπ‘£π‘£π›Ύπ›Ύπ‘€π‘€ πœ•πœ•π‘’π‘’οΏ½ πœ•πœ•π‘‘π‘‘ = π‘˜π‘˜π‘₯π‘₯ πœ•πœ•2𝑒𝑒� πœ•πœ•π‘₯π‘₯2 + π‘˜π‘˜π‘§π‘§ πœ•πœ•2𝑒𝑒� πœ•πœ•π‘§π‘§2 (5) Where π‘˜π‘˜π‘₯π‘₯ = π‘˜π‘˜π‘§π‘§ = 𝐾𝐾, we have With boundary conditions; U(0, z, t) = 0 U(x, 0, t) = 0 U(a, z, t) = 0 U(x, b, t) = 0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 222 And initial condition π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 0) = 100 Using the separation of variables method, Let equation (5) be written in the form of π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 𝑑𝑑) = 0 𝑀𝑀(π‘₯π‘₯, 𝑧𝑧) 𝜏𝜏(𝑑𝑑) and rearranged, we have πœ•πœ•2𝑀𝑀 πœ•πœ•π‘₯π‘₯2 + πœ•πœ•2𝑀𝑀 πœ•πœ•π‘§π‘§2 𝑀𝑀 = πœπœβ€² 𝜏𝜏 οΏ½ π‘šπ‘šπ‘£π‘£π›Ύπ›Ύπ‘€π‘€ 𝐾𝐾 οΏ½ For a function of x and z to be identically equal to a function of t for all x, z and t, both sides of this equation must be equal to a constant. For it to decay in time, we should anticipate a negative separation constant. πœ•πœ•2𝑀𝑀 πœ•πœ•π‘₯π‘₯2 + πœ•πœ•2𝑀𝑀 πœ•πœ•π‘§π‘§2 𝑀𝑀 = πœπœβ€² 𝜏𝜏 οΏ½ π‘šπ‘šπ‘£π‘£π›Ύπ›Ύπ‘€π‘€ 𝐾𝐾 οΏ½ = βˆ’βˆ2 (6) Equation (6) can be separated into space and time For time we have; πœπœβ€² + 𝜏𝜏𝐻𝐻 ∝2= 0 (7) Where; 𝐻𝐻 = π‘˜π‘˜ π‘šπ‘šπ‘£π‘£π›Ύπ›Ύπ‘€π‘€ For space, we obtain; πœ•πœ•2𝑀𝑀 πœ•πœ•π‘₯π‘₯2 + πœ•πœ•2𝑀𝑀 πœ•πœ•π‘§π‘§2 +∝2 𝑀𝑀 = 0 And can be further separated into the product; 𝑀𝑀(π‘₯π‘₯, 𝑧𝑧) = 𝑋𝑋(π‘₯π‘₯) 𝑍𝑍(𝑧𝑧) And dividing through by XZ, we have; American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 223 𝑋𝑋′′ 𝑋𝑋 = βˆ’π‘π‘β€²β€² 𝑍𝑍 βˆ’βˆ2= 𝛽𝛽2 (8) Where: πœ•πœ•2𝑋𝑋 πœ•πœ•π‘₯π‘₯2 = 𝑋𝑋′′ πœ•πœ•2𝑍𝑍 πœ•πœ•π‘§π‘§2 = 𝑍𝑍′′ 𝛽𝛽2 is the second separation constant and the sign associated with 𝛽𝛽2 remains to be verified. Thus; 𝑋𝑋′′ βˆ’ 𝛽𝛽2𝑋𝑋 = 0 (9) 𝑍𝑍′′ + (∝2+ 𝛽𝛽2)𝑍𝑍 = 0 (10) From the boundary conditions, we have; 𝑋𝑋(0) = 𝑋𝑋(π‘Žπ‘Ž) = 0 𝑍𝑍(0) = 𝑍𝑍(𝑏𝑏) = 0 The general solution of equation (9) is; 𝑋𝑋(π‘₯π‘₯) = π΄π΄π΄π΄π΄π΄π‘›π‘›β„Ž 𝛽𝛽π‘₯π‘₯ + π΅π΅π‘π‘π‘π‘π‘π‘β„Žπ›½π›½π‘₯π‘₯ (11) Applying the boundary conditions, When; π‘₯π‘₯ = 0, 𝐡𝐡 = 0 π‘₯π‘₯ = π‘Žπ‘Ž, 0 = 𝐴𝐴 sinh( 𝛽𝛽. π‘Žπ‘Ž) Implies that 𝑋𝑋(π‘₯π‘₯) = 0 π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 𝑑𝑑) = 0 Since U=0 does not satisfy the initial time condition, we conclude that the sign of the second separation constant was not chosen correctly. If we replace 𝛽𝛽2 with βˆ’π›½π›½2, then the separated ordinary differential equation becomes American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 224 𝒙𝒙′′ + 𝛽𝛽2π‘₯π‘₯ = 0 𝒛𝒛′′ + (βˆπŸπŸβˆ’ 𝛽𝛽2)𝑍𝑍 = 0 𝑋𝑋(π‘₯π‘₯) = 𝐴𝐴𝑐𝑐𝐴𝐴𝑛𝑛𝛽𝛽π‘₯π‘₯ + 𝐡𝐡𝑐𝑐𝑐𝑐𝑐𝑐𝛽𝛽π‘₯π‘₯ From applying the boundary conditions, we have, 𝐡𝐡 = 0 π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ sinπ›½π›½π‘Žπ‘Ž = 0 π›½π›½π‘Žπ‘Ž = π‘šπ‘šπ‘›π‘› 𝛽𝛽 = π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž Where; π‘šπ‘š=1, 2, 3, 4, 5......... Therefore, equation (11) becomes; 𝑋𝑋(π‘₯π‘₯) = 𝐴𝐴𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ (12) The general solution of equation (10) is 𝑍𝑍(𝑧𝑧) = C sin(π‘Œπ‘Œπ‘π‘) + 𝐷𝐷 cos(π‘Œπ‘Œπ‘π‘) From the boundary conditions, D = 0 Sin (Yb) = 0 Thus; π‘Œπ‘Œπ‘π‘ = 𝑛𝑛𝑛𝑛 π‘Œπ‘Œ = 𝑛𝑛𝑛𝑛 𝑏𝑏 Where; 𝑛𝑛 = 1, 2, 3,....... Therefore; 𝑍𝑍(𝑧𝑧) = 𝐢𝐢 sin οΏ½ 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑍𝑍� (13) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 225 The solution for equation (7) is 𝜏𝜏(𝑑𝑑) = π‘’π‘’βˆ’π»π»βˆ2𝑑𝑑 (14) Let; ∝𝟐𝟐= 𝛽𝛽2 + 𝑦𝑦2 Applying this to equation (10), we have that; ∝𝟐𝟐= 𝛽𝛽2 + 𝑦𝑦2 = οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž οΏ½ 2 + οΏ½ 𝑛𝑛𝑛𝑛 𝑏𝑏 οΏ½ 2 The three solutions of equation (12), (13), (14) can now be combined to yield π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 𝑑𝑑) = 𝑋𝑋(π‘₯π‘₯)𝑍𝑍(𝑧𝑧) 𝜏𝜏(𝑑𝑑) = 𝐴𝐴𝐢𝐢 sin οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ sin οΏ½ 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑍𝑍� π‘’π‘’βˆ’π»π»βˆ2𝑑𝑑 π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 𝑑𝑑) = 𝐸𝐸𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ . sin οΏ½ 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑧𝑧� π‘’π‘’βˆ’π»π»οΏ½οΏ½ βˆͺπœ‹πœ‹ π‘Žπ‘Ž οΏ½ 2 +οΏ½π‘›π‘›π‘Žπ‘Žπ‘π‘ οΏ½ 2 �𝑑𝑑 Where; 𝐸𝐸 = 𝐴𝐴𝐢𝐢 To satisfy the time condition a linear combination for all positive integers of m and n is required. π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 𝑑𝑑) = οΏ½ οΏ½πΈπΈπ‘šπ‘šπ‘›π‘›π‘π‘π΄π΄π‘›π‘› οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ . sin οΏ½ 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑧𝑧� π‘’π‘’βˆ’π»π»οΏ½οΏ½ βˆͺπœ‹πœ‹ π‘Žπ‘Ž οΏ½ 2 +οΏ½π‘›π‘›π‘Žπ‘Žπ‘π‘ οΏ½ 2 �𝑑𝑑 (15) ∞ 𝑛𝑛=1 ∞ π‘šπ‘š=1 The initial conditions require that π‘ˆπ‘ˆ(π‘₯π‘₯, 𝑧𝑧, 0) = 100 = οΏ½ οΏ½πΈπΈπ‘šπ‘šπ‘›π‘›π‘π‘π΄π΄π‘›π‘› οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ . sin οΏ½ 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑧𝑧� ∞ 𝑛𝑛=1 ∞ π‘šπ‘š=1 Using Fourier sine orthogonality relations to evaluate the coefficient πΈπΈπ‘šπ‘šπ‘›π‘› , we multiply by 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½π‘šπ‘šοΏ½πœ‹πœ‹ π‘Žπ‘Ž π‘₯π‘₯οΏ½and integrate from 0 to a to obtain οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑑𝑑π‘₯π‘₯ = οΏ½ οΏ½πΈπΈπ‘šπ‘šπ‘›π‘›π‘π‘π΄π΄π‘›π‘› οΏ½ 𝑛𝑛π‘₯π‘₯𝑧𝑧 𝑏𝑏 οΏ½οΏ½ sin οΏ½ π‘šπ‘šπ‘›π‘›π‘₯π‘₯ π‘Žπ‘Ž οΏ½ a 0 ∞ 𝑛𝑛=1 ∞ π‘šπ‘š=1 π‘Žπ‘Ž 0 sin οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ dx From orthogonality relation American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 226 οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ π‘Žπ‘Ž 0 sin οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑑𝑑π‘₯π‘₯ = οΏ½ 0,π‘šπ‘š β‰  π‘šπ‘šοΏ½ π‘Žπ‘Ž 2 ,π‘šπ‘š = π‘šπ‘šοΏ½ Therefore; οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ π‘Žπ‘Ž 0 𝑑𝑑π‘₯π‘₯ = οΏ½ οΏ½πΈπΈπ‘šπ‘šπ‘›π‘›π‘π‘π΄π΄π‘›π‘› οΏ½ 𝑛𝑛π‘₯π‘₯𝑧𝑧 𝑏𝑏 οΏ½ . π‘Žπ‘Ž 2 ∞ 𝑛𝑛=1 ∞ π‘šπ‘š=1 But; οΏ½ π‘Žπ‘Ž 2 = π‘Žπ‘Ž 2 ∞ 𝑛𝑛=1 Hence, we have; οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑑𝑑π‘₯π‘₯ = π‘Žπ‘Ž 2 οΏ½ πΈπΈπ‘šπ‘šπ‘›π‘›π‘π‘π΄π΄π‘›π‘› οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 οΏ½ ∞ π‘šπ‘š=1 π‘Žπ‘Ž 0 Multiplying by 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½π‘›π‘›πœ‹πœ‹π‘§π‘§ 𝑏𝑏 οΏ½ and integrating from 0 to b. οΏ½ οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 �𝑑𝑑π‘₯π‘₯ 𝑑𝑑𝑧𝑧 = οΏ½ π‘Žπ‘Ž 2 οΏ½πΈπΈπ‘šπ‘šπ‘›π‘›π‘π‘π΄π΄π‘›π‘› οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 οΏ½ 𝑑𝑑𝑧𝑧 ∞ 𝑛𝑛=1 𝑏𝑏 0 π‘Žπ‘Ž 0 𝑏𝑏 0 Applying the orthogonality relation οΏ½ οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 �𝑑𝑑π‘₯π‘₯ 𝑑𝑑𝑧𝑧 π‘Žπ‘Ž 0 𝑏𝑏 0 = π‘Žπ‘Ž 2 οΏ½ πΈπΈπ‘šπ‘šπ‘›π‘› 𝑏𝑏 2 ∞ π‘šπ‘š=1 Where, οΏ½ 𝑏𝑏 2 ∞ 𝑛𝑛=1 = 𝑏𝑏 2 Hence; οΏ½ οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šοΏ½π‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 �𝑑𝑑π‘₯π‘₯ 𝑑𝑑𝑧𝑧 π‘Žπ‘Ž 0 𝑏𝑏 0 = π‘Žπ‘Ž 2 . 𝑏𝑏 2 πΈπΈπ‘šπ‘šπ‘›π‘› = π‘Žπ‘Žπ‘π‘ 4 πΈπΈπ‘šπ‘šπ‘›π‘› Where; π‘šπ‘šοΏ½ = π‘šπ‘š American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 227 𝑛𝑛� = 𝑛𝑛 Therefore, οΏ½ οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 οΏ½ 𝑑𝑑π‘₯π‘₯ 𝑑𝑑𝑧𝑧 π‘Žπ‘Ž 0 𝑏𝑏 0 = π‘Žπ‘Žπ‘π‘ 4 πΈπΈπ‘šπ‘šπ‘›π‘› πΈπΈπ‘šπ‘šπ‘›π‘› = 4 π‘Žπ‘Žπ‘π‘ οΏ½ οΏ½ 100𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ π‘šπ‘šπ‘›π‘› π‘Žπ‘Ž π‘₯π‘₯οΏ½ 𝑐𝑐𝐴𝐴𝑛𝑛 οΏ½ 𝑛𝑛𝑛𝑛𝑧𝑧 𝑏𝑏 �𝑑𝑑π‘₯π‘₯ 𝑑𝑑𝑧𝑧 π‘Žπ‘Ž 0 𝑏𝑏 0 (16) Equation (15) is the exact solution to equation (5) with πΈπΈπ‘šπ‘šπ‘›π‘› computed from equation (16). 4. Case Study The soil for study is that of a highly compressible clay layer, 2m thick and is subjected to a vertical pressure of 100kPa that is maintained constant with time. Drainage is allowed from both the vertical and horizontal direction. From laboratory analysis carried out on soil sample obtained, the following data were obtained. Table 1: Soil Properties S/N Parameters Value 1 Moisture content 88.0 2 Specific gravity 2.59 3 Liquid limit 140 4 Plastic limit 37 5 Plasticity Index 103 6 Consolidation modulus, πΈπΈπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œ (MN/m2) 0.2 7 Coefficient of volume compressibility, π‘šπ‘šπ‘£π‘£ (MN/m2) 5.7E-010 8 Coefficient of consolidation Pressure, Cv, (m/S2) 3.4E-09 9 Compression index Cc 9.9366 10 Coefficient of permeability in the vertical direction, kz., (m/S2) 6.8E-11 The natural moisture content test result is 88%, with specific gravity 2.59, the Atterberg’s limit of 140%, plastic limit of 37% and plasticity index of 103%. The soil is highly plastic. The result for the pore water pressure using equation (15) at t = 10, 100, 600 and 1000 days for the two dimensional consolidation is presented in Figure 1. The result of the predicted pore water pressure using the developed analytical method showed a decrease in pore water pressure at various depth with time. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 228 Figure 1: Pore water pressure plot 5. Conclusion The one dimensional consolidation problem developed by Terzaghi has been expanded to a two dimensional to help simulate field condition. The validation of the equation, was carried out using consolidation parameters obtained from laboratory test carried out on a cohesive soil. The initial pore pressure of 100kPa can be seen to be reducing as the time increases. The equation can be used for two dimensional consolidation problem. Acknowledgement We are grateful to the management and staff of Fugro Nigeria Limited for sharing their wealth of experience with us during the course of this research. We also wish to thank Engr. Matthias Imomoh and Mr. Wole Phillips for their contribution towards the development of this manuscript. References [1]. K. R Arora. Consolidation of soil in Soil Mechanics and Foundation Engineering, 6th edition, A.K Jain, Nai Sarak, Delhi, 2004, pp. 256 - 305 [2]. R.D. Francesco. β€œExact Solution of Terzaghi’s Consolidation Equation and Extension to Two/Three- Dimensional Cases” Applied Mathematics, volume 4, pp. 713-717. 2013 [3]. L. Ho, B. Fatahi,. β€œAnalytical solution for the two-dimensional plane strain consolidation of an unsaturated soil stratum subjected to time-dependent loading” Compute Geotech, 67, pp. 1–16 Aug 2015 [4]. L. Ho, B. Fatahi, H. Khabbaz, β€œA closed form analytical solution for two-dimensional plane strain consolidation of unsaturated soil stratum”. Int. J. Number. Anal. Met. Geomech. vol 39, pp. 1665– 1692. 2015 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 Po re p re ss ur e (K pa ) Depth (m) 10 days 100 days 600 days 1000 days American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 56, No 1, pp 218-229 229 [5]. S. Inoue. β€œAn Example of Two-dimensional Consolidation Testing and Numerical Procedures for Embankment Materials” Irrigation Engineering and Rural Planning No.21, pp. 4-14. 1991 [6]. F. Oka, Adachi, Y. Okano. β€œTwo-Dimensional Consolidation Analysis Using an Elasto-Viscoplastic Constitutive Equation” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 10, pp. 1- 16. Aug 1986. [7]. J.Q. Su, Z. Wang. Two-dimensional consolidation theory of electro-osmosis. Geotechnique vol 53, no. 8, pp 759–763. Jan 2003. [8]. S.S. Razouki, A.A. Al-Zayadi. β€œDesign Charts for 2D consolidation under time-dependent Embankment Loading” Quarterly Journal of Engineering Geology and Hydrogeology, vol 36, pp. 246- 260, Apr 2003. The result of the predicted pore water pressure using the developed analytical method showed a decrease in pore water pressure at various depth with time.