{"id":232,"date":"2022-01-13T23:17:23","date_gmt":"2022-01-13T23:17:23","guid":{"rendered":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/effective-intergranular-stress-and-soil-parameters\/"},"modified":"2022-03-30T20:04:42","modified_gmt":"2022-03-30T20:04:42","slug":"effective-intergranular-stress-and-soil-parameters","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/effective-intergranular-stress-and-soil-parameters\/","title":{"raw":"2.1 Effective Intergranular Stress and Soil Parameters","rendered":"2.1 Effective Intergranular Stress and Soil Parameters"},"content":{"raw":"
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The theories of land subsidence are founded on basic principles of soil mechanics. Thus, the following discussion describes soil parameters, however, references to soil can be viewed as aquifer and confining bed material in a groundwater system.<\/p>\r\n

The soil is viewed as a set of grains in contact. Assume a degree of saturation equal to 1 (that is, full saturation). Make a (macroscopically) horizontal cross section through the soil intersecting the contact points (Figure\u00a011).<\/p>\r\n

\"A<\/p>\r\n

Figure\u00a011<\/strong>\u00a0<\/strong>\u2011\u00a0<\/strong>Schematic vertical cross\u2011section through a porous medium. The black dashed line is the crossing surface and the dotted orange line is the horizontal projection of the crossing surface.<\/p>\r\n

Consider a piece of such a section with area A<\/em> on a horizontal plane (dotted orange line, Figure\u00a011) and n<\/em> contact points (black arrows, Figure\u00a011). If F<\/em>zi<\/em><\/sub> is the vertical component of the force that the grains exchange through the i<\/em>th<\/em><\/sup> contact area (Figure\u00a011), we define \u201ceffective intergranular stress<\/em>\u201d \u03c3<\/em>z<\/em><\/sub> by Equation\u00a01.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\sigma _{z}=\\frac{\\sum_{i=1}^{n}F_{zi}}{A}[\/latex]<\/td>\r\n(1)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Equation\u00a01 shows that effective intergranular stress is the uniform stress over the unit horizontal projection of a crossing surface with n<\/em> contact points. Stress is a force per unit area and has dimensions of ML-1<\/sup>T-2<\/sup>, that is, the same dimensions as pressure. The effective intergranular stress is equivalent to the combined individual stresses, \u03c3<\/em>zi<\/em><\/sub>\u00a0=\u00a0F<\/em>zi<\/em><\/sub>\/A<\/em>i<\/em><\/sub> spread over the horizontal area, A<\/em>, with stress taken to be positive in compression such that the force is the same. That is,<\/p>\r\n

[latex]\\displaystyle \\sigma _{z}A=\\sum_{i=1}^{n}\\sigma _{zi}A_{i}[\/latex]<\/p>\r\n

namely:<\/p>\r\n

[latex]\\displaystyle \\sigma _{z}=\\frac{\\sum_{i=1}^{n}\\sigma _{zi}A_{i}}{A}=\\frac{\\sum_{i=1}^{n}F_{zi}}{A}[\/latex]<\/p>\r\n

Denote the geostatic stress by \u03c3<\/em>c<\/em><\/sub>, that is, the weight of a soil column applied to a unit horizontal area at a given depth. The weight of a soil column is the combined weight of the solids and the fluids in the pores. In the case of full saturation, \u03c3<\/em>c<\/em><\/sub> is equilibrated by \u03c3<\/em>z<\/em><\/sub> and the pore pressure p<\/em> as shown in Equation\u00a02. The fluid pressure is distributed over the unit area minus the area of grain contacts as expressed in the parentheses of Equation\u00a02.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\sigma _{c}=\\sigma _{z}+p\\left ( 1-\\sum_{i=1}^{n}A_{i}\\textup{cos}\\alpha _{i} \\right )[\/latex]<\/td>\r\n(2)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u03c3<\/em>c<\/em><\/sub><\/td>\r\n=<\/td>\r\ngeostatic stress (ML-1<\/sup>T-2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03c3<\/em>z<\/em><\/sub><\/td>\r\n=<\/td>\r\neffective intergranular stress (ML-1<\/sup>T-2<\/sup>)<\/td>\r\n<\/tr>\r\n
p<\/em><\/td>\r\n=<\/td>\r\npore pressure (ML-1<\/sup>T-2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03b1<\/em>i<\/em><\/sub><\/td>\r\n=<\/td>\r\nangle between the contact area, A<\/em>i<\/em><\/sub>, and the vertical<\/td>\r\n<\/tr>\r\n
A<\/em>i<\/em><\/td>\r\n=<\/td>\r\ncontact area normal to the force between grains (L2<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The contact area \u03a3(A<\/em>i<\/em><\/sub> cos\u03b1<\/em>i<\/em><\/sub>)<\/em> is much smaller than 1 (as explained in <\/a>Box 1<\/span><\/a>), thus the quantity within the parentheses is essentially 1, hence, on first approximation, Equation\u00a02 becomes Equation\u00a03.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n\u03c3<\/em>c<\/em><\/sub> = \u03c3<\/em>z<\/em><\/sub> + p<\/em><\/td>\r\n(3)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The geostatic load \u03c3<\/em>c<\/em><\/sub> is also called \u201ctotal vertical stress<\/em>\u201d. If \u03c3<\/em>c<\/em><\/sub> remains constant during pumping (this is essentially the case for a pumped confined aquifer because pores do not drain), a decrease of p<\/em> induces an equal increase of effective intergranular stress, \u03c3<\/em>z<\/em><\/sub>, under whose effect the pumped formation compacts. Of course, this is a preliminary analysis. A more complete study should take into account second order effects such as the forces of mutual attraction among the grains and the fluid surface tension as well as the gas pressure in partially saturated soils.<\/p>\r\n

To evaluate the compaction of a formation with decreased pore pressures we need to define a few dimensionless characteristic soil parameters:<\/p>\r\n\r\n