{"id":41,"date":"2021-10-02T23:21:34","date_gmt":"2021-10-02T23:21:34","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/viscous-flux\/"},"modified":"2022-01-10T17:56:34","modified_gmt":"2022-01-10T17:56:34","slug":"viscous-flux","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/viscous-flux\/","title":{"raw":"2.1 Viscous Flux","rendered":"2.1 Viscous Flux"},"content":{"raw":"
\r\n

The viscous flux appearing throughout this book is the volume flux calculated by Darcy\u2019s law. This calculation is most familiar in the context of water flow and appears in various forms throughout the GW-Project books. Here, we express Darcy\u2019s law in a form suitable for the calculation of the viscous contribution to the flow of the gas phase. As is the case for water and other fluids, the driving forces are the pressure gradient and the body force due to gravity. The latter may be important when the gas column of interest is very thick (Thorstenson and Pollock, 1989), but we include only the pressure gradient in the following developments and write Darcy\u2019s law as Equation\u00a01.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle v=-\\frac{k_{g}}{\\mu }\\frac{dp}{dl}[\/latex]<\/td>\r\n(1)<\/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
v<\/em><\/td>\r\n=<\/td>\r\nmacroscopic volume flux (specific discharge, L3<\/sup> \/ L2<\/sup> T = L\/T)<\/td>\r\n<\/tr>\r\n
p<\/em><\/td>\r\n=<\/td>\r\npressure of the gas (F\/L2<\/sup>)<\/td>\r\n<\/tr>\r\n
l<\/em><\/td>\r\n=<\/td>\r\ncoordinate along which the motion occurs (L)<\/td>\r\n<\/tr>\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity (FT\/L2<\/sup>) (which we treat as a constant in all subsequent developments)<\/td>\r\n<\/tr>\r\n
k<\/em>g<\/em><\/sub><\/td>\r\n=<\/td>\r\npermeability to gas (L2<\/sup>) (associated with the resistance to gas motion that arises solely from viscous shear at the pore scale)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The subscript, g<\/em>, distinguishes the gas permeability (k<\/em>g<\/em><\/sub>, also known as effective permeability) from the intrinsic permeability, k<\/em>, of the porous medium. This distinction is required because the co-existence of liquids in the pore space causes the gas permeability to be smaller than the intrinsic permeability, sometimes dramatically so (e.g., Brooks and Corey, 1966). The gas permeability k<\/em>g<\/em><\/sub> is equal to the intrinsic permeability when the porous medium is dry. The reduction in gas permeability due to the presence of liquid in the pore space is related to the concepts of effective and relative permeability as described by Corey (1994).<\/p>\r\n

Multiplication of v<\/em> by the total molar concentration of the gas C<\/em> (total moles of all components per unit volume), followed by use of the ideal gas law, p<\/em>\u00a0<\/em>=<\/em>\u00a0<\/em>RTC<\/em>, wherein R<\/em> is the gas constant (FL\/moles T) and T<\/em> is absolute temperature, gives Equation\u00a02 for the viscous mole flux of the gas phase.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N^{v}=-\\frac{k_{g}p}{\\mu RT}\\frac{dp}{dl}=-\\frac{k_{g}p}{\\mu }\\frac{dC}{dl}[\/latex]<\/td>\r\n(2)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The flux calculated by Equation\u00a02 is a contributor to the motion of the mixture as a whole. Being a component of the mixture, an individual species is carried along (advected) by the viscous contribution to the phase motion. The advection mole flux of an individual species due to the viscous flux is calculated according to [latex]N_{i}^{v}=x_{i}N^{v}[\/latex], i<\/em> = A, B<\/em> where x<\/em>i<\/em><\/sub> is the mole fraction of component i<\/em> in the mixture. The mole fraction is defined by x<\/em>i<\/em><\/sub> =<\/em> C<\/em>i<\/em><\/sub>\/C<\/em>, where C<\/em>i<\/em><\/sub> is the molar concentration of constituent i<\/em>. The sum of mole fractions over all constituents is always unity.<\/p>\r\n\r\n<\/div>","rendered":"

\n

The viscous flux appearing throughout this book is the volume flux calculated by Darcy\u2019s law. This calculation is most familiar in the context of water flow and appears in various forms throughout the GW-Project books. Here, we express Darcy\u2019s law in a form suitable for the calculation of the viscous contribution to the flow of the gas phase. As is the case for water and other fluids, the driving forces are the pressure gradient and the body force due to gravity. The latter may be important when the gas column of interest is very thick (Thorstenson and Pollock, 1989), but we include only the pressure gradient in the following developments and write Darcy\u2019s law as Equation\u00a01.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle v=-\frac{k_{g}}{\mu }\frac{dp}{dl}\"<\/td>\n(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n
v<\/em><\/td>\n=<\/td>\nmacroscopic volume flux (specific discharge, L3<\/sup> \/ L2<\/sup> T = L\/T)<\/td>\n<\/tr>\n
p<\/em><\/td>\n=<\/td>\npressure of the gas (F\/L2<\/sup>)<\/td>\n<\/tr>\n
l<\/em><\/td>\n=<\/td>\ncoordinate along which the motion occurs (L)<\/td>\n<\/tr>\n
\u03bc<\/em><\/td>\n=<\/td>\ndynamic viscosity (FT\/L2<\/sup>) (which we treat as a constant in all subsequent developments)<\/td>\n<\/tr>\n
k<\/em>g<\/em><\/sub><\/td>\n=<\/td>\npermeability to gas (L2<\/sup>) (associated with the resistance to gas motion that arises solely from viscous shear at the pore scale)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The subscript, g<\/em>, distinguishes the gas permeability (k<\/em>g<\/em><\/sub>, also known as effective permeability) from the intrinsic permeability, k<\/em>, of the porous medium. This distinction is required because the co-existence of liquids in the pore space causes the gas permeability to be smaller than the intrinsic permeability, sometimes dramatically so (e.g., Brooks and Corey, 1966). The gas permeability k<\/em>g<\/em><\/sub> is equal to the intrinsic permeability when the porous medium is dry. The reduction in gas permeability due to the presence of liquid in the pore space is related to the concepts of effective and relative permeability as described by Corey (1994).<\/p>\n

Multiplication of v<\/em> by the total molar concentration of the gas C<\/em> (total moles of all components per unit volume), followed by use of the ideal gas law, p<\/em>\u00a0<\/em>=<\/em>\u00a0<\/em>RTC<\/em>, wherein R<\/em> is the gas constant (FL\/moles T) and T<\/em> is absolute temperature, gives Equation\u00a02 for the viscous mole flux of the gas phase.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle N^{v}=-\frac{k_{g}p}{\mu RT}\frac{dp}{dl}=-\frac{k_{g}p}{\mu }\frac{dC}{dl}\"<\/td>\n(2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The flux calculated by Equation\u00a02 is a contributor to the motion of the mixture as a whole. Being a component of the mixture, an individual species is carried along (advected) by the viscous contribution to the phase motion. The advection mole flux of an individual species due to the viscous flux is calculated according to \"N_{i}^{v}=x_{i}N^{v}\", i<\/em> = A, B<\/em> where x<\/em>i<\/em><\/sub> is the mole fraction of component i<\/em> in the mixture. The mole fraction is defined by x<\/em>i<\/em><\/sub> =<\/em> C<\/em>i<\/em><\/sub>\/C<\/em>, where C<\/em>i<\/em><\/sub> is the molar concentration of constituent i<\/em>. The sum of mole fractions over all constituents is always unity.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-41","chapter","type-chapter","status-publish","hentry"],"part":86,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/41","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":8,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions"}],"predecessor-version":[{"id":397,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions\/397"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/86"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/41\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=41"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapter-type?post=41"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=41"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}