{"id":360,"date":"2020-10-24T19:19:49","date_gmt":"2020-10-24T19:19:49","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=360"},"modified":"2020-12-29T16:44:12","modified_gmt":"2020-12-29T16:44:12","slug":"applicability-of-darcys-law","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/applicability-of-darcys-law\/","title":{"raw":"4.5 Applicability of Darcy\u2019s Law","rendered":"4.5 Applicability of Darcy\u2019s Law"},"content":{"raw":"Darcy\u2019s law is a macroscopic law applied using average values of parameters for a representative portion of the porous media, the REV as discussed in Section 3 of this book. Darcy\u2019s law is applied to the flow of fluids through porous media. It describes a linear relationship between specific discharge and the hydraulic gradient. This relationship is valid for most all groundwater conditions. However, as the flow rate approaches zero or when high rates of flow occur in high hydraulic conductivity material like fractures or karst features, flow may not be linear and thus, Darcy\u2019s law does not appropriately represent groundwater flow.\r\n\r\nIt has been suggested that a more general formulation of the law may be posed as Equation 32.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle q=-K\\left(\\frac{dh}{dl}\\right)^m[\/latex]<\/td>\r\n(32)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n
m<\/em><\/td>\r\n=<\/td>\r\ncoefficient (dimensionless), if m<\/em> = 1, as in all the common situations, the flow law is linear, if m<\/em> is not equal to 1 then the equation describes non-linear conditions and should not be called Darcy\u2019s law. (Freeze and Cherry, 1979)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDarcy\u2019s law is strictly applicable for laminar flow of an incompressible fluid in a solid matrix (non-deforming) of porous medium in which the gradient of mechanical energy is the only driving force (Figure 29). It is applicable under steady state or transient conditions. Laminar flow occurs when a fluid flows in parallel lines with no disturbance between the lines. Laminar flow is by definition \u201cnot turbulent\u201d (Figure 29). Under turbulent flow conditions packets of water exhibit chaotic changes in velocity (for example, flow in a white-water stream).\r\n\r\n[caption id=\"attachment_364\" align=\"alignnone\" width=\"1024\"]\"Figure Figure 29 -<\/strong> Conceptualized laminar and turbulent flow at the pore scale. Small red dots represent a \u201cpacket of water\u201d that can be tracked along a flow path: a) laminar flow occurs when packets of water follow one another in a predictable manner, not getting ahead of or behind of their original position; and, b) turbulent flow occurs when motion of the fluid becomes chaotic (dashed arrows) and can no longer be described by equations of fluid mechanics for smooth flow. Particles do not strictly follow each other.[\/caption]\r\n\r\nTurbulent conditions begin to develop as flow velocity increases. The Reynolds number that characterizes the ratio of internal forces to viscous forces acting on fluid elements is often used to test for laminar or turbulent conditions (Equation 33).\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle R_{e}=\\frac{qd\\rho }{\\mu }[\/latex]<\/td>\r\n(33)<\/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
R<\/em>e<\/em><\/sub><\/td>\r\n=<\/td>\r\nReynolds number characterizes the ratio of internal to viscous forces acting on fluid elements (dimensionless)<\/td>\r\n<\/tr>\r\n
q<\/em><\/td>\r\n=<\/td>\r\nspecific discharge (L\/T)<\/td>\r\n<\/tr>\r\n
d<\/em><\/td>\r\n=<\/td>\r\ncharacteristic length (L)<\/td>\r\n<\/tr>\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\ndensity (M\/L3<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity (M\/(LT))<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe characteristic length term attempts to provide some information on the pore diameters available for flow. Authors suggest using an effective grain size (e.g., d<\/em>10<\/sub> finer than) (Todd and Mays, 2004), the mean pore dimension, mean grain diameter, or some function of the square root of the intrinsic permeability k<\/em> (Freeze and Cherry, 1979). Bear (1972) states \u201cDarcy\u2019s law is valid as long as the Reynolds number, based on average grain diameter, does not exceed some value between 1 and 10 (page 126)\u201d. Most authors on the topic agree that when R<\/em>e<\/em><\/sub> is less than one, laminar flow occurs, and Darcy\u2019s law is valid.\r\n\r\nFlow rates that exceed the upper limit of Darcy\u2019s law have been noted to occur in caverns and cavities of karstic limestones and dolomites, cavernous volcanic rocks (e.g., lava tubes), and some open framework boulder dominated deposits (in short, materials with large interconnected pores and extremely high hydraulic conductivities). Under most natural conditions, groundwater flows are laminar and Darcy\u2019s law is valid (Freeze and Cherry, 1979). However, turbulent flow may occur in a portion of a groundwater system when flow rates accelerate in the vicinity of high yield pumping wells and drains. Turbulent flow may also occur in fractures with large apertures when the system is stressed by pumping. The lower limit of Darcy\u2019s law is of little concern to most hydrogeologists as flow rates are extremely small.\r\n\r\nDarcy\u2019s law as presented here is not valid for compressible fluids. Fortunately, water has a relatively low compressibility. So, although water is not completely incompressible, this requirement can be relaxed to accommodate use of Darcy\u2019s law for the small compressibility of water. However, Darcy\u2019s law is not applicable if the density of the fluid varies due to differing pressure, temperature, and\/or high concentrations of dissolved constituents; it is not applicable if there are substantial differences in density from location to location within a flow system; and it is not applicable if thermal, chemical, or electrical gradients drive fluid flow. However, forms of flux equations based on Darcy\u2019s Law have been developed to accommodate density variations due to compressibility, non-uniform solute concentrations and non-uniform temperature. These equations are usually referred as representing Darcy\u2019s Law.\r\n\r\nAgain, fortunately, the density and viscosity changes of water are usually trivial at the low pressures and narrow range of temperatures occurring in most of the Earth\u2019s shallow subsurface making Darcy\u2019s law applicable.\r\n\r\nThe lengthy presentation of Darcy\u2019s Law in this section is purposeful. Darcy\u2019s Law is the key to understanding and interpreting groundwater flow in porous media. It provides the foundational relationships between head, gradients and hydraulic conductivities that hydrogeologists used every day to develop conceptual models of how natural groundwater systems work, generate groundwater budgets, identify source areas of groundwater contamination, manage groundwater supplies and quantify exchanges between surface water and groundwater systems. In the authors\u2019 experience, whenever a problem seems overwhelming or unclear, check Darcy\u2019s Law and be sure basic principles are being correctly applied. This advice has been valuable in resolving issues encountered by students attacking assigned problems as well as professional hydrogeologists and engineers managing large complex groundwater sites.","rendered":"

Darcy\u2019s law is a macroscopic law applied using average values of parameters for a representative portion of the porous media, the REV as discussed in Section 3 of this book. Darcy\u2019s law is applied to the flow of fluids through porous media. It describes a linear relationship between specific discharge and the hydraulic gradient. This relationship is valid for most all groundwater conditions. However, as the flow rate approaches zero or when high rates of flow occur in high hydraulic conductivity material like fractures or karst features, flow may not be linear and thus, Darcy\u2019s law does not appropriately represent groundwater flow.<\/p>\n

It has been suggested that a more general formulation of the law may be posed as Equation 32.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle q=-K\left(\frac{dh}{dl}\right)^m\"<\/td>\n(32)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
m<\/em><\/td>\n=<\/td>\ncoefficient (dimensionless), if m<\/em> = 1, as in all the common situations, the flow law is linear, if m<\/em> is not equal to 1 then the equation describes non-linear conditions and should not be called Darcy\u2019s law. (Freeze and Cherry, 1979)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Darcy\u2019s law is strictly applicable for laminar flow of an incompressible fluid in a solid matrix (non-deforming) of porous medium in which the gradient of mechanical energy is the only driving force (Figure 29). It is applicable under steady state or transient conditions. Laminar flow occurs when a fluid flows in parallel lines with no disturbance between the lines. Laminar flow is by definition \u201cnot turbulent\u201d (Figure 29). Under turbulent flow conditions packets of water exhibit chaotic changes in velocity (for example, flow in a white-water stream).<\/p>\n

\"Figure
Figure 29 –<\/strong> Conceptualized laminar and turbulent flow at the pore scale. Small red dots represent a \u201cpacket of water\u201d that can be tracked along a flow path: a) laminar flow occurs when packets of water follow one another in a predictable manner, not getting ahead of or behind of their original position; and, b) turbulent flow occurs when motion of the fluid becomes chaotic (dashed arrows) and can no longer be described by equations of fluid mechanics for smooth flow. Particles do not strictly follow each other.<\/figcaption><\/figure>\n

Turbulent conditions begin to develop as flow velocity increases. The Reynolds number that characterizes the ratio of internal forces to viscous forces acting on fluid elements is often used to test for laminar or turbulent conditions (Equation 33).<\/p>\n\n\n\n
<\/td>\n\"\displaystyle R_{e}=\frac{qd\rho }{\mu }\"<\/td>\n(33)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n
R<\/em>e<\/em><\/sub><\/td>\n=<\/td>\nReynolds number characterizes the ratio of internal to viscous forces acting on fluid elements (dimensionless)<\/td>\n<\/tr>\n
q<\/em><\/td>\n=<\/td>\nspecific discharge (L\/T)<\/td>\n<\/tr>\n
d<\/em><\/td>\n=<\/td>\ncharacteristic length (L)<\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\ndensity (M\/L3<\/sup>)<\/td>\n<\/tr>\n
\u03bc<\/em><\/td>\n=<\/td>\ndynamic viscosity (M\/(LT))<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The characteristic length term attempts to provide some information on the pore diameters available for flow. Authors suggest using an effective grain size (e.g., d<\/em>10<\/sub> finer than) (Todd and Mays, 2004), the mean pore dimension, mean grain diameter, or some function of the square root of the intrinsic permeability k<\/em> (Freeze and Cherry, 1979). Bear (1972) states \u201cDarcy\u2019s law is valid as long as the Reynolds number, based on average grain diameter, does not exceed some value between 1 and 10 (page 126)\u201d. Most authors on the topic agree that when R<\/em>e<\/em><\/sub> is less than one, laminar flow occurs, and Darcy\u2019s law is valid.<\/p>\n

Flow rates that exceed the upper limit of Darcy\u2019s law have been noted to occur in caverns and cavities of karstic limestones and dolomites, cavernous volcanic rocks (e.g., lava tubes), and some open framework boulder dominated deposits (in short, materials with large interconnected pores and extremely high hydraulic conductivities). Under most natural conditions, groundwater flows are laminar and Darcy\u2019s law is valid (Freeze and Cherry, 1979). However, turbulent flow may occur in a portion of a groundwater system when flow rates accelerate in the vicinity of high yield pumping wells and drains. Turbulent flow may also occur in fractures with large apertures when the system is stressed by pumping. The lower limit of Darcy\u2019s law is of little concern to most hydrogeologists as flow rates are extremely small.<\/p>\n

Darcy\u2019s law as presented here is not valid for compressible fluids. Fortunately, water has a relatively low compressibility. So, although water is not completely incompressible, this requirement can be relaxed to accommodate use of Darcy\u2019s law for the small compressibility of water. However, Darcy\u2019s law is not applicable if the density of the fluid varies due to differing pressure, temperature, and\/or high concentrations of dissolved constituents; it is not applicable if there are substantial differences in density from location to location within a flow system; and it is not applicable if thermal, chemical, or electrical gradients drive fluid flow. However, forms of flux equations based on Darcy\u2019s Law have been developed to accommodate density variations due to compressibility, non-uniform solute concentrations and non-uniform temperature. These equations are usually referred as representing Darcy\u2019s Law.<\/p>\n

Again, fortunately, the density and viscosity changes of water are usually trivial at the low pressures and narrow range of temperatures occurring in most of the Earth\u2019s shallow subsurface making Darcy\u2019s law applicable.<\/p>\n

The lengthy presentation of Darcy\u2019s Law in this section is purposeful. Darcy\u2019s Law is the key to understanding and interpreting groundwater flow in porous media. It provides the foundational relationships between head, gradients and hydraulic conductivities that hydrogeologists used every day to develop conceptual models of how natural groundwater systems work, generate groundwater budgets, identify source areas of groundwater contamination, manage groundwater supplies and quantify exchanges between surface water and groundwater systems. In the authors\u2019 experience, whenever a problem seems overwhelming or unclear, check Darcy\u2019s Law and be sure basic principles are being correctly applied. This advice has been valuable in resolving issues encountered by students attacking assigned problems as well as professional hydrogeologists and engineers managing large complex groundwater sites.<\/p>\n","protected":false},"author":1,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-360","chapter","type-chapter","status-publish","hentry"],"part":101,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/360","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":9,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/360\/revisions"}],"predecessor-version":[{"id":1175,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/360\/revisions\/1175"}],"part":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/101"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/360\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/media?parent=360"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapter-type?post=360"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=360"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/license?post=360"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}