{"id":97,"date":"2020-11-14T23:57:01","date_gmt":"2020-11-14T23:57:01","guid":{"rendered":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/?post_type=chapter&p=97"},"modified":"2020-12-12T23:22:31","modified_gmt":"2020-12-12T23:22:31","slug":"building-a-static-model-from-upscaled-properties","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/chapter\/building-a-static-model-from-upscaled-properties\/","title":{"raw":"8 Building a Static Model from Upscaled Properties","rendered":"8 Building a Static Model from Upscaled Properties"},"content":{"raw":"A more direct but also labor-intensive methodology is to populate the model using hydraulic properties \u201cupscaled\u201d from the finest scale data available. If cores are collected from a borehole, undisturbed soil or rock samples can be sent to a laboratory for porosity and hydraulic conductivity tests. This allows for correlation between lithology and hydraulic properties on the scale of inches. A representative volume is then built with a layering scheme following the major lithology types in the core. Each layer is assigned a \u201cblended\u201d hydraulic conductivity that represents a statistical average of laboratory measurements for that lithology. These layers are then translated to a bulk vertical (Kv) and horizontal (Kh) hydraulic conductivity (Figure 17).\r\n\r\n[caption id=\"attachment_99\" align=\"alignnone\" width=\"1024\"]\"Upscaling Figure\u00a017\u00a0-\u00a0<\/strong>Upscaling from depth discrete to bulk hydraulic conductivity (Brandenburg, 2020).[\/caption]\r\n\r\nBulk horizontal hydraulic conductivity is calculated as the arithmetic mean of the blended layers, as shown in Equation 3 for the example in Figure 17.\r\n\r\n\r\n\r\n
[latex]\\displaystyle Kh=\\frac{K_{1}b_{1}+K_{2}b_{3}+K_{3}b_{3}+K_{4}b_{4}+K_{5}b_{5}+K_{6}b_{6}+K_{7}b_{7}+K_{8}b_{8}+K_{9}b_{9}}{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}+b_{6}+b_{7}+b_{8}+b_{9}}[\/latex]<\/td>\r\n(3)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBulk porosity is also calculated was an arithmetic mean. Bulk vertical hydraulic conductivity is calculated as the harmonic mean of the blended layers, as shown in Equation 4 for the example in Figure 17.\r\n\r\n\r\n\r\n
[latex]\\displaystyle Kv=\\frac{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}+b_{6}+b_{7}+b_{8}+b_{9}}{\\frac{b_{1}}{K_{1}}+\\frac{b_{2}}{K_{2}}+\\frac{b_{3}}{K_{3}}+\\frac{b_{4}}{K_{4}}+\\frac{b_{5}}{K_{5}}+\\frac{b_{6}}{K_{6}}+\\frac{b_{7}}{K_{7}}+\\frac{b_{8}}{K_{8}}+\\frac{b_{9}}{K_{9}}}[\/latex]<\/td>\r\n(4)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn general, the harmonic mean is representative of K for layers perpendicular to the flow direction, while the arithmetic mean represents K for layers parallel to flow. While accurate in a volume immediately surrounding the borehole, some systematic method is still required to extend these results to the rest of the model volume. In practice, this is often accomplished by combining upscaling with stochastic modeling methods.","rendered":"

A more direct but also labor-intensive methodology is to populate the model using hydraulic properties \u201cupscaled\u201d from the finest scale data available. If cores are collected from a borehole, undisturbed soil or rock samples can be sent to a laboratory for porosity and hydraulic conductivity tests. This allows for correlation between lithology and hydraulic properties on the scale of inches. A representative volume is then built with a layering scheme following the major lithology types in the core. Each layer is assigned a \u201cblended\u201d hydraulic conductivity that represents a statistical average of laboratory measurements for that lithology. These layers are then translated to a bulk vertical (Kv) and horizontal (Kh) hydraulic conductivity (Figure 17).<\/p>\n

\"Upscaling
Figure\u00a017\u00a0–\u00a0<\/strong>Upscaling from depth discrete to bulk hydraulic conductivity (Brandenburg, 2020).<\/figcaption><\/figure>\n

Bulk horizontal hydraulic conductivity is calculated as the arithmetic mean of the blended layers, as shown in Equation 3 for the example in Figure 17.<\/p>\n\n\n\n
\"\displaystyle Kh=\frac{K_{1}b_{1}+K_{2}b_{3}+K_{3}b_{3}+K_{4}b_{4}+K_{5}b_{5}+K_{6}b_{6}+K_{7}b_{7}+K_{8}b_{8}+K_{9}b_{9}}{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}+b_{6}+b_{7}+b_{8}+b_{9}}\"<\/td>\n(3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Bulk porosity is also calculated was an arithmetic mean. Bulk vertical hydraulic conductivity is calculated as the harmonic mean of the blended layers, as shown in Equation 4 for the example in Figure 17.<\/p>\n\n\n\n
\"\displaystyle Kv=\frac{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}+b_{6}+b_{7}+b_{8}+b_{9}}{\frac{b_{1}}{K_{1}}+\frac{b_{2}}{K_{2}}+\frac{b_{3}}{K_{3}}+\frac{b_{4}}{K_{4}}+\frac{b_{5}}{K_{5}}+\frac{b_{6}}{K_{6}}+\frac{b_{7}}{K_{7}}+\frac{b_{8}}{K_{8}}+\frac{b_{9}}{K_{9}}}\"<\/td>\n(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In general, the harmonic mean is representative of K for layers perpendicular to the flow direction, while the arithmetic mean represents K for layers parallel to flow. While accurate in a volume immediately surrounding the borehole, some systematic method is still required to extend these results to the rest of the model volume. In practice, this is often accomplished by combining upscaling with stochastic modeling methods.<\/p>\n","protected":false},"author":1,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-97","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/chapters\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":3,"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/chapters\/97\/revisions"}],"predecessor-version":[{"id":171,"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/chapters\/97\/revisions\/171"}],"part":[{"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/chapters\/97\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/wp\/v2\/media?parent=97"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/pressbooks\/v2\/chapter-type?post=97"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/wp\/v2\/contributor?post=97"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/geologic-frameworks-for-groundwater-flow-models\/wp-json\/wp\/v2\/license?post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}