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Derived Geomechanical results ResInsight calculates several of the presented geomechanical results based on the native results present in the odb-files.
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Relative Results (Time Lapse Results) ResInsight can calculate and display relative results, sometimes also referred to as Time Lapse results. When enabled, every result variable is calculated as:
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<meta name="twitter:description" content="ResInsight computes several derived results. In this section we will explain what they are, and briefly how they are calculated.
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Derived Geomechanical results ResInsight calculates several of the presented geomechanical results based on the native results present in the odb-files.
|
||
Relative Results (Time Lapse Results) ResInsight can calculate and display relative results, sometimes also referred to as Time Lapse results. When enabled, every result variable is calculated as:
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$Value_{[t-b]} = Value_{[t]} - Value_{[b]}$">
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<meta property="og:description" content="ResInsight computes several derived results. In this section we will explain what they are, and briefly how they are calculated.
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Derived Geomechanical results ResInsight calculates several of the presented geomechanical results based on the native results present in the odb-files.
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||
Relative Results (Time Lapse Results) ResInsight can calculate and display relative results, sometimes also referred to as Time Lapse results. When enabled, every result variable is calculated as:
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<meta itemprop="description" content="ResInsight computes several derived results. In this section we will explain what they are, and briefly how they are calculated.
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Derived Geomechanical results ResInsight calculates several of the presented geomechanical results based on the native results present in the odb-files.
|
||
Relative Results (Time Lapse Results) ResInsight can calculate and display relative results, sometimes also referred to as Time Lapse results. When enabled, every result variable is calculated as:
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<li><a href="#derived-geomechanical-results">Derived Geomechanical results</a>
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<ul>
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<li><a href="#relative-results-time-lapse-results">Relative Results (Time Lapse Results)</a></li>
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<li><a href="#derived-result-fields">Derived Result Fields</a></li>
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<li><a href="#definitions-of-derived-results">Definitions of Derived Results</a></li>
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<li><a href="#case-constants">Case Constants</a></li>
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<li><a href="#compaction">COMPACTION</a></li>
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<li><a href="#st---total-stress">ST - Total Stress</a></li>
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<li><a href="#stm---total-mean-stress">STM - Total Mean Stress</a></li>
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<li><a href="#q---deviatoric-stress">Q - Deviatoric Stress</a></li>
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<li><a href="#dpn---shear-slip-indicator">DPN - Shear Slip Indicator</a></li>
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<li><a href="#gamma---stress-path">Gamma - Stress Path</a></li>
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<li><a href="#se---effective-stress">SE - Effective Stress</a></li>
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<li><a href="#sem---effective-mean-stress">SEM - Effective Mean Stress</a></li>
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<li><a href="#sa---stress-anisotropy">SA - Stress Anisotropy</a></li>
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<li><a href="#sfi">SFI</a></li>
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<li><a href="#dsm">DSM</a></li>
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<li><a href="#fos">FOS</a></li>
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<li><a href="#e---strain">E - Strain</a></li>
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<li><a href="#ev---volumetric-strain">EV - Volumetric Strain</a></li>
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<li><a href="#ed---deviatoric-strain">ED - Deviatoric Strain</a></li>
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<li><a href="#element-nodal-on-face">Element Nodal On Face</a></li>
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<li><a href="#sn---stress-component-normal-to-face">SN - Stress component Normal to face</a></li>
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<li><a href="#tph---horizontal-in-plane-shear-component">TPH - Horizontal in-plane shear component</a></li>
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<li><a href="#tnqv---horizontal-in-plane-shear-component">TNQV - Horizontal in-plane shear component</a></li>
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<li><a href="#tp---total-in-plane-shear">TP - Total in-plane shear</a></li>
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<li><a href="#tpinc---direction-of-tp">TPinc - Direction of TP</a></li>
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<li><a href="#faultmob">FAULTMOB</a></li>
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<li><a href="#pcrit">PCRIT</a></li>
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<li><a href="#pinc-and-pazi---face-inclination-and-azimuth">Pinc and Pazi - Face Inclination and Azimuth</a></li>
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<li><a href="#pore-compressibility">Pore Compressibility</a></li>
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<li><a href="#porosity-and-permeability">Porosity and Permeability</a></li>
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<li><a href="#mud-weight-window">Mud Weight Window</a></li>
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<h1 id="derived-results---geomechanical">Derived Results - Geomechanical</h1>
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<p>ResInsight computes several derived results. In this section we will explain what they are, and briefly how they are calculated.</p>
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<h2 id="derived-geomechanical-results">Derived Geomechanical results</h2>
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<p>ResInsight calculates several of the presented geomechanical results based on the native results present in the odb-files.</p>
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<h3 id="relative-results-time-lapse-results">Relative Results (Time Lapse Results)</h3>
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<p>ResInsight can calculate and display relative results, sometimes also referred to as Time Lapse results.
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||
When enabled, every result variable is calculated as:</p>
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<p>$Value_{[t-b]} = Value_{[t]} - Value_{[b]}$</p>
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<p>where:</p>
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<ul>
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<li>$b$ is the base time step,</li>
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<li>$t$ is the current time step</li>
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</ul>
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<p>Select the appropriate <strong>Base Time Step</strong> option in the <strong>Difference Options</strong> group to enable the time lapse result.</p>
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<p><a href="#R-image-82020922b5b5753205f0643d364ee359" class="lightbox-link"><img class="noborder lazy lightbox noshadow figure-image" loading="lazy" src="/images/appendix/DerivedRelativeResults.png" style=" height: auto; width: auto;"></a>
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<a href="javascript:history.back();" class="lightbox-back" id="R-image-82020922b5b5753205f0643d364ee359"><img class="noborder lazy lightbox noshadow lightbox-image" loading="lazy" src="/images/appendix/DerivedRelativeResults.png"></a></p>
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<p>Note: Relative Results calculated based on Gamma values and Stress Anisotropy are calculated slightly differently:</p>
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<p>Gamma:</p>
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<p>$Gamma_{i[t-b]} = \frac{ST_{i[t]} - ST_{i[b]}}{ POR_{[t]} - POR_{[b]} }$</p>
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<p>Stress Anisotropy:</p>
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<p>$SA_{ij[t-b]} = 2 * \frac{(ST_{i[t]} - ST_{i[b]}) - (ST_{j[t]} - ST_{j[b]})}{(ST_{i[t]} - ST_{i[b]}) + (ST_{j[t]} - ST_{j[b]})}$</p>
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<h3 id="derived-result-fields">Derived Result Fields</h3>
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<p>The calculated result fields are:</p>
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<ul>
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<li>Nodal
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<ul>
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<li>COMPACTION (Magnitude of compression)</li>
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</ul>
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</li>
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<li>Element Nodal and Integration Points
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<ul>
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<li>ST (Total Stress)
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<ul>
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<li>All tensor components</li>
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<li>Principals, with directions ($S_iinc, S_iazi$)</li>
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<li>STM (Mean total stress)</li>
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<li>Q (Deviatoric stress)</li>
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</ul>
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</li>
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<li>Gamma (Stress path)</li>
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<li>SE (Effective Stress)
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<ul>
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<li>All tensor components</li>
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<li>Principals, with directions</li>
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<li>SEM (Mean effective stress)</li>
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<li>SFI</li>
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<li>FOS</li>
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<li>DSM</li>
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</ul>
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</li>
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<li>E (Strain)
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<ul>
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<li>All tensor components</li>
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<li>EV (Volumetric strain)</li>
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<li>ED (Deviatoric strain)</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>Element Nodal On Face
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<ul>
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<li>Plane
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<ul>
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<li>Pinc (Face inclination angle)</li>
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<li>Pazi (Face azimuth angle)</li>
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</ul>
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</li>
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<li>Transformed Total and Effective Stress
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<ul>
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<li>SN (Stress component normal to face)</li>
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<li>TP (Total in-plane shear)</li>
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<li>TPinc (Direction of TP)</li>
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<li>TPH ( Horizontal in-plane shear component )</li>
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<li>TPQV ( Quasi vertical in-plane shear component )</li>
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<li>FAULTMOB</li>
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<li>PCRIT</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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<h3 id="definitions-of-derived-results">Definitions of Derived Results</h3>
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<p>In this text the label Sa and Ea will be used to denote the unchanged stress and strain tensor respectively from the odb file.</p>
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<p>Components with one subscript denotes the principal values 1, 2, and 3 which refers to the maximum, middle, and minimum principals respectively.</p>
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<p>Components with two subscripts however, refers to the global directions 1, 2, and 3 which corresponds to X, Y, and Z and thus also easting, northing, and depth.</p>
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<ul>
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<li>Inclination is measured from the downwards direction</li>
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<li>Azimuth is measured from the Northing (Y) Axis in Clockwise direction looking down.</li>
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</ul>
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<h3 id="case-constants">Case Constants</h3>
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<p>Two constants can be assigned to a Geomechanical case:</p>
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<ul>
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<li>Cohesion</li>
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<li>Friction angle</li>
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</ul>
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<p>In the following they are denoted s0 and fa respectively. Some of the derived results use these constants, that can be changed in the property panel of the Case.</p>
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<p><a href="#R-image-7164bbfee99773ae9b527160ae1cbec5" class="lightbox-link"><img class="noborder lazy lightbox noshadow figure-image" loading="lazy" src="/images/appendix/GeoMechCasePropertyPanel.png" style=" height: auto; width: auto;"></a>
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<a href="javascript:history.back();" class="lightbox-back" id="R-image-7164bbfee99773ae9b527160ae1cbec5"><img class="noborder lazy lightbox noshadow lightbox-image" loading="lazy" src="/images/appendix/GeoMechCasePropertyPanel.png"></a></p>
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<h3 id="compaction">COMPACTION</h3>
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<p>Compaction is the difference in vertical displacement (U3) between a grid node and a specified reference K layer.
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The reference K layer is specified in the property editor.</p>
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<p>For each node <i>n</i> in the grid, a node <i>nref</i> in the reference K layer is found by vertical intersection from the node <i>n</i>.</p>
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<p>$ If (Depth_n <= Depth_{nref}) $</p>
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<p>$ \space \space COMPACTION_n = -(U3_n - U3_{nref})$</p>
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<p>$ else $</p>
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<p>$\space \space COMPACTION_n = -(U3_{nref} - U3_n )$</p>
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<h3 id="st---total-stress">ST - Total Stress</h3>
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<p>$ST_{ii} = -Sa_{ii} + POR (i= 1,2,3)$</p>
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<p>We use a value of $POR=0.0$ where it is not defined.</p>
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<p>$ST_{ij} = -Sa_{ij} (i,j = 1,2,3 \text{ and i $\ne$ j})$</p>
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<p>$Sa_{ii}$ and $Sa_{ij}$ are the stresses calculated by Abaqus.</p>
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<p>$ST_i = \text{Principal value i of ST}$</p>
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<h3 id="stm---total-mean-stress">STM - Total Mean Stress</h3>
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<p>$STM = \frac{ST_{11} + ST_{22} + ST_{33}}{3} $</p>
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<h3 id="q---deviatoric-stress">Q - Deviatoric Stress</h3>
|
||
<p>$Q = \sqrt {\frac{3}{2} * ((ST_1 - STM)^2 + (ST_2 - STM)^2 + (ST_3 - STM)^2 }$</p>
|
||
<h3 id="dpn---shear-slip-indicator">DPN - Shear Slip Indicator</h3>
|
||
<p>Excess pore pressure parameter is defined as</p>
|
||
<p>$DPN = \frac{P_p - P_0} { \sigma_v - P_0 }$</p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li>$P_0$ is hydrostatic pore pressure,</li>
|
||
<li>$P_p$ is pore pressure (at the time of the incident) and</li>
|
||
<li>$\sigma_v$ total vertical stress ($ ST_{33} $).</li>
|
||
</ul>
|
||
<p>Hydrostatic pore pressure is</p>
|
||
<p>$ P_0 = \rho_w * TVDMSL * g $</p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li>$\rho_w$ is (average) density of formation water (default = 1.03),</li>
|
||
<li>TVDMSL is true vertical depth mean sea level and</li>
|
||
<li>$g$ is gravity.</li>
|
||
</ul>
|
||
<h3 id="gamma---stress-path">Gamma - Stress Path</h3>
|
||
<p>$Gamma_{ii} = \frac{ST_{ii}} {POR} (i= 1,2,3) $</p>
|
||
<p>$Gamma_{i} = \frac{ST_{i}} {POR} $</p>
|
||
<p>In these calculations we set Gamma to <em>undefined</em> if abs(POR) > 0.01 MPa.</p>
|
||
<h3 id="se---effective-stress">SE - Effective Stress</h3>
|
||
<p>$SE_{ij} = -Sa_{ij} (i,j = 1,2,3 \text{ where POR is defined})$</p>
|
||
<p>where $Sa_{ij}$ is the stress calculated by Abaqus.</p>
|
||
<p>$SE_i = \text{Principal value i of SE} $</p>
|
||
<h3 id="sem---effective-mean-stress">SEM - Effective Mean Stress</h3>
|
||
<p>$SEM = \frac{SE_{11} + SE_{22} + SE_{33}} {3} $</p>
|
||
<h3 id="sa---stress-anisotropy">SA - Stress Anisotropy</h3>
|
||
<p>$SA_{ij} = 2 \frac{ST_{i} - ST_{j}}{ ST_{i} + ST_{j}} (i,j = 1,2,3 \text{ and i $\lt$ j})$</p>
|
||
<p>The same expressions are available for effective stresses (where SE replaces ST in the equation above).</p>
|
||
<h3 id="sfi">SFI</h3>
|
||
<p>$$SFI = \frac{\frac{S0}{tan(fa)} + 0.5 * (SE_1 + SE_3) * sin(fa)} {0.5*(SE_1-SE_3)} $$</p>
|
||
<h3 id="dsm">DSM</h3>
|
||
<p>$DSM = \frac{tan(\rho)} {tan(fa)} $</p>
|
||
<p>where</p>
|
||
<p>$$ \rho = 2 * (arctan (\sqrt \frac{ SE_1 + a} {SE_3 + a}) \space – \frac {\pi} {4}) $$
|
||
$$ a = \frac {s0} {tan(fa)} $$</p>
|
||
<h3 id="fos">FOS</h3>
|
||
<p>$FOS = \frac{1}{DSM}$</p>
|
||
<h3 id="e---strain">E - Strain</h3>
|
||
<p>$E_{ij} = -Ea_{ij}$</p>
|
||
<h3 id="ev---volumetric-strain">EV - Volumetric Strain</h3>
|
||
<p>$EV = E_{11} + E_{22} + E_{33} $</p>
|
||
<h3 id="ed---deviatoric-strain">ED - Deviatoric Strain</h3>
|
||
<p>$ED = 2*\frac {E1-E3} {3} $</p>
|
||
<h3 id="element-nodal-on-face">Element Nodal On Face</h3>
|
||
<p>For each face displayed, (might be an element face or an intersection/intersection box face),
|
||
a coordinate system is established such that:</p>
|
||
<ul>
|
||
<li>Ez is normal to the face, named N - Normal</li>
|
||
<li>Ex is horizontal and in the plane of the face, named H - Horizontal</li>
|
||
<li>Ey is in the plane pointing upwards, named QV - Quasi Vertical</li>
|
||
</ul>
|
||
<p>The stress tensors in that particular face are then transformed to that coordinate system. The following quantities are derived from the transformed tensor named TS in the following:</p>
|
||
<h3 id="sn---stress-component-normal-to-face">SN - Stress component Normal to face</h3>
|
||
<p>$SN = TS_{33}$</p>
|
||
<h3 id="tph---horizontal-in-plane-shear-component">TPH - Horizontal in-plane shear component</h3>
|
||
<p>$TPH = TS_{31} = TS_{ZX} $</p>
|
||
<h3 id="tnqv---horizontal-in-plane-shear-component">TNQV - Horizontal in-plane shear component</h3>
|
||
<p>$TPQV = TS_{32} = TS_{ZY}$</p>
|
||
<h3 id="tp---total-in-plane-shear">TP - Total in-plane shear</h3>
|
||
<p>$TP = \sqrt {(TPH^2 + TPQV^2)} $</p>
|
||
<h3 id="tpinc---direction-of-tp">TPinc - Direction of TP</h3>
|
||
<p>Angle of the total in-plane shear relative to the Quasi Vertical direction</p>
|
||
<p>$TPinc = acos (\frac {TPQV} {TP}) $</p>
|
||
<h3 id="faultmob">FAULTMOB</h3>
|
||
<p>$FAULTMOB = \frac{TP}{tan(frictionAngle) * (TS_{ZZ} + \frac{cohesion}{tan(frictionAngle)} )}$</p>
|
||
<h3 id="pcrit">PCRIT</h3>
|
||
<p>$PCRIT = TS_{ZZ} - \frac{TP}{tan(frictionAngle)} $</p>
|
||
<h3 id="pinc-and-pazi---face-inclination-and-azimuth">Pinc and Pazi - Face Inclination and Azimuth</h3>
|
||
<p>These are the directional angles of the face-normal itself.</p>
|
||
<h3 id="pore-compressibility">Pore Compressibility</h3>
|
||
<h4 id="pore-compressibility-1">Pore Compressibility</h4>
|
||
<p>Pore compressibility between a reference state and the current stress state is defined as:</p>
|
||
<p>$ C_{p} = -\frac{ \alpha \Delta\epsilon_{vol}}{ \Delta P_p \phi_0} + \frac{1}{K_s} ( \frac{ \alpha } { \phi_0 } - 1) $</p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li>$ \alpha $ is the Biot coefficient,</li>
|
||
<li>$ \Delta\epsilon_{vol} $ is volumetric strain change (EV in ResInsight) between curret state and reference state,</li>
|
||
<li>$ \phi_0 $ is porosity on the Geostatic step,</li>
|
||
<li>$ \Delta P_p $ is change in pore pressure between current state and reference state,</li>
|
||
<li>$ K_s $ bulk modulus for the solid material (grain).</li>
|
||
</ul>
|
||
<p>The Biot porelastic coefficient ($\alpha$) defines the compressibility of sand grains: $\alpha = 1.0$ for incompressible grains,
|
||
and $\alpha < 1.0$ for compressible grains. $\alpha$ is not used for the initial (Geostatic) time step. The default value is 1.0, but values
|
||
per element can be imported as an <a href="/import/elementpropertytable/index.html">element property table</a>.</p>
|
||
<p>The bulk modulus for solid material is defined as:</p>
|
||
<p>$ K_s = \frac{ K_{fr} }{ 1 - \alpha}, K_{fr} = \frac{ E }{ 3(1-2\nu)} $</p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li>$ E $ is the elastic modulus (Young’s modulus) from element property table <a href="/import/elementpropertytable/index.html">MODULUS</a>.</li>
|
||
<li>$ \nu $ is Poisson’s ratio imported from element property table <a href="/import/elementpropertytable/index.html">RATIO</a>.</li>
|
||
</ul>
|
||
<h4 id="vertical-compressibility">Vertical Compressibility</h4>
|
||
<p>$ C_{v} = - \frac{ \Delta\epsilon_{\nu}}{ \alpha \Delta P_p } $</p>
|
||
<p>$ \Delta\epsilon_\nu $ is the vertical strain change between current state and reference state (E33 in ResInsight).</p>
|
||
<h4 id="vertical-compressibility-ratio">Vertical Compressibility Ratio</h4>
|
||
<p>$ C_{vr} = \frac{ C_v E(1-\nu) } { (1+\nu) ( 1 - 2\nu) } $</p>
|
||
<p>Vertical Compression Ratio is the ratio between the real vertical compression and the compression in a uniaxial strain case.
|
||
All parameters are described above.</p>
|
||
<h3 id="porosity-and-permeability">Porosity and Permeability</h3>
|
||
<h4 id="porosity">Porosity</h4>
|
||
<p>Porosity change is defined as either total change in porosity between initial (geostatic) state and current state,
|
||
or change in porosity between a reference state and the current state $ \Delta\phi $. The latter is given as</p>
|
||
<p>$ \Delta\phi = \phi_0(C_p \Delta P_p + \Delta\epsilon_{vol}) $</p>
|
||
<p>Here, $\Delta\phi_0$ is found from this equation with the reference state being the initial state (geostatic).
|
||
The current porosity is then given as</p>
|
||
<p>$ \phi = \phi_0 + \Delta\phi_0 $</p>
|
||
<p>with $ \phi_0 $ being the porosity at the initial state, $\Delta\phi_0$ is porosity change between initial (geostatic)
|
||
state current, $C_p$ is pore compressibility (between reference and current state), $\Delta P_p$ is change in pore pressure
|
||
and $\Delta\epsilon_{vol}$ is volumetric strain change.</p>
|
||
<h4 id="initial-porosity">Initial Porosity</h4>
|
||
<p>Porosity at the initial state:</p>
|
||
<p>$ \phi_0 = \frac{VOIDR} {1 + VOIDR} $</p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li>VOIDR is void ratio from Abaqus.</li>
|
||
</ul>
|
||
<h4 id="permeability">Permeability</h4>
|
||
<p>An expression for permeability is taken from Petunin (2011).</p>
|
||
<p>$ k = k_0( \frac{\phi}{\phi_0} )^A $</p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li>$k_0$ is the permeability at the initial state (unit: mD),</li>
|
||
<li>$\phi$ is porosity at current state,</li>
|
||
<li>$\phi_0$ is initial porosity,</li>
|
||
<li>$A$ is a constant</li>
|
||
</ul>
|
||
<h3 id="mud-weight-window">Mud Weight Window</h3>
|
||
<p>Mud Weight Window (MWW) represents the difference between the minimum and the maximum possible
|
||
mud weight between specific formation layers representing top and base of a fictitious well section.</p>
|
||
<p>To find MWW a two step procedure is needed:</p>
|
||
<ul>
|
||
<li>first find the limits per element,</li>
|
||
<li>determine the MWW for each element based on the vertical column.</li>
|
||
</ul>
|
||
<h4 id="finding-upper-and-lower-mud-weight-limit">Finding Upper and Lower Mud Weight Limit</h4>
|
||
<p>The upper mud weight limit (UMWL) and lower mud weight limit (LMWL) is found for each element intersected by the fictitious well.
|
||
The UMWL is either the fracture gradient (FG) or minimum horizontal stress (SHmin) for both sand and shale.
|
||
The LMWL is defined as the maximum of shear fracture gradient (SFG) and/or pore pressure in shale, and as pore pressure in sand.</p>
|
||
<p>The calculations for fracture gradient and shear fracture gradient, and the needed input, are described in detail
|
||
in <a href="/plot-window/wellborestabilityplots/index.html">Well Bore Stability Plots</a>.</p>
|
||
<h4 id="mud-weight-window-1">Mud Weight Window</h4>
|
||
<p>Thereafter, the combined use of mud weight limits for all elements between the top and base (for a given IJ)
|
||
determines the MWW parameters as will be further described below.</p>
|
||
<p>A reference element index ($K_{ref}$) represents the base or the top of the fictitious well. Then for a set
|
||
of elements with i = I, j = J and k = K to $K_{ref}$ the maximum LWML and the minimum UWML must be found from
|
||
these element values. Then the difference between the two defines the MWW parameter.</p>
|
||
<p>Thus for a vertical stack of elements $element_{ijk} = K \to K_{ref}$, $MWW_{ijk}$ is given as</p>
|
||
<p>$ MWW_{ijk} = maximum(LMWL) - minimum(UMWL) (k = K \to K_{ref}) $</p>
|
||
<p>Similar calculations are made below the reference layer, but then with the reference layer as the top layer.</p>
|
||
<p>In addition to the MWW parameter, the mud weight representing the middle of the drilling window (MWM) is calculated if MWW > 0.
|
||
Otherwise, MWM should is undefined.</p>
|
||
|
||
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<div id="R-footer" class="footerFooter showFooter"><a href="https://www.ceetronsolutions.com"><img src='https://resinsight.org//images/CeeSolLogoSmallForDarkBackground.png' style='width: 140px; '></a><br>
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||
ResInsight is developed by <br> <a href="https://www.ceetronsolutions.com">Ceetron Solutions</a>
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<p>in collaboration with</p>
|
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<a href="https://www.equinor.com">Equinor ASA</a>
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