Time-Lapse Marine CSEM Modeling in the Framework of Linear and Non-Linear Approximations
Authors: Bayat Maryam , Ghanati Reza , Lokmer Ivan 
Published: February 2023

&lt;p&gt;&lt;span id=&quot;page3R_mcid16&quot; class=&quot;markedContent&quot;&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;C&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;ontrol&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;led&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;-&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;source electromagnetic (CSEM) method&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;is widely used for submarine hydrocarbon reservoir &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;modeling.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;C&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;ompared to seismic reflection measurements,&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;this method is less accurate, but it&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;has reduced &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;processing&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;costs&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;The partial differential equation (PDE) and in&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;tegral equation (IE) methods are&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;usually &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;used for&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;numerical&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;forward modeling&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;of electromagnetic (EM)&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;fields&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;in the target area&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;This research has &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;utilized the&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;Fredholm integral equation of second order&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;for&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;the&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;forward modeling of a hydrocarbon &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;reservoir&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;F&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;ull&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;integral&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;equations&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;require&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;high&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;-&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;performance&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;computers&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;to &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;be solved&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;which&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;causes &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;unaffordable computational costs&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;when it comes to real data simulations.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;To avoid this problem, we use &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;effe&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;ctive &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;approximations&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;for solving IEs, thus&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;minimizing&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;the&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;costs and&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;increasing the speed of solving &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;electromagnetic problems. The&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;main&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;scope of this study is to&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;use&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;T&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;-&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;matrix approximation (TMA), Extended &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;Born approximation (EBA), and Born approximation (BA)&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;methods to solve the integral equation for 4D &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;(3 spatial&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;and 1 temporal dimensions)&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;forward&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;modeling&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;of a regular geometrical shape reservoir using &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;CSEM synthetic data.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;In hydrocarbon extractions, fluids are injected to increase pressure inside the &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;reservoirs. In this study, the lateral extent of injected fluid&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;is modeled to monitor small changes in electrical &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;conductivity,&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;while&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;the&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;conductivity&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;of&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;the&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;injected&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;fluid&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;and&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;hydrocarbon&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;remained&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;fixed.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;The &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;approximation results are compared with each other and&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;then&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;for their performance verification&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;inverse &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;modelin&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;g of the proposed methods is&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;implemented and then tested in MATLAB&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;. The &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;results&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;illustrate&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;that &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;the T&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;-&lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;Matrix approximation&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;is more accurate&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;and&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;has&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;a wider electrical conductivity application range&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;in &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;comparison with EBA and BA.&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;