Conditional Stochastic Simulations of Flow and  Transport with Karhunen-Loève Expansions, Stochastic  Collocation, and Sequential Gaussian Simulation

<div>Example from Section <a href="https://static.hindawi.com/articles/jam/volume-2014/652594/figures/#sec4.3">4.3</a>. Far left: permeability field used as a source of data for conditional simulations with KL-SC. Left: location of <svg height="12.4257pt" id="M567" style="vertical-align:-3.427751pt" version="1.1" viewbox="-0.0498162 -8.99795 40.8665 12.4257" width="40.8665pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z" id="g113-79"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="839" vert-adv-y="839"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,10.486,3.264)"><path d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z" id="g50-110"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="805" vert-adv-y="805"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,22.432,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,34.17,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> data points <svg height="10.5763pt" id="M568" style="vertical-align:-0.04990005pt" version="1.1" viewbox="-0.0498162 -10.5264 13.1319 10.5763" width="13.1319pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M494 0V32C450 38 426 53 393 105L296 260C330 305 354 338 375 361C411 396 421 401 476 410V442H293V410C332 402 337 394 323 371C311 351 295 324 274 293L227 371C213 393 218 402 252 410V442H16V410C70 402 80 391 108 344L196 199C166 159 140 123 115 95C81 54 61 43 7 32V0H196V32C156 41 150 53 167 81C182 105 199 138 219 166L269 79C287 49 283 38 245 32V0H494Z" id="g13-118"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="482" vert-adv-y="482"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.693,-5.984)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z" id="g50-43"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g></svg>. Right and far right: two realizations of conditional permeability field for KL-SC.</div>

Journal of Applied Mathematics

fig11

Figure 11

Figure 11: Conditional Stochastic Simulations of Flow and  Transport with Karhunen-Loève Expansions, Stochastic  Collocation, and Sequential Gaussian Simulation 

Figure 11 | Conditional Stochastic Simulations of Flow and  Transport with Karhunen-Loève Expansions, Stochastic  Collocation, and Sequential Gaussian Simulation 