Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media

<div>Distributions of <svg height="10.2124pt" id="M335" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z" id="g113-113"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="593" vert-adv-y="593"></glyph.data></g></svg>, <svg height="6.1673pt" id="M336" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.11985 6.1673" width="7.11985pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 57C137 57 126 69 139 124L195 366C210 431 205 448 182 448C155 448 89 413 23 350L36 326C73 354 103 376 112 376C118 376 118 365 113 340L61 118C54 90 52 68 52 51C52 0 75 -12 98 -12S151 -3 181 17C242 58 305 119 362 193H364L345 104C323 3 339 -12 359 -12C390 -12 464 35 515 96Z" id="g113-118"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="538" vert-adv-y="538"></glyph.data></g></svg>, and <svg height="6.02377pt" id="M337" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -5.81737 5.98461 6.02377" width="5.98461pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z" id="g185-40"></path><glyph.data ascent="989" descent="-303" horiz-adv-x="447" vert-adv-y="447"></glyph.data></g></svg>, <svg height="11.6445pt" id="M338" style="vertical-align:-3.29568pt" version="1.1" viewbox="-0.0498162 -8.34882 37.7378 11.6445" width="37.7378pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z" id="g185-47"></path><glyph.data ascent="989" descent="-303" horiz-adv-x="454" vert-adv-y="454"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,5.902,3.132)"><path d="M384 109C349 76 310 59 268 59C197 59 123 113 120 245C240 248 367 259 384 262C403 266 407 275 407 296C407 376 338 451 254 451H253C201 451 148 425 105 378C65 334 39 272 39 203C39 88 112 -12 235 -12C266 -12 339 5 402 84L384 109ZM230 411C285 411 319 365 319 313C319 298 313 293 294 293C236 291 180 290 125 290C141 370 185 411 230 411Z" id="g58-99"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="434" vert-adv-y="434"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,9.851,3.132)"><path d="M100 439H52L25 408L28 396H100V106C100 41 92 35 31 30V0H250V29C188 37 181 41 181 106V396H372V107C372 40 365 37 309 29V0H537V29C461 37 453 40 453 106V396H564C576 403 579 428 574 439H453V469C452 558 462 604 474 623S500 656 527 656C562 656 589 634 608 615C619 604 628 607 639 615C647 622 658 633 661 641C666 653 664 664 656 674C637 696 609 708 572 710C528 703 483 682 439 638C418 661 380 685 320 688C269 683 230 666 190 633C139 593 122 550 113 525S100 476 100 451V439ZM372 439H181V461C181 530 194 571 211 598C226 621 248 643 287 643C343 643 379 607 400 578C391 561 384 546 381 529C376 510 372 486 372 457V439Z" id="g58-26"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="572" vert-adv-y="572"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.061,0)"><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(.013,0,0,-0.013,31.324,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> m<sup>2</sup> s<sup>−1</sup>, <svg height="15.533pt" id="M339" style="vertical-align:-3.2956pt" version="1.1" viewbox="-0.0498162 -12.2374 70.9129 15.533" width="70.9129pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,8.338,-2.897)"><path d="M-21 591C-45 610 -87 648 -131 700L-155 684L-108 616H-397V566H-108L-155 498L-131 482C-87 534 -45 572 -21 591Z" id="g117-14"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="0" vert-adv-y="0"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="684" vert-adv-y="684"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,8.827,3.132)"><path d="M396 114C349 71 317 59 274 59C216 59 123 104 123 241C123 342 180 400 246 400C283 400 317 386 348 359C355 352 361 348 366 348C378 348 401 371 402 394C402 405 398 413 385 424C369 437 337 451 294 451H293C254 451 193 434 141 394C73 342 39 277 39 198C39 90 114 -12 243 -12C303 -12 371 31 415 90L396 114Z" id="g58-97"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="432" vert-adv-y="432"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,12.758,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z" id="g58-109"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="519" vert-adv-y="519"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,17.372,3.132)"><path d="M532 0V30C473 38 467 38 467 107V298C467 395 416 451 336 451C307 451 279 439 251 420C226 404 205 389 185 374V453C142 434 94 422 43 413V386C98 378 104 373 104 309V107C104 41 100 38 31 30V0H251V30C191 35 185 40 185 107V337C216 364 255 389 295 389C360 389 386 347 386 276V110C386 44 380 35 321 30V0H532Z" id="g58-108"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="557" vert-adv-y="557"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,22.358,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z" id="g58-113"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="376" vert-adv-y="376"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,25.734,3.132)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,32.811,0)"><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(.013,0,0,-0.013,44.074,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,48.572,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,54.812,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,59.955,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,66.253,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> m s<sup>−2</sup>, <svg height="14.7729pt" id="M340" style="vertical-align:-3.181499pt" version="1.1" viewbox="-0.0498162 -11.5914 53.7613 14.7729" width="53.7613pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z" id="g113-232"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,14.844,0)"><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(.013,0,0,-0.013,26.107,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,32.347,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,38.587,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,44.147,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,48.579,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg> m<sup>2</sup>, and <svg height="12.4698pt" id="M341" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 49.1343 12.4698" width="49.1343pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z" id="g113-232"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.212,0)"><path d="M368 703H309L44 -163H104L368 703Z" id="g113-48"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="412" vert-adv-y="412"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.574,0)"><path d="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z" id="g113-232"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,22.84,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,31.418,0)"><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(.013,0,0,-0.013,42.681,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg>.</div>

Geofluids

fig3

Figure 3

Figure 3: Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media 