Commit 1c154f4eaeac404e737204e4fb04edb8af1ff438

analytical solution consideration added to Harjoitustyo
Harjoitustyo/JY1_Harjoitustyo.lyx
(307 / 12)
  
309309\begin_layout Standard
310310Darcyn laki on muotoa
311311\begin_inset Formula \begin{equation}
312\phi v=-\frac{\kappa}{\mu}\nabla p\,,\label{eq:Darcy}\end{equation}
312v_{s}:=\phi v=-\frac{\kappa}{\mu}\nabla p\,,\label{eq:Darcy}\end{equation}
313313
314314\end_inset
315315
341341\begin_inset Formula $v$
342342\end_inset
343343
344 nesteen virtausnopeus ja
344 nesteen virtausnopeus (ilman väliainetta) ja
345345\begin_inset Formula $p$
346346\end_inset
347347
348348 paine.
349
350\begin_inset Formula $v_{s}$
351\end_inset
352
353:llä merkitään nk.
354 efektiivistä nopeutta.
349355 Jotta malli olisi ratkaistavissa tarvitaan jokin muu ehto muuttujille
350356\begin_inset Formula $v$
351357\end_inset
382382 ulottuvuudessa.
383383 Tällöin saadaan
384384\begin_inset Formula \begin{gather}
385\phi v=-\frac{\kappa}{\mu}p_{,x}\,,\nonumber \\
385v_{s}=\phi v=-\frac{\kappa}{\mu}p_{,x}\,,\nonumber \\
386386\rho_{,t}+\rho_{,x}v+\rho v_{,x}\,.\label{eq:1dModel}\end{gather}
387387
388388\end_inset
431431\begin_inset Formula $v$
432432\end_inset
433433
434:n lauseke massan säilymislain yhtälöön.
434:n lauseke massan säilymislain yhtälöön mallissa
435\begin_inset CommandInset ref
436LatexCommand ref
437reference "eq:1dModel"
438
439\end_inset
440
441.
435442 Oletetaan, että huokoinen aine ei ole täysin tiivis, jolloin
436443\begin_inset Formula $\phi>0$
437444\end_inset
449449\end_inset
450450
451451:llä voidaan jakaa.
452 Lisäksi oletettiin, että viskositeetti
452 Lisäksi oletetaan, että viskositeetti
453453\begin_inset Formula $\mu$
454454\end_inset
455455
456 on vakio.
456 ja permeabiliteetti
457\begin_inset Formula $\kappa$
458\end_inset
459
460 (väliaineelle ominainen vastus) ovat vakioita.
457461 Saadaan
462\begin_inset Formula \[
463\rho_{,t}+\rho_{,x}\left(-\frac{\kappa}{\mu}\frac{1}{\phi}p_{,x}\right)+\rho\left[(-\frac{\kappa}{\mu}\frac{1}{\phi}p_{,x})_{,x}\right]=0\,,\]
464
465\end_inset
466
467mistä edelleen
458468\begin_inset Formula \begin{equation}
459\rho_{,t}-\frac{1}{\mu}\frac{\kappa}{\phi}\rho_{,x}p_{,x}-\frac{\rho}{\mu}\left[(\frac{\kappa}{\phi})_{,x}p_{,x}+\frac{\kappa}{\phi}p_{,xx}\right]=0\,.\label{eq:1dTogether}\end{equation}
469\rho_{,t}-\frac{\kappa}{\mu}\frac{1}{\phi}\rho_{,x}p_{,x}-\rho\frac{\kappa}{\mu}\left[-\frac{1}{\phi^{2}}\phi_{,x}p_{,x}+\frac{1}{\phi}p_{,xx}\right]=0\,.\label{eq:1dTogether}\end{equation}
460470
461471\end_inset
462472
473Kertomalla puolittain funktiolla
474\begin_inset Formula $\phi$
475\end_inset
476
477 ja vakiolla
478\begin_inset Formula $\frac{\mu}{\kappa}$
479\end_inset
480
481 saadaan lopulta
482\begin_inset Formula \begin{equation}
483\frac{\mu}{\kappa}\phi^{2}\rho_{,t}-\phi\rho_{,x}p_{,x}+\phi_{,x}\rho p_{,x}-\phi\rho p_{,xx}=0\,.\label{eq:1dTogether2}\end{equation}
484
485\end_inset
486
463487Lisätään nyt yhtälöön
464488\begin_inset CommandInset ref
465489LatexCommand eqref
466reference "eq:1dTogether"
490reference "eq:1dTogether2"
467491
468492\end_inset
469493
501501.
502502 Tällöin
503503\begin_inset Formula \[
504\beta\rho p_{,t}-\frac{1}{\mu}\frac{\kappa}{\phi}\beta\rho(p_{,x})^{2}-\frac{\rho}{\mu}\left[(\frac{\kappa}{\phi})_{,x}p_{,x}+\frac{\kappa}{\phi}p_{,xx}\right]=0\,.\]
504\frac{\mu}{\kappa}\beta\phi^{2}\rho p_{,t}-\beta\rho\phi(p_{,x})^{2}+\phi_{,x}\rho p_{,x}-\phi\rho p_{,xx}=0\,.\]
505505
506506\end_inset
507507
508Yhtäpitävästi
509\begin_inset Formula \[
510p_{,t}-\frac{1}{\mu}\frac{\kappa}{\phi}(p_{,x})^{2}-\frac{1}{\beta\mu}\left[(\frac{\kappa}{\phi})_{,x}p_{,x}+\frac{\kappa}{\phi}p_{,xx}\right]=0\,.\]
508Yhtäpitävästi (jakamalla
509\begin_inset Formula $\rho$
510\end_inset
511511
512:lla)
513\begin_inset Formula \begin{equation}
514\frac{\mu\beta}{\kappa}\phi^{2}p_{,t}-\beta\phi(p_{,x})^{2}+\phi_{,x}p_{,x}-\phi p_{,xx}=0\,.\label{eq:1dFinalForm}\end{equation}
515
512516\end_inset
513517
514518Saimme toisen kertaluvun epälineaarisen differentiaaliyhtälön, missä parametri
535535Tutki ja esitä vaihtoehtoisia ratkaisutapoja.
536536
537537\end_layout
538
539\end_inset
540
541
542\end_layout
543
544\begin_layout Standard
545Tutkitaan yhtälöä
546\begin_inset CommandInset ref
547LatexCommand eqref
548reference "eq:1dFinalForm"
549
550\end_inset
551
552.
553 Etsitään aluksi ratkaisua paineelle aaltorintamamuotoisena
554\begin_inset Formula $p(x,t)=f(x+ct)$
555\end_inset
556
557, missä
558\begin_inset Formula $c$
559\end_inset
560
561 on vakio.
562 Merkitään muuttujaa
563\begin_inset Formula $s=x+ct$
564\end_inset
565
566.
567 Tällöin
568\begin_inset Formula \[
569p_{t}=f'(s)\cdot\frac{\text{d}s}{\text{d}t}=cf'\,,\quad p_{x}=f'\,,\quad p_{xx}=f''\,.\]
570
571\end_inset
572
573Sijoitetaan nämä yhtälööön
574\begin_inset CommandInset ref
575LatexCommand eqref
576reference "eq:1dFinalForm"
577
578\end_inset
579
580:
581\begin_inset Formula \[
582\frac{c\mu\beta}{\kappa}\phi^{2}f'-\beta\phi(f')^{2}+\phi_{,x}f'-\phi f''=0\,.\]
583
584\end_inset
585
586Koska yhtälössä esiintyy
587\begin_inset Formula $f$
588\end_inset
589
590:n ensimmäistä ja toista derivaattaa, tehdään sijoitus
591\begin_inset Formula $h(s)=f'(s)$
592\end_inset
593
594, jolloin saadaan ensimmäisen kertaluvun epälineaarinen differentiaaliyhtälö
595
596\begin_inset Formula $h:$
597\end_inset
598
599lle (joka vastaa painegradienttia)
600\begin_inset Formula \begin{equation}
601\frac{c\mu\beta}{\kappa}\phi^{2}h-\beta\phi h^{2}+\phi_{,x}h-\phi h'=0\,.\label{eq:eqToAnalyse}\end{equation}
602
603\end_inset
604
605
606\end_layout
607
608\begin_layout Standard
609Tarkastellaan aluksi tilannetta, jossa
610\begin_inset Formula $p_{,x}=f'=h$
611\end_inset
612
613 on pieni.
614 Tällöin termi
615\begin_inset Formula $h^{2}\approx0$
616\end_inset
617
618, ja analysoitava yhtälö on muotoa
619\begin_inset Formula \begin{equation}
620\frac{c\mu\beta}{\kappa}\phi^{2}h+\phi_{,x}h-\phi h'=0\,.\label{eq:Analysis1}\end{equation}
621
622\end_inset
623
624Jos lisäksi oletamme, että
625\begin_inset Formula $c\mu\beta/\kappa$
626\end_inset
627
628 on pieni, saadaan nollannen kertaluvun approksimaatioksi
629\begin_inset Formula \[
630\phi_{,x}h-\phi h'=0\,.\]
631
632\end_inset
633
634Tästä
635\begin_inset Formula \[
636\frac{\phi_{,x}}{\phi}=\frac{h'}{h}=\alpha\]
637
638\end_inset
639
640missä
641\begin_inset Formula $\alpha$
642\end_inset
643
644 on jokin mielivaltainen vakio.
645 Tästä ratkaisuna saadaan
646\begin_inset Formula \[
647\phi(x,t)=C_{1}(t)e^{\alpha x}\,,\quad h(s)=C_{2}e^{\alpha s}=C_{2}e^{\alpha(x+ct)}\,.\]
648
649\end_inset
650
651
652\end_layout
653
654\begin_layout Standard
655Tarkastellaan vielä yhtälöä
656\begin_inset CommandInset ref
657LatexCommand eqref
658reference "eq:Analysis1"
659
660\end_inset
661
662 ja tilannetta, jossa korroosiota ei tapahdu eli
663\begin_inset Formula $\phi$
664\end_inset
665
666 on vakio ja siten
667\begin_inset Formula $\phi_{,x}=0$
668\end_inset
669
670.
671 Saadaan
672\begin_inset Formula \[
673\frac{c\mu\beta\phi}{\kappa}h-h'=0\,.\]
674
675\end_inset
676
677Merkitään
678\begin_inset Formula $a=c\mu\beta\phi/\kappa$
679\end_inset
680
681.
682 Tällöin ratkaisu on
683\begin_inset Formula \[
684h(s)=C_{3}e^{as}=C_{3}e^{a(x+ct)}\,.\]
685
686\end_inset
687
688
689\end_layout
690
691\begin_layout Standard
692Tarkastellaan sitten alkuperäistä yhtälöä
693\begin_inset CommandInset ref
694LatexCommand eqref
695reference "eq:eqToAnalyse"
696
697\end_inset
698
699 ilman oletusta
700\begin_inset Formula $h$
701\end_inset
702
703 on pieni.
704 Tarkastellaan tilannetta, jossa aallon etenemisnopeus
705\begin_inset Formula $c$
706\end_inset
707
708 on hyvin pieni ja porositeetin muutos
709\begin_inset Formula $\phi_{,x}$
710\end_inset
711
712 on hyvin pieni.
713 Tällöin
714\begin_inset CommandInset ref
715LatexCommand eqref
716reference "eq:eqToAnalyse"
717
718\end_inset
719
720 saa muodon
721\begin_inset Formula \[
722\beta h^{2}+h'=0\,.\]
723
724\end_inset
725
726Tämän ratkaisu on
727\begin_inset Formula \[
728h(s)=\frac{1}{\beta(s+C_{4})}\,.\]
729
730\end_inset
731
732Tästä voidaan ratkaista myös
733\begin_inset Formula $f$
734\end_inset
735
736.
737 Saadaan
738\begin_inset Formula \[
739f(s)=\frac{1}{\beta}\ln(s+C_{4})+C_{5}\,.\]
740
741\end_inset
742
743Eli
744\begin_inset Formula \[
745p(x,t)=\frac{1}{\beta}\ln(x+ct+C_{4})+C_{5}\,.\]
746
747\end_inset
748
749
750\end_layout
751
752\begin_layout Standard
753Lisätään edelliseen tapaukseen verrattuna oletus, että
754\begin_inset Formula $\phi_{,x}$
755\end_inset
756
757 on merkittävä.
758 Tällöin tutkittava yhtälö on
759\begin_inset Formula \[
760\phi h'+\beta\phi h^{2}-\phi_{,x}h=0\,.\]
761
762\end_inset
763
764Tämä on funktiokertoiminen Bernoullin yhtälö.
765 Tehdään siis perinteinen sijoitus
766\begin_inset Formula $h=1/w$
767\end_inset
768
769, jolloin
770\begin_inset Formula $h'=-w'/w^{2}$
771\end_inset
772
773, ja yhtälö tulee sijoituksen ja
774\begin_inset Formula $-w^{2}$
775\end_inset
776
777:lla kertomisen jälkeen muotoon
778\begin_inset Formula \[
779\phi w'-\beta\phi+\phi_{,x}w=0\,.\]
780
781\end_inset
782
783Yhtäpitävästi
784\begin_inset Formula \[
785w'+\frac{\phi_{,x}}{\phi}w=\beta\,.\]
786
787\end_inset
788
789Tai
790\begin_inset Formula \[
791\frac{\phi_{,x}}{\phi}=\frac{\beta-w'}{w}=\gamma\,.\]
538792
539793\end_inset
540794
Harjoitustyo/JY1_Harjoitustyo.pdf
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