. . .- -. ✌✎ ✍✎✎ ●✍ ‘ - ...+~q.o ......
.
The Vaporization-Condensation Phenomenon in a Linear Heat Wave
CHIEH CHU SINCLAJR RESEARCH, INC. MEMBER AIME TULSA, OKIJL Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021
ABSTRACT I Martin et al. 3 suggested char the vaporization- condensation phenomenon is one of the main mech- A.theoretical investigatioebas beerrmade oftbe imisms of rhe heat-wave process, along with thermal 4 forward combustion process using a one.dimensional expansion and viscosity reduction. Wilson et al. linear raathematical model, taking into consideration reported the existence of a steam plateau peveral the effect of the vapon’zation-condensate’on which inches in lengrh in their small-scale tube-run occurs on the leading edge of the heat wave. This experiments. However, this important phenomenon rwrk involves the solution of five coupled partial- has never been taken into consideration in the S-11 ~ dif fererztial equations, Beside .s the vapor~zatiorz- nurnerovs theoretical an alyse e by various authors. corzaknsation phenomenon, these equations account The purpose of this work was to study the thermal for conduction, convection, combustion, heat loss, aspects of a linear heat wave, taking into con- dif~usion and bulk fluid flow, sideration the vapori zarion- condensation on the For the one-dimensional linear model studied.. ‘the leading edge of che wave, to determine the effect vaporizat ion-condensation phenomenon does no t of this phenomenon on the temperature profile of induce appreciable change in the. temperature at the reservoir, and CO investigate how this effect the combustion front; and its primary effect is to varies when other process variables are changed. create a steam plateau and to increase the length of the heated zone ahead of the comb~” ~ front, TIiEORY This effect becomes more pronounce. at lower We consider a reservoir of porous medium of pressures, higher porositieq or reduced gas satura- Cross-sectional area A,. extending from x = O to tions. T+e peak temperature and the temperature x = L This reservoir contains, aside from the solid .’ pro file on the leading)edge o/ the beat wave stabilize matrix itseIf, a gas phase and a “combined liquid after a ,certain period. The length oj tb e steam phase” which is a combination of two immiscible bank iemains practically constakt, although the liquid phases — namely, an oil phase and a water length of the water bank increases as the heat phqse. The oil present in the reservoir is ,assumed watd advances, to consist of three fractions, a honcondensable gas, a, nondistillable residuum, and a vaporizable oil INTRODUCTION ‘ fraction which may be present in both the gaa phase and Iiquid phase. Before che heat-wave The existence of die vaporization-condensation process begins, preheating has taken place and has phenomenon in the Heat-wave prnce ss and the imparted an initial temperature distribution TO(X) “important role pI Lyed by the phenomenon have to the reservoir. At the atert of the process, a been recognized by several investigators. Kuhn stream of oxygen-bearing gas is. introduced through and Kochl stated that steam plateaus were frequently the face at x = 0. This gas supports the combustion observed oti the temperature records of their experi- “of the residual fueI and supplies the heat through- ments. The steam plateaus were attributed primarily [- out ‘the process. to $he vaporization and subsequent condensation of In ofder.to develop attractable system of equations, the intetstitiaI water’ existing in the oil ‘sand. che model properties are postulated as follows: Szasz 2 suggested that both lijhter hydrocarbons (1) the system is isbtropic and homogeneous: (2) and water, are _vaporiz~d on the leading edge of the :. there is- a uniform fuel.. content throughout the ~ ~-heat wave, carried forward-in the gas stream, ‘and reserv6fi; (3)” combustion taliea~placg ‘“inja”moving- 1 then condensed to create banks of oil and water. vertical planer region of infinitesimal thickness; , Original manummlutreceived fn society of Petroleum Engineers (4) the temperatures of the fluids arid’ the solid I office Juty 11, 1963. Revised manuscript received Feb. 19, matrix are identical; (5). radiation is negligible; / ]g64,, paper presented at the SPE Annual Fell Meeting, heM, 1 , in New Orleana,.Oct. 6.9,. 1963* i (6) the gas and liquid saturations are uniform lRefercnoea siven at end of #aPer. throughout the. reservoir and temain ‘unchanged ,. .-— I (’ (-.”- .$ UNE,.++64 ‘... . . --...!! . . . /,, SE. .. ..- ,---- .“. - ... -...... 1..- :-.. .;.,... ‘. ..- . ..., .. -’.” ...... ’.... .’.:’...... ,- ,“-” - ,. .. ., ~ ~.- ;“”. ... ““:._ .:_”;..: :?;:”.: ‘:...,--- ::;-: ,.; .:,,.,;’ ’.-:””::;,.,”:’. “’”,: :“;”, ,::,””:: .;:;:;;::. . . . .- . . . . . #. ...,._
K throughout the. process; (7) pressure drop through . & ~~go the system is negligible; (8) volumetric rate of - k, CLO) gas flow is constant; (9) bulk fluid flow of the liquid phase is negligibl% (10) diffusion in the In this equation, the factor (1- VMW C Lw) waa in- liquid phase is negligible; and (11) water of cluded to account for the volume occupied by water combustion ia negligible. in the liquid phase. i3y defining In describing the forward’ combustion proce SS, five partial-differential equations are required. These are two equations of continuity for the s~ e=-— vapori zabie oil fraction (one for the gas phase and Sg another for. the liquid phase ), two similar equations of continuity for water and an equation of energy. we have No equation of motion was included in this work. Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 EQUATION OF CONTINUITY — GAS A [cL~I - VM. CJ].L CZoCLO + b. Cgo PHASE, VAPORIZABLE OIL FRACTION “ea
., . ...0 ...... (2)
The initial condition is
,, EQUATION OF CONTINUITY — By defining GAS, PHASE, WATER 9’ “A+sg a’c L+2C qidc Sg * = t+&-&-”- --&
Kmoke ao = $ s. +%(%-%;
In thjs equation,’ the driving force for vaporization and condensation does not depend on the iiquid phase concentration of water. By defining we have . . ‘, 1,
●
acgo 6’Zcgo ac go { — = Do’>~~-v — at ax K mw Pv “ Cw = _$SgRT + a. CLO + b. Cgo ...... (1) we have The initial and boundary conditions are 32CW acgw dcgw + bwcgw + CW i,c. , @t =() —=DW— — Cgo = Cgoi (4 (3 &2 ‘v & .. B,C.1’ @x=O c~o = Cgol ...... -. .. (3) ‘ /, digo The initial and boundary conditions are: B.C.2 ex=i -= o ax I.C., @t = O Cgw =’C&j(~) .
EQUATION OF CONTINUITY — Ll@JI12 .. B.c.l @x=o c gzu= Cgwl ‘ PHASE, VAPORIZABLE’ OIL FRACTION ‘ acgw, =o . : B.C.2 @x = L “b ‘ -z-’ t S6C!ETV .QF PE.TR12&EU~ .ENCXNSERS” JOURNAL. . . .:. .. :,.”,””! -“”” ,, . ..:., ..—...... -...” . L ,.. .: .,.- - . ..--,.. . -: -4--- ...... —.
EQUATION OF CONTINUITY — where LIQUID PHASE, WATER e(l - VJ, UJCLU) ) m,n SLY=“LW K mw Pm = T b- 3 d~-vMwcLwm,n+l) - a. *
or and
dcLw 60 At —Ccbwcgw +cw ...... *.. (4) Qm = e dt e(l-Vmw CLw ~,n+l’ -ao ‘f
The initial ~onditioni~s . The initial condition is Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021
Lc. f?t=o c LW = CLfUi(x) LC. c = cLwi l~m~M Lw m, 1 EQUATION OF ENERGY Similarly, Eqi 1 gives ~2 T p. c, aT VgPg Cg ~ —— —. c -c gom,n+ ~ gem,n k at= Z- k dx ,1 ( ). At c -2 Cgo + ego = ‘“m+l,n+l m,n-tl m- l,n+ 1 - ‘Pp~}(’-x+’)T++’Do ‘“ (1X)2 ...... ’ . (5) ( ,)
where c - Cgo ‘“m+l,n+l m- 1,17+i -v ‘t UOCLO ( 2AX ) m,n+l ...
The initial and boundary conditions are Substitution of the value of CLOm, n+ ~ from Eq. 2A into the above equation, followed by rearranging, Lc. @t=o T= Ti (%) gives , ,. 1 ,- B.C.1 “@x=O T=TI -A C i mo gom+~,n+l+B~cgom~+~-&o Cgom- ‘,”+ ~ =
&T .“O — B.C.2 @x=L F ~o ...... * *.., .“. .(lA) ax ,’ Eqs. 1 and 2 can be solved numerically by the where implicit finite-difference method. Eq. 2 is trans- — formed to ~ ~=+-. D At —.....—v At (k) 2’3.% .
‘(l-VMWCLW~,n+)CL.m,”+i’( l-VMWcL~,n) cLOm,n -. e J .At Do& [’ 1 B = l- boLQ+2-- aoQm At ..-— mo — (&c) 2 — .. -.. ..,:...... ?. ao.cb*,n+l + !0 Cgom,n+l -- -. .- ...... -. .: ,. !... Do <3t ‘ vAt ~ E .— +— This gives ‘o (AX)2 2AX ~ CLO r ‘% cLo; ,n + Qm Cgo -::’,-: . (2A) .m, n+l m,n+ 1 “and -- .r . ,-: ...:.: ... . =87 F =C ~om “+ aOPm At Chjm, n The initial condition is #to , The initial and boundary conditions are I*C. CLwm, ~= CLrui (~) l ...... +.. .**O .* **.. (5A) For the numerical ,solution of Eqs. 3 and 4, early att~mpts to apply the implicit finite-difference method proved to be unsuccessful. Since the rate of vaporization of water does not “depend upon the The parameters in this equation ate defined as follows: liquid-phase concentration, in certain cases where I Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 the prevailing temperature is high enough to induce Am= l-a a very high rate of vaporization, the liquid-phase , , Bm=2+y +71 concentration becomes negative. This is physically I E =l+a impossible, but the mathematical equations cannot, m .— recognize ‘the absurdity. The remedy of this situation calls for the application of corrections as soon as Fm=— “AX)2 ~. [C Lom,n,l (l-VM~ CLwm,n+l) the absurdity is found. The implicit finite-difference k.At I method, when applied, gives vslues of Cgom, closely tied together with the corresponding v&~~ - CLom,n (l-VMw CL~m,n )1 + ~&Lwmn+l .— of Cgom-l “7 , ‘d Cgom +l,n + 1“ This makes it --. difficult td apply corrections where nece sssty. Based on this considera;rion, the method of charac- (m+ ~)Ax teristics 12 was used.. “Lwm,n) dsL + y ‘m,n + ~ s III applying the niethod of characteristics to Eq. I (m- ~)lb 3, rhe second-order diffusion term was neglected inasmuch as this term exerts only a very small effect on the re suiting concentration pfofile whereas its inclusion in the e quation entails, ,a large-amount ., of iteration. Eq. 3 is therefore simplified to where acgW acgw vgPgcgAx —= bwCgw+cw a= dt ‘v dx I 2k This gives VgPgNO g Ax = Cgwm-vu + At (Bw Cgw + ‘cw)R ‘P= ,k T Ckwm,ni-l Ax’* ...... 0 (3A) P. C. (+)2 ‘Y= where R denotes the average condition along the k At characteristic base curve connecting the points At - —, n) on the x, t plane. The b ‘p (Ax)* ‘m’n+l) ‘d ‘m ‘Ax q = initial and boundsty conditions are k,. and B.C. = cgw~ Cgw I..n I ‘With the omission of the second-order term from Eq. 3, the boundary condition B,.C. 2 was discarded to The initial and boundary conditions are...... — .._& avoid redundance o“frestraining’ conditions’.” Eq. ”4 ...... I ~;’-, ““““4 ~m, ~ can be. approximated -by =T~(%’Z). 1~m ~hf B.C. 1 Tl, n = T1 l Ecw. 1A and 5A can be solved by a Gaussian are listed in Tables 3 and 4, respectively. Control elimination procedure described by B“ruce, et al. 13 runs were made in which the vaporization= concfensation phenomenon was neglected. PROCEDURE Additional computer tuns were made in which some ~ other process variables were varied. These variables A computer program was written and executed on are the thermal conductivity and fuel content of the a digital computer. The program consists of iive parts: (1) calculation of the temperature profile; ‘ (2) calculation of the concentration profiles of TABLE 1 — PROCESS VARIABLES water in the gas and liquid phases; (3] calculation Ru’n of the concentration profiles of the vaporizable oil Number p% r# Sg Btu/hr?u It-” F Remarks fraction in the gas and liquid phases; (4) adjustment —— —— of the concentrations: and (5) integration of the 1 400,0 0,3 0.3 c Basic canditbrs heat content of the system. 2 40.0 od 0,3 0 Change in prossure. 3 400.0 0.2 O*3 o Changs In porosity Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 In the system of equations describing the process, 4 400.0 0.3 O*2 o Chenge In gas saturation the three-phase fluid flow aspect of the problem was 5 400,0 O*3 0,3 0.02 Heat loss included not taken into consideration, This omission occa- sionally caused a local accumulation of gas and ,, liquid beyond the. amount that can be physically TABLE 2- PROCESS PARAMETERS contained in the system. In order to circumvent this difficulty of excessive local accumulation without considerable y complicating the rreatment of ‘e 0.24 Btu/lb-OF the problem, the following numerical procedure Do 0,0068 sq ft/hr 0,369 lb/sq ft.hr was used. Vgfg 5500 Btu/lb 02 The sum of the gas phase concentrations of water {mW(T) 1713+ 6.96T hr-l and the vaporizable oil fraction at one point was KmO(T) 2253 + 6.33T hr_1 compared with the maximum allowable concentration k 0.8 Btu/hr.ft-OF at that point corresponding to the prevailing pressure and temperature. Any surplus was transferred to the w next spatial increment. Similar treatment was made Pvual ‘ Exp(15.47:8600/~) pd. for any autplus in the [iquid-phase concentrations ‘Mw 0.?89 a ft-lb.mele beyond a certain preassigned maximum. this way, 1,25 lb/cu ff In : 14830 Btu/lb-mole the original postulate about the negligible bulk A: 16300 Btu/lb.male liquid flow was discarded in favor’ of a better p=.= 30 Btu/cu ft “F representation of the actual situation. It was, how- cd 003 lb fuel/lb 02 ever, here postulated chat the heat content of the, gas or Iiqui& thus transferred was not high enough TABLE 3 — BOUNDARY CONDITIONS to change the temperature of the next spatial increment. This procedure wd’s included in Pat Variable selected Values 4 of the. program. T, O°F’ “ e’ In Part 5 of the program, the increased he~t 0 lb. mole/cu ft %wf content during a certain time interval was checked c O lb.mole/cu It with the heat generation, The discrepancy between do1 the two was found to be in the neighborhood of 1 per cent. TABLE 4 - INITIAL CONDITIONS ‘ The resuIts of five computer runs are reported D;stance (ft) Temp. ~F) _ Cwwentratimrs_ (-lb.mele/cu_\It here, for the ,cases where the vaporization- x condensation phenomenon was ,included. In these 0, runs, a” slightly simplified scheme was’ used in .0-10 ,0 ‘o 0.00 ‘which the factor ( l-VJ~ ~ CL J in Eq. Z>was replaced. 12 190 0 0 “0. by unity. Comparison of the results between, the ,14’”” 380.0 o 0“ o more”’rigorous scheme represented by thk unmodified $ 16 S30 o 00 0 18 640 0 00 0 Eq. 2 and this slightIy simplified scheme showed 20 700 0 00 0’ no appreciable difference m the temperature pro- ’22 570 0 0,0”0 files. The process variabIes which. were varied for 24 490 0.038 1.5 0,0008 0.02 the various runs are lis~ed in Table 1. Orher 26 410 0,039 1,8 0,0042. 0,18 . .’–. . process parameters ate””tabd-ated_ in TabIe 2. The “’28” ““ 180 - 0,052,:- 1.8 0,0~57. -.oi,8’ –;:._.., mass transfer coefficients K ~w and K no were 30 50 - 0,033 108 0,0036 ‘ 0,18. ,“ estimated: by ine’ans of the la-factor qpproach, 14 32 10 0.00008 1,8 0.00001 0, IB modified by a factor representing the estimated 34-52 0 0,00008 1.8 0,000004 0.18 ~ 54 0 0,00008 1.2,- 0.000004 0.18 : ., ,gas-liquid interracial tires.per uqit volume ‘of the .~ ., system:’ The boundary and initia~- conditions used” 0, ,, 0JJOO08 1,0 0.000004-0.16.., 58.120 ,0 0,00008 1.0 !“-0;000004 0,10 : “-“ . . J ,. ;..,. .-. . .-, . . . JUNE, ”1964 ..’ . .. . . ‘::.: ‘.; .s ? ., i-” .’ -- -. .. —-- .- — .- .— -. ., —-. . ., reservoir, and the mass velocity and oxYgen con- no valid temperature distribution data from actual centration of the injected gas. The~nclusion of the field operations have been published, Second, the vaporization-condensation phenomenon did not major portion of the low gradient region seems to alter the general trend of the effects of these be at a temperature lower than the boiling point of variables on the temperature profiles. Computer water under the prevailing total pressure. Examina- runs were also made in which the vaporization- tion of the observed temperature profiles reported condensation of the vaporizable oil fraction alone by .Wilson et u1.4 revealed similar results. These was included. The results showed that the two phenomena can both be adequately explained by hydrocarbon-phase change had much less influence the partial pressure effect of the gaseous components, on the temperature distribution than the aqueous- In the typical case depicted in Fig. 1, the steam phase change. The results of these computer runs bank extends ro only 9 ft, whereas the water bank ate not reported in detail here. covers a distance of 60 ft. hiost of rhe water in the water bank is at the same temperature as the reser- voir itself. RESULTS AND DISCUSSION Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 INTERRELATION OF TEMPERATURE EFFECT OF THE VAPORIZATION. AND CONCENTRATION PROFILES CONDENSATE ON PHENOMENON TWO temperature profiles are presented in Fig. 2, Fig. 1 shows a typical temperature profile and The dotted line represents a hypothetical case where typical concentration profiles of water and the the effect of the vaporization-condensation phenom- vaporizable oil fraction in the gas and liquid phases. enon is negligible: The soIid line was obtained In the temperature profile shown in this figure, using the same set of. reservoir and there is a region of low temperature gradient on the operating conditions, taking into consideration the additional leading edge of the heat wave. Precisely in this effects ofvapokization and condensation. Comparison region of low temperature gradient, the steam of theyc t~c muves shows chat the vaporization- concentration in the gas phase jump’ti from zero to condensation phenomenon does not induce any a peak concentration and then decreases. In this same region, the water concentration in the liquid appreciable change in the temperature at the com- bustion front. As to the region ahead of the com- phase builds up from zero to the maximum con- . . bustion front, the tcmperatur~ is. incrertsed when centration that can be retained by the reservoir. vaporization and .condensatirm are Since the region of low temperature gradient coincides con sidered, Hence, one effect of the vaporization-condensation with the regioh of coexistence of water and steam, phenomenon is to increase the length of the heated. this region demotes what is commonly referred to as zone ahead of the combustion front. the steam plateau. Fig,” 3 shows two curves representing the tempera- There are, however, two things to-be noticed in ture history of a point at x = 40 ft.” Here again, the connection with the steam plateau. First, although dotted line corresponds to a case where the effects the region of low temperature gradient extends for of vaporization “and condensation are neglected. a distance of several feet, the region of constant The solid line was obtained using the same set temperature, in cases where it exists, only covers of conditions, taking into account the vaporization- a short distance. Thus, the conventional picture condensation phenomenon. In this typical case, the of a steam plateau which extends horizontally for temperature at the poini concerned reached ,2ooF a long distance may not be always valid. Wilson above the datum about ,six days ahead in et ‘al, q showed a flat,,region, several inches long, kime due to the effect of the vaporization-condensation ‘! in their observed temperature profile. Unfortunately, , T-rd . y 6oo’- ?41 ,? * 1. ,. -L--—-J---.1 .-. ; 1 1 , -+-la & OslO’’” 152025 30364045 .’11s 1s0 MWANCE , X , tt.- . ! ‘.”. ‘. . ,,.. ,., ,“” ;). . ,: FIG. 1 - TYPICAL TEkEkAT_tiE ‘ A;D CONCEkTtiTiON “PRbFILE9. 9 ------ phenomenon. However, tpis ,point reaches its TIME-DEPENDENCE OF TEMPERATURE highest temperature at the same time, unaffected the by the inclusion of the vaporization-condensation The three curves shown in Fig. 4 represent phenomenon. ,4 (Run No,l) ~ _WITH VAPORIZATION-CONDENSATION .--. --WITHOUT VAPORIZATION-CONDENSATION :1000 ------b . g Soo - ~ 600 - Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 a w 400 - ; ~ 200 - PO l...... ~ o’ 10 20 30 . 40 50 60 DISTANCE , X , ft. FIG. 2= EFFECT OF VAPORIZATION-CONDENSATION PHENOMENON ON TEMPERATURE PROFILE. (Run No.!) !, & wITH VAPORIZATION-CONDENSATION ‘ ● . ‘.wITHW vAPoRizA710N - C0NDEN5Ali0N 1?1000 - .------—-- k . g Soo - ~ 8600 1’ 2 ,& 400 [2: ., i“”’~=4’[’,,oo PI&3 Soo ,. o 200 400 / ,“ TIME.....—.. t!..—hrs. FIG. 3 —EFFECTOF ~APORIZATIO”N-CONDENSATIONPHENOMENONON TEMPERA~REHIS~RY. 1 I . I /, , ~ ~looo - 800 “ i ~ 2500h@ ~ 600- ~ 3750 hrs. -, a I 110. 120’ ‘ —.— . .—-. -. .. —..- . .. . . * temperature profiles of the same system at three po@ts 20 ft ,apart, It can be seen that the curve for differerit times. After a certain initial period, the the 60-ft point can be superposed on the 80-ft temperature at the combustion front stabilizes and point curve, upon displacement along the time %is. does not vary with time any longer. Furthermore’, the ,,t~rnce, any point in ;he system tindergoes the ssme shape of the temperature profile on the leading temperature history irrespective of its location, edge of the heat wave also stabilizes. The length p~vided that it is located far &ough from the in- of the steam bank remains practically constant jection point not to be primarily influenced by the although the length of the water bank increases as initi rd conditions. the heat wave advances, Wilson et al.4 mentioned Our finding about the stabilization of the teinper- that as time progresses the length of the steam ature at the combustion front and of the temperature plateau increases. Since this statement was based profile on the leading edge of the heat wave is still on their experiments c airied out in a short tube and true under nonadiabatic conditions. This is shown only for a brief period of time, it may well represent in Fig. 6 where the product of heat-loss coefficient the phenomenon in the initial period of the process and perimeter shape factor, bp, was assumed to be during which no stabilization has yet beeri attained. U.OZ Btu/hr-cu ft-°F. The use ot the heat-loss Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 As to our finding about the constant length of the coefficient to account for the heat 10Ss to the over- steam plateau after this initial period, its validity burden and underburden was based on the assumption can only be substantiated or di sproved by actual that the heat loss at any point in the reservoir is temperature distribution data from 1arge-scale, Iong- solely a function of the temperature at that point. duration field operations. This same technique was previously used by T3erry Fig. 5 shows the temperature history of three and Parrish.15 d .- . TIME , hr. FIG. 5 — ‘i’EMPERATURE HISTORY AT VARIOUS POINTS (RUN 0 $ :. . hp -0.02 Btu/hr. ff?‘F. ,, J’looo - L. . x 800 -’ F s 6 2 w 400 - p a ...... w 0, 30 ; “. 40 50 60 70 , PISTANCE , X , ff. :. ,. ). ,. . .. . , . FIG, 6,‘:- “TtitiERATuRE PR&LE&’” titii “NoN-ADIABATIC i20NDx~oN5 (RW 5): ,...... —-..—.....—. . . . -. .- FACTORS INFLUENCING THE compared with the solid matrix. It would be expected VAPORIZATION-CONDENSATION PHENOMENON ~ thet- the effect of vaporization and condensation is more pronounced at a higher porosity. This is pre- Process variables such as the totai pressure of cisely so, as indicated by the curves shown in Fig, the system; the porosity of the reservoir, and the 8. gas and liquid saturations in the porous space Similarly, Fig. 9 shows ‘the temperature: profiles directly influence the vaporization- condensation gas phenomenon, at two different gas saturations. A lower Fig. 7 shows temperature profiles at two different saturation and hence a higher liquid saturation seem pressures. The solid line represents the case where co induce a iarger “effect cf vaporization and con- the total pressure Pt is 400 psia and the dotted line densation. where the total pressure is 40 psia, Comparison of these two curves clearly indicates that the steam ‘&N&LUSIONS plateau occurs at a lower temperature and extends The following conclusions have been reached for over a longer distance wh~is the pressure is reduced. the one-dimensional’ linear mathematical model Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 This result agrefs with the experimental results studied. of W&son et al. L The vaporization-condensation phenomenon In Fig. 8, the temperature profiles at two different does not induce appreciable change in the temper- porosities are compared. Higher porosity means ature at the combustion front. The primary effect higher fluid content and hence higher relative impor- of the vaporization-condensation phenomenon is to tance of the fluid contained in the reservoir as —. *create a steam plateau and to increase the length tl. — ,*----\ /’) ‘\ Soo -’ \ P+ c 4oo’pslo 600 - ----”-- Pt = 40 Psio I \ \ \ 200 - \ ------0 I 0 10”, 20 30 .40 50 ( 01STANCE ,: X , ft.: 1 FIG. 7 — TEMPERATURE PROFILES AT DIFFERENT PRESSURES (RUNS 1 AND 2). ,-” *0.3 ------$ _0.2 ( 4 + ,’ ...- . .“ ...... ,, ...... iNSTANiX, x, ft. ,, -. . . — .- — —. —.● of the heated zone ahead of the combustion front. K ~ = Mass transfer coefficient, hr -1 2. The temperature profile on the leading edge ‘ k”= Thermal conductivity, Btu/hr-ft-°F of a linear heat wave stabilizes after a certain initial period. The length of the steam bank remains ke = Vaporization equilibrium constant, dimen. practically constant although the length of the sionless ~water bank increases as the. heat wave a’dvances. L = Length of the system, ft 3. Lower pressures, higher porosities or reduced M = Maximum number of m gas saturations correspond to a more pronounced effect Of the vaporization-condensation phenomenon m= Spatial Iocation on the temperature distribution. N = Maximum number of n No = Oxygen content of gas, lb ~ /lb gas This work represents the first attempt to include the vaporization-conden satic$n phenomenon in a n = Time leveI theoretical anal ysis. Some of the properties post u- P = Perimeter shape factor, sq ft heat loss area/ Iated for the model may have influenced the con- Cu ft Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 clusions. A more realistic model should incIude, P:.= Total pressure, psia among other things, the fluid-flow aspects of the process. Furthermore, a corresponding treatment Pv = Vapor pressure, psia > should be made using a radial model. These q = Gas flow rate, cu ft/hr considerations will increase the number and com- R = Gas constant, psia cu ft/lb-moIe-°F plexit y of the partial differential equations to be solved. S = Saturation T = Temperature, F NOMENCLATURE Ta = Temperature, R t = Temporal coordinate, hr A = Cross-sectional area, sq ft vii ~ = Molar volume of water, cu fc/lb-mole c = Heat capacity, Btu/lb -% Vg = Superficial gas velocity, ft/hr c go = Coricentration of the vaporizable oil fraction in the gas phase, lb-moles/cu ft W = Fuel contenr, lb/cu ft c gw = Concentration of steam in the gas phase, x = Spatial coordinate, ft lB-inoles/cu ft x ~ = Location of combustion front, ‘ft c LO= Concentration of rhe vaporizable oil fraction ‘ in the liquid oil phase, lb-moles/cu ft GREEK LETTERS I CLW = Concentration of water in the combined .3 = Incremental sign liquid phase, lb “moles/cu ft 8 = Dirac delta distribution, ft-l D = Diffusivity, sq ft/ht A = La~ent heat of vaporization, ~tu/lb-m&e g =’ Heat generation constant, Btu/lb 02. p = Density, lb/cu ft consumed . ~ A Porosity h = Heat loss coefficient, Btu/hr-sq ft ‘F : - a =’ Fuel-oxygen raqio 1’ ...... 4 DISTANCE , X , ft. I ..” ““ . %BSCRIPTS .,- Radial Movement of a Cylindrical Heat Source — Application to the Thermal Recovary Process”, g = Gas phase ~rUnS,, AIME (1959) volt 216, 115. 6. Hailey, H. R. end Larkin, 1% K.: “Heat Conduction i = Initial in Underground Combustion’), Trans., AXMII(1959) L = Liquid phase vo~ 216, 123, ra .= Spatial location 7* Etaiky, W R. and LarkIn, P, K.: 68@nductiOn- Convection in ‘Underground Combustion t’, Trans., n = Time level AIME (1960) Voi. 219, 320. o = Oil s. Selig, F. and Couch, & J.:$Wnt@rdiache Verbrennung sia &f&derungs-methode”, b S: erTs icb% sches R = Average condition along the characteristic lngenieur=Arcbiu., Bd. XV, Heft 1.4 (1961). base curve 9, Baker, P. E,: t/Temperat~re profiles in Underground s = Solid Combustion~t, Sot. Pet. l%g, JOW. (March, 1962) vol. 2, 21. w = Water 10, C@I, 12. J. and.. Selig, F.: “Die Bedeutu~g dea Wermetranaportee fur in-eitu-Verbrennungsvorgange”, Downloaded from http://onepetro.org/spejournal/article-pdf/4/02/85/2157681/spe-680-pa.pdf by guest on 27 September 2021 Era&t urad Kohle. Etdgas.Petrocbemie (JuIY, 1962) REFERENCES vol. 15, 515,, IL Chu, C.: t~Two.Dimensional Anatysia of a Radial 1. Kuhn, C. S. end Koch, R. L.: “Newest Method of Heat Wave)$, Jour. Pet. Tech, (Ott., 1963) 1137. increasing 011 Recovery?~, Oil and Gas ]. (Aug, 10, 12; Hiidebrand, F. B.: Aduanced Calculus /or ihgineers, 1953) 92. Prentice-Hall, fnc., Englewood Ciiffs, N. J. (1949). Szaa=, s. E,: JIOII Rtscavary by Thermal Methods “I 2. 13 Bruce, G. H., Peaceman, D. W,, Rachford, ”H. H., The Mines Magas+e (Nov., 1960) 32, Jr. and Rice, J. D.: c~Celculation of Unsteady-state 3. Martin, W. L., Alexander, J. D. and Dew, J. N.: Gas Flow through Porous Media”, Trans., AIME ~~pmcess Variables of In-Situ Combustion’r, Trans., (1953) vol. 198, 79. AIME (1958) V04 213, 2S. 14. Hougen, 0. A. and %taon, K. M.: Chernfca/ Process 4 Wilson, L, &, Reed, R. L., Reed, D. W,, Clay, R. R. Principks, Vol. III, John Wiley & Sons, Inc., New and. Harrison, N. H.: t~~me Ef feets of pressure on York (1949). Forward end Reverse Combustion$’, Sot. Pet, E%g. 15, Berry, V. J., Jr, and P&rieh, D. R.: “A Theoretical ,jOUf, (1963) Vol. 2, 127. Analysis of Heat Flow in Reverse Combustion’, S, Ramey, H, J,, Jr,: ~CTrensient Heat COtldUCtiOtl ~~g Twns., AIME (1960) Vol. 219, 124. *** ,. ,, /. < ...... 2..’ ...... -# ,, .: . . ,. . .