B g. B w. C w Water compressibility (pressure -1 ) G i G p m N N p. P Pressure R p Cumulative producing gas-oil ratio (st.vol./st.

Transcription

1 illustrate the simlest ssible mdel e can have fr analysis f reservir behavir, e ill start ith derivatin f s-called. his tye f mdel excludes fluid fl inside the reservir, and cnsiders fluid and rck exansin/cmressin effects nly, in additin, f curse, t fluid injectin and rductin. First, let us define the symbls used in the material balance equatins: ymbls used in material balance equatins Frmatin vlume factr fr as (res.vl./st.vl.) Frmatin vlume factr fr il (res.vl./st.vl.) Frmatin vlume factr fr ater (res.vl./st.vl.) C r re cmressibility (ressure - ) C Water cmressibility (ressure - )!! Cumulative as injected (st.vl.) G i G m N N Cumulative as rduced (st.vl.) Initial as ca size (res.vl. f as ca)/(res.vl. f il zne) Oriinal il in lace (st.vl.) Cumulative il rduced (st.vl.) ressure R Cumulative rducin as-il rati (st.vl./st.vl) = G / N R s b W e W i W lutin as-il rati (st.vl. as/st.vl. il) Gas saturatin saturatin Water saturatin emerature ulk vlume (res.vl.) re vlume (res.vl.) Cumulative aquifer influx (st.vl.) Cumulative ater injected (st.vl.) Cumulative ater rduced (st.vl.)! Density (mass/vl.)! rsity hen, the lack fluid hase behavir is illustrated by the fllin fiures: Fluid hase behavir arameters (lack ) R s Deartment f etrleum Enineerin and Alied Gehysics rfessr Jn Klee Nreian University f cience and echnly Auust 7, 004

2 ! +! R density:! = Water cmressibility: C s =!( )( ) Water vlume chane: = e! ( c ) c Finally, e need t quantify the behavir f the res durin ressure chane in the reservir. he rck cmressibility used in the fllin is the re cmressibility, and assumes that the bulk vlume f the rck itself des nt chane. re vlume behavir Rck cmressibility: C r! = ( )( )! r rsity chane:! =! e! ( + c ) c r he material balance equatins are based n simle mass balances f the fluids in the reservir, and may in rds be frmulated as flls: rincile f material cnservatin! Amunt f fluids resent%! Amunt f %! Amunt f fluids remainin% in the reservir initially ( fluids rduced = in the reservir finally (st. vl.) ' (st. vl.) ' (st. vl.) ' We ill define ur reservir system in terms f a simle blck diaram, ith an initial reservir stae befre rductin/injectin starts, and a final stae at hich time e uld like t determine ressure and/r rductin. lck diaram f reservir il rductin: N as rductin: RN ater rductin: W Initial stae () Final stae () Gas Gas Water Water as injectin: Gi ater injectin: Wi aquifer influx: We Deartment f etrleum Enineerin and Alied Gehysics rfessr Jn Klee Nreian University f cience and echnly Auust 7, 004

3 3 he t staes n the blck diaram are reflected in the fluid hase behavir lts as flls: Initial and final fluid cnditins R s Nte: If a as ca is resent initially, then the initial ressure is equal t the bubble int ressure N, e ill aly the abve material balance equatin t the three fluids invlved, il, as and ater: Equatin : material balance r yieldin! resent %! remainin % in the reservir! % in the reservir ( rduced = initially finally (st. vl.) (st. vl.) ' ' (st. vl.) ' N - N = / N N = (! ) Equatin : Water material balance r yieldin! Water resent %! Water remainin% Water Water in the reservir! %! %! Aquifer% in the reservir rduced injected influx initially ( + + = finally (st. vl.) (st. vl.) (st. vl.) (st. vl.) ' ' ' ' (st. vl.) ' / - W + W i + W e= / ( = * + m N )* % ( ) ' ( i e ) % ' + +! -! + W W W,- Deartment f etrleum Enineerin and Alied Gehysics rfessr Jn Klee Nreian University f cience and echnly Auust 7, 004

4 4 Equatin 3: Gas material balance r yieldin! lutin as %! Free as % resent in the resent in the! Gas %! Gas % rduced injected reservir initially + reservir initially ( + (st. vl.) (st. vl.) (st. vl.) ' (st. vl.) ' ' '! lutin as %! Free as % resent in the resent in the = reservir finally + reservir finally (st. vl.) ' (st. vl.) ' NR s + mn / - R N = (N-N )R s + / ) + = N (R s R s ) + m( - * % )( % '( N (R R s ) + G + i., + / + ( ) In additin t these three fluid balances, e have the fllin relatinshis fr fluid saturatins and re vlume chane: Equatin 4: um f saturatins + + =.0 Equatin 5: re vlume chane = ( +cr! ) Deartment f etrleum Enineerin and Alied Gehysics rfessr Jn Klee Nreian University f cience and echnly Auust 7, 004

5 5 y cmbinin the 5 equatins abve, and ruin terms, e btain the material balance relatinshis, as shn bel: HE COMLEE LACK OIL MAERIAL ALANCE EQUAION: here (, ) ( ) F = N E + me + E + W + W + G f i e i rductin terms are [ ( ) ] F = N + R! R + W s il and slutin as exansin terms are ( ) ( ) E =! + R! R s s as ca exansin terms are E % =! ' and rck and ater cmressin/exansin terms are E ( m) C + C f, =! +! r Deartment f etrleum Enineerin and Alied Gehysics rfessr Jn Klee Nreian University f cience and echnly Auust 7, 004