Material Balance Equations

Transcription

1 TG450 Reservir Recvery Techniques 07 T illustrate the simlest ssible mdel e can have f analysis f reservir behavi, e ill start ith derivatin f s-called. This 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 g Fmatin vlume fact f gas (res.vl./st.vl.) Fmatin vlume fact f il (res.vl./st.vl.) Fmatin vlume fact f ater (res.vl./st.vl.) C r e cmressibility (ressure - ) C Water cmressibility (ressure - ) Δ Cumulative gas injected (st.vl.) G i G m N N Cumulative gas rduced (st.vl.) Initial gas ca size (res.vl. f gas ca)/(res.vl. f il zne) Original il in lace (st.vl.) Cumulative il rduced (st.vl.) ressure R Cumulative rducing gas-il rati (st.vl./st.vl) = G / N R s g T b W e W i W ρ φ lutin gas-il rati (st.vl. gas/st.vl. il) Gas saturatin Oil saturatin Water saturatin Temerature ulk vlume (res.vl.) e vlume (res.vl.) Cumulative aquifer influx (st.vl.) Cumulative ater injected (st.vl.) Cumulative ater rduced (st.vl.) Density (mass/vl.) sity Then, the lack Oil fluid hase behavi is illustrated by the flling figures: Fluid hase behavi arameters (lack Oil) R s g Oil density: ρ ρ = + ρ R g s Negian University f cience and Technlgy August, 07

2 TG450 Reservir Recvery Techniques 07 Water cmressibility: C = ( )( ) Water vlume change: = e ( c ) T c Δ Δ Finally, e need t quantify the behavi f the es during ressure change. The rck cmressibility used in the flling is the e cmressibility, and assumes that the bulk vlume f the rck itself des nt change. e vlume behavi Rck cmressibility: C r φ = ( )( ) φ r sity change: φ = φ e φ ( + c Δ) T c Δ The material balance equatins are based n simle mass balances f the fluids, and may in ds be fmulated as flls: rincile f material cnservatin Amunt f fluids resent Amunt f Amunt f fluids remaining initially fluids rduced = finally (st. vl.) (st. vl.) (st. vl.) We ill define ur reservir system in terms f a simle blck diagram, ith an initial reservir stage befe rductin/injectin starts, and a final stage at hich time e uld like t determine ressure and/ rductin. lck diagram f reservir r il rductin: N gas rductin: RN ater rductin: W Initial stage () Final stage () Gas Gas Oil Oil Water Water gas injectin: Gi ater injectin: Wi aquifer influx: We Negian University f cience and Technlgy August, 07

3 TG450 Reservir Recvery Techniques 07 3 The t stages n the blck diagram are reflected in the fluid hase behavi lts as flls: Initial and final fluid cnditins g R s Nte: If a gas 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, gas and ater: Equatin : Oil material balance yielding Oil resent Oil remaining Oil rduced = initially finally (st. vl.) (st. vl.) (st. vl.) N - N = / N N = ( ) Equatin : Water material balance yielding Water resent Water remaining Water Water Aquifer 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 Negian University f cience and Technlgy August, 07

4 TG450 Reservir Recvery Techniques 07 4 Equatin 3: Gas material balance lutin gas Free gas resent in the resent in the Gas Gas rduced injected reservir initially + reservir initially + (st. vl.) (st. vl.) (st. vl.) (st. vl.) lutin gas Free gas resent in the resent in the = reservir finally + reservir finally (st. vl.) (st. vl.) yielding NR s + mn / g R N + G i = (N N )R s + g / g g = N (R s R s ) + m( ) g N (R R ) + G s i ( g ) In additin t these three fluid balances, e have the flling relatinshis f fluid saturatins and e vlume change: Equatin 4: um f saturatins + + g =.0 Equatin 5: e vlume change = ( +cr Δ ) Negian University f cience and Technlgy August, 07

5 TG450 Reservir Recvery Techniques 07 5 y cmbining the 5 equatins abve, and gruing terms, e btain the material balance relatinshis, as shn bel: THE COMLETE LACK OIL MATERIAL ALANCE EQUATION: here (, ) ( ) F = N E + me + E + W + W + G g f i e i g rductin terms are [ ( ) ] F = N + R R + W s g il and slutin gas exansin terms are ( ) ( ) E = + R R s s g gas ca exansin terms are E g g = g and rck and ater cmressin/exansin terms are E ( m) C + C f, = + r Δ Negian University f cience and Technlgy August, 07

6 TG450 Reservir Recvery Techniques 07 6 MATERIAL ALANCE EQUATION FOR A CLOED GA REEROIR The material balance equatin f a clsed gas reservir is very simle. Alying the mass balance rincile t a clsed reservir ith 00% gas, e may derive the general eguatin G g = (G G ) g here G is gas initially in lace, G is cumulative gas rductin, and g is the fmatin-vlume-fact f gas. ince g is given by the real gas la g = (cnstant) Z (here temerature is assumed t be cnstant) the abve material balance equatin may be reritten as G Z = (G G ) Z Z This equatin reresents a straight line relatinshi n a G = 0, and thrugh G at e may get an estimate f G. Z = ( G G ) Z Z vs. G. lt. The line asses thrugh = 0. y making a best-fit straight line t measured data, and extralate, Z at The straight-line relatinshi is very useful in estimating the initial vlume f gas-in-lace (G ) frm limited rductin histy. Negian University f cience and Technlgy August, 07