Non-Darcy Porous-Media Flow According to the Barree and Conway Model: Laboratory and Numerical- Modeling Studies

Transcription

1 Non-Darcy Porous-Mea Flow Accorng to the Barree an Conway Moel: Laboratory an Numercal- Moelng Stues Btao La, Jennfer L. Mskmns, an Yu-Shu Wu, SPE, Colorao School of Mnes Summary Ths paper presents the results of our new expermental stues conucte for hgh flow rates through proppant packs, whch show that the Barree an Conway (4) flow moel s capable of overcomng lmtatons of the Forchhemer non-darcy equaton at very hgh flow rates. To quantfy the non-darcy flow behavor usng the Barree an Conway moel, a numercal moel s evelope to smulate non-darcy flow. In aton, an analytcal soluton s presente for steay-state lnear non-darcy flow an s use to verfy the numercal-smulaton results. The numercal moel ncorporates the Barree an Conway moel nto a general-purpose reservor smulator for moelng multmensonal, sngle-phase non-darcy flow n porous an fracture mea an supplements the laboratory fnngs. The numercal moel s then use to perform senstvty analyss of the Barree an Conway flow moel s parameters an to nvestgate transent behavor of non-darcy flow at an njecton well. Introucton The objectves of ths paper are () to present expermental ata from our current laboratory stues of hgh flow rates through proppant packs an () evelop mathematcal-moelng tools to quantfy such hgh-flow-velocty, non-darcy-flow behavor. Our expermental results show that non-darcy flow occurs at hgh flow rates an that the conventonal Forchhemer moel may not be suffcent to escrbe the observe hgh-flow-rate behavor. On the other han, the Barree an Conway (4) flow moel s foun to be able to match the entre range of observe ata from low to hgh flow rates. The moelng tools evelope nclue both analytcal an numercal approaches for smulatng sngle-phase non-darcy flow wth the Barree an Conway moel. In aton, the numercal moel s use to perform parameter-senstvty analyss an to obtan nsght nto transent non-darcy flow wth the Barree an Conway flow moel. Darcy s law (Darcy 856) s the founaton for stues of flu flow n porous mea. Accorng to Darcy s law, the pressure graent ( P/ L) can be relate to the flu vscosty µ an superfcal velocty v through a constant k (permeablty), as emonstrate n Eq. : P L v () k Forchhemer (9) observe evatons from the lnearty of Darcy s Law at hgh flow rates. Forchhemer expane Darcy s lnear form nto a quaratc flow equaton that s now commonly referre to as Forchhemer s equaton (Eq. ): P v L k v () Copyrght Socety of Petroleum Engneers Ths paper (SPE 6) was accepte for presentaton at the SPE Rocky Mountan Petroleum Technology Conference, Denver, 4 6 Aprl 9, an revse for publcaton. Orgnal manuscrpt receve for revew February 9. Revse manuscrpt receve for revew 6 February. Paper peer approve March. Even wth ths atonal quaratc term, Eq. may not aequately escrbe all of Forchhemer s ata; therefore, he eventually ae an atonal cubc term (Eq. 3) to try to account for these evatons. Other authors have also note the nablty of Forchhemer s equatons to escrbe all ata sets (Carmen 937; Fan et al. 987; Kececoglu an Jang 994; Montllet 4; Barree an Conway 4). The effect of these screpances can have a major mpact on the assessment of flow rate or pressure strbuton for a gven porous meum, as shown n Fg. (a typcal Forchhemer plot, n whch X -axs s P, an Y -axs s L ). Fg. shows an example of expermental ata that evates from the Forchhemer plot. Fg. also shows that s not a constant wth ncrease flow rate. P v L k v v 3, (3) where non-darcy coeffcent, /m; an flu ensty, kg/m 3. Jones (97) specfcally note that large evaton from the lnear Forchhemer plot occure for core samples wth large values of the term k. He shows that the Forchhemer plot becomes concave ownwar, leang to the observaton of hgher apparent permeablty than those precte by Forchhemer s equaton at hgh velocty. Also, at hgher velocty, the -factor enote for nertal force s not a constant for porous mea. Base on extensve expermental results an fel observatons, Barree an Conway (4) propose a new, more general moel for non-darcy flow n porous mea that oes not rely on the assumptons of a constant permeablty or a constant -factor. In ther moel, Darcy s law s stll assume to apply, but the absolute permeablty s replace by the apparent permeablty, shown n Eq. 4: P L v, (4) k app where k app s the apparent permeablty an s efne as k k k k ( mn) (5) app mn E ( N ) Re Eq. 4, along wth Eq. 5, s calle the Barree an Conway flow moel n ths paper. In Eq. 5, k app becomes constant (plateau behavor) at low (.e., k ) an hgh Reynols numbers (N Re ). The plateau behavor at low Reynols numbers was expermentally valate by several authors (.e., Fan et al. 987) before Barree an Conway (4). On the other han, the plateau behavor at hgh Reynols numbers s a hypothess, an t s partly valate by Lopez (7) an La (). The exponental coeffcent E n Eq. 5 escrbes the overall heterogenety of the test samples. Its relaton wth sortng of a porous meum s examne by La (). The better the sortng, the closer the value of E s to.. For sngle-seve proppant (the proppants reman n the specfe seve n a seve-analyss experment), the value of E s equal to 7 March SPE Journal

2 Y Therefore, t can be euce that for a homogeneous porous meum, the value of E s equal to unty. In ths work, for the proppant packs teste, whch can be consere homogeneous, E s foun to be equal to unty (Barree an Conway 4; La ). In Eq. 5, N Re s efne by v N Re, where k app apparent rate-epenent permeablty, k mn mnmum permeablty at hgh rate, k constant Darcy permeablty, N Re Reynols number, ensty, g/cm 3, v superfcal velocty, cm/s, µ vscosty, an nverse of the characterstc length. The mensonless form of Eq. 5, when the E equals unty, can be wrtten as Eq. 6: kmn kapp kmn k (6) k k N β-factorslope (m) X ( ) Re k/y(x) Fg. Typcal Forchhemer plot where the -factor s the slope of the lne an the permeablty (k) s the ntercept. Note that the changng slope of the ata ncates that the -factor s not a constant n porous mea, base on the expermental ata (Martns et al. 99). In Eqs. 5 an 6,, nverse of the characterstc length, s relate to the mean partcle sze (Barree an Conway 4). Also, s correlate to the closure stress an the crushng an falure of the proppant grans. There are several avantages to, or benefts of, usng the Barree an Conway equaton. Frst, t proves a sngle equaton to escrbe the entre range of flow veloctes through porous mea teste. Ths s of great beneft to quanttatve or moelng stues because a sngle equaton can be programme to moel flow wthout neeng to artfcally set swtchover ponts from Darcy- to non-darcy-flow regmes. Secon, because the equaton escrbes the entre range of flow veloctes, physcal-transton zones are also honore an capture as flow veloctes ncrease or ecrease. The moel also proves for a plateau area or a constant apparent permeablty at hgh rates, whch has been suggeste an moele by other authors (Ergun 95; Fan et al. 987). In partcular, our expermental ata show that the equaton evelope by Barree an Conway s able to overcome many of the rawbacks that numerous authors have ponte out when conuctng Forchhemer analyss whle stll honorng the bascs of Darcy an Forchhemer flow behavor uner lower flow rates. Laboratory Expermental Data an Results A large number of experments wth proppant packs have been carre out usng a sngle-phase ntrogen non-darcy-flow apparatus evelope by Lopez (7). The apparatus uses a cell system consstng of a -cm-long Tygon tube (to account for embement) wth an nlet ameter of.95 cm. The cell s flle wth proppant an fve pressure ports, space 5 cm apart, whch are nstalle along the pack (see Fg. ). The assemble proppant pack s place n a hgh-pressure vessel that can apply varous closure stresses from 34.5 MPa usng a hyraulc ol system. The nlet gas-flow rate ranges from 5 g/sec, whch covers most gas-proucton rates encountere n the fel. Note that the gas-flow rate n ths work s three tmes hgher than the flow rate n Barree an Conway (4) (the hghest N Re n ths paper s 3, whle the hghest N Re n Barree an Conway s work s approxmately ). As an example, the proceure of the test s llustrate uner 36- atm (,-ps) confnng-stress contons. Frst, a 36-atm (,- ps) confnng stress s apple to the test sample by a hyraulc ol pump. When the confnng stress s stable, ntrogen gas wth a purty greater than 99% s njecte nto the proppant pack, wth an nlet gas pressure of.5 atm (,8 ps). When the gas-flow rate reaches steay-state contons, the fve port pressures, temperature nse the proppant pack, an the gas mass-flow rate are recore by a ata-aquston system. When the ata correcton s fnshe, the nlet-gas pressure s lowere to 9 atm (,6 ps) whle keepng the confnng stress constant, an ntrogen gas contnues to be njecte nto the proppant pack. The port pressures, the temperature, an mass-flow rate of gas are then agan recore. Ths proceure s repeate untl the gas-nlet pressure reaches the lower lmt (for nstance, 6.8 atm or ps). Proppant samples teste to ate nclue a we range of commonly use szes an types, nclung ceramcs an natural sans, rangng from /6 to mesh. Expermental ata are analyze usng a regresson metho for the Forchhemer quaratc an cubc methos an the Barree N Pressure ports # # #3 #4 #5 Proppant N 3 cm 5 cm 5 cm 5 cm 5 cm 3 cm Confnng stress 6 cm Fg. Schematc of the proppant pack use n the non-darcy-flow test system. Proppant s place n a -cm-long Tygon tube wth an nlet ameter of.95 cm. Fve pressure ports are nstalle along the pack wth a spacng of 5 cm, whch proves a pressure profle across the pack, not just at the nlet an outlet. Confnng stresses are apple to the pack usng hyraulc ol. March SPE Journal 7

3 8 Data Forchhemer qua Forchhemer cubc Barree an Conway P/x, atm/cm m/a, g/(s*cm ) Fg. 3 Results of pressure graent vs. mass-flow rate for a ceramc /4 proppant uner a confnng stress of 7.5 MPa. The expermental ata (blue amons) agree wth the Barree an Conway moel (re) from low to hgh flow rates. The Forchhemer quaratc correlaton (green) overestmates the pressure rop, whle the Forchhemer cubc correlaton (blue) unerestmates the pressure rop at hgh gas-flow rates. an Conway technque (Eqs. through 4). Because contnuous an sgnfcant varatons of pressure-graent an gas-property parameters are expecte along the pack, all calculatons an analyses are performe at the mpont of the pack. The pressure graent of the mpont of the cell s calculate usng the numercal-fferentaton technque known as central-fference-ervatve formulae (Grffths an Smth 6), as follows: P ( L h P P P P ) P3 where P L pressure graent at the mpont of the cell, P pressure at Port, P pressure at Port, P 3 pressure at Port 3, P 4 pressure at Port 4, P 5 pressure at Port 5, an h stance between each port. A etale explanaton regarng how to conuct the ata reucton s prove n Lopez (7). An example showng the comparson among the expermental results an the three flow moels s shown n Fg. 3, where the x-axs s normalze for mass-flow rate nstea of flu velocty. The ata plotte n Fg. 3 are taken from a ceramc /4 proppant uner a confnng stress (cp) of 7.5 MPa (4, psg). The expermental ata obtane from the laboratory tests are tabulate n Table. The gas mass-flow rate ( m ) measurement n these experments s base on the real-gas law (Lopez 7), for whch a known-volume gas tank s use. The key parameters n each flow moel (for nstance, -factor n the Forchhemer quaratc equaton; n the Forchhemer cubc equaton; an k mn, k mn /k, an n the Barree an Conway moel) are obtane by usng the back-regresson analyss (curve-fttng) technque. In Table, P through P 5 represent the fferent ports (see Fg. ) at whch pressure values are measure urng the experments; P t an T t are the ntal pressure an temperature n the gas tank, respectvely; an P tf an T tf are the fnal pressure an temperature n the gas tank, respectvely. The fel unts n the measurement wll be converte to SI unts n the calculaton. As can be seen n Fg. 3, the expermental ata agree qute well wth the Barree an Conway moel across the entre flow-velocty range from low to hgh gas-flow rates. The Forchhemer quaratc correlaton overestmates the assocate pressure rop, an the Forchhemer cubc equaton evates from the expermental ata at hgh gas-flow rate. All sample ata taken to ate show smlar agreement wth the Barree an Conway equaton across the observe flow spectrum. Fg. 4 s a summary plot of all the test ata taken to ate usng the mensonless form of the Barree an Conway moel (Eq. 6). Fg. 4 emonstrates that all of the expermental ata collapse nto one curve n agreement wth the Barree an Conway moel, as woul be expecte n a mensonless form. One plateau of the logose equaton format s clearly observe at low Reynols numbers. When converte to fel unts, the test ata shown n Fg. 4 cover fel gas-proucton rates from less than 77. m 3 / to more than m 3 /, emonstratng that the moel s accurate across the ntervals of nterest for the nustry. Mathematcal Moel In orer to complement the laboratory-ata results an quantfy non-darcy-flow behavor n reservors, numercal an analytcal moelng s neee n general. We conser an sothermal reservor system consstng of one sngle-phase flu (gas or lqu) n porous or fracture mea. Governng Equaton. In an sothermal system contanng one flu, one mass-balance equaton s neee to escrbe the flu flow n porous mea an mass conservaton of the flu, leang to Eq. 7: ( ) ( v ) q, (7) t where s the effectve porosty of the meum, s the ensty of the flu uner reservor contons, q s the snk/source term of the phase per unt volume of formaton, an v s the superfcalflow-velocty vector, efne n the followng (Eq. 8), an extene Barree an Conway s moel n a vector form for multmensonal flow: 7 March SPE Journal

4 TABLE EXPERIMENTAL DATA OBTAINED FROM THE CERAMIC /4 PROPPANT-PACK TESTS UNDER A CONFINING STRESS OF 4, ps (7.5 MPa) Test P (psg) P (psg) P 3 (psg) P 4 (psg) P 5 (psg) T ( F) P t (psg) T t ( F) P tf (psg) P T tf Δt x ( F) (s) (atm/cm) m/a (gm/s-cm ) k k mr v, (8) ( kmr ) v where k mr s the relatve mnmum relatve permeablty, efne as the the rato of mnmum permeablty to Darcy s permeablty. s the flow-potental graent, as efne n Eq. 9: ( ) P g D, (9) where P s the pressure of the flu, g s gravtatonal acceleraton, an D s the epth from a atum. Bounary an Intal Contons. Bounary an ntal contons are neee to complete the mathematcal escrpton for non-darcy flow n reservors. For sngle-phase flow, the ntal status of the flow system s escrbe by the ntal conton or pressure spatal strbuton. As wth Darcy flow, there are three types of bounary contons at wells or outer bounares for non-darcy flow: () frst-type or Drchlet bounary (.e., constant or tmeepenent pressure); () flux-type or Neuman bounary, epenng on proucng or njecton conton; an (3) a more general thr type of mxe pressure an flux bounary for multlayere well bounares for general proucton or njecton wells. Numercal Formulaton an Soluton The flow-governng equatons, Eqs. 7 an 8, for sngle-phase non- Darcy flow of gas or lqu n porous mea as escrbe by the Barree an Conway moel are hghly nonlnear, an n general, nee to be solve numercally. In ths work, the methoology for usng a numercal approach to smulate the non-darcy flow conssts of the followng three steps: () spatal scretzaton of the mass-conservaton equaton; () tme scretzaton; an (3) teratve approaches to solve the resultng nonlnear, screte algebrac equatons. A mass-conservng scretzaton scheme, base on control volume or ntegral fnte fference (Pruess et al. 999), s use an scusse here. March SPE Journal 73

5 .E 77. m 3 /.E- η 83,68. m 3 /.E-.E-3.E-.E-.E.E.E.E3.E4 N Re Fg. 4 Dmensonless plot of all teste proppants usng the Barree an Conway moel. As can be seen, all the expermental ata collapse onto one curve that matches closely to the theoretcal Barree an Conway moel (shown as a ashe black an yellow lne). One ata plateau s clearly observe at low Reynols numbers, an transton zones are capture. The test ata cover fel proucton rates from less than 77. m 3 / to more than m 3 /. The control-volume approach proves a general spatal scretzaton scheme that can represent a D, D, or 3D oman usng a set of screte meshes. Tme scretzaton s carre out usng a backwar, frst-orer, fully mplct fnte-fference scheme. Specfcally, the non-darcy-flow equatons, as scusse prevously, have been mplemente nto a three-phase reservor smulator (Wu, ). As mplemente numercally, Eq. 7 s scretze n space usng an ntegral fnte-fference or control-volume scheme for a porous an/or fracture meum wth an unstructure gr. The tme scretzaton s carre out wth a backwar, frst-orer, fnte-fference scheme. The screte nonlnear equaton for gas or lqu flow at Noe s shown n Eq. : n { ( ) n V n ( ) } n flow j Q, () t j where the superscrpt n enotes the prevous tme level, n s the current tme level, V s the volume of Element (porous or fracture block), t s the tmestep sze, contans the set of neghborng elements (j) (porous or fracture) to whch Element s rectly connecte, flow j n s the mass flow term for the flu between Elements an j, an Q s the mass snk/source term at Element for the flu. Base on the D flow rate, Eq. A-4, usng the Barree an Conway moel (Appenx A), the mass flow term (flow j n ) n Eq. for the non-darcy flow between Blocks an j s evaluate by Eq. : flow j ( ) j ( kk ) j / j A j ( j ) ( kk ) j / j j ( 4 ( k ) ) j / j () where the subscrpt j/ enotes a proper averagng or weghtng of propertes at the nterface between the two Elements an j, A j s the common nterface area between the connecte Blocks or Noes an j, s the stance from the center of Block to the common nterface of Blocks an j, an the flow-potental term n Eq. s efne as n Eq. : P j g D / () Note that n Eq., a screte equaton of mass conservaton of the flu has the same form regarless of the mensonalty of the moel oman [.e., t apples to D, D, or 3D analyses of flow through porous or fracture mea (Wu )]. Numercal-Soluton Scheme. Eq., the screte nonlnear equaton, s solve fully mplctly wth a Newton-Raphson teraton metho. Let us wrte the screte nonlnear Eq. n a resual form as Eq. 3: R { } V t n n ( ) n ( ) flow j j Q n (,, 3,, N), (3) where N s the total number of noes/elements/grblocks of the gr. Eq. 3 efnes a set of (N) couple nonlnear mass-balance equatons that nee to be solve smultaneously. In general, one prmary varable per noe s neee n the Newton teraton for solvng one equaton per noe. We select flu pressure as the prmary varable, an treat all the rest of the epenent varables, such as vscosty, porosty, an ensty, as seconary varables, whch are calculate from the prmary varable at each noe an at each teraton. In terms of the prmary varable, the resual equaton, Eq. 3, at a Noe s regare as a functon of the prmary varables not only at Noe, but also at all ts rectly neghborng noes j. The Newton-Raphson teraton scheme leas to to Eq. 4: 74 March SPE Journal

6 NRe τ.e5 τ8.e4 τ6.e4 τ4.e4 τ.e4 analytcal solutons an then use them to verfy numercal-moel results. The governng equaton (Eq. 7) for D lnear, horzontal, steay-state flow s smplfe as Eq. 6: ( v) (6) The flow rate accorng to the Barree an Conway moel becomes: P P P kk kk L L 4 k L v (7).E.E 4.E 6.E 8.E n R ( xj, p) n x p R xm p x ( ), (, ) j (,, 3,, N), (4) where x j s the prmary varable at Noe an all ts rect neghbors; p s the teraton level; an,, 3,, N. The prmary varables n Eq. 4 nee to be upate after each teraton, as shown by Eq. 5: x x x (5) p p p Pressure Graent (Pa/m) Fg. 5 Relatonshp between Reynols number (N Re ) an pressure graent (Pa/m) accorng to the Barree an Conway moel. Relatonshps for fferent values wth k mn 9.869E m. The rest of the nput parameters use are lste n Table. The Newton-Raphson teraton process contnues untl the resuals, R k,n, or changes n the prmary varables, x p, over an teraton are reuce below preset convergence tolerances. In aton, the numercal metho s use to construct the Jacoban matrx for Eq. 4, as outlne n Forsyth et al. (995). At each Newton-Raphson teraton, Eq. 4 represents a system of N lnearze algebrac equatons wth sparse matrces, whch are solve by a lnear teratve matrx-equaton solver. Treatment of Intal an Bounary Contons. Drchlet bounary contons are hanle wth the nactve cell or bg-volume metho, as normally use n the TOUGH coe (Pruess et al. 999). In ths metho, a constant-pressure noe s specfe as an nactve cell or wth a huge volume, whle keepng all the other geometrc propertes of the mesh unchange. For flux or Neuman bounary contons, multlayere wells, an Cauchy or mxe bounary contons, a general hanlng proceure s scusse n Wu et al. (996) an Wu (). Moel Verfcaton an Applcaton Ths secton presents one verfcaton an one applcaton example to emonstrate the usefulness of the propose numercal approach n moelng non-darcy flow n reservors. Frst, we erve steay-state-flow The soluton for steay-state ncompressble flu flow s m m ( L x) A A Px ( ) P, (8) m k A kk an the soluton for steay-state, slghtly compressble flu flow s ( cp ) ± m m P c c A A ( x L) ( cp) c ( P ) P m k A kk (9) The etale evelopment of these two solutons, Eqs. 8 an 9, s prove n Appenx B. Non-Darcy-Flow Behavor. Fg. 5 presents several characterstc curves of flow rate n terms of Reynols number vs. pressure graent accorng to the Barree an Conway moel, an shows obvous nonlnear flow behavor. The plots n Fg. 5 are generate usng Eq. 7 for D flow wth the parameters gven n Table. In examnng the non-darcy flow behavor between the curves of Fg. 5, we have foun that the parameter s more senstve than other parameters for the normal range of pressure graents. As the value of ecreases, the flow becomes more nonlnear, as shown n Fg. 5. In aton, Fg. 6 shows the effect of the secon Barree an Conway moel parameter, k mn, on the flow rate vs. pressure graent, showng the nonlnear behavor as k mn ecreases. Moel Verfcaton. Here, we use the analytcal soluton (Eq. 8) of D steay-state flow to check the numercal-smulaton results. In numercal scretzaton, a D lnear reservor formaton m long, wth a unt cross-sectonal area, s represente by a D unform lnear gr of, elements wth x. m. The parameters use for ths comparson are lste n Table 3. We compare two cases wth fferent mnmum-permeablty an nverse-characterstc-length values, where Case has k mn. arces an (/m) an Case has k mn. arces an (/m). In the two scenaros, the pressure at the outlet bounary s mantane at 7 Pa, an a constant mass proucton rate s propose at x for both the analytcal an numercal solutons. The numercal calculaton s carre out untl steay state s reache. TABLE PARAMETERS USED IN FIGS. 5 AND 6 FOR PLOTTING RELATIONSHIPS BETWEEN FLOW RATE AND PRESSURE GRADIENT Parameter Value Unt Darcy permeablty k Darcy Mnmum permeablty k mn. Darcy Vscosty μ. Pa s Densty, kg/m 3 Characterstc length τ,, /m March SPE Journal 75

7 N Re k mn.e 3 k mn 8.e 4 k mn 6.e 4 k mn 4.e 4 k mn.e 4.E.E 4.E 6.E 8.E Pressure Graent (Pa/m) Fg. 6 Relatonshp between Reynols number (N Re ) an pressure graent (Pa/m) accorng to the Barree an Conway moel. Relatonshps for fferent k mn values wth 6.e4 m. The rest of the nput parameters use are lste n Table. Fg. 7 shows the comparson results from the two solutons an ncates that excellent results are obtane from the numercal smulaton, as compare to the analytcal soluton. Fg. 7 also shows that the pressure strbutons for the two scenaros are lnear along the lnear-flow recton for the ncompressble, steay-state flow cases. Ths s because the steay-state flow velocty s consstent everywhere; thus, the apparent permeablty s also constant (Eq. 4), so we have a lnear pressure profle. Moel Applcaton. The applcaton example presents a raalflow problem usng the numercal moel to calculate transent pressure at an njecton well. The reservor formaton s a unform, raally nfnte system (approxmate by r e m n the numercal moel) of m n thckness, an s represente by a D raal gr of, raal ncrements wth a r sze that ncreases logarthmcally away from the well raus (r w. m). The formaton s ntally at a constant pressure of 7 Pa an s subjecte to a constant volumetrc-njecton rate of m 3 / at the well, startng at t. Parameters use for the smulaton stuy are lste n Table 4. Fg. 8 presents the smulate transent-pressure responses at the well an a comparson for the four cases wth a combnaton of mnmum-permeablty an characterstc-length values of Case wth Darcy flow (.e., k mn k ); Case wth k mn. arces an (/m); Case 3 wth k mn. arces an (/m); an Case 4 wth k mn arcy an (/m). In Fg. 8, the lowest, sol, black curve shows the results for Case (or Darcy flow) for comparson. The uppermost, sol-pnk curve shows the results for Case, ncatng the largest ncrease n njecton pressure or the hghest flow resstance cause by the non-darcy flow because of the smaller values of the characterstc length an mnmum permeablty k mn. The sol-blue-crcle curve, the secon from the bottom, s the result of Case 3, showng a very small fference from the Darcy-flow case because of usng a large ( ). The green-trangle curve, the secon from the top, s for Case 4, also showng a large njecton-pressure ncrease because of a smaller ( ) use n ths case. It s very nterestng to note that n all the four cases of Fg. 8, the pressure responses have a lnear relatonshp wth tme on the semlog plot, except wth the very early tme. In aton, the four semlog straght lnes are parallel to one another. Ths behavor ncates that () the mpact of non-darcy flow on pressure transents s equvalent to a constant flow resstance, superpose onto the pressure change of Darcy flow, an () the Darcy permeablty k can be estmate usng the slope of semlog straght lnes from pressure-rawown or -bulup curves. Conclusons Laboratory ata from hgh-flow-rate tests through proppant packs show that the Barree an Conway moel s able to escrbe the entre range of flow veloctes from low to hgh flow rates uner tests, whle the Forchhemer moel fals to cover the hgh en of flow rates. The expermental ata set nclues a matrx of proppant types an szes an spans the range of gas-flow rates n most fels. In an effort to quantfy non-darcy-flow behavor n porous an fracture mea an to complement the laboratory ata, a mathematcal/numercal moel s evelope by ncorporatng the Barree an Conway moel nto a general-purpose reservor smulator. The numercal formulaton for ths ncluson s base on unstructure grs of control volume. In aton, several analytcal solutons uner steay-state lnear-flow contons are erve an use to verfy the numercal-smulaton results for the steay-state lnear-flow case. These analytcal solutons can be use to estmate non-darcy flow-moel parameters from laboratory ata. As an example of applcaton, the numercal moel s apple to evaluate the transent non-darcy flow behavor at an njecton well. The numercal-moelng results ncate that the parameter of nverse characterstc length s more senstve than other non-darcy moel parameters, whle the mpact of the mnmumpermeablty plateau s shown only at extremely large flow rates or pressure graents. The analytcal solutons an the numercal smulators from ths work can be use for analyzng non-darcyflow behavor n laboratory, near-well flow, an fel stues. Nomenclature A cross-sectonal area of flow, m A j common nterface area between the connecte Blocks or Noes an j c compressblty of porous mea, /Pa stance from the center of Block to the common nterface of Blocks an j D epth from a atum E exponental constant, mensonless TABLE 3 PARAMETERS USED FOR MODE L VERIFICATION CHECKING NUMERICAL- SIMULATION RESULTS AGAINST THE ANALYTICAL SOLUTION, AS SHOWN IN FIG. 7 Parameter Value Unt Cross secton area A m Darcy permeablty k Darcy Mnmum permeablty k mn.,. Darcy Vscosty μ. Pa s Characterstc length τ,,, /m Densty, kg/m 3 Mass proucton rate m 5 kg/s Pressure at outer bounary P 7 Pa 76 March SPE Journal

8 Pressure (Pa).E7 8.E6 6.E6 4.E6.E6 Ana, τ.e6 Ana, τ.e5 Num., τ.e6 Num., τ.e Dstance (m) Fg. 7 Comparson between the analytcal an numercal solutons for D steay-state flow n a lnear system. The analytcal soluton for Case s shown as a sol blue lne, whle the numercal soluton for Case s shown as purple squares. The analytcal soluton for Case s shown as a sol green lne, whle the numercal soluton for Case s shown as re trangles. Moel parameters use are lste n Table 3. g gravtatonal-acceleraton constant h stance between each port, cm k permeablty, arces k app apparent rate-epenent permeablty, arces k constant Darcy permeablty, arces k mn mnmum permeablty at hgh rate, arces k mr mnmum permeablty relatve to Darcy permeablty, fracton L length, m m flu mass-flow rate, gm/sec M molecular gas weght N total number of noes/elements/grblocks of the gr N Re Reynols number, mensonless P pressure at Port, atm P pressure at Port, atm P 3 pressure at Port 3, atm P 4 pressure at Port 4, atm P 5 pressure at Port 5, atm P ntal pressure, Pa P t ntal pressure n the gas tank, psg P tf fnal pressure n the gas tank, psg q snk/source term TABLE 4 PARAMETERS USED FOR SIMULATION OF TRANSIENT PRESSURE CHECKING NUMERICAL-SIMULATION RESULTS AGAINST THE ANALYTICAL SOLUTION, AS SHOWN IN FIG. 8 Parameter Value Unt Darcy permeablty k Darcy Mnmum permeablty k mn.,. Darcy Intal porosty φ. Vscosty μ. Pa s Reference ensty, kg/m 3 Volumetrc njecton rate q, m 3 / Total compressblty of flu an rock C T 6. /Pa Well raus r w. M Formaton thckness h M Intal-formaton-gas pressure P 7 Pa Injecton Pressure (Pa).35E7.3E7.5E7.E7.5E7 τ,, k mn.e7.e-4.e-.e.e.e4.e6 Tme (hour) Darcy flow τ,, k mn. τ,, k mn. Fg. 8 Transent-pressure responses smulate at an njecton well. Case gves the bottom black sol curve for Darcy flow (.e., k mn k ); Case gves the top sol pnk curve, [.e., k mn. arces an (/m)]; Case 3 gves the sol-bluecrcle curve [.e., k mn. arces an (/m)]; an Case 4 gves the sol-green-trangle curve [.e., k mn arcy an (/m)]. The smulaton parameters use are lste n Table 4. Q flu volumetrc-flow rate, m 3 /s Q mass snk/source term at Element for the flu r raus stance, m r s skn-zone raus, m r w wellbore raus, m R gas unversal constant T temperature, K T t ntal temperature n the gas tank, F T tf fnal temperature n the gas tank, F v superfcal velocty, m/s V volume of Element z gas z-factor P/ L potental graent, Pa/m non-darcy coeffcent, /m or /cm t tmestep sze µ flu vscosty, pose or Pa.s flu ensty, kg/m 3 ntal ensty, kg/m 3 nverse of the characterstc length, / m effectve porosty of the meum flow-potental graent March SPE Journal 77

9 Acknowlegments The authors wsh to thank the members of the Fracturng, Aczng, Stmulaton Technology (FAST) Consortum locate at the Colorao School of Mnes an the Stmlab Proppant Consortum for ther support. Yu-Shu Wu woul also lke to thank the support from Snopec Inc. of Chna through the Natonal Basc Research Program of Chna (6CB4). References Barree, R.D. an Conway, M.W. 4. Beyon Beta Factors: A Complete Moel for Darcy, Forchhemer, an Trans-Forchhemer Flow n Porous Mea. Paper SPE 8935 presente at the SPE Annual Techncal Conference an Exhbton, Houston, 6 9 September. org/.8/8935-ms. Carmen, P.C Flu Flow Through Granular Bes. Trans. Inst. Chem. Engrs. Lonon 5: Darcy, H.P.G The Publc Fountans of the Cty of Djon (Les Fontanes publques e la vlle e Djon), trans. P. Bobeck. Dubuque, Iowa: Kenall Hunt Publshng Co. ( March 4). ISBN Ergun, S. 95. Flu Flow Through Packe Columns. Chem. Eng. Prog. 48 (): Fan, R.M., Km, B.Y.K, Lam, A.C.C., an Phan, R.T Resstance to the Flow of Flus Through Smple an Complex Porous Mea Whose Matrces Are Compose of Ranomly Packe Spheres. J. Flus Eng. 9: Forchhemer, P.F. 9. Wasserbewegung urch Boen. Zetschrft es Verenes eutscher Ingeneure 45 (5): Forsyth, P.A., Wu, Y.S., an Pruess, K Robust numercal methos for saturate-unsaturate flow wth ry ntal contons n heterogeneous mea. Av. Water Resour. 8 (): org/.6/39-78(95)-j. Grffths, D.V. an Smth, I.M. 6. Numercal Methos for Engneers, secon eton. Boca Raton, Flora: Chapman an Hall/CRC. Jones, S.C. 97. A Rap Accurate Unsteay-State Klnkenberg Permeameter. SPE J. (5): SPE-3535-PA. Kececoglu, I. an Jang, Y Flow Through Porous Mea of Packe Spheres Saturate Wth Water. Transactons of ASME 6: 64. J. Flus Eng. 6 (): La, B.T.. Expermental Measurement an Numercal Moelng of Hgh Velocty Non-Darcy Flow Effects n Porous Mea. PhD ssertaton, Colorao School of Mnes, Golen, Colorao. Lopez. H.D. 7. Expermental Analyss an Macroscopc an Porelevel Flow Smulatons to Compare Non-Darcy Flow Moels n Porous Mea. PhD ssertaton, Colorao School of Mnes, Golen, Colorao. Martns, J.P., Mlton-Tayler, D., an Leung, H.K. 99. The effects of Non-Dary Flow n Proppe Hyraulc Fractures. Paper SPE 79 presente at the SPE Annual Techncal Conference an Exhbton, New Orleans, 3 6 September. MS. Montllet, A. 4. Flow Through a Fnte Packe Be of Spheres: A Note on the Lmt of Applcablty of the Forchhemer-Type Equaton. J. Flus Eng. 6 (): Pruess, K., Olenburg, C., an Mors, G TOUGH User s Gue, Verson.. Report LBNL-4334, Contract No. DE-AC3-76SF98, Lawrence Berkeley Natonal Laboratory, Berkeley, Calforna (November 999). Wu, Y.-S.. A vrtual noe metho for hanlng wellbore bounary contons n moelng multphase flow n porous an fracture mea. Water Resour. Res. 36 (3): WR9336. Wu, Y.-S.. Numercal Smulaton of Sngle-Phase an Multphase Non- Darcy Flow n Porous an Fracture Reservors. Transport n Porous Mea 49 (): Wu, Y.-S., Forsyth, P.A., an Jang, H A consstent approach for applyng numercal bounary contons for subsurface flow. J. Contam. Hyrol. 3 (3): Appenx A Relatonshp of Flow Rate vs. Pressure Graent Wth the Barree an Conway Moel The flow rate of D lnear flow wth the Barree an Conway equaton (Barree an Conway 4) for flu velocty s erve an prove n Eqs. A- through A-4. P v x (a-) k mr k kmr v Ths s smply a mofe Eq. A-: P v (a-) x kmr kmr v kmr k v Wrtng Eq. A- n a quarautc form gves P v k k x v k P x (a-3) Solvng for v n Eq. A-3, we obtan P P P kk kk 4 k v (a-4) Note that Eq. A- s applcable to both ncompressble an compressble flu flow, accorng to the Barree an Conway moel. Appenx B Steay-State-Flow Solutons for Incompressble an Slghtly Compressble Lqus an Gas We conser D, steay-state, non-darcy flow n a lnear porous meum accorng to the Barree an Conway moel. The governng equaton s ( v) (B-) The lnear-flow system wth a length of L an cross-sectonal area A s subject to () constant mass flux m an () constant pressure (P P ) at the outlet, x L. A soluton of Eq. B- wth the two contons s P P P kk kk 4 k x x m A (B-) Eq. B- can be further smplfe as m k A kk P m m x A A (B-3) Note that Eq. B-3 s a general soluton an s applcable for both lqu an gas flow accorng to the Barree an Conway non- Darcy moel. Incompressble Flu-Flow Soluton. For ncompressble snglephase lqu ( ), ntegratng Eq. B-3 an usng the bounary conton at x L, we obtan m k A kk P m m x A A m k A kk P m m A A L an Eq March SPE Journal

10 Slghtly Compressble Flu-Flow Soluton. For slghtly compressble sngle-phase lqu flow, [ cp ( P)] (B-4) Substtutng Eq. B-4 for Eq. B-3 an completng the ntegraton, we have the soluton (Eq. 9) for slghtly compressble flu flow. Gas-Flow Soluton. For sngle-phase gas flow, the ensty s escrbe by the real-gas law PM zrt (B-5) Substtutng Eq. B-5 nto Eq. B- an performng the ntegraton leas to P m k A kk PM zrt P m m A A x (B-6) P Then, the steay-state-flow soluton for sngle-phase gas s gven by P m k A kk M P m m P RT z A A x (B-7) P x Btao La s an assstant professor n the Petroleum Engneerng Department at the Unversty of Lousana at Lafayette n Lafayette, Lousana. She has more than 8 years of laboratory research experence an teaches a varety of courses, nclung phase behavor, reservor engneerng, an petroleum economcs. Her research nterests nclue reservor characterzaton, reservor rock mechancs, an multphase flow n porous mea. La hols an MS egree n geotechncal engneerng an a PhD egree n petroleum engneerng. She serves as a revewer for SPE journals an Petroleum Scence an Engneerng. Jennfer L. Mskmns s an assocate professor n the Petroleum Engneerng Department at the Colorao School of Mnes (CSM) n Golen, Colorao. Mskmns s the founer an Drector of the Fracturng, Aczng, Stmulaton Technology (FAST) Consortum at CSM. She teaches a varety of courses nclung completons an stmulaton classes, geologc fel camps, an petroleum economcs. Mskmns hols BS, MS, an PhD egrees n petroleum engneerng. She serve as the Executve Etor for the SPE Proucton & Operatons Journal from 8 an was a SPE Dstngushe Lecturer. Yu-Shu Wu s a professor an CMG Reservor Moelng Char n Petroleum Engneerng at CSM. He s also a guest scentst at the Earth Scences Dvson of the Lawrence Berkeley Natonal Laboratory. At CSM, he teaches reservor engneerng courses, supervses grauate stuents, an conucts research n the areas of multphase flu an heat flow n porous mea, enhance ol recovery (EOR) operaton, CO geosequestraton, reservor smulaton, enhance geothermal systems, an unconventonal hyrocarbon reservors. He hols BS an MS egrees n petroleum engneerng from unverstes n Chna, an MS an PhD egrees n reservor engneerng from the Unversty of Calforna at Berkeley. He serves as an Assocate Etor for SPE Journal. March SPE Journal 79