Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands 1

Transcription

1 PETROPHYSICS, VOL. 47, NO. 6 (DECEMBER 2006); P ; 21 FIGURES, 5 TABLES Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands 1 Jesús M. Salazar 2, Carlos Torres-Verdín 2, Faruk O. Alpak 2,3, Tarek M. Habashy 4, and James D. Klein 5 This paper describes the suc cess ful appli ca tion of a new petrophysical inver sion method to esti mate per me - abil ity from bore hole array induc tion mea sure ments. We con sider mea sure ments acquired in a North Lou i si ana tight gas sand for ma tion sub ject to water-base mud-fil - trate inva sion. The inver sion meth od ol ogy incor po rates the phys ics of two-phase immis ci ble dis place ment and salt mix ing between the invad ing water-base mud fil trate and con nate water. More over, the inva sion model hon ors the phys ics of mudcake growth as well as the petrophysical prop er ties that gov ern the pro cess of two-phase three-com - po nent flow. The out come of the inver sion is the abso lute per me abil ity for each flow sub unit within a gas-bear ing production zone. Rock for ma tions under con sid er ation con sist of low-per me abil ity amal gam ated sands. Array induc tion mea sure ments exhibit sig nif i cant ver ti cal fluc tu a tions within an indi vid ual fluid pro duc tion unit. In view of this, the esti ma tion of per me abil ity is designed to con sider the ABSTRACT effect of the num ber of lay ers and of their thick ness when describ ing a fluid pro duc tion unit. We show how the pro - gres sive addi tion of flow sub units improves the match of array induc tion mea sure ments within the lim its of ver ti cal res o lu tion. Accu rate recon struc tions of layer-by-layer per me abil ity are pri mar ily con strained by the avail abil ity of a-pri ori infor ma tion about time of inva sion, rate of mud-fil trate inva sion, over bal ance pres sure, cap il lary pres sure, and rel a tive per me abil ity. Sen si tiv ity anal y ses show that the esti mated val ues of per me abil ity prop erly repro duce the mea sured array induc tion logs even in the pres ence of small changes of rel a tive per me abil ity, cap il - lary pres sure, poros ity, and Archie s param e ters. The esti - mated val ues of per me abil ity agree well with those of core mea sure ments acquired from other wells in the same gas-bear ing for ma tion. Keywords: array induc tion, per me abil ity, inva sion, inva sion mod el ing, tight gas sands INTRODUCTION Tight sands com prise impor tant accu mu la tions of nat u - ral gas. Sim i lar to con ven tional oil- and gas-bear ing for ma - tions, tight gas sands are asso ci ated with com plex geo log i - cal and petrophysical sys tems that include heterogeneities at all spa tial scales. How ever, unlike con ven tional oil and gas res er voirs, tight gas sands usu ally exhibit unique gas stor age and pro duc tion char ac ter is tics. Effec tive pro duc - Manu script received by the Edi tor Feb ru ary 22, 2006; revised manu script received Octo ber 25, Orig i nally pre sented at the SPWLA 46th Annual Log ging Sym po sium, New Orleans, LA, June 26 29, 2005, paper FF. 2 The Uni ver sity of Texas at Aus tin, Depart ment of Petro leum and Geosystems Engi neer ing, 1 Uni ver sity Sta tion Mail Stop C0300, Aus tin, TX ; s: salazarjm@mail.utexas.edu, cverdin@mail.utexas.edu, Omer.Alpak@shell.com 3 Cur rently with Shell Inter na tional E&P 4 Schlumberger-Doll Research, Math e mat ics and Modeling Depart ment, 36 Old Quarry Road, Ridgefield, CT 06877; thabashy@ ridgefield.oil field.slb.com 5 ConocoPhillips, 600 N. Dairy Ash ford Road, Houston,TX 77079; Jim.D.Klein@conocophillips.com 2006 Soci ety of Petrophysicists and Well Log Ana lysts. All rights reserved. December 2006 PETROPHYSICS 527

2 Salazar et al. tion of these resources is only pos si ble with an accu rate descrip tion of key res er voir param e ters, par tic u larly per me - abil ity, cap il lary pres sure, rel a tive per me abil ity, and water sat u ra tion (Newsham and Rush ing, 2002). The rock for ma - tions con sid ered in this paper cor re spond to North Lou i si - ana low-per me abil ity tight gas sand units sub ject to water-base mud-fil trate inva sion. The main goal of the study is to esti mate abso lute per me - abil ity by quan ti fy ing the influ ence of mud-fil trate inva sion on bore hole array induc tion logs. This pro cess requires the avail abil ity of well logs, fluid prop er ties, and core mea sure - ments. Core lab o ra tory data are not avail able for the for ma - tion under anal y sis. Instead, we make use of lab o ra tory mea sure ments from nearby fields pen e trat ing the same geo - log i cal inter val to cal i brate the petrophysical prop er ties cal - cu lated from well logs and other empir i cal rela tion ships (S.A. Holditch Asso ci ates, 1988 and Luffel et al., 1991). Core mea sure ments are used to syn the size the petrophysical model nec es sary to sim u late the phys ics of mud-fil trate inva sion. Such a model is cal i brated against the exist ing suite of wire line logs, espe cially array induc - tion logs. Flow sub units defined from poros ity and the ini - tial guess of abso lute per me abil ity are taken as hor i zon tal lay ers to sim u late the pro cess of mud-fil trate inva sion with a two-dimen sional chem i cal flow sim u la tor that includes the effect of salt mix ing between mud fil trate and con nate water. We gen er ate spa tial dis tri bu tions of elec tri cal resis - tiv ity from the sim u lated cross-sec tions of water sat u ra tion and salt con cen tra tion using Archie s equa tion. In turn, these spa tial dis tri bu tions are used to sim u late array induc - tion imager tool (AIT 1 ) mea sure ments acquired across the for ma tion. Numer i cal sim u la tion of mud-fil trate inva sion has been attempted pre vi ously to sim u late bore hole elec tro mag netic mea sure ments and to esti mate petrophysical prop er ties. Semmelbeck and Holditch (1988) mod eled con ven tional induc tion resis tiv ity tools in syn thetic low-per me abil ity homo ge neous for ma tions. Tobola and Holditch (1991) and Yao and Holditch (1996) used a his tory-match ing approach to esti mate per me abil ity based on the one-dimen sional (radial) sim u la tion of time-lapse array induc tion resis tiv ity logs for water-base mud invad ing a gas-sat u rated low-per - me abil ity for ma tion. Salazar et al. (2005) esti mated per me - abil ity in sev eral field exam ples using the con cept of petrophysical flow units and resis tiv ity match ing of a ver ti - cally het er o ge neous tight gas sand for ma tion under the effect of water-base mud-fil trate inva sion. Non lin ear inver - sion of bore hole induc tion resis tiv ity mea sure ments has been applied suc cess fully using the phys ics of water-base 1 Mark of Schlumberger mud-fil trate inva sion for the esti ma tion of per me abil ity from one-dimen sional (radial) frac tional flow dis tri bu tions (Ramakrishnan and Wilkinson, 1999), as well as for the esti ma tion of poros ity and per me abil ity in two-dimen sional axisymmetric rock for ma tions (Alpak et al., 2006 and Torres-Verdín et al., 2006). In this paper, we esti mate abso lute per me abil ity using the inver sion algo rithm devel oped by Alpak et al. (2006), where esti ma tion of per me abil ity for each layer in a two-dimen sional rock for ma tion is posed as a non lin ear minimization prob lem. We con sider lim it ing val ues of rock and fluid prop er ties to quan tify the impact of the spa tial dis - tri bu tions of water sat u ra tion and salt con cen tra tion on array induc tion mea sure ments. Sen si tiv ity anal y ses are also per formed to assess the impact of the assump tions made about mud prop er ties, mudcake growth, time of inva sion, cap il lary pres sure, rel a tive per me abil ity, and fluid vis cos ity on the esti mated val ues of per me abil ity. The first stage of the study con sists of defin ing rock types by relat ing geo log i cal frame work, lithofacies, and petrol ogy to poros ity and per me abil ity. The sec ond stage of the work inte grates the rock type model with for ma tion eval u a tion data to define res er voir com part ments and flow units (Salazar et al., 2005). We inte grate logs and core mea - sure ments to extend the rock-type model, and to com pute con tin u ous stor age and flow capac ity spe cific to a given pro duc tion unit. The third stage of the work quan ti fies the influ ence of mud-fil trate inva sion on the spa tial dis tri bu tion of flu ids in per me able rocks around the wellbore. Over bal ance pres - sure along with the low poros ity of the rock con trib utes to deep inva sion of mud fil trate. In-situ gas sat u ra tion of the for ma tion ranges from 40 to 60%, with the remain ing pore space occu pied by irre duc ible water sat u ra tion. The salin ity of mud fil trate is between 1,500 to 3,800 ppm, whereas the salin ity of the con nate water is approx i mately 160 kppm. After 5 days of inva sion, spa tial dis tri bu tions of elec tri cal resis tiv ity are cal cu lated from the sim u lated spa tial dis tri - bu tions of water sat u ra tion and salt con cen tra tion. Sub se - quently, we sim u late array induc tion mea sure ments and com pare them against actual field mea sure ments. Sim u la tions of mud-fil trate inva sion con tinue with the only free param e ter being aver age abso lute per me abil ity per flow unit. All the remain ing petrophysical param e ters required by the sim u la tion are either cal cu lated from well logs or extrap o lated from the core mea sure ments. We use a mod i fied Timur-Tixier per me abil ity equa tion (Balan et al., 1995) to com pute ini tial val ues of abso lute per me abil ity. These ini tial per me abil ity val ues are pro gres sively adjusted in the sim u la tion of mud-fil trate inva sion until an accept - able match is reached between the mea sured and sim u lated array induc tion curves. In the final stage, we per form 528 PETROPHYSICS December 2006

3 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands TABLE 1 Sum mary of Archie s param e ters and rock and fluid prop er ties used to esti mate water sat u ra tion and porosity. Variable Units Value Archie s tortuosity fac tor a Archie s cemen ta tion expo nent m Archie s sat u ra tion expo nent n Con nate water resis tiv 210 F ohm m 0.02 Inva sion water resis tiv 210 F ohm m 0.56 Matrix den sity g/cm Shale den sity g/cm Water den sity g/cm Water-hydro car bon den sity (mix ture) g/cm automatic inver sion of per me abil ity from the AIT appar ent resis tiv ity curves using as a start ing model the results obtained from man ual adjust ment of layer per me abil ity val ues. GEOLOGICAL DESCRIPTION The North Lou i si ana tight gas sand for ma tion under anal y sis con sists of very fine- to fine-grained sand stone, shale, and some sandy, fossiliferous oolitic lime stone (Finley, 1984). Pri mary sed i ments orig i nated from a major influx of terrigenous clastic depos its dur ing the Early Cre - ta ceous. Flu vial depo si tion by at least two major rivers was respon si ble for sed i ment accu mu la tion. The rocks in this for ma tion are tex tur ally mature quartz arenites and subarkose sands (McGowen and Har ris, 1984). Inter pre ta - tion of a large area of the for ma tion sug gests global sed i - ment depo si tion by a sys tem of coalesc ing del tas prograding from the west, north west, and north (Finley, 1984). In gen eral, geo log i cal cross-sec tions through the for ma tion show a thick, sand-dom i nated wedge of sed i - ments mainly con sist ing of braided stream depos its. Such braided stream facies were reworked by marine trans gres - sion when the del tas entered the marine envi ron ment (Finley, 1984). ASSESS MENT OF WATER SAT U RA TION AND EFFEC TIVE POROS ITY We focus the petrophysical anal y sis to a 20.5-ft thick inter val. The for ma tion con sists of a rel a tively clean gas-sat u rated clastic sequence with con nate water salt con - cen tra tion in the order of 160 kppm. There fore, we use Archie s (1942) equa tion to com pute water sat u ra tion with - out spe cific adjust ments for the pres ence of shale. Water sat u ra tion is given by S n w R R w t a m, (1) where is effec tive poros ity, R w is con nate water resis tiv ity, R t is true for ma tion resis tiv ity, a is the tortuosity fac tor, and m and n are the cemen ta tion (lithol ogy) and sat u ra tion expo - nents, respec tively. Effec tive poros ity is com puted using a dual-fluid dual-min eral non lin ear model given by Rwxo a b n ( ) m R xo ( 1 C ) C sh, sh ma sh where b is bulk den sity, ma is matrix (quartz) den sity, sh is shale den sity, 1 is mud fil trate (fresh water) den sity, 2 is the den sity of the gas/mud-fil trate mix ture esti mated from log inter pre ta tion charts (Schlumberger, 1991), R wxo is the invaded zone water resis tiv ity, R xo is the invaded zone resis - tiv ity, and C sh is vol u met ric shale con cen tra tion cal cu lated with a lin ear trans for ma tion of the gamma-ray log. Appli ca tion of Thomas and Stieber s (1975) methodology to log-computed volumetric shale concentration and poros ity indi cated that the dom i nant dis tri bu tion of clay in the for ma tion was in the form of dis persed shale. We com puted an ini tial guess of effec tive poros ity ( o ) from shale-cor rected neu tron and den sity poros ity via the for mula (2) 1 o N sh 2 D sh 2 ( ) ( ), (3) 2 where N sh and D sh are shale-cor rected neu tron and den sity poros ity, respec tively. Equa tion (2) is solved by min i miz ing the dif fer ence between mea sured bulk den sity ( b ) and bulk den sity com puted with the ini tial guess of poros ity, i.e., min ( i ) 2 ( bi bi ), [ 01, ]. (4) The min i mum of equa tion (4) yields the best esti mate of effec tive poros ity. Water sat u ra tion is com puted with equa - tion (1). Table 1 describes the petrophysical and fluid param e ters used to apply this meth od ol ogy. In order to assess the value of true for ma tion resis tiv ity devoid of inva - sion and shoul der-bed effects, we sim u lated the AIT appar - ent resis tiv ity mea sure ments assum ing a hor i zon tal layer under go ing pis ton-like inva sion. The sim u lated AIT mea - sure ments were com pared to field mea sure ments and the val ues of for ma tion elec tri cal resis tiv ity adjusted until reach ing a good match between sim u la tions and mea sure - ments. This strat egy yielded reli able val ues of ini tial water sat u ra tion (S w ). Fig ure 1 is a plot of lithol ogy, resis tiv ity, and poros ity logs across the depth inter val under anal y sis. December 2006 PETROPHYSICS 529

4 Salazar et al. Fig ure 2 shows the log of water sat u ra tion com puted with Archie s equa tion and resis tiv ity val ues obtained with for - ward mod el ing of AIT appar ent resis tiv ity mea sure ments. We believe that the rel a tively high val ues of cal cu lated S w (40%-60%) are due to irre duc ible water sat u ra tion in the for ma tion that could be due to pres ence of clay and microporosity. Fluid pro duc tion in the well is mainly gas, hence con firm ing that the val ues of com puted water sat u ra - tion cor re spond to irre duc ible fluid. INITIAL MODEL FOR ABSOLUTE PERMEABILITY A per me abil ity value (ini tial guess) is nec es sary to ini tial - ize the sim u la tion of the pro cess of mud-fil trate inva sion. Core mea sure ments were only avail able from nearby gas fields that pen e trated the same for ma tion. Such mea sure - ments were used to con struct the ini tial guess of per me abil - ity. Sev eral approaches were con sid ered to cal cu late the best initial estimate of permeability (Balan et al., 1995 and Haro, 2004). Sensitivity analysis indicated that the generalized Timur-Tixier equa tion prop erly repro duced the inter play between porosity, permeability, and irreducible water saturation of core mea sure ments. Accord ingly, the rela tion ship between permeability (k, md), poros ity (, frac tion), and irre - ducible water saturation (S wirr, frac tion) is given by B k A, (5) S where A, B, and C are con stants to be deter mined from a multilinear regres sion anal y sis. Klinkenberg-cor rected per me abil ity and poros ity were obtained from the avail able core mea sure ments at 2,000 psi of over bur den pres sure. We used gas-water cap il lary pres - sure curves obtained from the same core mea sure ments to esti mate irre duc ible water sat u ra tion. This pro ce dure yielded the fol low ing equa tion for per me abil ity: C wirr k S wirr. (6) The degree of con fi dence for this equa tion is in the order of 60% based on the cor re la tion coef fi cient of the multilinear regres sion. We applied equa tion (6) to well logs using poros ity from equa tion (2) and irre duc ible water sat u - ra tion 1 com puted from (Dewan, 1983) 1 Equa tion (7) is only used as an input of equa tion (6) to com pute the ini tial guess of per me abil ity. Based on the expla na tion given in the pre vi ous sec tion, ini tial water sat u ra tion is con sid ered equal to irre duc ible water sat u ra tion. There fore, the remain der of our anal - y sis is car ried out assum ing such a sat u ra tion con di tion. FIG. 1 Well logs avail able for petrophysical assess ment. The top of the inter val under anal y sis is at 0 ft and the base at 20.5 ft. The left-hand track describes the gamma-ray log indi cat ing a clean sand inter val, the mid dle track shows the AIT appar ent resistivity measurements along with the simulated apparent resis tiv ity curves (AIT10 and AIT90), and the right-hand track dis plays poros ity from neu tron and bulk den sity logs with the continuous magenta curve describing effective porosity ren - dered by the two-fluid two-min eral non lin ear model. FIG. 2 Syn the sized petrophysical logs. The left-hand track describes the com puted shale con cen tra tion, the mid dle track describes water sat u ra tion in the flushed and vir gin zones along with an estimate of irreducible water saturation (calculated via Dewan s (1983) meth od ol ogy, and only used to com pute an ini - tial guess of per me abil ity), and the right-hand track shows the ini tial guess of per me abil ity obtained from a mod i fied Timur-Tixier model. 530 PETROPHYSICS December 2006

5 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands TABLE 2 Sum mary of aver age petrophysical prop er ties for the sin gle-layer case. Variable Units Value Thick ness ft 20.5 Effec tive poros ity frac tion Water saturation frac tion Shale concentration frac tion Abso lute Per me abil ity md S wirr C sh tsh, (7) where t is total poros ity and tsh is total shale poros ity. An alter na tive way to esti mate S wirr is via the gen eral rela tion - ship S wirr = C sh + / t, where and are con stants to be deter mined. We used equa tion (7) since it does not require knowl edge of addi tional empir i cal con stants and S wirr can be obtained directly from well logs. More over, the objec tive of equa tion (6) is to obtain only an esti mate of per me abil ity that will be used as an ini tial guess for the inver sion algo - rithm. Table 2 is a sum mary of the aver age petrophysical prop er ties cal cu lated for the inter val under study. Fig ure 2 dis plays the results of the petrophysical anal y sis with the main curves ren dered by the meth od ol ogy explained above. Finally, Fig ure 3 is a semilog cross-plot of per me abil ity ver - sus poros ity, indi cat ing that the esti mated per me abil ity in the well under con sid er ation exhib its a cor re la tion with poros ity sim i lar to that of core mea sure ments only in the low-poros ity zone. t 106 logarithmically spaced radial nodes that include shorter radial steps in the near wellbore region. Rock prop er ties are con sid ered con stant in the radial direc tion. In a sin gle-layer case, the ver ti cal grid con sists of thin numer i cal lay ers to ensure high accu racy in the esti ma tion of the flow rate of mud-fil trate invad ing the for ma tion. We make use of a Lorenz plot (Gunter et al., 1997) to iden tify indi vid ual flow units in the for ma tion under anal y - sis. Fig ure 4 shows one such plot, describ ing the rela tion - ship between cumu la tive poros ity and cumu la tive per me - abil ity as a func tion of res er voir thick ness. The vari able nature of this plot indi cates the pres ence of sev eral indi vid - ual flow sub units within the for ma tion. Spe cif i cally, each seg ment with a con stant slope in the plot iden ti fies an indi - vid ual flow sub unit with con stant petrophysical prop er ties. Steep slopes in the same plot are asso ci ated with high val - ues of flow capac ity. Fig ure 5 is an exam ple of the finite-dif fer ence grid used in the sim u la tion of mud-fil trate inva sion. Capillary pressure and relative permeability We use Brooks-Corey (Corey, 1994) two-phase equa - tions to assign water-gas cap il lary pres sure curves to each petrophysical layer. Accord ingly, the first imbi bi tion cycle of the cap il lary pres sure curve is given by P P e p ( 1 S ), (8) k c c o N where P c is cap il lary pres sure, P co is the coef fi cient for cap il - SIMULATION OF MUD-FILTRATE INVASION We use a multiphase chem i cal flow sim u la tor to cal cu - late the flow rate of mud-fil trate inva sion in a man ner sim i - lar to that of a water injec tion pro cess. The soft ware used is a mod i fied ver sion of the code UTCHEM devel oped by The Uni ver sity of Texas at Aus tin (Delshad et al., 1996). This soft ware is a finite-dif fer ence sim u la tor referred to as INVADE (Wu et al., 2004 and Wu et al., 2005) that was spe - cif i cally devel oped to solve the par tial dif fer en tial equa - tions and bound ary con di tions of immis ci ble cylin dri cal flow cou pled with mudcake growth. Radial grid and ver ti cal flow units Sim u la tion of mud-fil trate inva sion is per formed assum - ing cylin dri cal flow and per me abil ity iso tropy. A two-dimen sional finite-dif fer ence grid is con structed with FIG. 3 Graphical comparison of the initial guess of permeability (Timur-Tixier) and the per me abil ity of rock-core sam ples acquired from two dif fer ent wells in the same for ma tion. December 2006 PETROPHYSICS 531

6 Salazar et al. lary pres sure, e p is the pore-size dis tri bu tion expo nent, is poros ity, k is per me abil ity, and S N is the nor mal ized wet ting phase sat u ra tion, given by and e w rw rw N k k 0 S, (10) S N S w S wr 1 S S where S wr and S nwr are the resid ual wet ting and non-wet ting phase sat u ra tions, respec tively. Water-gas rel a tive per me - abil ity curves in the sat u rated zone are also esti mated via the Brooks-Corey equa tions, namely, FIG. 4 Mod i fied Lorenz plot used to iden tify indi vid ual flow subunits. The stor age capac ity curve [Cum( h )] does not exhibit sig nif i cant changes. How ever, the flow capac ity curve [Cum(k H )] dis plays abrupt changes in slope thereby indi cat ing the pres - ence of indi vid ual flow sub units within the same pro duc tion interval. wr nwr (9) 0 k k ( 1 S ), (11) rnw rnw N where k rw and k rnw are wet ting and non-wet ting rel a tive 0 0 permeabilities, k rw and k rnw are rel a tive per me abil ity end points, and e w and e nw are empir i cal expo nents for each fluid phase. We esti mated the three param e ters included in the Brooks-Corey equa tion (expo nents, coef fi cients and end points) from core mea sure ments acquired in a nearby field from the same rock for ma tion. Fig ure 6 is a graph i cal rep re - sen ta tion of cap il lary pres sure and rel a tive per me abil ity curves cal cu lated with the equa tions described above. INVADE results At the out set, the sim u la tion of mud-fil trate inva sion is per formed assum ing a homo ge neous flow unit. Petrophysical prop er ties of this flow unit are described in Table 2, and Table 3 describes the cor re spond ing mud fil - trate, mudcake and for ma tion flu ids prop er ties. The sim u la - tion of the pro cess of mud-fil trate inva sion requires accu - rate knowl edge of the time dur ing which the well was exposed to inva sion. Accord ing to field reports, the inter val under anal y sis was exposed to inva sion for a time period between three and six days before the acqui si tion of resis - tiv ity logs. Sev eral sim u la tion cases for three, four, five, and six days of inva sion were con sid ered to obtain the best esti mate e nw FIG. 5 Graph i cal descrip tion of the finite-dif fer ence grid used in the sim u la tion of mud-fil trate inva sion and array induc tion mea - surements. FIG. 6 Graph i cal descrip tion of the Brooks-Corey water-gas relative permeability and capillary pressure curves used to per - form the sim u la tion of mud-fil trate inva sion. 532 PETROPHYSICS December 2006

7 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands TABLE 3 Sum mary of mudcake, fluid, and for ma tion prop - er ties used in the sim u la tion of the pro cess of mud-fil trate invasion. Variable Units Value Mudcake reference permeability md 0.03 Mudcake reference porosity frac tion 0.30 Mud Solid Frac tion frac tion 0.06 Mudcake max i mum thick ness cm 1.02 Mudcake com press ibil ity expo nent frac tion 0.40 Mudcake expo nent mul ti plier frac tion 0.10 Mud hydro static pres sure psi 5,825 Ini tial for ma tion pres sure psi 5,000 Mud fil trate vis cos ity cp 1.00 Gas viscosity cp 0.02 Mud fil trate den sity g/cm Gas den sity g/cm Mud fil trate salt con cen tra tion ppm 3,600 Connate water salt concentration ppm 160,000 Wellbore radius cm Max i mum inva sion time days 5.00 For ma tion Tem per a ture F 210 For ma tion outer bound ary m 262 Residual water saturation frac tion 0.40 Residual gas saturation frac tion 0.10 of the rate of mud-fil trate inva sion. Fig ure 7 describes the tran sient behav ior of the flow rate of mud fil trate, mudcake thick ness, and pres sure across mudcake. The flow rate decays monotonically with time before reach ing its steady-state value. In low per me abil ity rock for ma tions (~ 1 to 5 md), lab o ra tory exper i ments (Dewan and Chenevert, 2001) and numer i cal sim u la tions (Wu et al., 2005) have proved that despite the fact that the flow rate is rel a tively high at the onset of the inva sion, it quickly tends to steady-state behav ior after the mudcake is com pletely formed. The max i mum mudcake thick ness is reached approx i mately 15 hours after the onset of inva sion. Once the mudcake is com pletely formed, the over bal ance pres - sure sta bi lizes at approx i mately 800 psi and the flow rate becomes con stant. The com puted flow rate is aver aged over the entire sim u - la tion time and input as a time-con stant value to sim u late the pro cess of inva sion. For the case of a ver ti cally het er o - ge neous flow unit the flow rate is esti mated sep a rately for each indi vid ual flow sub unit (petrophysical layer). A five-day inter val was cho sen as the opti mum time of inva - sion after study ing sim u la tion results for dif fer ent time intervals. For a long time of inva sion (more than three days), mudcake is already formed and the aver age flow rate is dom - i nated by its early time behav ior (less than 15 hours after the onset of inva sion). There fore, the error in the esti ma tion of flow rate for extended inva sion times is neg li gi ble. FOR WARD MOD EL ING AND INVER SION ALGO RITHMS The for ward prob lem con sists of sim u lat ing the pro cess of two-phase flow of water-base mud fil trate invad ing a par tially gas-sat u rated for ma tion. This prob lem is mod eled as con vec tive trans port of aque ous and hydro car bon phases, and com po nents of water, hydro car bon, and salt con cen tra tion (Alpak et al., 2003). Iso ther mal con vec tive mis ci ble trans port is assumed for the salt com po nent while dif fu sion between mud fil trate and con nate water is neglected. Upper, lower, and exter nal bound aries of the for - ma tion impose no-flow con di tions. A con stant flow rate, obtained as the time aver age of the flow rate yielded by INVADE (the con stant line shown in the upper panel of Fig - ure 7), is imposed at the bore hole wall for each numer i cal layer as a fixed source con di tion. Mod el ing of multi-phase and multi-com po nent fluid-flow is per formed with the ECLIPSE 1 com mer cial finite-dif fer ence res er voir sim u la - tor. Petrophysical prop er ties such as poros ity, ini tial water sat u ra tion, and ini tial esti mate of per me abil ity obtained from the petrophysical assess ment, along with core-cal i - brated Brooks-Corey cap il lary pres sure and rel a tive per me - abil ity curves are the main rock prop er ties input to the sim - u la tions (Table 2). In addi tion, mudcake prop er ties, hydro - 1 Mark of Schlumberger FIG. 7 Time evo lu tion of (a) flow rate of mud-fil trate inva sion, (b) mudcake thick ness, and c) pres sure across the mudcake. December 2006 PETROPHYSICS 533

8 Salazar et al. static and for ma tion pres sure, and fluid prop er ties are input to model the pro cess of mud-fil trate inva sion. Table 3 sum - ma rizes the for ma tion, fluid, and mudcake prop er ties used in the numer i cal sim u la tions described in this paper. The main out puts of the sim u la tion are the spa tial dis tri bu - tions of water sat u ra tion and salt con cen tra tion. Water sat u ra - tion is trans formed into elec tri cal resis tiv ity via the Archie (1942) equa tion. On the other hand, the spa tial dis tri bu tion of salt con cen tra tion is used to com pute equiv a lent val ues of water resistivity (R w ) using the equa tion doc u mented by Bigelow (1992) and Hallenburg (1998) with the cor re spond - ing Arps (1953) temperature conversion factor, namely, R w , (12) [ NaCl] T 6.77 where T is res er voir tem per a ture in o F, and [NaCl] is salt con cen tra tion in ppm. Equa tion (12) is an aver age approx i - ma tion of the three equa tions doc u mented by Worthington et al. (1990). They used least-squares regres sion to fit the val ues given on tables of brine con duc tiv i ties as a func tion of spe cific val ues of NaCl-brine con cen tra tion. Equa tion (12) matches the three resis tiv ity zones defined in Worthington et al. s work with an error below 2% for salin - ity between 500 and 100,000 ppm. For salt con cen tra tion between 100,000 to 230,000 ppm the error is 2-10%. In the res er voir under con sid er ation, salt con cen tra tion ranges between 1,500 and 160,000 ppm. There fore, we adopted equa tion 12 to con vert salt con cen tra tion to water resis tiv ity. Array induction resistivity modeling The next stage is the for ward mod el ing of array induc - tion mea sure ments from the spa tial dis tri bu tion of for ma - tion resis tiv ity. This requires the fre quency-domain solu - tion of Maxwell s equa tions. We per formed the numer i cal sim u la tion with the SLDMINV finite-dif fer ence algo rithm advanced by Druskin et al. (1999). This soft ware pro vides multi-fre quency sim u la tions of array induc tion mea sure - ments via the Spec tral Lanczos Decom po si tion Method (Alpak et al., 2003). Inver sion method for the esti ma tion of per me abil ity Inver sion of per me abil ity from array induc tion mea sure - ments is posed as the minimization of a qua dratic objec tive func tion sub ject to model con straints (Alpak et al., 2006). In this paper, we adopt the qua dratic objec tive func tion C( x) [ { e( x) } ( x x p ) ]. (13) 2 where 2 is the tar get data mis fit, and e(x) is the vec tor of data resid u als con structed as the nor mal ized dif fer ence between the mea sure ments and their sim u la tions, i.e., e( x) 2 M ( x) S j 1 m j 1 j 2, (14) In the above expres sions, M is the num ber of mea sure - ments, m j denotes the j-th mea sure ment, and S j is the cor re - spond ing sim u lated mea sure ment for the vec tor of unknown model param e ters, x. The lat ter vec tor is given by T x [ x1,, x N ], (15) where the supscript T rep re sents the trans pose and N is the num ber of unknown param e ters, in this case layer-by-layer val ues of abso lute per me abil ity. In equa tion (13), the pos i - tive sca lar fac tor is used as a sta bi li za tion param e ter (also called a Lagrange mul ti plier) to assign rel a tive impor tance to the two addi tive terms of the objec tive func tion. Finally, the vec tor x p con tains ref er ence val ues of layer per me abil ity used to bias the search for the min i mum of the qua dratic objec tive func tion. In our case, vec tor x p is con structed with the ini tial guess of layer per me abil ity val ues ren dered by log anal y sis (equa tion (6)). In this paper, the mea sure ment vec tor con sists of val ues of AIT appar ent resis tiv ity sam pled at 0.25-foot inter vals. There are five appar ent resistivities per sam pling depth and a total of 277 sam pling points in the depth inter val from ft to ft. Con se quently, the total num ber of input mea sure ments is 1385 includ ing mea sure ments above and below the flow unit shown in Fig ure 1 ( ft). Based on mea sure ment noise con sid er ations, the tar get mis - fit in equa tion (13) is set to We assume that the loca tions of layer bound aries are known from the pre vi ously described mod i fied Lorenz plots. More over, the inverted model param e ters are con - strained to remain within their phys i cal bounds using a non - lin ear trans for ma tion (Habashy and Abubakar, 2004). Minimization of the objec tive func tion is per formed with a Gauss-New ton method that enforces a back track ing line search algo rithm along the descent direc tion. This minimization strat egy guar an tees a monotonic reduc tion of the data mis fit from iter a tion to iter a tion. The choice of the Lagrange mul ti plier is adap tively linked to the con di tion num ber of the Hessian matrix of the qua dratic objec tive func tion (Alpak et al., 2004). ESTIMATION OF PERMEABILITY: FIELD EXAM PLE We con sider the field exam ple described in Fig ure 1 to test our inver sion meth od ol ogy. A homo ge neous sin PETROPHYSICS December 2006

9 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands gle-layer case was used as ini tial model to cal i brate the sim - u la tion of mud-fil trate inva sion and the resis tiv ity inver - sion. Sen si tiv ity anal y sis was per formed to assess the influ - ence of sev eral petrophysical prop er ties on the inverted per - me abil ity until obtain ing the best for ward model rep re sent - ing the for ma tion. Sub se quently, a het er o ge neous multi-layer case was con sid ered to esti mate layer-by-layer permeabilities within the fluid pro duc tion unit. Base case: homo ge neous pro duc tion unit Ini tially, we per form the sim u la tion assum ing that the fluid pro duc tion unit is a homo ge neous and iso tro pic sin - gle-layer for ma tion. To this end, ini tial water sat u ra tion was obtained from Archie s equa tion, poros ity from a non lin ear two-fluid two-mineral model, and the ini tial per me abil ity guess from the mod i fied Timur-Tixier equa tion. The coef fi - cient for cap il lary pres sure, P co, is set to 1.1 psi.darcy 1/2, and the expo nent for cap il lary pres sure, e p, is equal to 1.8. The water end-point for relative permeability, k rwo, is set to 0.9 and o for gas, k rnw, is set to 0.3. Finally, the water expo nent, e w, is set to 2.0 and the expo nent for gas, e nw, is set to 2.5. Fig ure 6 dis plays the water-gas cap il lary pres sure and rel a tive per me - abil ity curves. As observed from the rel a tive per me abil ity curves of Fig ure 6, crit i cal water sat u ra tion 1, S wcr, is on the order of 58% for this homo ge neous-for ma tion case. 1 Crit i cal water sat u ra tion is here defined as the value of water sat u - ra tion when rel a tive permeability of the wet ting phase is equal to that of the non-wet ting phase. Fig ure 8 shows spa tial dis tri bu tions (radial and ver ti cal direc tions) of water sat u ra tion, salt con cen tra tion, and elec - tri cal resis tiv ity within the rock for ma tion yielded by the sim u la tion of mud-fil trate inva sion. The assumed time of inva sion is five days. Inva sion reaches a radial dis tance of approx i mately 7 ft from the wellbore. Such a rel a tively deep inva sion sub stan tially influ ences the array induc tion mea sure ments. The cross-sec tions shown in Fig ure 8 are input to SLDMINV to sim u late array induc tion imager tool (AIT) mea sure ments with an ini tial guess of abso lute per - me abil ity equal to 1.5 md. Sim u la tion of AIT mea sure ments is per formed at every 0.25 ft ver ti cally along the bore hole. Results of such a sim u la tion are shown in Fig ure 9 along with the gamma ray log. On aver age, the sim u lated deep appar ent resis tiv ity (AIT 90) fol lows a sim i lar trend to that of the field data. Con versely, the shal low est appar ent resis - tiv ity curve (AIT 10) fol lows the trend of the field data only along the clean est inter val of the res er voir (4-12 ft) which is also the one that exhib its the larg est resis tiv ity val ues. Inter - me di ate appar ent resis tiv ity curves (AIT 20, AIT 30, and AIT 60) on aver age do not fol low the same trend of the field data. A radi ally smoother spa tial dis tri bu tion of elec tri cal resis tiv ity is nec es sary to match the inter me di ate AIT appar ent resis tiv ity curves. Two of the most impor tant param e ters that con trol the shape of the radial pro file of inva sion are cap il lary pres sure and rel a tive per me abil ity. Con se quently, a sen si tiv ity anal y sis is nec es sary to assess the influ ence of those two for ma tion param e ters on the sim - u lated spa tial dis tri bu tions of elec tri cal resis tiv ity. FIG. 8 Results of the sim u la tion of mud-fil trate inva sion at five days after the onset of inva sion. The cross-sec tions describe water saturation, salt concentration, and formation resistivity for the case of a ver ti cally homo ge neous fluid pro duc tion unit. FIG. 9 Com par i son of the sim u lated and mea sured array induc tion appar ent resistivities (right-hand panel) for the base case. The left-hand panel shows the gamma-ray log. December 2006 PETROPHYSICS 535

10 Salazar et al. Sensitivity of array induction measurements to capillary pressure and relative permeability We performed simulations of array induction measurements after enforcing small perturbations to the water-gas capillary pressure and relative permeability curves. Small perturbations (±10%) were performed to the capillary pressure coefficient, Pco, as well as to the wetting phase exponent, ew, for relative permeability. No significant changes were observed in either the spatial distributions of electrical resistivity or the simulated array induction measurements resulting from these perturbations. Thus, we proceeded to enforce yet larger perturbations (approximately -70% and +300%) of the same parameters. Figure 10 shows the simulated array induction measurements obtained for the cases when Pco = 0.35 psi.darcy1/2 (low capillary pressure) and Pco = 3.7 psi.darcy1/2 (high capillary pressure). Although such changes in capillarity are rather high, they scarcely affect the simulated apparent resistivity curves. One of the best ways to observe the effect of changes in relative permeability is by modifying the critical water saturation (Swcr). Figure 11 shows the simulated array induction curves for the cases when ew = 20.0 (Swcr» 80%) and ew = 2.0 (Swcr» 35%), in that order. In the second case (Swcr» 35%), it was necessary to change the values of irreducible and initial water saturation to achieve such an extreme shape in the relative permeability curves. For high values of FIG. 10 Numerical simulation of array induction measurements: sensitivity to capillary pressure. The left-hand panel shows simulation results obtained for the case of a rock formation whose capillary pressure is 70% lower than that of the base case. The right-hand panel shows simulation results obtained for the case of a rock formation whose capillary pressure is 300% higher than that of the base case. 536 critical water saturation we observe less splitting of the intermediate apparent resistivity curves. This observation indicates that the invasion front is relatively sharp and, therefore, does not entail a good agreement between the mea sured and sim u lated array induc tion appar ent resistivities. A more pronounced splitting of the simulated apparent resistivity curves is observed for the case of low critical water saturation, thereby resulting in a better agreement between the measured and simulated array induction curves. However, we remark that for a gas-saturated reservoir such as the one considered in this paper, it is unlikely to encounter relative permeability curves with such a low value of Swcr. Therefore, the base case value (ew = 2.0, Swcr» 58%) was used to perform the inversion. One important observation from this analysis is that large variations of relative permeability cause appreciable changes in the shape of the spatial distribution of electrical resistivity. Capillary pressure is also an important factor to be considered in the analysis. However, our sensitivity analysis shows that the effect of capillary pressure on the shape of the invasion profile is negligible compared to that of relative permeability. It is also pertinent to mention that neither of these two properties affects the radial length of invasion or the electrical resistivity in the near-borehole and virgin zones. Only the transition zone remains influ- FIG. 11 Numerical simulation of array induction measurements: sensitivity to relative permeability. The left-hand panel describes simulations performed for a strongly water-wet sand with critical water saturation 45% higher than that of the base case. The right-hand panel shows simulation results obtained for the case of critical water saturation 35% lower than that of the base case. PETROPHYSICS December 2006

11 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands enced by the changes of relative permeability and capillary pressure. Sensitivity of array induction measurements to permeability, porosity, Archie s parameters, and initial water saturation We performed sensitivity analyses to the initial guess of absolute permeability (k = 1.5 md) to assess its influence on the AIT apparent resistivity curves. Figures 12 and 13 show simulated apparent resistivity curves for the cases of 1 (base case, k = 1.5 md), 10, 0.1, and 100 times the initial guess of permeability. We observe a significant influence of small values of absolute permeability on the simulated resistivity curves. Specifically, the shallow apparent resistivity (AIT 10) increases with high values of permeability while the deep apparent resistivity (AIT 90) slightly decreases. A larger splitting of the intermediate apparent resistivity curves is also observed for high values of permeability. Figures 12 and 13 also indicate that the shallow apparent resistivity curves remain the most sensitive to changes of absolute permeability. Therefore, we conclude that absolute permeability strongly affects the spatial distribution of electrical resistivity. This observation applies not only to the electrical resistivity of the transition zone, but also to the radial length of invasion. Perturbations of porosity and Archie s parameters produce a uniform parallel shift of the five simulated apparent resistivity curves when plotted on a logarithmic scale. This effect contrasts with that due to perturbations of permeabil- FIG. 12 Numerical simulation of array induction measurements: sensitivity to the initial guess of permeability. The leftand right-hand panels describe apparent resistivity curves simulated for the cases of 1 and 10 times the permeability of the base case, respectively. December 2006 ity, where the shift of the simulated apparent resistivity is not uniform for all the AIT curves. For low values of cementation exponent (m), the simulated values of apparent resistivity uniformly decrease with respect to those of the base case. A similar situation occurs either with a decrease of the saturation exponent (n) or with an increase of porosity. For the problem at hand, we found that the simulated AIT apparent resistivity curves exhibited higher sensitivity to m than to either n or porosity. However, when the perturbations of m, n, and porosity were below 10% with respect to the values assumed for the base case, we observed no appreciable differences on the simulated apparent resistivity curves. Similar to the case of Archie s parameters, changes in initial water saturation entail a uniform parallel shift of the simulated AIT apparent resistivity curves. As expected from Archie s equation, low values of initial water saturation yield high values of resistivity and high values of initial water saturation yield low values of resistivity. For changes below 10% of initial water saturation the simulated AIT apparent resistivity curves do not change appreciable with respect to those of the base case. Based on this analysis, we conclude that large variations of initial water saturation (>10%) have an important influence on the shape of the radial profile of invasion. Furthermore, the influence of initial water saturation on the radial shape of invasion is the largest in the virgin zone. FIG. 13 Numerical simulation of array induction measurements: sensitivity to the initial guess of permeability. The leftand right-hand panels describe apparent resistivity curves simulated for the cases of 0.1 and 100 times the permeability of the base case, respectively. PETROPHYSICS 537

12 Salazar et al. Sensitivity of the estimated permeability to mudcake prop er ties and time of inva sion We per formed addi tional sen si tiv ity anal y ses to study the influ ence of var i ous mudcake param e ters on the esti - mated per me abil ity. Vari a tions of mudcake prop er ties entail vari a tions in the flow rate of mud-fil trate inva sion. Mudcake per me abil ity, max i mum mudcake thick ness, and mudcake com press ibil ity expo nent are the three param e ters that pri mar ily gov ern the flow rate of inva sion. These param e ters affect both the shape of the inva sion pro file and the radial length of inva sion. The higher the mudcake per - me abil ity the higher the flow rate; the lower the mudcake thick ness and com press ibil ity expo nent the higher the flow rate. Fig ure 14 describes the influ ence of mudcake per me - abil ity on the flow rate of inva sion. We observed that vari a - tions above 20% in these prop er ties entailed vari a tions over 15% of the flow rate of inva sion. In turn, these vari a tions of flow rate entailed vari a tions of one order of mag ni tude or less of the inverted val ues of for ma tion per me abil ity. Fig ure 15 dis plays the sim u lated AIT appar ent resis tiv ity curves for the base case assum ing two val ues of mudcake per me - abil ity. Per tur ba tions in other inva sion param e ters such as mudcake poros ity, mud solid frac tion, and mudcake expo - nent mul ti plier scarcely affected the cal cu lated flow rate of mud-fil trate inva sion. Time of inva sion was another impor tant param e ter con - sid ered in our sen si tiv ity anal y sis. We found that the effect of time of inva sion on the esti mated val ues of per me abil ity was sim i lar to that asso ci ated with vari a tions of mudcake per me abil ity assum ing a con stant time of inva sion. For dif - fer ences of time of inva sion of one day or less we observed vari a tions of half an order of mag ni tude of the esti mated per me abil ity. Vari a tions of the esti mated val ues of per me - abil ity remained mar ginal for vari a tions of the time of inva - sion of six hours or less. Base-case permeability inversion We first con sider the base for ma tion model. All petrophysical prop er ties are assumed known with the excep tion of abso lute per me abil ity. The ini tial per me abil ity guess (k = 1.5 md) entered to the inver sion was obtained from the mod i fied Timur-Tixier equa tion. Five Gauss-New - ton iter a tions were nec es sary to achieve con ver gence of the minimization pro cess, yield ing a value of abso lute per me - abil ity equal to 55.4 md. Fig ure 16 shows the sim u lated array induc tion mea sure ments asso ci ated with the inverted value of abso lute per me abil ity. The sim u lated AIT appar ent resis tiv ity curves fol low a trend sim i lar to that of the mea - sure ments. A clear split ting of the inter me di ate curves is also notice able. We observe a better agree ment between the mea sured and sim u lated AIT appar ent resis tiv ity curves as a result of the inver sion. The next stage of the anal y sis con - sists of improv ing the ver ti cal agree ment between the mea - sured and sim u lated AIT appar ent resis tiv ity curves by pro - gres sively increas ing the num ber of flow sub units in the rock for ma tion. This pro ce dure yields layer-by-layer esti - mates of abso lute per me abil ity. FIG. 14 Time evo lu tion of the flow rate of mud-fil trate inva sion for three val ues of mudcake per me abil ity. The base case cor re - sponds to k mc = md. In gen eral, vari a tions of k mc lower than 20% do not cause appre cia ble vari a tions of flow rate. FIG. 15 Numerical simulation of array induction measure - ments: sen si tiv ity to mudcake per me abil ity. The left- and right-hand pan els describe appar ent resis tiv ity curves sim u lated for the cases of vari a tions of -17% and +17% the k mc of the base case, respec tively. 538 PETROPHYSICS December 2006

13 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands TABLE 4 Summary of average petrophysical properties for the case of a multi-layer formation. h, ft f, frac. Sw, frac ko, md General case: vertically heterogeneous flow unit In order to account for vertical heterogeneities in the flow unit, we subdivided it into multiple horizontal layers. The modified stratigraphic Lorenz plot shown in Figure 4 was used to identify layer boundaries. This plot indicated that the maximum number of flow units was eight. Simulations of mud-filtrate invasion and array induction measurements were performed for cases with 2, 3, 4, 5, 6, 7, and 8 layers. In this paper we report inversion results only for the case of highest vertical resolution (8 layers). Initial values of porosity, water saturation, and permeability were calculated in similar fashion to the base forma- FIG. 16 Numerical simulation of array-induction measurements yielded by the inversion of absolute permeability. The left-hand panel is a cross-section of electrical resistivity and the right-hand panel compares the simulated and measured AIT apparent resistivity logs. December 2006 tion case and averaged across each flow subunit. Table 4 describes the thickness and petrophysical properties of each layer used in the description of the formation. Mudcake and mud filtrate properties remained the same as in the base case. However, flow rates of mud-filtrate invasion were computed specifically for each flow unit. The rate of mud-filtrate invasion is largely controlled by rock properties in low permeability formations (Wu et al, 2005). Because capillary pressure remains a function of irreducible water saturation, porosity, and permeability, the flow rate of mud-filtrate invasion is slightly different for each flow unit. Relative permeability, on the other hand, depends on irreducible water saturation and the Brooks-Corey parameters. Therefore, we assumed the same shape of relative permeability curves for all the layers but included layer-dependent variations of critical water saturation. Figure 17 describes the spatial distributions of water saturation, salt concentration, and electrical resistivity resulting from the simulation of mud-filtrate invasion. The invasion front reaches radially deeper zones in the formation when permeability is high. Figure 18 compares the corresponding simulated AIT apparent resistivity curves with the measured curves. We simulated the AIT apparent resistivity curves shown in Figure 18 using the initial permeability guess. As observed, the match between measured and simulated AIT apparent resistivity curves is not acceptable. The latter observation indicates that the best estimate of absolute permeability has not been reached from forward mod- FIG. 17 Cross-sections of water saturation, salt concentration, and electrical resistivity simulated for the case of a vertically heterogeneous flow unit. The time of invasion is five days. Layer permeabilities were assigned using initial permeability values obtained from petrophysical analysis (starting guess). PETROPHYSICS 539

14 Salazar et al. eling. Additional changes of permeability for each layer are needed to reach a better agreement between simulated and measured apparent resistivity curves. Estimation of layer-by-layer permeabilities We proceed to adjust the layer permeability values to improve the match between simulated and measured AIT apparent resistivity curves. This adjustment is guided by vertical variations of the resistivity curves. Simulation of mud-filtrate invasion is performed after each manual change followed by numerical simulation of array induction measurements. The process is repeated as many times as needed to reach an acceptable match between the simulated and measured AIT apparent resistivity curves. Figure 19 describes the spatial distributions of water saturation, salt concentration, and electrical resistivity simulated after several multiple changes of layer permeabilities. Figure 20 compares the simulated and measured AIT apparent resistivity curves. We observe that the simulated deepest (AIT90) and shallowest (AIT10) apparent resistivity curves agree well with the corresponding measured curves. However, the agreement between the intermediate apparent resistivity curves is not acceptable. We conclude that automatic permeability inversion is needed to improve the agreement between simulations and measurements. General case of permeability inversion FIG. 18 Numerical simulation of the array induction measurements: initial reservoir and fluids properties for the case of a multi-layer heterogeneous formation. FIG. 19 Cross-sections of water saturation, salt concentration, and electrical resistivity simulated for the case of a vertically heterogeneous fluid production unit. The time of invasion is five days. Layer permeabilities were manually assigned to reach an acceptable agreement between simulated and measured AIT apparent resistivity logs. 540 We implemented the inversion of layer permeabilities using as initial guesses the permeability values rendered by the manual adjustment process described in the previous section. The inversion was conducted in two steps. Initially, the permeability values yielded by the manual resistivity match ing were taken as the start ing guesses. Eight Gauss-Newton iterations were necessary to achieve convergence of the inversion. Permeability values obtained from FIG. 20 Simulation of array induction measurements: layer permeabilities were manually assigned to reach an acceptable agreement between simulated and measured AIT apparent resistivity logs. PETROPHYSICS December 2006

15 Estimation of Permeability from Borehole Array Induction Measurements: Application to the Petrophysical Appraisal of Tight Gas Sands TABLE 5 Sum mary of per me abil ity val ues inverted from array induc tion mea sure ments for the case of a multi-layer for ma tion. Dif fer ent col umns describe the per cent change of per me abil ity with respect to that of the ini tial guess. Resis tiv ity Matching % of Inver sion % of (md) change (md) change Uncertainty Low Low Low Low Low Mod er ate Very High High this inver sion were used to re-com pute the flow rate of mud fil trate with INVADE and then used as the new ini tial guesses for the sub se quent inver sion. Table 5 describes the val ues of layer per me abil ity yielded by the sec ond step of the inver sion after 8 iter a tions along with the cor re spond ing per cent age change when com pared to the Timur-Tixier per - me abil ity val ues. The devi a tion of the inverted permeabilities with respect to the ini tial guesses is as low as one-half order of mag ni tude and as high as four orders of mag ni tude. Table 5 gives a qual i ta tive indi ca tion of the level of uncer tainty of each esti mated value of per me abil ity (right-most col umn) based on the sen si tiv ity of the sim u - lated appar ent resis tiv ity curves to a small per tur ba tion of per me abil ity. Fig ure 21 shows the spa tial dis tri bu tion of elec tri cal resis tiv ity yielded by the inver sion, whereas Fig ure 22 com - pares the sim u lated and mea sured appar ent resis tiv ity curves. Even though most of the sim u lated AIT appar ent resis tiv ity curves (in sev eral lay ers) agree well with the field curves, addi tional work is needed to assess more real - is tic val ues of abso lute per me abil ity for those lay ers that exhibit mod er ate to high lev els of uncer tainty. We remark that lay ers close to the top and base of the fluid pro duc tion unit are highly influ enced by shoul der-bed effects. Some of the dif fer ences between sim u lated and mea sured appar ent resis tiv ity curves could be due to azi muthal vari a tions of elec tri cal resis tiv ity neglected by the sim u la tion model. Uncer tainty anal y sis Sen si tiv ity anal y ses were car ried out to assess the uncer - tainty of the esti mated flow rate of mud-fil trate inva sion. The inverted abso lute per me abil ity decreased by two orders of mag ni tude when the aver age flow rate was made equal to twice the value of the flow rate for the ini tial case. How ever, for 5 and 10 times the value of the ini tial flow rate, the inverted val ues of per me abil ity remained the same as those obtained for the case of 2 times the ini tial flow rate. For the case at hand, it is very unlikely for such large errors to be made in the esti ma tion of flow rate as the lat ter is mainly con di tioned by for ma tion per me abil ity. We observed that FIG. 21 Cross-section of electrical resistivity simulated for the vertically heterogeneous fluid production unit model after the inver sion of abso lute per me abil ity from array induc tion logs. The time of inva sion is five days. FIG. 22 Comparison of the measured and simulated array induc tion logs yielded by the inver sion of abso lute per me abil ity for the case of a ver ti cally het er o ge neous fluid pro duc tion unit. December 2006 PETROPHYSICS 541