Petrophysical Reservoir Characterization and Uncertainty Analysis

Transcription

1 Petrophyscal Reservor Characterzaton and Uncertanty Analyss Luz G. Loures Consortu or Research n Elastc Wave Exploraton esology, The Unversty o Calgary, Calgary, Alberta, Canada loures@ucalgary.ca and Fernando Moraes Unversdade Estadual do Norte Flunense, Macaé, Ro de Janero, Brazl ABTRACT A ethodology o petrophyscal reservor characterzaton s presented. The goal o ths ethodology s to provde a coplete soluton or reservor propertes deternaton, ncludng estaton and uncertanty analyss. The strengths o ths reservor characterzaton ethodology are: ) uncertanty analyss and ) ntegraton o ultple geophyscal data-sets, rock physcs analyss and pror noraton n a straghtorward way provded by the Bayesan raework. The nerence proble reported on n ths paper s orulated to solve the proble o porosty estaton. The sources o noraton are pre-stack sesc data, well log data and core saples. The waveor elastc pre-stack nverson s ncorporated n ths ethodology to access the porous edu physcal property noraton ro pre-stack sesc data. Geostatstcal odelng s ncorporated to access the porous edu spatal varablty noraton ro well-log data. The ethodology s pleented consderng a reservor coposed o block cells. One posteror pd s coputed or each reservor cell. Two cell volues contan the nal result. One s constructed wth the odes o the posteror pds related to each cell and represents the estated porosty odel, and the other s constructed wth the condence nterval o the posterors pds and represents a easure o the related uncertantes. Ths paper s gong to descrbe the general theory o the ethodology, the practcal pleentaton and results o tests wth a 3-D pre-stack sesc data set ( D Blackoot data). INTRODUCTION Despte the portant decsons that are ade ater the reservor quantcaton, the current reservor characterzaton practces oten al to account or the uncertantes assocated wth each pece o noraton used n ths process. oe uncertanty exaples are related to: dsturbances n the sesc data; data processng and nverson perored to obtan sesc attrbutes, the

2 dscrete earth odel and the rock physcs odels relatng attrbutes to petrophyscal propertes. Bascally, the goal s to copute a argnal posteror dstrbuton or the average porosty n each cell o the dscrete reservor odel. In ths case, we have dvded the data noraton nto two classes o data: -carryng noraton propagated ro the well locatons, whch are the experental varogra and -one carryng noraton ro the surace sesc data, whch are the sesc attrbutes. The waveor sesc nverson s ncorporated nto ths Bayesan orulaton va an elastc Bayesan nerence work ro pre-stack sesc data. Ths step ollows the work o Gouvea and cales (1998). The result o ths sesc elastc nerence s a jont noral posteror pd or the elastc paraeters o a 1-D layered edu gven an nput sesc gather. A lkelhood uncton or the sesc attrbutes that represents the beles about porosty ater the knowledge o the pre-stack sesc data s constructed wth the paraeters o these posteror pds (.e. the axu a posteror and the covarance atrx) together wth rock physcs odels and petrophysc observatons. To constructng a class o lkelhood that carres noraton propagated ro the well we dene the experental varogra as a data-set that carry the noraton regardng the spatal varablty o porosty. Fnally, the posteror pd or porosty s the product o these two lkelhoods and the pror pd. Ths posteror pd represents the beles about the cell porosty gven the knowledge o the pre-stack sesc data, whch carry the porous edu elastc noraton, and the knowledge o well-log data, whch carry the porous edu spatal varablty noraton. Ths text s gong to descrbe the theoretcal developent and soe tests. Bayes Theore BAYEIAN FORMULATION Consder the reservor odel coposed by a set o N block. The proble conssts o akng nerences about edan porosty or each cell: φ, =1, N, usng a set o data d and pror noraton I, whch s any noraton obtaned ndependently ro the data. Followng the Bayesan approach o nerence the soluton s gven by the posteror dstrbuton p(φ d,i). Ths uncton s the noralzed product o the pror pd and lkelhood uncton;

3 p ( φ d ) l(d φ) r( φ I ), (1) where r( φ I ) s the pror pd, l( d φ ) s the jont pd or the data, also known as the lkelhood uncton. In order to consder the posteror pd as the soluton o an nverse proble, the lkelhood ust be dened (.e. the relaton between the data d and the paraeter φ exsts and s known); and there are copatbltes between the pror understandng o the odel and the nal results,.e. l( d φ ) >0 or soe φ where r( φ I ) >0. Now t s necessary to dene the lkelhood uncton and the pror dstrbuton to access the posteror dstrbuton. Lkelhood Functon Ths work ollows standard steps to construct the lkelhood uncton, whch s suarzed by: - selectng the datasets whch carry noraton about φ; ndng atheatcal expressons relatng each type o porosty and I- denng statstcal odels (pds) or data dstrbutons, based on data uncertanty. Followng these three steps, let us dene the data set. Ths ethodology denes two types o ndependent data set: - a data-set o densty, p-wave and s- wave velocty sesc attrbutes, represented by R 3N assocated wth the N cells o the reservor odel and - a dataset carryng noraton about the spatal varablty o reservor porosty, whch s represented by v R L. v s a set o experental porosty-porosty varogra values coputed ro well log data. Consderng v and as statstcally ndependent datasets, the lkelhood uncton should be expressed as the product o two ndependent dstrbutons: l( d φ ) = l(v, φ) = l1(v φ) l( φ) ; () Data v dstrbuton l 1 (v φ) : Assung addtve errors n the data v, t should be wrtten as v = 1 ( φ ) + e1, (3) where e s a rando varable representng a set o addtve and ndependent errors. The error e ncorporates the uncertantes, whch are assocated wth the porosty estates n the well, the dscrete earth odel and the atheatcal unctons 1 that relates these data wth porosty. The odelng operator 1 s the varogra uncton ro geostatstcs. Next step s to establsh the crtera to select the probablty densty odels or l 1 (v φ) data pd. We chose to use the prncple o axu entropy to assgn probabltes and assue that the rst and second order oent noraton s

4 sucent to descrbe the errors. Accordng to these choces, the l 1 (v φ) data pd s noral and should be expressed as l (v 1 1, σ 1 ) = exp ) L 1 πσ1 σ 1 T [ v ( φ) ] [ v ( φ ], 1 φ 1 (4) and σ 1 consdered the unknown data error varance. Ths choce crteron to construct a lkelhood pd wll be used as a standard crteron elsewhere n ths work. esc attrbutes data dstrbuton l (s φ) : Let s rho, s p and s s be a set o vectors representng densty, p-wave and s-wave velocty sesc attrbutes respectvely. These data vectors should be represented as the su o a uncton o porosty, whch are deternstc varables, and an error coponent, whch s a probablstc varable s s s rho p s = ( φ ) + e 3 4 = ( φ ) + e = ( φ ) + e 3 4,, (6) The error e, =,3,4 ncorporate the uncertantes, whch are assocated wth the process o data acquston, data processng, the elastc nverson, the dscrete earth odel, and assocated wth the atheatcal unctons, =1,,3 that relate these data wth porosty. The next step s to dene the relatonshps between data vectors and porosty, represented by the unctons 1, and 3. For densty, we use 1 = ρ + φ( ρ ρ ); (7) where ρ s the atrx densty and ρ s the pore lud densty and are consdered unknown paraeters. For and 3 the choce s the rock physcs odels studed by Han et al (1986). ( φ ) = a + bφ + cγ ; (8) ( ) = a + b φ + γ ; 3 φ 3 3 c3 (9) where γ s the unknown clay content and a, b and c or =1, are the unknown regresson coecents. The set o equatons (6), ater the substtutng equatons (7), (8) and (9), should be treated as ult-regresson odel wth auto-correlated errors (Zelner, 1996). In rock physcs and elsewhere we encounter sets o regresson equatons and t s oten the case that the dsturbances are correlated. It s portant that non-ndependence o dsturbances ters be taken nto account n akng nerences. I ths s not done, nerences ay be greatly aected.

5 The l pd s dened as a noral dstrbuton and the regresson coecents o the rock physcs odels and the atrx and lud densty are unknown. We ade the choce to elnate these unknown paraeters o the Bayesan orulaton by a argnalzaton process and to ncorporate the noraton ro laboratory petrophyscal experents ollowng the Bayesan technque to predct a pd or a new observaton gven an old observaton. Followng ths Bayesan technque, the predctve pd or a vector o uture observaton s, whch s assued to be generated by the ultple regresson process speced by the set o equatons (6), s derved ater the knowledge o an old set o observatons, whch s assued to have the sae statstcal propertes o s. For now, consder an avalable set o sesc attrbutes related to the reservor cells and the assocated covarance atrx C s. Consder the exstence o hgh resoluton estates o porosty and clay volue X =[φ γ ] n soe poston where core saples are avalable. Let s consder = [s rho s p s s ] the sesc attrbutes estated or the cells at these core sapled poston. A ultple regresson process wth the set o equatons (6) can generate the vector, where the regresson coecents and the atrx and lud denstes are unknown varables but the petrophyscal propertes porosty φ and clay content γ, whch represent the control varables are known. The pd or these unknown paraeters (regresson coecents and atrx and lud densty), gven the sesc attrbutes and the assocated petrophyscal estates X should be represented by l + (a, b, c, ρ, ρ C s,, X ), or =1,. Now, let s consder a cell at soe poston o the reservor space wth the unknown porosty and clay volue x = [φ γ]. The jont pd or the sesc attrbutes s = [s rho s p s s ] and the unknown a, b, c ρ, and ρ (or =1,) assocated wth ths cell gven the knowledge o sesc attrbutes and the assocates core saples petrophyscal estates X s represented by l (s, a, b, c, ρ, ρ C s,,x,x). Ths pd should be decoposed as the product o two ndependent pds: l(s, a, b, c, ρ, ρ C,, X,x) =. =1,. (10) l (s,c, a, b, c, ρ, ρ,x) l ( a, b, c, ρ, ρ C,, X ) + One way o dervng the predctve pd s to wrte down the jont pd l (s, a, b, c, ρ, ρ,x,x), =1, and argnalze (.e. ntegrate) wth respect to a, b, c, ρ and ρ (or =1,) to obtan the argnal pd or s, whch s the predctve pd. Pror dstrbuton Ths work consders the denton o a non-data base pror dstrbuton (NDBP) descrbed by Jereys (1936) to access the pror dstrbuton. A NDBP s derved ro theoretcal knowledge o the physcal edu and ro the nvestgator s background experences. Followng the Bayesans ost conservatve practce to

6 dene the pror pd, consder that wth all prevous avalable noraton, the only vertable stateent s that porosty should all between 0 and 1 nterval. The use o a boxcar uncton s consstent wth expressng ths pror noraton. Posteror dstrbuton Wth the Bayes Theore appled, the posteror dstrbuton should be p( φ, γ, a, b, c, ρ, ρ v, s,c,, X, I ). = 1, (11) l (v φ) l (s, a, b, c, ρ, ρ C,, X,x) r( φ I ) 1 The regresson coecents and the atrx and lud densty are not the target o nvestgaton. Petrophyscal property nerence and assocated uncertanty s all that atters n ths proble. These unnterestng paraeters are elnated by ntegraton o the jont posteror. Ths procedure, whch s known n statstcs as argnalzaton o the jont dstrbuton, s appled and the argnal posteror pd s obtaned: p( φ, γ v, s,c d,, X, I ) p( φ, γ, a, b, c, ρ, ρ v, s,c d,, X, I ) da db dc ;=1,. (1) dρ dρ All nerence questons can be addressed to the posteror. For exaple, one can use the ean, edan or ode as estates or the nterval porosty and the standard devaton or condence ntervals as easure o uncertanty. REAL DATA EXAMPLE The Blackoot area was chosen or tests. The Blackoot area s located 15 k southeast o trathore, Alberta, Canada. The target rocks are ncsed valley-ll sedents, whch conssts o the lower Cretaceous sandstone o Glaucontc Foraton at the Blackoot Feld. In the Blackoot area the Glaucontc sands thckness vares ro 0 to over 35, at the depth o approxately 1,580 eters (between 1.0 s and 1.1). It s subdvded nto three phases o valley ncson. The lower and upper ebers are ade o quartz sandstone, 0.18 porosty average, whle the ddle eber conssts o low porosty tght lthc sandstone. The sesc elastc nerence: a ull waveor nverson: The rst step n ths ethodology s a sesc elastc nerence, whch provdes an elastc odel o the edu and the assocated covarance atrx C s. Ths work ollows the ethodology presented by Gouvea and cales (1998), whch developed a Bayesan orulaton or the pre-stack sesc nverse proble to estate elastc veloctes and densty or elastc pedances. In ther work, all uncertantes are descrbed by noral dstrbutons, but careul consderaton s taken to construct the covarance atrces.

7 Ths elastc nverson ethodology consders a 1D reservor wth n-layers. The layers thcknesses do not vary durng the nverson process. Elastc veloctes (P and -wave veloctes) and denstes ro layers o a target nterval are nverted ro a sesc gather. Accordng to the Bayes Theore the general orulaton o ths sesc nverse proble, can be wrtten as p( d,i ) l(d ) r( I ), (13) where p ( d, I ) s the resultng posteror pd or the sesc attrbutes and l (d ) and r ( I ) are, respectvely, the lkelhood and the pror pd. represents the elastc attrbutes and d the pre-stack sesc data. In denng the probablty odels, one proble arses due to the non-lnearty o the orward sesc proble d=g(s), where the g s the sesc odelng operator dened by the elastc relectvty ethod (Muller, 1985). Even when the pror pd and lkelhood are Gaussan, the posteror pd p( d) cannot be obtaned n closed or. The standard soluton s to rst to obtan optu estates Ŝ, then a Gaussan approxaton or p ( d) s constructed on the bass o the lnearzaton o the orward proble around pont Ŝ. When l (d ) and r ( I ) are both Gaussan, axzng probablty densty s equvalent to nzng the exponental arguent o the Gaussan, leadng to standard non-lnear least square proble. The process o optzaton by conjugate gradent used n Gouvea and cales (1998), and adapted by the purpose o ths work, can be suarzed by the ollowng expresson: + ηδ, (14) n+ 1 = n where δ n s the drecton o the nth teraton step and η s the step length. The gradent o the objectve uncton s gven by 1 d n Θ = G(s)C (d - g()), (15) where G() s the atrx o Frechét dervatves o the orward ter g(s) and C d s the data varance atrx. Ater copletng the optzaton process, the resultng Gaussan approxaton or the sesc attrbutes around Ŝ can be expressed by 1 p( d) exp - T [ ˆ 1 ] [ ˆ C ], s (16) where the covarance atrx s C s =[G T C d -1 G] -1 whch the Frechet dervatves evaluated at Ŝ.

8 The vertcal coponent o the 1995 Blackoot 3D-3C data-set was selected or ths exaple. A prevous processng has ollowed wth a pre-stack Krchho Te Mgraton. The requency range o the vertcal coponent s ro 0 hz to 115 hz and the bn nterval s 30 x 30. The physcal odel conssts o layers o 5 eters thckness nsde o the target nterval. The nverson ethodology was appled to each grated age gather o a sub volue o ths data-set. The Fgure 1 shows a secton o the elastc odel obtaned or the target nterval. Ths secton conssts o result o the nverson perored on all age-gathers o cross-lne 130. Fgure shows the poston o ths cross-lne (red lne) and well postons. Fgure 1: esc attrbutes densty, p-wave velocty and s-wave velocty, respectvely the ages ro top, eddle and botto. Fgure : The base ap o the area wth the poston o soe well logs and the cross-lne 130 (red lne). The petrophyscal nerence: The porosty nerence s appled on the target nterval along the cross-lne represented by Fgure 1. To copute l the clay content γ s consdered an a pror known paraeter. These paraeters are estated or the cells at nter well space usng the clay content estaton ro the gaa ray log and a varographc odelng. Then the posteror dstrbuton or porosty, expressed by Equaton (1), can be rewrtten as p( φ, γ = γ v, s,c,, X ). d The core saples data analyss ro the wells 1608 and 0808 were used to construct the atrx X and the elastc odel obtaned ro the age gather adjacent o these wells to construct the atrx. Usng a horzontal ovng wndow, runnng across the reservor secton, a dstrbuton l (Equaton 10) s calculated or a cell n the centre poston o each wndow (the data vector s s the sesc attrbutes ro cells allng nsde the wndow). The Fresnel Zone s consdered or denng the denson o the wndow. In the sae way, l 1 (Equaton 4) s also coputed or each cell poston.

9 Fnally, both dstrbutons are cobned by the applcaton o Equaton (11) to yeld one posteror pd or each cell o the reservor. Two volues o the dscretzed reservor represent the nal results. One shows the ode o the posteror pds, representng the nal estated porosty odel, and another shows the length o 0.95 posteror probablty centered at the ode, representng the assocated uncertanty odel. Three derent tests are perored usng derent aounts o noraton: usng only the varogra data v: p( φ v,, I ) l 1 ( φ, γ = γ v) r( φ I ) ; usng only the sesc attrbute data (s): p( φ, γ = γ s,c,, X, I ) l (s, a, b, c, ρ, ρ C,, X, x) r( φ I ) ; - usng both data types (v and s),.e. p( φ, γ, a, b, c, ρ, ρ v, s,c,, X, I ) l (v φ) l (s, a, b, c, ρ, ρ C,, X,x) r( φ I ) 1 The Fgure 3 shows the odel estated by the ode o these three tests respectvely Fgure 3a, b and c. On Fgure 3a we can see that the odel estated by the varogra data has relatvely hgh porosty values and shows us a sooth horzontal varaton. On Fgure 3b shows the channels wth porosty varyng between 0.10 and 0.. The odel showed on Fgure c s slar that on Fgure b. The Fgure 4 shows the length o the centered nterval havng 0.95 probabltes, correspondng to each one o the estates n Fgure 3. Ths gves a easure o the spread o the posteror pd and the resoluton or porosty o each cell o the reservor. Fgure 4a shows that the data v s ore noratve or cells on the rght porton o the secton (south). Ths porton has ore wells than the let porton o the secton (north, check the well postons on Fgure ). Fgure 4b shows that the noraton about porosty contaned n the sesc attrbutes s hoogeneously dstrbuted across the secton. Fro Fgure 4c we conclude that the an source o noraton coes ro the sesc data. CONCLUION The tests show reasonable porosty odels obtaned ro the ode o the posteror pds. The assocated uncertanty, represented by the length o 0.95 probablty ntervals, consstently vares dependng on the aount o noraton avalable. The varogra ttng procedure allowed descrbng the noraton about the porosty ro the wells at nter-well locatons. For the nverson o sesc attrbutes alone the level o uncertanty vares hoogeneously across the

10 odel. When cobnng varogra and attrbute data, we observe that the an source o noraton s the sesc data. Fgure 3: FIG 1: Iages o the reservor secton representng the odes o the Fgure 4: FIG 13: Iages o the reservor posteror dstrbutons or porosty by the use secton representng the length o the 0.95 o varogra data (a), sesc attrbutes (b) probablty nterval o the posteror and both datasets (c). dstrbutons obtaned ro the nverson o varogran values (a), sesc attrbutes data (b) and both datasets (c). ACKNOWLEDGEMENT We wsh to thanks the sponsors o CREWE, to CNPq; a Braslan Federal Agency or scentc and technologcal developent. We would lke to thank Wence P. Gouvea and the ebers o CREWE, specally or Han-xng Lu or help wth data processng and or the Proessors Gary Margrave or helpul dscussons. REFERENCE Gouvea, W. and cales, J. A., 1998, Bayesan sesc waveor nverson: Paraeter estaton and uncertanty analyss, J. Geophys. Res., 103, Jereys, H., 1936, Theory o Probablty: Oxord Unversty Press, London. Han, D. H., Nur, A. and Morgan, D., 1986, Eects o porosty and clay content n wave veloctes n sandstones. Geophyscs, 51, Muller, G., 1985, The relectvty ethod: a tutoral: Journal o Geophyscs, 58, Zellner, A., An Introducton to Bayesan Inerence to Econoetrcs, Wlley Intercence.