SUMMARY INTRODUCTION. Figure 1: An illustration of the integration of well log data and seismic data in a survey area. Seismic cube. Well-log.

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1 Perophyscal propery esmaon from sesmc daa usng recurren neural neworks Moaz Alfarraj, and Ghassan AlRegb Cener for Energy and Geo Processng CeGP, Georga Insue of Technology SUMMARY Reservor characerzaon nvolves he esmaon perophyscal properes from well-log daa and sesmc daa. Esmang such properes s a challengng ask due o he non-lneary and heerogeney of he subsurface. Varous aemps have been made o esmae perophyscal properes usng machne learnng echnques such as feed-forward neural neworks and suppor vecor regresson SVR. Recen advances n machne learnng have shown promsng resuls for recurren neural neworks RNN n modelng complex sequenal daa such as vdeos and speech sgnals. In hs work, we propose an algorhm for propery esmaon from sesmc daa usng recurren neural neworks. An applcaons of he proposed workflow o esmae densy and p-wave mpedance usng sesmc daa shows promsng resuls compared o feed-forward neural neworks. INTRODUCTION Reservor characerzaon RC s he process of esmang perophyscal properes of he subsurface usng nformaon obaned from well-log, core, and sesmc daa. The goal of RC s o esmae perophyscal properes such as porosy, densy and permeably a any locaon and deph n a reservor. RC s a complex process due o he non-lneary and heerogeney of he subsurface. There s no clear mappng from sesmc daa o well-logs, and even f such mappng exss mgh no generalze well beyond he sudy area. Smply saed, he RC problem s fndng a funconal approxmaon from sesmc daa o well-log daa so ha log daa can be generalzed beyond well locaon o he enre reservor area. From a machne learnng perspecve, he goal s o ran an esmaon model on he sparsely avalable well-logs and her correspondng sesmc daa as llusraed n Fgure 1 such ha can esmae one or several well-logs properes a a gven locaon and deph/me usng sesmc daa a he same locaon. Then, he model can be used o generae a propery volume for he enre reservor area. Well-log Well 1 Sesmc Survey Sesmc cube Well 2 Well 3 Well 4 p Tme p Inpu: Sesmc cube Tme Oupu: Well-log Fgure 1: An llusraon of he negraon of well log daa and sesmc daa n a survey area. Alhough hs problem mgh seem o be a perfec seup for regresson algorhms such as suppor vecor regresson SVR, decson rees, and feed-forward neural neworks, here are many challenges ha preven such algorhms o fnd a proper mappng ha can be generalzed for an enre survey area. One of he challenges s he lack of daa from a gven survey area on whch a model can be raned, as we are lmed o he number of drlled wells n an area. For hs reason, such regresson algorhms need o have a lmed number of parameers and a good regularzaon mechansm n order o preven over-fng and o be able o generalze beyond he ranng daa. In addon, here are wo common mehods o model he problem so ha regresson algorhms can be used. The frs mehod s o rea each daa pon n a well-log n deph as an ndependen sample and ry o esmae s value from he correspondng sesmc daa samples. Ths mehod fals o capure he emporal dynamcs of well-log daa ha s he dependency of a daa pon a a gven deph on he daa pons before and afer. An alernave approach s o esmae he enre well-log a once from he correspondng sesmc daa o ncorporae he emporal dependency n deph/me of perophyscal properes. However, hs approach severely lms he amoun of daa from whch he algorhm can learn; because each well-log n hs scheme s reaed as a sngle ranng sample. Wh a lmed amoun of daa samples, common machne learnng algorhms wll fal o generalze beyond he ranng daa. Furhermore, sesmc daa are capured a lower resoluon han ha of well-log daa whch make hs problem even more dffcul. In order o remedy hs ssue, a daa preprocessng sep s requred before aempng o ran any machne learnng algorhms Chak e al., Several aemps have been made usng machne learnng and sascal learnng ools such as arfcal neural neworks, and suppor vecor regresson o solve he RC problem Al-Anaz and Gaes, 2012; Chak e al., 2015; Gholam and Ansar, 2017; Chak e al., The leraure shows grea promse for machne learnng algorhms for propery esmaon. However, mos regresson algorhms rea daa samples ndependenly such ha a predcon s made solely from he npu daa wh no nfluence from he oupus from daa pons before or afer he arge pon. Well-log daa exhb ner-log correlaons, such ha logs may follow ceran nrnsc paerns due o conssency n lhology n a gven sudy area. Furhermore, welllogs also exhb ner-log emporal correlaons,.e. correlaons beween propery samples for a gven deph range. In hs sudy, we propose he use of recurren neural neworks RNNs o capure he aforemenoned correlaons of wells logs n a gven survey area by modelng well-log daa as sequences n deph/me. The proposed workflow s raned and valdaed usng well-logs and her correspondng sesmc daa from he Neherlands offshore F3 block.

2 FEED-FORWARD AND RECURRENT NETWORKS Despe he success of feed-forward neural neworks for varous learnng asks, hey have her lmaons. Feed-forward neural neworks have an underlyng assumpon ha daa pons are ndependen and hus he nernal sae of he neworks s cleared afer a daa sample s processed whch would be fne, unless daa s no ndependen whch s he case for sequenal daa. Recurren neural neworks are a class of arfcal neural neworks ha can capure emporal dynamcs of sequenal daa lke me seres, audo and vdeo. Unlke feed-forward neural neworks, RNNs have a hdden sae ha can be passed beween sequence samples whch serves as memory allowng hem o capure very long emporal dependences n sequenal daa. RNNs have ofen been ulzed o solve many problems n language modelng and naural language processng NLPMkolov e al., 2010, speech and audo processng Graves e al., 2013, and vdeo processng Ma e al., A sngle layer feed-forward neural nework produces an oupu y whch s a weghed sum of npu feaures x followed by an acvaon funcon a non-lneary lke he sgmod or hyperbolc angen funcons,.e. y = σ Wx + b where x and y are he npu and oupu feaure vecors of he h sample, respecvely, σ s he acvaon funcon, W and b are he learnable weghs marx and bas vecor, respecvely. The same equaon s appled for all daa samples ndependenly o produce oupus. In addon o he affne ransformaon and non-lneary, RNNs nroduce a hdden sae varable ha s compued usng he curren npu and he hdden sae varable from he prevous sep, h = σ y = σ W xh x + W hh h 1 + b h, 1 W hy h + b y where x, y and h, are he npu, oupu, and sae vecors a me sep, respecvely, W s and b s are nework weghs, and bas vecors respecvely. For me = 0, he hdden sae varable s se o h 0 = 0. Fgure 2 shows a sde-by-sde comparson beween a feed-forward un and a recurren un. Feed-forward nework Hdden recurren nework Hdden Addonally, hey have more parameers o learn compared o feed-forward neworks. The problem was solved usng backpropagaon hrough me BPTT algorhms Werbos, 1990, whch urns gradens no a long produc of erms usng he chan rule. Theorecally, RNNs are supposed o learn longerm dependences from her hdden sae varable. However, even wh BPTT, RNNs faled o learn long-erm dependences manly because he gradens end o eher vansh or explode for long sequences as hey were backpropagaed hrough me. New RNN archecures wh more sophscaed acvaon funcons have been proposed o overcome he ssue of vanshng gradens usng gaed uns. Examples of such archecures are Long Shor-Term Memory LSTM Hochreer and Schmdhuber, 1997 and he recenly proposed Gaed Recurren Uns GRU Cho e al., Such archecures have been shown o capure long-erm dependency and perform well for varous asks such as machne ranslaon and speech recognon. In hs paper, we ulze GRUs n our proposed model o enhance he esmaon of perophyscal properes from sesmc daa. Gaed Recurren Uns GRUs supplemen he smple RNN descrbed above by ncorporang a rese-gae and an updae-gae varables whch are nernal saes ha are used o evaluae he long-erm dependency and keep nformaon from prevous mes only f hey are needed. The forward sep hrough a GRU s gven by he followng equaons, u = sgmod W xu x + W hu h 1 + b z r = sgmod W xr x + W hr h 1 + b r ĥ = anh W xĥ x + bĥ1 + r h = 1 u h 1 + u ĥ W hĥ h 1 + bĥ2 where z and r are he updae-gae, and rese-gae vecors, respecvely, ŷ s he canddae oupu, W s and b s are he learnable parameers, and s he elemen-wse produc. Noe ha n addon o he oupu sae, he GRU nroduces wo new sae varables, updae-gae u and rese-gae r, whch conrol he flow of nformaon from one me sep o anoher, and hus hey are able o capure long-erm dependency. Fgure 3 shows an example of a GRU nework unfolded hrough me. Noe ha all GRU s n an unfolded nework share he same parameers. 2 Hdden Hdden % %' %' %'* Inpu Oupu %, GRU GRU GRU GRU Fgure 2: An llusraon of feed-forward and recurren neworks. When RNNs were frs proposed n 1980s, hey were hard o ran because hey nroduced a dependency beween daa samples whch made he gradens more dffcul o compue. % %' %' %'* " # " # " # " # Fgure 3: Gaed Recurren Un GRU unfolded hrough me.

3 METHOD Daa Preprocessng Well-logs are acqured a a much hgher vercal resoluon han sesmc arbues whch requres a preprocessng sep n order o successfully ran an esmaon model and guaranee s convergence. One approach o preprocessng he daa s o regularze he logs by smoohng such ha boh he logs and sesmc arbues have comparable nformaon conen Chak e al., Ths s done by flerng log daa wh a low-pass fler o mach frequency conen of sesmc daa. Ths sep reduces he varaon of log daa n a small me wndow so ha he model can capure he overall rend of logs raher han he small hgh frequency varaons. Furhermore, he daa samples are normalzed such ha each log race has a zero mean and a un sandard devaon whch s a common sep before ranng a machne learnng model. Proposed Model In order o capure he ner- and nra- log correlaons as well as o esablsh a funconal approxmaon from sesmc o log daa, we propose a smple 2-layer recurren neural nework, namely a GRU, followed by a lnear regresson layer. As we have dscussed above, oupus of he GRU are a funcon of an affne ransform of he npus plus bas, whch can be seen as feed-forward nework by self. In addon, ulzes he updae-gae and rese-gae varables o mprove he nework s oupus a a gven me sep based on he neworks prevous saes. The proposed workflow s shown n Fgure 4. Sesmc cube Daa Preprocessng $ " # %& " # Gaed Recurren Un GRU %& " # %+ " # Gaed Recurren Un GRU %+ " # Regresson Layer! " # Tme ms Esmaed log Fgure 4: The proposed workflow wh 2 layer GRU and a regresson layer. For a gven well log, a sesmc cube s exraced around he well locaon o be used as an npu o ran he model. The sesmc cube s of sze p p T where p s he number of sesmc races n he nlne and crosslne drecons, and T s he number of samples n a race. Le x R p p T be he sesmc cube a locaon, and y be he log race a he same locaon. The model processes he daa sequenally n me such ha npus he sesmc slce a me, x R p p, and he sae varables of boh GRUs a me 1, h1 1 and h2 1, n order o compue he oupu sae varables a me. The regresson layer hen akes h2 and compues he esmaed propery a me, ỹ. If he sample o be predced s he frs sample n he log = 0, sae varables are se o zero. The process s hen repeaed o esmae he enre propery race. Durng he ranng of he model, when all he N logs n he ranng daase have been esmaed as ỹ, = 1,...,N, hey are compared o he measured log y, = 1,...,N usng Mean Squared Error MSE loss funcon. The error s hen used o compue he gradens and o correc he model s parameers usng BPTT. Afer proper ranng, he model s performance s assessed on he valdaon daase by compung he Pearson correlaon coeffcen beween he esmaed logs and he measured logs. The Pearson correlaon coeffcen s compued as, y ȳ ρ = y ȳ 2 EXPERIMENTAL EVALUATION ȳ ỹ ỹ ỹ 2. 3 The daase conans 4 wells, F021, F032, F034, and F061 from he Neherlands offshore F3 block. For each of he wells, we exraced a sesmc cube of 7 7 races cenered a he well p = 7 as n Fgure 1. The proposed workflow s hen raned usng sesmc cubes as npus and a sngle propery log from he well-log daa. In our expermens, we raned wo dencal neworks, one o esmae densy and he oher o esmae p- wave mpedance, boh of he neworks are smlar o he one shown n Fgure 4. Due o he small sze of he daase, ranng regularzaon s needed o ensure ha he model does no over-f o he ranng daa. One such echnque s early soppng n whch he ranng s sopped afer a small number of epochs. More ranng epochs wll defnely mprove he performance of he model on he ranng daase, bu he model wll fal o generalze. In addon, we used daa augmenaon by usng mulple roaons of he sesmc cubes along he me axs o ncrease he number of he ranng samples. The model n Fgure 4 wh a 2 layer, 32-feaure hdden sae varable GRU was esed on he daase descrbed above. In addon, he same daase was used o ran a 2-layer, 32-neuron feed-forward neural nework. The performance of he models s hen assessed usng 4-fold valdaon, where hree of he wells are used for ranng and he remanng well s used for esng. The process s repeaed 4 mes, and he resuls are averaged for all expermens. The resuls are summarzed n Table 1. The resuls show ha even wh a small daase, he recurren neural nework can esmae log daa from sesmc daa wh much hgher correlaon han he feed-forward nework. Noe ha he feed-forward nework was no able o ran properly on such a small daase. Feed-forward Recurren Propery Tranng Valdaon Tranng Valdaon P mpedance Densy Table 1: Correlaon coeffcen beween esmaed and measured properes.

4 Fgure 5 shows a scaer plo of he measured densy and he esmaed densy usng he proposed workflow for ranng and valdaon daases. We can see ha he esmaed densy vares almos lnearly wh respec o he measured densy. Fgure 6 shows examples of esmaed densy logs usng he proposed workflow. 650 Tranng log Esmaed Measured 650 Valdaon log Esmaed Measured s worh nong ha a problem as dffcul as propery esmaon mgh need a more complex and deeper learnng model; however, he number of model parameers ncrease wh complexy and hus much more daa s requred o ran such models properly. The goal of hs expermen was o show he power of recurren neural neworks for propery esmaon by ulzng her emporal dependences, compared o he feedforward neural neworks whch rea daa samples ndependenly. 3 Tranng log Tme ms ,000 Tme ms , ,050 1,050 Esmaed densy Normalzed ,100 1,150 1,200 1,250 1,300 1,100 1,150 1,200 1,250 1, Esmaed densy Normalzed Measured densy Normalzed Valdaon log Measured densy Normalzed Fgure 5: Scaer plos of measured densy and esmaed densy from he ranng and valdaon daases Normalzed densy Normalzed densy Fgure 6: An example of measured densy and esmaed densy logs from he ranng and valdaon daases. CONCLUSIONS In hs paper, we proposed a machne learnng algorhm for well-log propery esmaon from sesmc daa usng recurren neural neworks. The proposed workflow was valdaed usng 4-fold valdaon for densy and p-wave mpedance esmaon from sesmc daa. Alhough he ranng was carred ou on a small daase, he valdaon resuls ndcae a grea poenal of recurren neural neworks for reservor characerzaon. Wh a larger daase for ranng, he model could be used o generae propery volumes for a survey area from sesmc daa. ACKNOWLEDGEMENTS Ths work s suppored by he Cener for Energy and Geo Processng CeGP a Georga Insue of Technology and Kng Fahd Unversy of Peroleum and Mnerals KFUPM.

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