Using the Perpendicular Distance to the Nearest Fracture as a Proxy for Conventional Fracture Spacing Measures

Transcription

1 Usng the Perpendcular Dstance to the Nearest Fracture as a Proxy for Conventonal Fracture Spacng Measures Erc B. Nven and Clayton V. Deutsch Dscrete fracture network smulaton ams to reproduce dstrbutons of fracture spacng, devaton from local fracture orentaton, fracture ntensty, the number of fracture ntersectons and fracture length. Ideally, ths smulaton would compare target hstograms of true fracture spacng as measured by conventonal means to hstograms of fracture spacng n the models; however, for computatonal reasons, dfferent measures may be preferred. The perpendcular dstance between nearest neghbour fractures s one proxy for fracture spacng. Ths study uses smulaton studes to explore the valdty of usng the perpendcular dstance as a proxy for fracture spacng. Introducton Prevous work has resulted n the creaton of DFNSIM (Nven & Deutsch 2010a; Nven & Deutsch 2010b), whch s a program for dscrete fracture network (DFN) smulaton. DFNSIM s able to generate DFNs that honour hstograms of fracture spacng and relatve orentaton of fractures that are generated from outcrop or borehole data. DFNSIM generates an optmzed DFN by addng and removng fractures from the DFN at random and observng the effect of the change on an objectve functon. Among other factors, the objectve functon compares a hstogram of a measure of fracture spacng for the DFN to a target hstogram based on observed data. Fracture spacng s the orthogonal dstance between fracture planes and can be measured as the dstance between fractures along a scan lne, or down a borehole (Makel 2007; Twss & Moores 1992). For the purposes of ths artcle, fracture spacng measured n ths way s referred to as the true fracture spacng. An early verson of DFNSIM calculated relatve hstograms of true fracture spacng usng the prevously descrbed scan lne method. The methodology was as follows (See Fgure 1): 1. Vst a random locaton wthn the modellng area of nterest. 2. Propagate an magnary scan lne through the DFN at an orentaton that s perpendcular to the average fracture orentaton. 3. Measure the spacng between ntersectons of the fractures and the magnary scan lne. 4. Repeat N tmes (say N=1000), vstng a new locaton each tme. In ths way, DFNSIM would calculate true fracture spacng smlar to how a geologst mght. However, experence has shown that optmzng a DFN requres anywhere from 10 3 to 10 7 teratons dependng, on the requred fracture ntensty. Recalculatng the hstogram of fracture spacng (whch requres hundreds or potentally thousands of magnary samplng lnes) each tme the DFN s permuted becomes computatonally expensve. Thus, an alternatve measure of fracture spacng (or proxy) was requred. The methodology for DFN smulaton, encoded n DFNSIM, proposes that the average perpendcular dstance between a fracture and ts nearest neghbour s used as an alternatve measure of fracture spacng. Fgure 2 shows the calculaton of the average perpendcular dstance between a fracture and ts nearest neghbour. Snce fracture spacng s normally measured perpendcular to the fractures, the dstance between nearest neghbours s measured normal to the fracture planes. Note, that the perpendcular dstance between a fracture and ts nearest neghbour stll meets the defnton of fracture spacng from the lterature (Makel 2007; Twss & Moores 1992).The perpendcular dstances (d p1 and d p2 ) between each fracture s centrod and the other fracture are calculated. Note that d p1 d p2. Thus, the average of d p1 and d p2 s taken to be the perpendcular dstance to the nearest fracture. If the magnary ray from one centrod does not ntersect the fracture, the ntersecton s taken where t would have been f the fracture was nfnte n extents

2 Ths purpose of ths study s to establsh the valdty of usng the average perpendcular dstance to the nearest fracture as a proxy for the true fracture spacng measured usng scan lnes. Perpendcular Dstance as a Proxy for Fracture Spacng A DFN wth 1000 fractures was smulated n an area of nterest that s 1000 x 1000 m. All fractures have the same orentaton and are the same sze (30 m n length). A relatve hstogram of true fracture spacng was calculated and s shown n Fgure 3. The mean fracture spacng s 32 m. For the same fracture network, the average perpendcular dstance to the nearest fracture can be calculated as descrbed n the prevous secton. An addtonal challenge s that the average perpendcular dstance to the nearest fracture depends upon the bandwdth that s chosen (for more nformaton on the bandwdth, please see (Nven & Deutsch 2010b)). There s an nverse relatonshp between perpendcular dstance to the nearest neghbour and bandwdth. As the bandwdth ncreases, the average perpendcular dstance decreases. The relatonshp between bandwdth and mean perpendcular dstance to the nearest fracture s shown n Fgure 4. The hstogram of average perpendcular dstance for the DFN wth 1000 fractures, calculated usng a bandwdth of 30 m, s shown n Fgure 5. The mean perpendcular dstance to the nearest fracture s 8.1 m. One hundred DFN realzatons were smulated by usng the same settngs and varyng only the random number seed. For each realzaton the average true fracture spacng and average perpendcular dstance to the nearest neghbour were calculated. Fgure 6 shows a scatterplot of the average fracture spacng versus the average perpendcular dstance for each of the 100 DFN realzatons. The scatterplot shows zero correlaton between the average true fracture spacng and the average perpendcular dstance per realzaton. The effect of ncreasng and decreasng bandwdth was nvestgated but was not found to make a dfference to the results. Fracture Spacng Hstogram Reproducton Fgure 7 shows a map of lneaments from an area of Northern Alberta (Pana et al. 2001). The fgure shows two fracture sets, one trendng southwest-northeast and the other trendng southeast-northwest. The southwest-northeast fracture set was dgtzed for ths example. In total, there are 425 fractures from the SW-NE set that were dgtzed. The rght sde of Fgure 7 shows one of the fracture sets dgtzed. The relatve hstogram of true fracture spacng was calculated usng 1000 magnary scan lnes and s shown n Fgure 8 (shown wth blue damonds). Meanwhle, Fgure 9 shows the hstogram of perpendcular dstance to the nearest fracture for the dgtzed lneaments (also shown wth blue damonds). Next, the dgtzed lneaments are taken as the truth and the lneaments were modelled usng DFNSIM. Target hstograms of perpendcular dstance to the nearest fracture are bult from the dgtzed lneaments. The target, ntal DFN and fnal optmzed DFN hstograms of perpendcular dstance to the nearest fracture are shown n Fgure 9. Note the excellent ft between the target and fnal optmzed DFN hstograms of perpendcular dstance to the nearest fracture. The true fracture spacng hstograms for the ntal and fnal DFNs are also calculated, whch are shown n Fgure 8 along wth the target hstograms. The true spacng of the fnal DFN s only a slght mprovement over the ntal random DFN. An attempt was made to calculate optmzed DFNs usng dfferent bandwdths to calculate the perpendcular dstances n hopes of mprovng the hstogram match wth no success. Reducng the Problem to One Dmenson Fgure 10 shows a depcton of fractures perpendcular to a one-dmensonal lne at coordnate locatons x ( = 1,..., n). The ntersectons between the fractures and the lne, whch could be seen as a scan lne, defnes the true fracture spacngs: s, 2,..., = x x 1 = n (1) 106-2

3 The perpendcular dstances between fractures are also calculated as the mnmum of: 1) the dstance to the prevous fracture, or 2) the dstance to the next fracture. Or, mathematcally: p = mn( x x, x x ) (2) Note that the perpendcular dstance p s always less than or equal to s. 10,000 fractures were smulated randomly along the one-dmensonal lne, and then were sorted by x-coordnate locaton. The sorted fracture centrod locatons allow calculaton of spacng between them as n Equaton (1). The perpendcular dstances between each fracture and ts nearest neghbour are also calculated as n Equaton (2). Next, the average true spacng, along wth the average perpendcular dstance s calculated for ths realzaton of fractures. The process s repeated, untl 20 realzatons of fractures are generated, calculatng the mean true spacng and mean perpendcular dstance to the nearest neghbour for each realzaton. Fgure 11 shows the relatonshp between the mean values of true spacng and perpendcular dstance for the 20 realzatons. There s zero correlaton between the average true fracture spacng per realzaton and the average perpendcular dstance to the nearest neghbour. Indvdual dstances and spacngs, rather than the average spacng and perpendcular dstances, are examned next. Fgure 12 shows the relatonshp between ndvdual values of perpendcular dstance to the nearest fracture and true spacng between fractures for one of the 20 realzatons. Ths s a scatter plot of p vs. s where =2,,n. Note earler, p s. The correlaton between s and p s 0.5. Whle the correlaton of 0.5 between the true spacngs and the mean perpendcular dstances s sgnfcant, ths case only consders one dmenson (fractures on a lne). The relatonshp shown n Fgure 12 has a lower correlaton n two or three dmensons. Perpendcular Dstance to Nearest Fractures n Opposte Drectons It was thought that there mght be an mprovement n the correlaton of true spacngs and perpendcular dstances by consderng an alternatve measure of perpendcular dstance. For the next study, the average of two perpendcular dstances n opposte drectons was consdered. Fgure 13 shows the calculaton of ths modfed perpendcular dstance that consders opposte drectons. If the fractures occur along a 1 dmensonal lne, ths amounts to: [ x x 1] + [ x+ 1 x] x+ 1 x 1 D = average( x x 1, x+ 1 x) = =, = 1,...,1000 (3) 2 2 A smulaton study was conducted where 1000 fractures, f, are smulated along a 1D lne ( f, = 1,...,1000 ). Fracture locatons and orentatons are generated accordng the followng rules: x 1 = 0.5 x = x 1 + normnv(rand(), µ =1, σ=0.5), =2,...,1000 o.e. the prevous value s added to a value randomly selected from a normal dstrbuton wth mean = 1 and standard devaton = 0.5. All fractures are orented perpendcular to the lne The dea here s to smulate fractures on a one-dmensonal lne that have sem-regular spacng. The hstogram of true fracture spacng s shown n Fgure 14. Next, the perpendcular dstances that consder both drectons (Equaton (3)) are calculated. Fgure 15 shows the scatter plot of true spacngs versus perpendcular spacngs consderng opposte drectons. The correlaton coeffcent s 0.71, whch s an mprovement over the perpendcular dstance n one drecton. Then, the 1000 non-random fractures are used as the Truth and a modfed verson DFNSIM s used to model those fractures. The target hstogram of average perpendcular dstrbuton n opposte drectons s bult by the modfed verson of DFNSIM fractures are smulated wth the target hstogram of average perpendcular dstance n 106-3

4 opposte drectons. The target, ntal DFN and fnal DFN relatve hstograms of average perpendcular dstance n opposte drectons are shown n Fgure 5. Note the excellent match between the target and fnal DFNs. Fgure 17 shows the correlaton between true spacng and average perpendcular dstance for the ntal DFN (r = 0.707). Fgure 18 shows the same correlaton for the fnal optmzed DFN (r = 0.57). Note the drop n correlaton between the ntal and fnal DFNs. For one fnal fgure, consder Fgure 19, whch shows the fracture spacng for the truth fractures generated at the begnnng, along wth the true spacng of the ntal DFN and fnal DFN. The fnal DFN s an mprovement over the ntal DFN n terms of true fracture spacng, but s not even close to matchng the truth. Conclusons and Future Work Early attempts at DFN smulaton desgned to reproduce hstograms of true fracture spacng usng the scan lne methodology faled due to computatonal lmtatons. As a result, the perpendcular dstance between a fracture and ts nearest neghbour was ntroduced. Several smulaton studes were conducted to assess the valdty of usng the perpendcular dstance measure as a proxy to the true fracture spacng. There are dfferences between the two measures, but the proxy perpendcular dstance measure s deemed reasonable for dscrete fracture smulaton.. Future work should nclude re-vstng DFN smulaton usng the hstogram of true fracture spacng. Whle early attempts faled, there may be some effcences that can be mplemented n the code to reduce the computatonal burden. For example, t may be possble to re-use the same scan lnes wth every perturbaton n the DFN. As well, t may be possble to pre-calculate all the spacngs once, whch can then be saved n memory and re-used as necessary. References Makel, G.H., The modellng of fractured reservors: constrants and potental for fracture network geometry and hydraulcs analyss. Geologcal Socety, London, Specal Publcatons, 292(1), pp Avalable at: Nven, E.B. & Deutsch, C.V., 2010a. A New Approach to DFN Smulaton, Paper 102. In Twelfth Annual Report of the Centre for Computatonal Geostatstcs. Edmonton, Alberta: Centre for Computatonal Geostatstcs, Unversty of Alberta, pp Nven, E.B. & Deutsch, C.V., 2010b. DFNSIM: A New Program for Smulatng Dscrete Fracture Networks, Edmonton, Alberta: Centre for Computatonal Geostatstcs, Unversty of Alberta. Pana, D.I., Waters, J. & Grobe, M., GIS complaton of structural elements n Northern Alberta, Release 1.0, Alberta Energy and Utltes Board - Alberta Geologcal Survey. Twss, R.J. & Moores, E.M., Structural Geology, New York: Freeman and Company

5 Random Scanlne Fracture Spacngs Fgure 1: Fracture spacngs are calculated along a random scan lne, that s perpendcular to the average fracture orentaton. A nearby fracture Note: Fracture projecton to establsh ntersecton and perpendcular dstance d p2 Fracture under consderaton Fracture Centrod Fracture Centrod d p1 d = p, ave d p1 + d 2 p2 Fgure 2: The procedure for calculatng the perpendcular dstance to the nearest fracture Relatve Frequency Fracture Spacng (m) Fgure 3: Relatve hstogram of fracture spacng for a DFN wth 1000 fractures

6 Mean Perpendcular Dstance (m) Bandwdth (m) Fgure 4: The nverse relatonshp between bandwdth and mean perpendcular dstance to the nearest fracture for a DFN wth 1000 fractures. Relatve Hstogram Perpendcular Dstance to the Nearest Fracture (m) Fgure 5: Relatve hstogram of perpendcular dstance to the nearest fracture for a DFN wth 1000 fractures, usng a bandwdth of 30 m. 20 Average Perp. Dst. to Nearest Fracture per Realzaton R² = Average True Spacng per Realzaton Fgure 6: The mean fracture spacng versus the mean perpendcular dstance for each of 100 DFN realzatons

7 Fgure 7: Lneaments measured from satellte magery. Rght: Dgtzed lneament set. Relatve Hstogram Spacng of Dgtzed Fractures Spacng of Intal DFN Spacng of Fnal DFN Lnement Spacng (kms) Fgure 8: True fracture spacng hstograms for the dgtzed lneaments, ntal DFN and fnal (optmzed) DFN. Relatve Frequency Target Dst. Intal Dst. Fnal Dst Perp. Dst. to Nearest Lneament (kms) Fgure 9: Target, ntal DFN and fnal DFN hstograms for perpendcular dstance to the nearest fracture

8 Fgure 10: 11 one-dmensonal fractures, smulated along a lne. Mean Spacng Per Realzaton Mean Perpendcular Dstance Per Realzaton Fgure 11: Relatonshp between mean spacng and mean perpendcular dstance per realzaton. Spacng R² = Perpendcular Dstance to the Nearest Fracture Fgure 12: Relatonshp between ndvdual values of perpendcular dstance and spacng

9 Fgure 13: New method for calculatng the perpendcular dstance to the nearest neghbours. The perpendcular dstance s the average of the two dstances n opposte drectons. Relatve Frequency True Fracture Spacng Fgure 14: Relatve hstogram of true fracture spacng. 2.5 Avg. Perp. Dst. n Opposte Drectons R² = Actual Fracture Spacng Fgure 15: Correlaton between true fracture spacng and average perpendcular dstance n opposte drectons

10 1.40 Relatve Hstogram Target Intal Fnal Avg. Perp. Dstance n Opposte Drectons Fgure 16: Target, ntal DFN and fnal DFN hstograms of average perpendcular dstance n opposte drectons. Avg. Perp. Dst. n Opposte Drectons R² = Actual Fracture Spacng Avg. Perp. Dst. n Opposte Drectons R² = Actual Fracture Spacng Fgure 17 (left): True fracture spacng versus average perpendcular dstance n opposte drectons for the ntal DFN. Fgure 18 (rght): for the fnal DFN. Relatve Frequency Truth Intal DFN Spacng Fnal DFN Spacng True Fracture Spacng Fgure 18: True fracture spacng for the orgnal fractures (.e. the truth), the ntal DFN and the fnal DFN