gradual deformation - Stochastic methods for history...

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Unformatted text preview: Stochastic methods for history matching subject to geological constraints ERE 284 Gradual deformation In general: gradual deformation guarantees geological consistency and allows seismic conditioning by construction 2 methods: • Linearly combining geostatistical realizations • Perturbing random numbers Linearly combining geostatistical realizations Two geostatistical models generated with Seq. Gauss. method Y(1) X cos(rD) Y=reservoir model Transformed into Normal scores + Y(2) X sin(rD) = Y(rD) • Has same mean and var • Has same variogram Idea: find rD that matches best production data Procedure I te ration 1 realizations I te ration 2 I te ration 3 I te ration 4 Algorithm 1. Generate two reservoir models using SGS 2. Transform the models into standard Gaussian pdf 3. Start outer loop 3.1 start Inner loop A. choose initial rD in [0,2π ] B. combine two realizations C. transform back into the data histogram D. run flow simulation E. Calculate objective function F. Set update rD value goto step B, unless done 4. Done when history match SGS=sequential Gaussian simulation When hard data is present Y(1) α1 Y(2) α2 Y Y(3) * 3 realization are required to honor any hard data Combine realizations Y=α1 Y(1) + α2 Y(2) + α3 Y(3) with parameters: 12 α1 = + cos rD 33 12 π α2 = + sin − + rD 33 6 α3 = α3 Simulator Obj . Function = f c w −f o w 12 π + sin − − rD 33 6 History matching in the reservoir modeling workflow Upscaled seed realization History matched coarse model Downscaled model match downscale Upscale upscale Downscaling simulated high permeability layer (non-uniform) High permeability layer maintained in upscaling Initial seed realization High permeability layer as informed by well-log data Final non-uniform model Flow Simulation Coarsen while HM Z(1) α1 Z (2) Non-uniform Z upscale sim ulator O bj . Fun. = f wc − f wo α2 α3 Z(3) Upscaling and HM m atch • Fast m atch • No posterior upscaling or downscaling Fine scale and coarse scale are kept in parallel during each iteration Case ­ 1 Injector Producer 0.6 0.5 0.4 fw 0.3 0.2 0.1 0 0 100 200 300 400 500 Time, days •Fixed Pressure •Mob. Ratio 1 Hard data at every 5 blocks •Range (horizontal) •Range (vertical) •log-mean •log-variance •kz/kx : : : : : 30 blocks 1 block ~4 ~1 0.3 Case ­ 1 2.5 Objective function 2 1.5 1 0.5 0 0 5 10 15 20 25 30 Number of iterations Obj. function decrease for 2 realizations Realization-1 Realization-2 Case ­ 1 0.6 0.5 0.4 0.3 0.2 reference initial_1 initial_2 initial_3 iteration_20 fw 0.1 0 0 100 200 300 400 500 600 time, days Case ­ 1 Reference After: Iteration 1 Iteration 5 Iteration 20 fine coarse Case ­ 1 Flow simulations on fine grid Hard data conditioned realizations Case ­ 1 Flow simulations on coarse grid History matched realizations Gradual deformation of sequential simulations Any sequential simulation r equires * A random path * A series of probability models (ccdf) at each node u = (x, y, z) * A vector v of random numbers to draw values k(u) at all u How to deform gradually a realization: 1. Perturb ccdfs, see previous talk, or 2. Perturb random numbers v v = G( y 1 ), or y 1 = G −1 ( v ) G = standard Gaussian cumulative distribution v(r) = G( y 1 cos r + y 2 sin r), y 1 , y 2 = vector of N(0,1) deviates use v(r) to generate k ( r ) ( u) Gradual deformation: SGS 100.000 r = ­π t = -180 100.000 r = ­π/2 t = -9 0 r = 0 t = -0 2.500 2.000 1.500 1.000 0.5000 0.0 100.000 2.500 2.000 1.500 1.000 0.5000 North North 0.0 -0.5000 -1.000 -1.500 -2.000 -2.500 -0.5000 -1.000 -1.500 -2.000 -2.500 0.0 0.0 0.0 r = π/6 t = 30 E a st 100.000 0.0 North 0.0 r = π/3 t = -30 E a st 100.000 0.0 r = π/2 t = 60 E ast 100.000 100.000 100.000 100.000 2.500 2.000 1.500 1.000 0.5000 2.500 2.000 1.500 1.000 0.5000 0.0 North North 0.0 -0.5000 -1.000 -1.500 -2.000 -2.500 -0.5000 -1.000 -1.500 -2.000 -2.500 0.0 0.0 E ast 100.000 0.0 0.0 NOTE: Random path constant for all realizations E ast 100.000 0.0 North 0 .0 E a st 100.000 The wheel of fortune 90 -y1 180 0 y1 270 Optimal blend -Y1 -180 -150 -120 -Y2 -90 -60 -30 Y1 0 Phase 30 60 Y2 90 120 150 -Y1 180 -y2 Objective function Gradual deformation: SNESIM 100.000 r = 0 t=0 Training image 150.000 facies 4 facies 3 North facies 2 North 0.0 0.0 facies 1 E a st 400.000 0.0 0.0 E a st 100.000 r = π/3 6 t=10 100.000 100.000 r = π/1 8 t=5 100.000 r = π/2 t=90 facies 4 facies 4 facies 4 facies 3 facies 3 facies 3 North North facies 2 North facies 2 facies 2 facies 1 facies 1 facies 1 0. 0 0.0 E a st 100.000 0. 0 0.0 NOTE: Random path constant for all realizations Ea s t 100.000 0. 0 0.0 E a st 100.000 Advantage * Works well for any type of geology * Can be used to perturb a model within zones Zonal deformation Geological body Zone 2 r2 v(r2 ) Zone 1 r1 v(r1 ) Zone 4 Recall: Sequential Simulation. Zone 4 r4 v(r4 ) 1. Random Path Zone 3 2. ccdfs r3 v(r3 ) 1. Random numbers I v( r1 ) = G( y 1 cos r1 + y 2 sin r1 ) v( r2 ) = G( y 1 cos r2 + y 2 sin r2 ) v( r3 ) = G( y 1 cos r3 + y 2 sin r3 ) v( r4 ) = G( y 1 cos r4 + y 2 sin r4 ) ⇒ v(r1 , r2 , r3 , r4 ) = {v(r1 ), v(r2 ), v(r3 ), v (r4 )} Fracture case Reference mo del 50.000 50.000 In itia l Guess 10 00.00 10 00.00 80 0.000 80 0.000 60 0.000 60 0.000 North 40 0.000 North 40 0.000 20 0.000 20 0.000 0.0 0 .0 0.0 0. 0 0. 0 East 50 .00 0 0. 0 East 50 .00 0 before geostat, iter 1 50.000 Kfracture=1500 mD Kmud = 50 mD 50.000 after geostat , iter 1 10 0.000 (A) Re ference model 50 .0 00 50 .0 00 (B) Initial Guess 1 000 .00 1 000 .00 8 00.0 00 8 00.0 00 6 00.0 00 6 00.0 00 4 00.0 00 4 00.0 00 2 00.0 00 2 00.0 00 Complete results North 0 .0 0.0 0 .0 North 0 .0 0 .0 East 5 0.00 0 0 .0 East 50.0 00 50 .0 00 (C ) be fore ge ostat, iter 1 1 00.0 00 50 .0 00 (D) a fter ge ostat, iter 1 N orth N orth Iteration 1 5 0.00 0 0 .0 f aci es 1 f aci es 0 0.0 East 5 0.00 0 0 .0 0 .0 East 50.0 00 ( E) before geostat, ite r 2 50 .0 00 50 .0 00 (F ) aft er geost at, iter 2 6 0.00 0 Nor th Nor th Iteration 2 5 0.00 0 0 .0 f aci es 1 f aci es 0 0.0 East 5 0.00 0 0 .0 0 .0 East 50.0 00 (G) before geostat , iter 5 50 .0 00 50 .0 00 (H) a fter ge ostat, iter 5 6 0.00 0 North North Iteration 5 5 0.00 0 0 .0 f aci es 1 f aci es 0 0.0 East 5 0.00 0 0 .0 0 .0 East 50.0 00 Fractional flow 1 0.8 0.6 fw 0.4 0.2 0 target initial 0 20 0 400 600 Time 800 1000 fw 0.4 0.2 0 target iteration 1 0 200 400 600 Time 800 1000 1 0.8 0.6 fw 0.4 0.2 0 target iteration 2 0 20 0 400 600 Time 800 1000 fw 0.4 0.2 0 target iteration 5 0 200 400 600 Time 800 1000 1 0.8 0.6 1 0.8 0.6 9­spot case 200. 2 Location of wells 5 160. Injectors 120. 3 Zones ? 80. 40. 0. 1 0. 40. 80. 120. 160. 4 200. Producers Reference/Initial (A) Reference model 20 0.000 20 0.000 (B) Initial guess 1 000 .0 0 1 000 .0 0 8 00.00 0 8 00.00 0 6 00.00 0 6 00.00 0 North 4 00.00 0 North 4 00.00 0 2 00.00 0 2 00.00 0 0 .0 0 .0 0 .0 0 .0 0 .0 East 20 0.0 00 0 .0 East 20 0.0 00 20 0.000 (C) Before geostat, iter 1 (D) After geostat, iter 1 20 0.000 Zoning (B) Initial guess 20 0.0 00 1 000 .0 0 1 000 .0 0 8 00. 00 0 8 00.00 0 6 00. 00 0 6 00.00 0 4 00. 00 0 North 4 00.00 0 2 00. 00 0 2 00.00 0 0 .0 0 .0 0 .0 0 .0 East 20 0.0 00 (D) After geostat, iter 1 20 0.0 00 1 000 .0 0 1 000 .0 0 8 00. 00 0 8 00.00 0 Comparedto reference (A) Reference model 20 0.0 00 20 0.0 00 (B) Initial guess 1 000 .0 0 1 000 .0 0 8 00.00 0 8 00.00 0 6 00.00 0 6 00.00 0 North 4 00.00 0 North 4 00.00 0 2 00.00 0 2 00.00 0 0 .0 0 .0 0 .0 0 .0 0 .0 East 20 0.0 00 0 .0 East 20 0.0 00 (C) Before geostat, iter 1 20 0.0 00 20 0.0 00 (D) After geostat, iter 1 1 000 .0 0 1 000 .0 0 8 00.00 0 8 00.00 0 (A) Reference model 20 0.000 20 0.000 (B) Initial guess 1 000 .0 0 1 000 .0 0 8 00.00 0 8 00.00 0 6 00.00 0 6 00.00 0 4 00.00 0 4 00.00 0 Complete results North 2 00.00 0 North 2 00.00 0 0 .0 0 .0 0 .0 0 .0 0 .0 East 200.0 00 0 .0 East 200.0 00 (C) Before geostat, iter 1 20 0.000 20 0.000 (D) After geostat, iter 1 1 000 .0 0 1 000 .0 0 8 00.00 0 8 00.00 0 Iteration 1 0 .0 6 00.00 0 6 00.00 0 North 4 00.00 0 North 4 00.00 0 2 00.00 0 2 00.00 0 0 .0 0 .0 0 .0 0 .0 East 200.0 00 0 .0 East 200.0 00 ( E) Before geostat, iter 4 20 0.000 20 0.000 (F ) After geostat, iter 4 1 000 .0 0 1 000 .0 0 8 00.00 0 8 00.00 0 Iteration 4 0 .0 6 00.00 0 6 00.00 0 North 4 00.00 0 North 4 00.00 0 2 00.00 0 2 00.00 0 0 .0 0 .0 0 .0 0 .0 East 200.0 00 0 .0 East 200.0 00 0.8 Fractional flow Well 1 1 0.8 0.6 fw 0.4 0.2 0 target initial iteration 4 1 0.8 0.6 0.4 0.2 0 target initial iteration 4 fw Well 2 0.6 fw 0.4 0.2 0 target initial iteration 4 0 200 target initial iteration 4 400 time Well 3 600 1 0.8 0.6 fw 0.4 0.2 0 0 0 200 target initial iteration 4 400 time Well 3 600 0 200 400 time Well 4 600 200 400 time Well 5 600 1 0.8 0.6 fw 1 0.8 0.6 fw 0.4 0.2 0 target initial iteration 4 fw 1 0.8 0.6 0.4 0.2 0 target initial iteration 4 0.4 0.2 0 0 200 1 400 time Well 5 600 0 200 400 time 600 0 200 400 time 600 ...
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