Methods fo histoy matching unde geological constaints Jef Caes Stanfod Univesity, Petoleum Engineeing, Stanfod CA 9435-222, USA Abstact Two geostatistical methods fo histoy matching ae pesented. Both ely on the sequential simulation pinciple fo geneating geologically sound ealizations. The fist method elies on petubing the sequential simulation though the petubation of the conditional distibution models; the second method elies on the petubation of andom numbes. We show that both appoaches ae geneal in the sense that a lage vaiety of geological scenaios can be geneated while histoy matching. Howeve, the conditional pobability method is moe efficient due to the ability to change the andom path duing the histoy matching pocedue. We demonstate these methods on two synthetic examples: a fist example demonstates how histoy matching can be pefomed unde a taining image based geological model constaint using multiple-point geostatistics; a second example shows how a combinations with an existing steamline-based histoy matching algoithm can povide efficient histoy matching yet maintaining geological consistency. Intoduction In the pactice of esevoi modeling, histoy matching efes to the pocess of obtaining a esevoi model based on a wealth of flow data such as well pessues, factional flow, flow mete o RTF data. Howeve the name histoy matching is pooly chosen since the objective is not to meely fit such data, but to calibate a pemeability model that is not only consistent to flow data but also to othe types of infomation. While histoic dynamic data is an impotant constaint to the esevoi model, it is often not fully constaining, a lage degee of feedom still emains. This equies the assumption of a pio model, o in esevoi tems, an assumption of geological heteogeneity o choice of geological scenaio. Many types of geological heteogeneities may lead to equally adequate histoy matched models. Howeve, neglecting to state a pio model is equivalent to assuming a maximally smooth pio hence often leads to ovely smooth pemeability models that have no pediction powe. Many authos have established this poblem but no single method exists that can histoy match unde a wide vaiety of geological scenaios. Most algoithms ae limited by the assumption of vaiogam-based spatial vaiability, o wose ely stongly on the assumption of multi- Gaussianity, which is aely adequate in heteogeneous esevois exhibiting stong connectivity o cuvi-linea coelations. In this pape we pesent two, simila, appoaches that meet the objective of epoducing a wide vaiety of geological constaints while histoy matching. Ou methodology is based on the concept of building esevoi models using the sequential simulation pinciple, which is eviewed fist. 8 th Euopean Confeence on the Mathematics of Oil Recovey Feibeg, Gemany, 3-6 Septembe 22
2 Sequential Simulation Sequential conditional simulation (eutsch and Jounel, 998) is one of the most used geostatistical algoithms. The eason fo this is CPU speed, obustness and the ability to geneate a wide vaiety of geological scenaios conditioned to static infomation. The pinciple is simple: any had (diect) infomation is assigned to a egula o iegula gid of choice, then each node along a pedefined andom path is simulated sequentially by andom dawing fom a conditional distibution model. Thee elements ae equied. A andom path visiting each node u 2. A set of sequentially estimated conditional distibutions (ccdf), temed P(A B) fom which values ae dawn. Each P(A B) models the conditional pobability of the value to be simulated, namely A, conditional to both had data and peviously simulated values, temed as B. 3. A seies of andom numbes v used to daw fom the ccdf models. The type of the distibution detemines the type of geological heteogeneity epoduced. Fo example in sequential Gaussian simulation (sgsim, eutsch and Jounel, 998) the ccdf P(A B)is Gaussian with mean and vaiance detemined by simple Kiging of the had data and peviously simulated nodes. Sgsim only equie the maginal distibution and covaiance function to be known, hence is limited in tems of the geological heteogeneity it epoduces (maximum entopy). A ecent new sequential simulation algoithm, temed snesim (single nomal equation simulation, Stebelle, 2), elies on a taining image fo the quantification of geological pattens. The method is staightfowad: the conditional pobability P(A B) is modeled by scanning the taining image fo eplicates of the joint data event (A,B). The ccdf P(A B) is estimated by calculating the fequency of the event A occuing given the event B occus. An efficient pocedue fo pefoming such scanning and estimation is outlined in Stebelle (22). Figue e shows an example taining image of ellipsoidal shapes, while Figues (a,b,c) ae thee ealizations constained to 5 well data. This method allows geneating a wide vaiety of geological scenaios, including non-object shapes (Caes and Zhang, 22) as well as conditioning to a wide vaiety of seismic data (Caes et al., 2; Jounel, 22). Petubing sequential simulations Any histoy matching method elies on a petubation mechanism that moves some geostatistical ealization to histoy match. Howeve, such petubation should not destoy the geological concept intended by the applied sequential simulation algoithm, be it vaiogam o taining image-based. Ou aim is to petub sequential simulation ealizations by petubing eithe. The pobability models P(A B) used to daw simulated values. 2. The seies of andom numbes used to daw fom P(A B) The fome is a novel appoach, the latte has been poposed by Hu et al. (2), but a new gadient-based vesion is pesented.
3 Petubing the pobabilities P(A B) Methodology Petubations of the conditional pobabilities should be such that the esulting sequential simulation honos the assumed pio geological concept expessed in the taining image. Sequential simulation can hono a wide vaiety of spatial constaints, such as the coelation stuctue between the vaiable simulated and some soft data (e.g. seismic) though co-kiging models o block aveage constaints though block-kiging. Howeve, sequential simulation cannot integate diectly with poduction data, which is obseved at a point (well) and moeove vaies in. The appoach taken in this pape is to tun poduction data into a spatial constaint and use sequential simulation to integate that constaint into a geologically sound ealization. Conside fo simplicity and claity of pesentation a binay indicato andom function I(u) which is defined as follows: if facies occus I( u) = if facies occus efine a non-stationay Makov chain on the entie set of I(u) "u, stating fom an ealization i (o) ( u ), geneating iteations ( i ) ( u) till convegence (histoy match). The poduction data (pessues, flow) is denoted by. We define a 2x2 tansition matix as follows ) ( ) ( ) P(I ( u) =,I ( u) = ) = P(I ( u) = ) = P(A) ) ( ) ( ) P(I ( u) =,I ( u) = ) = P(I ( u) = ) = ( P(A)) ) ( ) ( ) P(I ( u) =,I ( u) = ) = P(I ( u) = ) = ( P(A)) ) ( ) ( ) P(I ( u) =,I ( u) = ) = P(I ( u) = ) = ( P(A)) The tansition matix quantifies the pobability of switching state (=facies) given only the poduction data (not the geology). It is paameteized by a single paamete that be feely chosen. Note that in case = the tansition fom facies to facies is simply based on the pio P(A), i.e. the popotion of facies, which is equivalent to stating that a maximum change should be made at u, while = entails that no change should be made. Fom the tansition pobabilities we can easily deive the following conditional pobability ( ) P(A ) = ( ) i ( u) + P(A) which is the conditional pobability of dawing facies at any location u, given the poduction data and is expessed as a mixtue of the cuent ealization and the maginal popotion P(A) of facies. depends on the poduction data and is found though a simple optimization pocess as follows. Fist, we geneate ealization of the andom function I(u) using a sequential simulation method, e.g. snesim. Howeve, we use the pobability model P(A ) as a soft data set by ecombining the single conditional pobability models P(A B) andp(a ) into a joint conditional pobability P(A B,) using a Jounel s atio independence hypothesis (Jounel, 22): 8 th Euopean Confeence on the Mathematics of Oil Recovey Feibeg, Gemany, 3-6 Septembe 22 ()
4 x b d a -P(A B, ) -P(A B) -P(A ) -P(A) = with x =,b =,d =,a = b a P(A B, ) P(A B) P(A ) P(A) a P(A B, ) = = + x a+ bd The joint conditional pobability P(A B,) is used in sequential simulation as the conditional ) distibution, instead of P(A B). enote as i ( u) a single ealization dawn using P(A B,), then we find by solving the optimization poblem = min{ (f (t) f (t,i ( u))) opt o s ) 2 w w t wells w o s ) whee f w(t) is the obseved flow and pessue data and f w(t,i ( u)) the simulated flow and ) pessue on a given ealization i ( u). Typically five flow simulations ae equied to find the ) optimal value. The esulting ealization i opt ( u) will match bette the poduction data, yet opt will still hono the same geological pio model. The latte is guaanteed though Eq. (2) which combines the pobability P(A B) detemined fom the taining image with P(A ) which depends on the poduction data. To povide a lage space of possible ealizations, the andom path is changed at each iteation. In case of multiple (K) facies, each facies class s k can be modeled using a diffeent indicato andom vaiable then obtain a set of conditional pobabilities if facies class sk occus I( u,s k ) = else P(A ) = ( )i ( u,s ) + P(A ) with A ={i ( u,s ) = } ( ) ) k k k k k (3) (2) Application: Global petubations based on taining images The snesim algoithm is used to geneate a x efeence model in Figue A based on the taining image of Figue E. A full 5-spot configuation is used, wate injecting into oil at constant ate, poduces ae bottom-hole pessue contolled. Mobility atio is 5. The pemeability is 5m fo facies and 5 m fo facies, and is assumed known. The injecto is located in the middle, poducing well in bottom left, well 2 in bottom ight, well 3 in top left, well 4 in top ight. Figue (A) shows a good connection between the injecto and poducing wells and 4, a poo connection to poducing wells 2 and 3, exhibiting late beakthough.. Figue shows that a good match is obtained afte 8 iteations (about 4 flow simulations). The model and histoy matched model is shown in Figue B,C. The histoy matched ealization in Figue C maintains the geological continuity of elliptical shapes common to the efeence and model.
5 Petubation of andom numbes Methodology Hu et al. (2) popose to petub sequential simulations by petubing andom numbes. This method is equally geneal as the method above, although less efficient. Conside two andom vectos Y and Y 2 whose components ae all standad nomal and mutually independent. Fom these two vectos, a new vecto Y( ) is constucted as follows Y( ) = Ycos + Y2sin Consequently, Y( ) is also standad nomal and has mutually independent components, fo all possible. Based on Y( ),one defines a vecto V( )asfollows V( ) = G( Y( )) with G the standad nomal cumulative distibution function. All components of V( ) ae theefoe unifomly distibuted. Hence, fo vaious values one obtains a seies of vectos of unifom andom numbes v( ) that ae petubations of v() = v. Since any sequential simulation method uses a set of unifom andom numbes, one can geneate ealizations k ( u) that ae petubations of the model k ( u ). To obtain a histoy match using this method a simila objective function as Eq. (3) can be minimized. The method is often temed gadual defomation, although this tem is only stictly applicable in the case of sequential Gaussian simulation. Fo discete type vaiable (facies), in which case one would use the snesim and sisim algoithms, the petubation is actually a-gadual, i.e. the objective function O( ) is not a smooth function of the paamete. This method can be applied to any type of sequential simulation, howeve, is less efficient due to the fact that the andom path emains fixed duing the entie pocedue. This slows the etieval of the full space of ealizations available. A vaiable andom path, such as used in the pobability petubation method can seach moe efficiently the space of all possible ealizations. Application: Steamline histoy matching Applying the petubation of andom numbes method equies hundeds of flow simulation if applied to the example in Figue. This is due to the fixing of the andom path in the entie pocedue as outlined above. To avoid this amount of flow calculation, we apply a steamlinebased histoy matching method pesented in Wang and Kovscek (2). Wang and Kovscek pesent a histoy matching algoithm based on matching of flights (TOFs) along a set of steamlines in a black oil model. Fo a given poduce, the factional flow of the poduce is function of the -of-flight along all steamlines connecting to the poduce. Fo each poduce, the obseved factional flow is conveted into a set of -offlights fo all steamlines enteing that poduce. Since the of flight of a steamline can be witten as a diect function of the hamonic aveage of the pemeability along that steamline, the poblem of histoy matching is tuned in a poblem of matching hamonic aveages along steamlines. enote by k SL (k m ( u)) a hamonic aveage along a steamline SL m that depends on the pemeability field k ( u ). To histoy match, we solve the following optimization poblem 8 th Euopean Confeence on the Mathematics of Oil Recovey Feibeg, Gemany, 3-6 Septembe 22
6 with kslm = min{o( ) = (k k (k ( u ))) } (4) opt 2 SLm SLm all SLm the hamonic aveages detemined fom the poduction data (see Wang and Kovscek, 2). To solve this optimization poblem, no flow simulations ae equied. In ode to povide additional flexibility, the pemeability model can be petubed pe. A could fo example be defined as the set of all pemeabilities that ae hit by steamlines that ente a given poduce, hence thee ae as many s as poduces. Howeve, the petubation should be such that the geological model continuity is maintained acoss s. Instead of petubing pemeabilities diectly, andom numbes ae petubed diffeently in each, each petubation paameteized with an individual (Hu et al., 22). This guaantees geological continuity acoss s. A gadient-based optimization method can then be used to find the optimal set, =,,N (Caes, 22) of the poblem in Eq. (4). To demonstate the impotance of the zoning pocedue we develop a 9 spot case with 4 injectos and 5 poduces, as shown in Figue 2A. The efeence model is a 2x2 field shown in Figue 2A and is geneated using "sgsim". An guess is geneated in Figue 2B. A steamline simulato is un on the model to obtain TOF and steamline geomety. Using a simple intepolation pocedue, we constuct a set of s-of-influence pe poduce, as shown in Figue 2G. Using this zoning, we attach a paamete to each and find jointly set of paametes, =,,N that match best the hamonic aveages. The esult is shown in Figue 2. Fo compaison, Figue 2C shows a simila match to the hamonic aveages, but not accounting fo any geology. Afte fou iteations, a satisfactoy match is found, see Figue 2F. Note that the s vay with each iteation, Figue 2H shows the s afte the final iteation, which ae diffeent fom Figue 2G. Instead of having to evaluate possibly hundeds of flow simulations, we obtain a histoy match using only fou steamline simulations, yet a satisfactoy match is obtained as shown in Figue 2I. Refeences Caes, J., Avseth, P., and Mukeji, T., 2 Geostatistical integation of ock physics, seismic amplitudes and geological models in -Sea tubidite systems, The Leading Edge, 2, 3. Caes, J. and Zhang, T., 22. Multiple-point Geostatistics: a quantitative vehicle fo integating geologic analogs into multiple esevoi models, In: AAPG Memoi: Integation of outcop and moden analog data in esevoi models, (eds) Gamme, G.M et al., 22. Caes, J., 22. Efficient gadual defomation using a steamline-based poxy method. Stanfod Cente fo Resevoi Foecasting, Annual Meeting, no 5. Stanfod, Califonia, USA. Wang, Y. and Kovscek, T., 2. Steamline appoach fo histoy matching poduction data. SPE Jounal, 5,4. eutsch, C.V. and Jounel, A.G., 998. GSLIB: Geostatistical Softwae Libay, Oxfod Pess. Hu, L.Y., Blanc, G. and Noetinge, B., 2. Gadual defomation and iteative calibation of sequential stochastic simulations. Math. Geol., 33, 4. Jounel, A.G., 22. Combining knowledge fom divese souces: an altenative to taditional data independence hypotheses, Math. Geol., 34, 5. Stebelle, S., 22. Conditional simulation of complex geological stuctues using multiple-point statistics, Math. Geol., 34,.
7 (A) Refeence model.8.7.6.5.4.3 Well efeence match init ().35.3 5.5. Well 2 efeence match init. 5 2 3 4 steps (days) 2 3 4 steps (days) East (B) Iteation 8.5.4 Well 3 efeence match init.7.6.5 Well 4 efeence match init.3.4 No th 5 taining image East Initial model No th.3.. 2 3 4 steps (days) (E) 2 3 4 steps (days) (C) East 5 East Figue : (A) Refeence model, (B) Histoy matched model stating fom (C) Initial ealization, () histoy matched esults, (E) Taining image used 8 th Euopean Confeence on the Mathematics of Oil Recovey Feibeg, Gemany, 3-6 Septembe 22
8 poduces (A) Refeence model (B) Initial guess (G injectos 8 6 4 East East (C) Befoe geostat, ite ()Aftegeostat,ite 8 6 4 East East (E) Befoe geostat, ite 4 (F) Afte geostat, ite 4 (H 8 6 4 East East Well Well 2.8.8 (I).6.4.6.4.8 Well 5 2 4 6.8.6.4 Well 3 2 4 6 Well 4.8.6.4.6.4 2 4 6 2 4 6 2 4 6 Figue 2: (A) efeence model and location of wells, (B) model, (C) steamline petubations of the model without accounting fo geology, () with geology, (E) last iteation, not accounting fo geology, (F) histoy matched ealization, (G) s pe poduce based on the model, (H) s of the final model, (I) factional flow esults.