Uncertainty Quantification of a Fractured Reservoir

Using Distance-based Method and Streamline Simulation

Summary report

08/20/2010

Changhyup Park

1. RESEARCH OBJECTIVES

Evaluate the factors affecting cumulative oil production in fractured reservoir

Determine the applicability of distance-based method on fractured reservoir

Sensitivity analysis using distance-based method comparing with typical experimental

design

Sampling reliability of DKM (Distance Kernel Method)

Correlation between 3DSL and ECLIPSE runs

2. DESCRIPTION OF WORKFLOW

Fig.1 Workflow for uncertainty quantification in this study.

3. RESULTS & DISCUSSION

3.1 Model Generation using S-GeMS

The synthetic discrete fracture models in 2D system(100x100x1, ftzyx 10 ) are

generated using ‘Ti Generator’ of S-GeMS. We use two different objects and two distribution

type(i.e. density map and random distribution, see Fig. 2). The density map was made using

‘SGSIM(Sequential Gaussian Simulation)’ module as well.

The object, 1O is various and distributed diagonally with the constraints in Table 1 while the

property of the other, 2O is fixed and located dominantly in the horizontal direction; their

constraints are 1) distribution map, 2) the length, 3) the ratio of matrix and fracture volume, and

4) the orientation, respectively. As shown in Table 1, the number of cases is 72(i.e. 2x2x3x6=72)

and each case has 10 different generations and thereby there are 720 different models.

(a) (b)

Fig. 2 Map to explain (a) the distribution characteristics of 1O and 2O , and (b) the density map.

Table 1. Constraints used in generating the models

Constraints Distribution Fracture length Volume proportion

(Matrix: Fracture)

Fracture orientation

(z axis, degree)

Number of case 2 2 3 6

Input variables

▪ Density map(Fig.2(b))

▪ Random dist.

Length of O1

▪ lx=Tr(15,25,30)

ly=Tr(3,4,5)

▪ lx=Tr(25,35,40)

ly=Tr(4,5,6)

▪ 0.7: 0.3

▪ 0.5:0.5

▪ 0.3:0.7

Object 1, O1

▪ Tr(0,30,60)

▪ Tr(30,60,90)

▪ Tr(60,90,120)

▪ Tr(90,120,150)

▪ Tr(120,150,180)

▪ Tr(0,90,180)

Fixed value

Length of O2

lx=Tr(5,10,20)

ly=Tr(1,2,3)

Ratio of O1 and O2

(O1:O2)=2:1

Object 2, O2

▪ Tr(0,5,10)

* Tr(min, mode, max) means triangular distribution with minimum, mode, and maximum value.

Limitations of model generation

Validation of generated models

I did not verify whether the model properties are similar to that of input parameters, statistically.

For example, the volume proportion between matrix and fracture was directly used in sensitivity

analysis but I am not sure whether the ratio was same or not. It means our fracture volume might

be underestimated since the generation would stop when the fracture volume exceeds the

threshold. To fix this matter, I should have examined it by putting the values together in

‘DPNUM’.

Representativeness of input variables on the real fractured reservoir

To depict the effect of fracture orientation, I used 6 cases but the 5 cases have 30 degree

difference and the last is fully random. When we determine the flow direction dominant, we have

to use only 5 cases not fully randomized one. If the random orientation affects the flow response

significantly, we could not catch its effect correctly.

In this problem, I only used the triangular distribution but it is difficult to say that it can fully

consider the characteristics of fracture distribution. For instance, many field surveys have

reported that the orientation of real fractured reservoir follows the Fisher distribution (i.e. the

large fractures have their main orientation and gathered the smaller in their direction)

In conclusion, we must pay more attention to choose the statistical parameters in more realistic

manner.

3.2 Streamline Simulation using 3DSL

To produce oil as much as possible, I placed 5 water injection wells and 5 producers at less

fractured zones (Fig. 3). I assumed the constant bottom hole pressure, 50000 kPa in injection

wells and the fluid rate, 20 sm3/day in production wells (see Appendix A). Complete penetration

in z direction was assumed. Each well location was summarized in Table 2.

Fracture and matrix had its own permeability ( k ) and porosity ( ) like md 10000Fk ,

md 100Mk , 02.0F , and 2.0M (The subscript F and M represent fracture and matrix,

respectively). The mass transfer rate ( ) between fracture and matrix was 0.05.

Fig. 3 Well placement for injecting water and for producing the fluids (oil and water).

Table 2. Well location used in this study.

Well type Location (x,y)

Injection well INJ1=(7,3) , INJ2=(3,5), INJ3=(10,95), INJ4=(67,98), INJ5=(97.5)

Production well PROD1=(34,20), PROD2=(34,80), PROD3=(53,53), PROD4=(78,80), PROD5=(87,34)

Fig. 4 illustrated the relative permeability curve for (a) fracture and (b) matrix, respectively. The

former was straight line while the latter for matrix followed the Corey equation to get a smoother

one (an exponent 2 for water and 3 for oil). Any effect of capillary pressure was not considered

in the fracture but it was in the matrix as shown in Fig. 5.

Fig. 6 depicted the water saturation profile via streamline simulation as time went by. The

hydraulic channeling through some fractures was observed realistically.

Limitation of Streamline simulation

Relationship between spatial distribution of fractures and flow path

The interrelationship between the spatial distribution of fractures and the real flow path was not

investigated. Though the fast movement of water through fractures was observed, the role of

each fracture was not examined on the transport, that is to say, the connectivity in a fracture

network. It can be much helpful to construct the backbone structure using the spatial distribution

of fractures. If we set the weight on each fracture according to its effect on the transport, the

sensitivity analysis might be complicated but accurate more.

(a) (b)

Fig. 4 Relative permeability curve for (a) fracture and (b) matrix.

Fig. 5 Plot of capillary pressure used in the fluid transport through matrix.

Fig. 6 One example of water saturation profile demonstrated by StudioSL.

3.3 Robustness of DKM (Distance Kernel Method)

Two dissimilarity matrices were examined; one was based on FOPR (Field Oil Production Rate;

Eq. 1) and the other was WOPR (Well Oil Production Rate; Eq. 2).

n

t

jiij

tPMax

tP

tPMax

tPd

0

2

))((

)(

))((

)( (1)

5

1 0

2

,,

))((

)(

))((

)(

well

n

t well

wellj

well

welli

ijtPMax

tP

tPMax

tPd (2)

where ijd is the distance between i th and j th reservoir model, )(tP is the oil production rate at

time t, and the subscript well meant the production well.

From the distance map demonstrated in 2D Euclidean space (see Fig. 7), Eq. 2 showed well-

sorted result rather than Eq. 1. Eq. 1 resulted out one dimensional distribution relatively but

represented well the characteristics of spatial distribution affecting the cumulative field oil

production (Fig. 8). Since we placed the wells in the matrix (i.e. the less fractured zone in density

map), the less fractures between an injection well and a production well revealed more oil

production. The leftist side in Fig 8 represented the maximum oil production so that the less

existence of fractures among wells resulted out the more oil production.

(a) (b)

Fig. 7 Multi-dimensional scaling with 7 clusters using (a) Eq. 1 and (b) Eq. 2 in 2D space.

Fig. 8 Multi-dimensional scaling using Eq. 1: each point represented a reservoir model.

It was difficult to determine the superiority of the definition of distance by comparing the error

of quantiles( qE in Eq. 3) that was sum of difference of P10, P50, and P90 between the whole

dataset and the centroids selected by DKM.

)()()()()()(3

1909050501010

tPtPtPtPtPtPN

E DKMentireDKMentireDKMentire

ts

q (3)

where tsN is the number of time step.

Fig. 9 illustrated the error of quantiles when using Eq. 1 and Eq. 2, respectively. The red dot

line meant 0.05% of maximum field oil production. Although the error using Eq. 1 showed more

stable but because of random seeds to select the centroid in DKM, the trajectory can be

changeable. More than 15% samples from 720 models) would be enough to show the reliable

accuracy. The data of Fig. 9 were from streamline simulation so it needed to re-examine

ECLIPSE runs.

(a) (b)

Fig. 9 Error of quantiles by changing the number of clusters: (a) Eq. 1 and (b) Eq. 2. But these

used the results from 3DSL runs.

Limitation of DKM

Difficulty to determine the better distance equation

The distance map based on FOPR showed the relationship more clear between the spatial

characteristics and the oil production but the other was not examined the connectivity among

wells. The reliability of selecting the model by clustering was similar to each other as shown in

Fig. 9.

Difficulty to distinguish the individualized well response with respect to the spatial

distribution between connected wells

Because we had many wells and did not investigate the connectivity between interconnected

wells, it was difficult whether the fracture distribution reflected the production response, directly.

It needed to confirm the relationship between various distributions of distance map (see Fig. 10)

and its flow response as well as the spatial distribution.

(a) (b) (c) (d) (e)

Fig.10 Distance map based on individualized response of (a) PROD1, (b) PROD2, (c) PROD3,

(d) PROD4, and (e) PROD5.

3.4 Sensitivity analysis based on RSM (Response Surface Methodology)

The comparison of typical experimental design with DKM was carried out to validate the

robustness of DKM on extracting the factors affecting the production significantly. Eq. 2, well-

based distance was used in DKM.

Two response surface equations were as follows

n

i

ii xy1

0 (4)

ji

n

i ij

ij

n

i

ii xxxy

11

0 (5)

In Eq. 4 and 5, y was the cumulative oil production volume (sm3) at 9000 days, x was the input

parameter normalized in Table 3, n was the number of input parameters (n=4), and was the

coefficient. Normalized x was determined along the number of case as written in Table 1.

Fig. 11 illustrated the sensitivity of input parameters affecting the cumulative oil production,

which was assumed as the true for it was by all data (720 models). In the case of linear terms,

distribution map (M) and fracture proportion (F) affected significantly the oil production, and in

interaction terms, their interaction (MF) did as well. In a word, distribution map and fracture

proportion were important factors on the amount of oil production at 9000 days.

Table 3. Factors and their range in experimental design

Input parameter, x (Indicator in Fig. 10) Normalized value

Distribution map (M) Random map [-1], Density map [+1]

Length of object 1, O1 (L) Small in Table 1 [-1], Large [+1]

Volume proportion of fracture (F) Fracture volume proportion = (0.3, 0.5, 0.7)= [-1, 0, +1]

Fracture orientation (O) [-1, -0.6, -0.2, 0.2, 0.6, +1]

(a) (b)

Fig. 11 Sensitivity of parameters on cumulative oil production at 9000 days when using all

dataset, 720 models: in the case of (a) linear terms (Eq. 4) and (b) interaction ones (Eq. 5).

To test the robustness of DKM, 16 models were selected by typical experimental design and

also by the center of cluster in multi-dimensional scaling. Typical experimental design used the

minimum and the maximum x value in Table 3 and thereby the number of selection was 16 (= 2 (# of input parameter)

= 24). It was the reason why the number of cluster in DKM was 16. Because of

randomness on scattering the initial locations in DKM, the process selecting 16 models was

repeated up to 50.

Fig. 12 and Fig. 13 showed box plots of 50 runs in the case of linear terms and that of

interaction ones, respectively. Fig. 12(a) and Fig. 13(a) were of typical experimental design and

the others were by DKM. The blue line represented the significant level. As shown in Fig. 12 and

13, the sensible parameters by both methods were distribution map (M) and fracture proportion

(F) as same as the true of all data. It revealed the reliability of DKM on selecting the

representative samples that could cover the population.

(a) (b)

Fig. 12 Box plots of RSM with linear terms, Eq. 4: (a) experimental design and (b) DKM.

(a) (b)

Fig. 13 Box plots of RSM with interaction terms, Eq. 5: (a) experimental design and (b) DKM.

However, some selections by DKM might fail to determine the sensitivity of parameters on

response. As shown in Fig. 12(b) and 13(b), DKM had the risk to ignore fracture proportion as

the important factor. On the other hand, Fig. 12(a) and 13(b) mentioned that there was no risk of

this problem in experimental design. This phenomenon would be severe at RSM with interaction

terms since the range of in DKM was smaller (i.e. ± 20 in DKM and ± 40 in experimental

design, see Fig. 13).

It was one reason why the number of cluster in DKM was insufficient. As mentioned at 3.3,

around 15% (about 100 clusters) of all models would be enough to explain the characteristics of

the population. 16 clusters were much small and could show the uncertain result. Fig. 14

described the error changes of linear terms as increasing the number of cluster. As the number of

cluster increased, the significant level converged the constant (Fig. 14(a)) and the average

absolute error ( E in Eq. 6) to the true value (Fig. 11(a)) was reduced (see Fig. 14(b)).

1

0

1 n

ientire

i

DKM

i

entire

i

nE

(6)

In Eq. 6, n was the number of input factors (n=4) and the superscript entire

meant the true value in

Fig. 11(a).

(a) (b)

Fig. 14 Convergence trend as increasing the number of clusters in DKM: (a) significant level and

(b) average absolute error.

Another reason might be found at the clustering methodology. Let’s see Fig. 7(b), the distance

map in 2D space. The reservoir models got together and made the dense zone at the south-west

direction while the others were much sparse. Our clustering scheme does not consider the density

of clouds. To cover the all data, current clustering was good but, at the view of model selection

in sensitivity analysis, if being able to choose more models in dense area, the robustness of

sensitivity analysis could be improved. In conclusion, as described in Fig. 15, each point

represented its reservoir model and gathered at the left side showing the higher response. If we

wanted to determine which factors affected the higher response, it needed to select more models

in the dense zone.

Fig. 15 Distribution of y (cumulative oil production at 9000 days) in 3D Euclidean space.

Limitations of sensitivity analysis

Weight problem of centroid in clustering

Let’s assume that there was two zones separated by the degree of density in the distance map.

What is the better way to select the centroid? If we set the adequate weight with respect to the

density among the neighboring points, is it possible to obtain the more reliability with small

number of clusters?

It have to examine again using ECLIPSE runs

The contents in this section were by using the results of streamline simulation. We have to

examine again with Eclipse runs.

Joint Modeling not RSM

This work based on RSM produced the averaged calculation similar to the linear slope method.

To catch the outlier or determine the red link (significant fracture that determines whether the

flow occurs or not), more robust approach in sensitivity analysis is needed. Its necessity is larger

in fractured reservoir because of completely different response with respect to the connectivity of

fracture network.

3.4 ECLIPSE run

720 DPSP (Dual Porosity and Single Permeability) models were run and then it compared with

the results of streamline simulation. Fig. 16 showed one comparison of cumulative production

using 3DSL and ECLIPSE. The trajectories were much similar to each other. It showed the

reliability of streamline simulation in 3DSL.

(a) (b)

Fig. 16 One example of comparing (a) cumulative oil production and (b) cumulative water

production between 3DSL and ECLIPSE.

Three evaluations were studied; one was cumulative oil production at 9000 days, another was

the distance of cumulative oil production at 9000 days between model i and j (Fig. 17(a)), and

the third was the distance matrix of cumulative oil production at 1000, 2000, 3000, 4000, 5000,

6000, 7000, 8000, and 9000 days (Fig. 17(b)). The correlation coefficients were summarized in

Table 4.

(a) (b)

Fig. 17 Schematic diagram to define the distance: the difference (a) of cumulative oil production

at 9000 days and (b) at 9 time-steps.

Table 4. Correlation coefficient for three different evaluations

Number of data Correlation coefficient

Cumulative oil production (sm3) at 9000 days 720 0.9703

Fig. 17(a) 258840 0.9238

Fig. 17(b) 720 x 9 0.9356

The correlation between 3DSL and ECLIPSE was good so that the confidence of results in Fig. 9

were established. Fig. 18 illustrated the error of quantiles (see Eq. 3) using the results of

ECLIPSE runs. The perturbation was larger than that of 3DSL in Fig. 9 but as increasing the

number of clusters, the error was significantly reduced and converged below 0.05% of maximum

oil production.

Fig. 18 Trajectory of the error of quantiles using ECLIPSE runs as increasing the number of

clusters in DKM.

Limitations of ECLIPSE runs

DPDP(Dual Porosity Dual Permeability) model

DPDP model was needed not DPSP for obtaining the realistic responses of fractured reservoir. In

this work, we used DPSP model at both 3DSL and ECLIPSE so that their trajectories were

matched well. To examine the applicability of DKM and 3DSL in fractured reservoir, DPDP in

ECLIPSE and DPSP in 3DSL had to be conducted.

Effects of random seeds in clustering

Because of random seeding, Fig. 9 and 18 were needed to redraw using box plot.

4. SUMMARY

DKM shows good robustness in selecting the samples covering the population but the

accuracy in sensitivity analysis can depend on the number of clusters. To obtain the

reliable applicability with small number of time-consuming full simulation, the choice of

centroids is important. Selecting more samples in dense area can be one solution but

additional examination was necessary.

The spatial connectivity of fracture network should compare with the response since all

fractures do not participate in flow transport significantly. The role of each fracture on

the transport has to be investigated.

Density map and the volume ratio of fracture and matrix are significant factors in

determining the higher cumulative oil production.

5. FUTURE WORKS

Run the sensitivity analysis again using the results of ECLIPSE not those of 3DSL

Upgraded approach in sensitivity analysis not RSM

Careful selection of input parameters in fractured reservoir

Comparison of DPDP in ECLIPSE and DPSP of 3DSL

Possibility of the distance map in sensitivity analysis