part 5 simulation ứng dụng địa thống kê trong tìm kiếm thăm dò dầu khí

ài giảng ứng dụng địa thống kê trong tìm kiếm thăm dò dầu khí. giúp sinh viên hiểu biết sâu hơn về những phương pháp như mô phỏng ngẫu nhiên, tất định, gauss . Từ đó làm cơ sở trong việc xây dựng mô hình địa chất 3D trong phần mềm petrel

St h ti Si l ti

Stochastic Simulation

… all I want is ONE answer

Flow Chart of Steps in Stochastic

Modeling petrophysics

Simulation grid

(modified from Tyler and others, 1994)

Reservoir production simulation(modified from Tyler and others, 1994)

Petrophysical Property Modeling: p y p y g Prerequisites

W k ithi “h ” lith f i / k t l ifi ti

„ Work within “homogeneous” lithofacies/rock-type classification

„ Sequence stratigraphic framework

„ Clean data: positioned correctly, manageable outliers, grid spacing is appropriate

„ Need to understand special features and “special” data:

„ Need to understand special features and special data:

… trends

… production data

… seismic data

„ Considerations for areal grid size:

… practical limit to the number of cells

… need to have sufficient resolution so that the upscaling is meaningful

… this resolution is required even when the wells are widely spaced (simulation

… this resolution is required even when the wells are widely spaced (simulation algorithms fill in the heterogeneity)

„ Work with “grid nodes” We assign a property for the entire cell knowing that there are “sub-cell” features

St h ti Si l ti Wh ?

Stochastic Simulation, Why?

„ From a limited number of samples we are asked to build a complete numerical

model of attributes such as permeability and porosity

„ There are many ways to build a model which is consistent with the available sampled data

„ Whatever procedure we choose to build our model, we must consider the p ,

following:

… the distribution of values

… the variability of value

… the connectedness of extreme values

… our ability to assess the impact on our uncertainty

E ti ti Si l ti

Estimation versus Simulation

„ Estimation is locally accurate and smooth, appropriate for visualizing trends, inappropriate for flow simulation where extreme values are important, and does pp p p ,not assess global uncertainty

„ Simulation reproduces histogram, honors spatial variability (variogram),

appropriate for flow simulation, allows an assessment of uncertainty with

alternative realizations possible

Properties • Honors Wells

• Honors Histogram

• Honors Variogram

• Honors Wells

• Honors Variogramg

Image • Noisy

• Same variabilityeverywhere

• Smooth away fromwells

Use • Flow Simulation • Mapping

• Uncertainty Calculation • Volumetricspp g

St h ti Si l ti

Stochastic Simulations

„ Simulation differs from estimation (kriging in two ways)

… Kriging provides a Best Local Estimate without regard to resulting spatial

variability In simulation, global features and statistics take precedence over local accuracy

… Kriging provides a single numerical model (BEST) Simulation provides many alternative models (Each is a good representation of the reality in some global sense)

St h ti Si l ti

Stochastic Simulation

„ We want to utilize the measure of uncertainty that is returned in the kriging process

„ We want the simulation to honor our variogram and our data at the well locations

„ We want to build numerous, equiprobable reservoir models to capture our uncertainty

i th d t

in the data

„ We would like to flow-simulate these alternate reservoir models to obtain a

distribution of flow response variables

Si l ti R i H t it Simulating Reservoir Heterogeneity

„ Major heterogeneity's and large scale structure should be modeled first The large scale structures has a major influence on volumetrics and fluid movement

„ If the reservoir consists of a mixture of different statistical populations, the geometry

of the populations should be simulated first and then filled with continuous fields of

of the populations should be simulated first and then filled with continuous fields of properties

„ Reproduction of the spatial continuity of extreme values should be given priority (flow barriers and channels)

„ which one do you retain?

„ which one do you retain?

„ How many realizations should be generated?

… The answer depends on:

„ Aspect of uncertainty or the statistic being quantifiedp y g q

„ Precision with the uncertainty assessment is required

„ Presence of correlation between the realization

… Few realizations are required to assess an average statistic such as the average

it 20 li ti i it li bl 500 li ti ill t i ifi tl iporosity 20 realizations is quite reliable 500 realizations will not significantly improve our estimate A large number of realizations are required too assess an extreme percentile of a distribution At least 200 realizations would be required to determine the 1% percentile of the pore volume distribution

… More realizations lead to less uncertainty

Comparing Alternative Stochastic

Comparing Alternative Stochastic Models

„ Generate plausible realizations in a reasonable amount of time

„ Honor as much pertinent information as possible

„ Explore the largest space of uncertainty

Conditional Simulation of Flow

Conditional Simulation of Flow

… Alternative: Stream Line Simulation

„ An important use of conditional simulations is for determining the scaling law for

displacements in heterogeneous media and deriving effective flow properties for use

in coarse grid simulations

Si l ti M th d Simulation Methods

„ Gaussian Methods

… Sequential Gaussian Simulation

… Truncated Gaussian Simulation

Conditional Simulation Example

using Well and Seismic Data

Well Data

Ri k M Risk Map

Variogram

Model

Conditional Simulation Kriging or

Why Sequential Gaussian

Why Sequential Gaussian

Simulation?

„ Gaussian simulation is used because it is extraordinarily straightforward to

establish conditional distributions: shape of all conditional distributions is

Gaussian (normal) and the mean and variance are given by kriging

1 Transform data to “normal space”

2 Establish grid network and coordinate system (Z rel-space)

3 Decide whether to assign data to the nearest grid node or keep separate

4 Determine a random path through all of the grid nodes

(a) search for nearby data and previously simulated grid nodes

(b) construct the conditional distribution by kriging

(c) draw simulated value from conditional distribution

5 Back transform and check results

⇒ Price of mathematical simplicity is the characteristic of maximum spatial entropy, i.e., low and high values are disconnected Not appropriate for permeability

… Does not handle litholgic or indicator variable very well

… single variogram model

… single variogram model

… impossible to specify different spatial correlation characteristics for extreme values

… requires a normal model

… extreme values are very poorly correlate

Step 1 - Do Normal Scores

Step 1 Do Normal Scores

z

Step 2 - Assign data values to

Step 2 - Assign data values to

closest grid nodes

Step 3 - Establish random path

St 4 Vi it ll

Step 4 - Visit every cell once

„ Informed cells shown in yellow

„ Do kriging, generate conditional distribution

„ Populate current cell, Move on

St 4 Vi it ll

Step 4 - Visit every cell once

„ Do kriging, generate conditional distribution

„ Populate current cell, Move on

S ti l I di t Si l ti

Sequential Indicator Simulation

„ An indicator is simply a data transform

„ An indicator variable is a binary variable that has only two possible values - 0 or 1

„ Indicators allow different ranges of the data to be modeled with different variogram

f ti

functions

Wh I di t ?

Why Indicators?

„ Advantages

… easy to understand and use

… robust data representation with regard to extreme values

… allows us to analyze and model different measures of spatial continuity

(variograms) for different thresholds (cutoffs) or categories (facies) of the data

… provides a complete distribution of indicator valued (cdf) Instead of a single mean value at each estimation pointp

… allows for the integration of hard and soft data when applying estimation of simulation techniques

Cutoff Indicator Coding of g

Sparse Data

Indicators Can Be Interpreted as Probabilities:

I di tPermeability

Indicator

k < 70 = 0

k > 70 = 1

Like any Other Numerical Variable, an Indicator Variable Can be Analyzed Statistically:

Mean = Number of 1’s / Total Number of Samples

= Proportion of Type 1

= Probability of Encountering Type 1The Mean Provides a Complete Description of the Univariate Distribution

Indicator Cutoff Maps for Porosity

Indicator Cutoff Maps for Porosity Data

1st cutoffPhi < 12%

Map KeyColor Indicator

= 1

2nd cutoff

= 0

Phi < 18%

Indicator Variograms for Porosity

Indicator Variograms for Porosity

Cutoff Maps

1st cutoff

Range Ellipses Ellipse Parameters

Alpha Anisotropy RatioIndicator Map

1st cutoffPhi < 12%

Phi < 18%

Indicator Bin Maps for Porosity

Indicator Bin Maps for Porosity

Data

1st bin0% < Phi < 12%

2nd bin

Map KeyColor Indicator

= 112% < Phi < 18%

= 0

3rd binPhi > 18%

Indicator Variograms for Facies

Category Maps

Ellipse Parameters

Al h A i t R ti

1st categoryFacies 1

Range Ellipses Alpha Anisotropy Ratio

2nd categoryg yFacies 2

3 category

Detailed Steps in Sequential

Detailed Steps in Sequential

Indicator Simulation

1 Establish grid network and coordinate system (Zrel-space)

2 Assign data to the nearest grid node (take the closest of multiple data assigned tosame node)

3 Determine a random path through all of the grid nodes

(a) find nearby data and previously simulated grid nodes

(b) construct the conditional probabilities by kriging

(c) draw simulated value from conditional distribution

4 Check results

(a) honor data?

(b) honor global proportions?

(c) honor variogram?

(d) look reasonable

Step 1

Assign Data to Grid Nodes

Why?

„ Explicitly honor data - data values will appear in final 3-D model

„ Improves the CPU speed of the algorithm: searching for previously simulated nodes

„ Visit each cell once and only once in random order

„ Can do this in many ways:

… draw a random number and multiply it by N

… sort an array of random numbers while carrying an

… array of indices capitalize on the limited period length of linear congruential generators

„ Skip over cells (actually grid nodes) that already have a value

„ Skip over cells (actually grid nodes) that already have a value

Step 3a

Find Nearby “Informed” Nodes

„ “Informed” nodes refers to both data-nodes and nodes that have been informed earlier in the random path

„ Typically use spiral search to identify the close nodes

„ Limit the number of nodes actually considered:

… octant search

… maximum per octant (say 4)

… maximum number

Step 3b

Construct Conditional Distribution

„ Conditional distribution is constrained by:

… global proportion of each lithology type

… local data

… local data

… “local” proportion from secondary data such as seismic

„ Calculate by kriging the binary indicator transform for each rock type

STEP 3c

Construct Conditional Distribution

Construct Conditional Distribution with Kriging

„ Given n nearby data values k(u i ),i=1, ,n how do we calculate the conditional

[ )

; ( ) ( )

(

λ λ

„ Determine weights λα(u), α=1, ,n by the well known “normal system” or kriging.

„ Kriging weights account for two things:

… clustering of the data locations

… closeness of the data to the location being considered

… closeness of the data to the location being considered

Draw a Simulated Value

„ Since the conditional probabilities were estimated by kriging with a given

variogram γg γkk(h), k=1, ,K, the simulated values, taken all together, will reproduce ( ) g pthose variograms ,γk(h), k=1, ,K

Example: Indicator Simulation of Example: Indicator Simulation of

Stochastic Realization of Facies

in a Cross Section Through Eolian Sandstone

No Vertical Exaggeration Individual Blocks are 5 Feet by 50 Feet

(modified from Cox and Others, 1994)

Stochastic Realization of Permeability y

in a Cross Section Through Eolian Sandstone

No Vertical Exaggeration gg Individual Blocks are 5 Feet by 50 Feet

Permeability - Kh

max

(modified from Cox and Others, 1994)

<0.1 md 0.1 - 0.5 md 0.5 - 2.5 md 2.5 - 15 md >20 md

Obj t B d Si l ti

Object Based Simulation

B l Si l ti

Boolean Simulation

„ Facies objects are sequentially added to a background until the desired volume proportion of the facies is achieved

„ the size and shaped of the objects are chosen from a distribution function

characteristic of the depositional environment

„ Step by step procedure

… select a random position by drawing random coordinates for a point

… Mark the point with the attributes (size, shaped, orientation…) of the chosen p ( p )facies drawn from a distribution function characterizing the facies

… continue until the desired volume fraction is achieved

Wh St ti ti l Di t ib ti ? Why use Statistical Distributions?

„ As input to stochastic modeling packages to constrain the likely dimensions of sandbodies that cannot be correlated deterministically across a field

„ To aid deterministic correlation's by suggesting which sands are likely to extend fieldwide

„ To suggest how sandbody dimensions vary systematically with sequence

stratigraphic architecture

Sandstone Body Width Vs Sandstone Body Width Vs Thickness by Environment

Sandstone Body Width

Sandstone Body Width

Distributions by Environment

Boolean Simulation of Sand Channels

Conditioning Data

Boolean Simulation of Sand Channels

Sand Shale

Boolean Simulation of Sand Channels

Sand bodies randomly located

Honoring Well Data

Boolean Simulation of Sand Channels

Sand bodies randomly located

to coincide with sands in wells

Boolean Simulation of Sand Channels

Random sand body conflicts with

Interwell Bodies

Boolean Simulation of Sand Channels

Random sand body conflicts withwell and must be dropped or moved

Boolean Simulation of Sand Channels

Final Realization

Sand bodies added until net to gross

Boolean Simulation of Sand Channels

Sand bodies added until net-to-grossratio reaches desired target

Calculating Channel Orientation

Horizontal and Vertical Proportion

Horizontal and Vertical Proportion Curves

„ While the net/gross determines the total volume of channel objects in the reservoir, the proportion curve is used to identify the vertical and horizontal locations of these objects

Ch l P t

Channel Parameters

Channel Objects in a Gridded Channel Objects in a Gridded Model

C i li G idd d Obj t Curvi-linear Gridded Objects

w k

Object Orientation and Location

Object Orientation and Location Parameters

Splay Orientation and Location

Splay Orientation and Location Parameters

7

v

V ti l P ti C

Vertical Proportion Curves

Ch l S l d L

Channels, Splays and Levees

Well is Off Center for Thick Well is Off Center for Thick Channel

S i i Seismic

S i i U d f C diti i

Seismic was Used for Conditioning

Seismic was not Used for Seismic was not Used for Conditioning