Data types

In essence, the purpose of ERT is to pass uncertain parameter values to a forward model and then store the resulting outputs. Forward models include all necessary pre-processing and post-processing steps, as well as the computational model (e.g., a physics simulator like ECLIPSE) that produces the predictions. Consequently, ERT must be able to read and write data in a format compatible with the forward model.

Data managed by ERT are organized into distinct data types, each of which will be detailed in this chapter. The data types in ERT can be classified based on two main criteria:

  1. Dynamic behaviour: whether the data type is a static input to the simulator, such as porosity or permeability, or an output of the simulation.

  2. Implementation: this includes the type of files it can read and write, how it is configured, and so forth.

Note: All data types share a common namespace, meaning that each keyword must be globally unique.

Scalar parameters with a template: GEN_KW

This section describes the distributions built into ERT that can be used as priors. For detailed description and examples on how to use the GEN_KW keyword, see here.

The algorithms used for updating parameters expect normally distributed variables. ERT supports other types of distributions by transforming normal variables as outlined next.

  1. ERT samples a random variable x ~ N(0,1) - before outputing to the forward model this is transformed to y ~ F(Y) where the distribution F(Y) is the correct prior distribution.

  2. When the prior simulations are complete ERT calculates misfits between simulated and observed values and updates the parameters; hence the variables x now represent samples from a posterior distribution which is Normal with mean and standard deviation different from (0,1).

The transformation prescribed by F(y) still “works” - but it no longer maps to a distribution in the same family as initially specified by the prior. A consequence of this is that the update process can not give you a posterior with updated parameters in the same distribution family as the Prior.

Reproducibility

When ERT samples values there is a seed for each parameter. This means that if ERT is started with a fixed RANDOM_SEED each prior that is sampled will be identical. When running without a random seed ERT will output which random seed was used, so it is possible to reproduce results as long as that is kept.

This section only applies if a fixed seed is used:

  • If the ensemble size is increased from N -> N+1 the N first realizations will be identical to before

  • Parameter order is irrelevant

  • Parameter names are case sensitive, PARAM:MY_PARAM != PARAM:myParam

NORMAL: Gaussian Distribution

The NORMAL keyword allows assigning a Gaussian (or normal) prior to a variable. It requires two arguments: a mean value and a standard deviation.

Syntax

VAR NORMAL <mean_value> <standard_deviation>

Parameters

  • <mean_value>: The mean of the normal distribution.

  • <standard_deviation>: The standard deviation of the normal distribution.

Example

For a Gaussian distribution with mean 0 and standard deviation 1 assigned to the variable VAR:

VAR NORMAL 0 1
../../_images/normal.png

Notes

The NORMAL keyword is integral for scenarios demanding priors that reflect typical real-world data patterns, as the Gaussian distribution is prevalent in many natural phenomena.

LOGNORMAL: Log Normal Distribution

The LOGNORMAL keyword is used to assign a log normal prior to a variable. A variable is considered log normally distributed if the logarithm of that variable follows a normal distribution. The logarithm in question is the natural logarithm. If \(X\) is normally distributed, then \(Y = e^X\) is log normally distributed.

Log normal priors are especially suitable for modeling positive values that exhibit a heavy tail, indicating a tendency for the quantity to occasionally take large values.

Syntax

VAR LOGNORMAL <log_mean> <log_standard_deviation>

Parameters

  • <log_mean>: The mean of the logarithm of the variable.

  • <log_standard_deviation>: The standard deviation of the logarithm of the variable.

Example

Histogram from values sampled from a lognormal variable specified with log-mean of 0 and log standard deviation 1.

VAR LOGNORMAL 0 1
../../_images/lognormal.png

TRUNCATED_NORMAL: Truncated Normal Distribution

The TRUNCATED_NORMAL keyword is utilized to assign a truncated normal distribution to a variable. This distribution works as follows:

  1. Draw a random variable \(X \sim N(\mu,\sigma)\).

  2. Clamp \(X\) to the interval [min, max].

Syntax

VAR TRUNCATED_NORMAL <mean> <standard_deviation> <min> <max>

Parameters

  • <mean>: The mean of the normal distribution prior to truncation.

  • <standard_deviation>: The standard deviation of the distribution before truncation.

  • <min>: The lower truncation limit.

  • <max>: The upper truncation limit.

Example

VAR TRUNCATED_NORMAL 2 0.7 0 4
../../_images/truncated_ok.png

UNIFORM: Uniform Distribution

The UNIFORM keyword is used to assign a uniform distribution to a variable. A variable is considered uniformly distributed when it has a constant probability density over a closed interval. Thus, the uniform distribution is fully characterized by it’s minimum and maximum values.

Syntax

VAR UNIFORM <min_value> <max_value>

Parameters

  • <min_value>: The lower bound of the uniform distribution.

  • <max_value>: The upper bound of the uniform distribution.

Example

To assign a uniform distribution spanning between 0 and 1 to a variable named VAR:

VAR UNIFORM 0 1
../../_images/uniform.png

Notes

It can be shown that among all distributions bounded below by \(a\) and above by \(b\), the uniform distribution with parameters \(a\) and \(b\) has the maximal entropy (contains the least information).

LOGUNIF: Log Uniform Distribution

The LOGUNIF keyword is used to assign a log uniform distribution to a variable. A variable is said to be log uniformly distributed when its logarithm displays a uniform distribution over a specified interval, [a, b].

Syntax

VAR LOGUNIF <min_value> <max_value>

Parameters

  • <min_value>: The lower bound of the log uniform distribution.

  • <max_value>: The upper bound of the log uniform distribution.

Example

To assign a log uniform distribution ranging from 0.00001 to 1 to a variable:

VAR LOGUNIF 0.00001 1
../../_images/loguniform.png

Notes

The log uniform dstribution is useful when modeling positve variables that are heavily skewed towards a boundary.

CONST: Dirac Delta Distribution

The CONST keyword ensures that a variable always takes a specific, unchanging value.

Syntax

VAR CONST <value>

Parameters

  • <value>: The fixed value to be assigned to the variable.

Example

To assign a value of 1.0 to a variable:

VAR CONST 1.0
../../_images/const.png

DUNIF: Discrete Uniform Distribution

The DUNIF keyword assigns a discrete uniform distribution to a variable over a specified range and number of bins.

Syntax

VAR DUNIF <nbins> <min_value> <max_value>

Parameters

  • <nbins>: Number of discrete bins or possible values.

  • <min_value>: The minimum value in the range.

  • <max_value>: The maximum value in the range.

Example

To create a discrete uniform distribution with possible values of 1, 2, 3, 4, and 5:

VAR DUNIF 5 1 5
../../_images/dunif.png

Notes

Values are derived based on the formula: \(\text{min} + i \times (\text{max} - \text{min}) / (\text{nbins} - 1)\) Where \(i\) ranges from 0 to \(\text{nbins} - 1\).

ERRF: Error Function-Based Prior

The ERRF keyword allows creating prior distributions derived from applying the normal CDF (involving the error function) to a standard normal variable. Note that the CDF is not necessarily the standard normal, as SKEWNESS and WIDTH corresponds to its negative mean and standard deviation respectively. This allows flexibility in creating distributions of diverse shapes and symmetries.

Syntax

VAR8 ERRF MIN MAX SKEWNESS WIDTH

Parameters

  • MIN: The minimum value of the transform.

  • MAX: The maximum value of the transform.

  • SKEWNESS: The asymmetry of the distribution.

    • SKEWNESS < 0: Shifts the distribution towards the left.

    • SKEWNESS = 0: Results in a symmetric distribution.

    • SKEWNESS > 0: Shifts the distribution towards the right.

  • WIDTH: The peakedness of the distribution.

    • WIDTH = 1: Generates a uniform distribution.

    • WIDTH > 1: Creates a unimodal, peaked distribution.

    • WIDTH < 1: Forms a bimodal distribution with peaks.

Examples

  1. For a symmetric, uniform distribution:

    VAR ERRF -1 1 0 1
../../_images/errf_symmetric_uniform.png

  1. For a right-skewed, unimodal distribution:

    VAR ERRF -1 1 2 1.5
../../_images/errf_right_skewed_unimodal.png

Notes

Keep in mind the interactions between the parameters, especially when both SKEWNESS and WIDTH are adjusted. Their combination can result in a wide range of distribution shapes.

DERRF: Discrete Error Function-Based Distribution

The DERRF keyword is a discrete version of the ERRF keyword. It is designed for creating distributions based on the error function but with discrete output values. This keyword facilitates sampling from discrete distributions with various shapes and asymmetries.

Syntax

VAR DERRF NBINS MIN MAX SKEWNESS WIDTH

Parameters

  • NBINS: The number of discrete bins or possible values.

  • MIN: The minimum value of the distribution.

  • MAX: The maximum value of the distribution.

  • SKEWNESS: The asymmetry of the distribution.

    • SKEWNESS < 0: Shifts the distribution towards the left.

    • SKEWNESS = 0: Produces a symmetric distribution.

    • SKEWNESS > 0: Shifts the distribution towards the right.

  • WIDTH: The shape of the distribution.

    • WIDTH close to zero, for exampe 0.01: Generates a uniform distribution.

    • WIDTH > 1: Leads to a unimodal, peaked distribution.

    • WIDTH < 1: Forms a bimodal distribution with peaks.

Examples

  1. For a discrete symmetric, uniform distribution with five bins:

    VAR_DERRF1 DERRF 5 -1 1 0 1
../../_images/derrf_symmetric_uniform.png

  1. For a discrete right-skewed, unimodal distribution with five bins:

    VAR_DERRF2 DERRF 5 -1 1 2 1.5
../../_images/derrf_right_skewed.png

TRIANGULAR: Triangular Distribution

The TRIANGULAR keyword is used to define a triangular distribution, which is shaped as a triangle and is determined by three parameters: minimum, mode (peak), and maximum.

Syntax

VAR TRIANGULAR MIN MODE MAX

Parameters

  • XMIN: The minimum value of the distribution.

  • XMODE: The location (value) where the distribution reaches its maximum (or peak).

  • XMAX: The maximum value of the distribution.

Description

The triangular distribution is a continuous probability distribution with a probability density function that is zero outside the interval [XMIN, XMAX], and is linearly increasing from XMIN to XMODE and decreasing from XMODE to XMAX.

Example

To define a triangular distribution with a minimum of 1, mode (peak) of 3, and maximum of 5:

VAR_TRIANGULAR TRIANGULAR 1 3 5
../../_images/triangular.png

Loading GEN_KW values from an external file

The default use of the GEN_KW keyword is to let ert sample random values for the elements in the GEN_KW instance, but it is also possible to tell ert to load a precreated set of data files. This can for instance be used as a component in an experimental design based workflow. When using external files to initialize the GEN_KW instances you supply an extra keyword INIT_FILE:/path/to/priors/files%d which tells where the prior files are:

GEN_KW  MY-FAULTS   MULTFLT.tmpl   MULTFLT.INC   MULTFLT.txt    INIT_FILES:priors/multflt/faults%d

In the example above you must prepare files priors/multflt/faults0, priors/multflt/faults1, … priors/multflt/faultsn which ert will load when you initialize the ensemble. The format of the GEN_KW input files can be of two varieties:

  1. The files can be plain ASCII text files with a list of numbers:

1.25
2.67

The numbers will be assigned to parameters in the order found in the MULTFLT.txt file.

  1. Alternatively values and keywords can be interleaved as in:

FAULT1 1.25
FAULT2 2.56

in this case the ordering can differ in the init files and the parameter file.

The heritage of the ERT program is based on the EnKF algorithm, and the EnKF algorithm evolves around Gaussian variables - internally the GEN_KW variables are assumed to be samples from the N(0,1) distribution, and the distributions specified in the parameters file are based on transformations starting with a N(0,1) distributed variable. The slightly awkward consequence of this is that to let your sampled values pass through ERT unmodified you must configure the distribution NORMAL 0 1 in the parameter file; alternatively if you do not intend to update the GEN_KW variable you can use the distribution RAW.

3D field parameters: FIELD

The FIELD keyword is used to parametrize quantities that span the entire grid, with porosity and permeability being the most common examples. For detailed description and examples see here.

2D Surface parameters: SURFACE

The SURFACE keyword can be used to work with surface from RMS in the irap format. For detailed description and examples see here.

Simulated data

The datatypes in the Simulated data chapter correspond to datatypes which are used to load results from a forward model simulation and into ERT. In a model updating workflow instances of these datatypes are compared with observed values and that is used as basis for the update process. Also post processing tasks like plotting and QC is typically based on these data types.

Summary: SUMMARY

The SUMMARY keyword is used to configure which summary vectors you want to load from the (Eclipse) reservoir simulation. In its simplest form, the SUMMARY keyword just lists the vectors you wish to load. You can have multiple SUMMARY keywords in your config file, and each keyword can mention multiple vectors:

SUMMARY  WWCT:OP_1  WWCT:OP_2  WWCT:OP_3
SUMMARY  FOPT FOPR  FWPR
SUMMARY  GGPR:NORTH GOPR:SOUTH

If you in the observation use the SUMMARY_OBSERVATION or HISTORY_OBSERVATION keyword to compare simulations and observations for a particular summary vector you need to add this vector after SUMMARY in the ERT configuration to have it plotted.

You can use wildcard notation to all summary vectors matching a pattern, i.e. this:

SUMMARY WWCT*:* WGOR*:*
SUMMARY F*
SUMMARY G*:NORTH

will load the WWCT and WWCTH, as well as WGOR and WGORH vectors for all wells, all field related vectors and all group vectors from the NORTH group.

General data: GEN_DATA

The GEN_DATA keyword is used to load text files which have been generated by the forward model. For detailed description and examples see here.

EnKF heritage

With regards to the datatypes in ERT this is a part of the application where the EnKF heritage shows through quite clearly, the datetypes offered by ERT would probably be different if ERT was made for Ensemble Smoother from the outset. Pecularites of EnKF heritage include:

  1. The FIELD implementation can behave both as a dynamic quantity, i.e. pressure and saturation, and static property like porosity. In ERT it is currently only used as a parameter.

  2. The parameter types have an internal pseudo time dependence corresponding to the “update time” induced by the EnKF scheme. This pseudo time dependence is not directly exposed to the user, but it is still part of the implementation and e.g. when writing plugins which work with parameter data managed by ERT you must relate to it.

  3. The time dependence of the GEN_DATA implementation. This is just too complex, there have been numerous problems with people who configure the GEN_DATA keywords incorrectly.