Pseudo-relative permeabilities estimation from reservoir performance history data

Heriot-Watt University

Institute of Petroleum Engineering

Supervisors

From HWU: Dr Gillian Pickup

From ASC TPU: Andrey V. Ryazanov

MSc Petroleum Engineering
Project Report 2004/2005

Mikhail A. Tuzovskiy

Pseudo-relative permeabilities
estimation from reservoir
performance history data

Mikhail A. Tuzovskiy 2 Heriot-Watt University
TABLE OF CONTENTS

Declaration ..................................................................................................................................... 4

Acknowledgements ........................................................................................................................ 5

Summary ........................................................................................................................................ 6

Introduction ................................................................................................................................... 7

Chapter 1 Mathematical Model ................................................................................................. 10

Setting up a problem 10
Relative permeabilities representation 11
Power functions 11
B-splines 12
Error Analysis 13
Multivariable penalized misfit functional 13
Approximation error 15
Automatic history matching algorithm specification 15
Initial guess 16
Eclipse model simulation 17
Minimization algorithm 17
Results 18
Chapter 2 Synthetic Model Calculations................................................................................... 19

Model specification 19
Automatic history matching algorithm performance 19
Representations competition 21
Sensitivity to the initial guess in the B-spline representation 22
Results 24
Chapter 3 “K” Oil Field Adaptation ......................................................................................... 26

Geological description 26
Model specification 26
History matching 28
Results 30
Conclusions .................................................................................................................................. 32

Suggestions for further work...................................................................................................... 33

Figures and Tables ...................................................................................................................... 34

Figures 34
Tables 47
References .................................................................................................................................... 49

Appendix ...................................................................................................................................... 51

MSc Petroleum Engineering 2004/2005 Project Report

A. B-splines 51
B. Error analysis 52
C. Knots choice for each type of initial guess 53



Declaration
“I Mikhail A. Tuzovskiy confirm that this work submitted for assessment is my own and

expressed in my own words. Any uses made within it works of another authors in any form (e.g.

ideas, equation, figure, text, tables, programs) are property acknowledged at the point of their

use. A list of the reference employed is included.

Signed: ………………
Date: ………………”



Acknowledgements
I would like to thank for the help and supporting during my individual project the following

people and organizations:

• My supervisors Andrey V. Ryazanov, Dr. Gillian Pickup for their recommendation;
• The staff of Institute of Petroleum Engineering of Heriot-Watt University for lectures;
• The staff of Heriot-Watt Approved Support Centre for guidance;
• The staff of Krasnoyarsk State University, especially PhD Alexandr A. Tuzovskiy, for
good scientific basis and support;
• Heriot-Watt Approved Support Centre for financial support and giving data;
• All students of our course for supporting;
• SPE and SPWLA organizations for provision to use e-library;



Summary
Pseudo-relative permeabi1ity curve is an important and basic data for design and analysis of

reservoir development. Reservoirs simulations based on such data reasonably can be applied for

examination of various recovery methods, field facilities requirements and placement.

Initially, estimates of relative permeability are obtained from laboratory experiments with

reservoir core samples through explicit methods [1]. However these estimates may not be very

representative of the entire reservoir and, thus, giving significant simulation errors.

In this work existing implicit approaches of relative permeabilities estimation were

researched and automatic history matching algorithm was developed to estimate pseudo-relative

permeability curves for “K” oil field. Reservoir model inputted with obtained pseudos shows

good agreement with field production data and give improvement over matches obtained using

upscaled permeability curves [2].

Power functions and B-spline representation of pseudo-relative permeability curves were

mainly subjected to researches. Implicit approach implies inversion of reservoir production

history data in order to obtain relative permeability curves (parameters of representation).

Solution of such inverse problem reduces to minimization of penalized multivariable functional.

Common features of inverse problems like ill-posedness, nonuniqueness of solution were

assessed and accommodated.

Next, synthetic reservoir model was set up and automatically matching algorithm for

adjusting representation parameters was primary tested. Further sensitivity studies were carried

out to determine optimum number of adjusted parameters, optimal choice of initial guess and

optimal measure of estimation error. B-spline representation of pseudos shows most promising

results.

Finally, developed automatically matching algorithm was used to obtain set of pseudo

relative permeability curves for “K” oil field.

Introduction
Pseudo-relative permeabi1ities are tables of numbers, which allows reproducing a fine-scale

simulation on a coarse grid, thus, is an important and basic data for design and analysis of

reservoir development. Reservoirs simulations based on such data reasonably can be applied for

examination of various recovery methods, field facilities requirements and placement. The

economical benefits which can be achieved by the use of such information are obvious:

improvement of sweep efficiency in water floods, oil recovery increase, etc.

Initially, estimates of relative permeability are obtained from laboratory experiments with

reservoir core samples through explicit methods. In such experiments one fluid (e.g., water) is

pumped through a sample of the reservoir rock that is saturated with another fluid or fluids (e.g.,

oil and connate water). The flow data typically consist of measurements of two time-dependent

quantities—the pressure drop across the core and the volume of displaced fluid recovered. The

methods of Johnson et al.' and Jones and Roszelle [1] are explicit methods in which saturation

values and corresponding relative-permeability values at the end of the core sample are estimated

directly. One disadvantage of the explicit method is that such physical effects as capillary

pressure cannot easily be taken into account in the interpretive procedure. Another disadvantage

is that the process of differentiating measured data tends to magnify the effects of error present in

the measured data [3], [4]. Additionally, the reservoir core samples represent only a very small

portion of the reservoir. Consequently, the estimates of relative permeabilities may not be very

representative of the entire reservoir. Samples maybe altered in coring and processing and thus

may not reflect in-situ conditions. Also, the flow regime characterized by the laboratory

experiment may not match that in the field [5]. Thus, during reservoir characterization the

assumption that the relative permeabilities are known functions can be a major source of

weakness [6].


In order to perform reservoir simulation which will give reasonable results, upscaling is

widely applied in recent years to convert highly detailed geological models to simulation grids.

There are a range of single- and two-phase upscaling techniques. The brief overview of existing

techniques is given here [7]. In this work upscaled relative permeability curves by Kyte and

Berry method [2] are compared with pseudo relative permeabilities deduces from reservoir

performance history data.

Alternatively, an implicit approach may be used. In this approach, relative-permeability

curves are chosen so that the quantities simulated with the mathematical model of the experiment

match [8]. Such approach is applicable then reservoir production data, such as well pressures and

water and oil production rates, will become available and can be used in estimating relative

permeability curves. An inverse process, called reservoir history matching, must be used for

estimating relative permeability curves from pertinent field data. In history matching, the

estimates of relative permeability curves are adjusted until the simulated data match the observed

field data [5]. The matching process can be manual or automatic, using special computer

algorithms. Manual history matching is based on trial-and-error process and strongly depends on

human factor having no general guidelines. Since, the best estimates of the reservoir properties

are not likely to be found by manual history matching. In automatic history matching, the

estimates can be chosen as these parameter values that minimize a penalized multivariable

functional, which is usually taken to be a weighted sum of squared differences between the

observed and calculated reservoir production data [9]. The history matching thus becomes a

mathematical minimization problem that can be solved with automated optimization algorithms

[5].

This work is a research in the field of automatically history matching approaches, which are

based on various representations of pseudo-relative permeabilities curves.

The first fundamental work in this field was by Watson et al. in 1980 [9]. They use optimal

control algorithm for automatic history matching, represented relative permeabilities with power


functions and estimated only the exponents in the power function representation. In further

research [9] sensitivity analysis and an error covariance analysis were employed to study the

feasibility and accuracy of estimating relative permeability parameters. Further approach was

presented by Johnson et al. in 1982 [11], A modified Gauss-Newton non-linear regression

programs was used to automatically vary input parameters for the simulation, again power

representation of pseudos was used. In 1986 Kerig and Watson [8] developed a procedure to

analyze the accuracy with which relative permeabilities may be estimated by parameter

estimation and firstly use cubic spline representation of pseudos. B-spline representation of

pseudo-relative permeabilities was firstly applied by Watson et al. in 1988 [4], auto matching

algorithm was based on sequential quadratic programming (SQP) technique. In addition whey

studies a method for choosing an appropriate number of partitions in B-spline representations.

Alternatively, Zhang [12] presented a modified power function representation for hydrophilic

reservoirs. Prior knowledge and pressure data was used to improve estimates of pseudos by

Watson [5]. A streamline approach of pseudo-relative permeabilities was applied by Kulkarni

[6].

The main objective of this work is development of automatic history matching algorithm

for pseudo-relative permeability estimation, which will incorporate all best results from the

relative permeability estimation studies, and further its application to «K» oil field sector in order

to obtain pseudo-relative permeabilities.

As a result of work, a set of pseudo-relative permeabilities for «K» oil field is deduced by

automatic history matching algorithm. Obtained pseudos estimates are consistent with the

reservoir production and yet conform to the prior estimates. Good history matches of reservoir

behavior were observed.



Chapter 1 Mathematical Model
Setting up a problem
The main problem of this work is to obtain a set of pseudo-relative permeabilities using

field history performance data. Water cut and Oil production rate data are used. Permeability

curves are specified in parametric form (Power functions, B-splines). To get desired curves,

parameters are adjusted until an optimum match is reached between theoretical and history data.

Such kind of problem is clearly inverse, thus needs some relevant discussion.

Inverse problems in reservoir characterization are ill posed. This implies that on of the

following conditions is not satisfied:

1. Solution existence

2. Solution uniqueness

3. Continuous solution dependence on initial data. This means that slight perturbations

in the data may create large oscillations in the solution.

In the solution of such kind of problems [13], solution existence is postulated. Such kind of

assumption seems quite applicable for field simulations, where existence of pseudos is obvious.

Generally, field production data can be noisy because of measurement errors. These errors may

cause oscillations in the relative permeability estimates. From the practical point of view

unsteadiness of solution means solution nonuniqueness and regularization is necessary to obtain

physically meaningful relative permeability estimates. This can be done by making allowance to

prior knowledge (laboratory data, etc.), by use of weighting coefficients and smoothing terms. A

good examination of the nature of ill-posed problems was done by Kulkarni et al [6]. They

systematically investigate the nonuniqueness associated with the inverse problem and

quantitatively evaluate the role of additional data such as pressure response in addition to water-

cut history at the wells using trade-off analysis. The main conclusions of their work are


appreciated in multivariable penalized misfit functional construction in the following sections of

this work.

Relative permeabilities representation
The accuracy with which the relative permeability curves may be estimated depends on the

parametric forms chosen to represent the curves. Because there is no reliable theoretical model to

describe relative-permeability curves, empirical functions are chosen.

Power functions
The most commonly used parametric representations for relative permeabilities are the

power functions:

e
k = ko S w (1)
rw rw nw

e
k = ko S o (2)
ro ro no

where krw is the endpoint relative permeability for water, Snw is the normalized water saturation:

S w − S wi
Snw =
1 − S or − S wi

and ew is a real number. Similarly, kro is the endpoint relative permeability for oil, Sno is the

normalized oil saturation, and eo is a real number. In this representation, the unknown parameters

are krw, kro, ew and eo. This relative permeability representation assumes a functional

relationship between relative permeability and saturation.

The advantage of the functional forms (1), (2) is that although they contain only four

parameters krw, kro, ew, eo they give the typically concave shape of the relative permeability

curves when ew, eo > 1. The problem then becomes one of estimating these four parameters,

rather than the complete curves krw(S), kro(S) [10]. This is computationally convenient because

only few parameters need be determined. However, this functional form is relatively stiff and

cannot represent many different shapes. Consequently, it cannot accurately represent many

relative-permeability curves. In many cases, its use can result in large errors in the estimated

relative permeabilities [8]. For this reasons less attention is devoted to a power function

representation in this work.

To obtain more accurate estimates of relative permeabilities, it is necessary to consider

more flexible functional forms that contain a greater number of parameters [8].

B-splines
With B-splines, polynomial splines of any order can be conveniently implemented in the

estimation procedure. This feature allows greater flexibility for incorporating inequality

constraints into the parameter-estimation process [4]. For more detailed description see Appendix

A.

With B-splines, the relative permeability curves may be written as follows:

n n
k rw = ∑ Pi w N i ,k ( Snw) k ro = ∑ Pi o N i ,k ( Sno) (3)
i =0 i =0

Pi w and Pi o , i = 0, n represent the parameters to be determined by the parameter-

estimation method. Commonly, the order of splines, k, is chosen to be four, resulting in the cubic

spline representation which was previously investigated [4] and further considered as optimal.

Appropriate number of B-spline partitions is another point to consider. The great advantage

with a B-spline representation, compared to a power function representation, is that any

continuous function can be approximated arbitrarily well by polynomial splines, provided that

sufficient number of knots are allowed [6]. Unfortunately, increasing the dimension of the

parameter space will cause the problem to be more ill conditioned. Watson et al. evaluated, based

on an error analysis, that for field-scale applications the relative permeability curves can take

irregular shapes and, consequently, the dimension of the B-spline space was chosen to be eight

for each relative permeability curve, resulting in 16 unknown parameters [4] (Figure 1). This

implies that we are using two exterior and two interior knots for each curve. However, if the


irreducible water saturation, Swi, and residual oil saturation, 1-Sor, are given exterior knots for

water and oil relative permeabilities coincide (Sw1=Sw5=Swi and Sw4=Sw8=1-Sor in Figure 1),

reducing number of unknown parameters to 12. Finally, krw1(Swi)=0 and kro4(1-Sor)=0, and for

B-spline representation it is necessary to estimate 10 parameters. And only 8 parameters are

needed to be found out if the end point relative permeabilities krw4(1-Sor) and kro1(Swi) are also

given.

Error Analysis
To assess the measure of accuracy with which relative permeabilities may be estimated by

parameter estimation methods special objective function is constructed.

Multivariable penalized misfit functional
An objective function J is defined as the weighted sum of squares of the differences

between the observed and calculated values of the reservoir pressure and water cut.

∑ (FWCT ) ∑ (FOPR )
1 2 1 2
J (β ) = hist
− FWCT calc ( β ) + hist
− FOPR calc ( β ) (4)
σ 2
w i σ 2
o i

where FWCT – field water cut, FOPR – field oil production rate for data element i. The weight

factors σ w , σ o are the variances of the error in the measurements of FWCT and FOPR. Vector β
2 2

represents the adjustable parameters.

In general, water pseudo-relative permeability is determinable with less precision, because

first production data is mostly influenced by oil flow, rather then water flow. “Water data”, data

which is influenced by water flow, is only becomes available after water cut. FWCT term in

functional (4) give response to water flow, while FOPR is represent oil flow. It is thought that

including of these two terms will improve quality of estimation of pseudo-relative permeabilities.

The history-matching problem consists of determining the unknown vector β so that J

is minimized. In case of power function representation of permeabilities, vector β is given by:


β = (krw, kro, ew, eo) (5)

While for B-spline representation in case of known end-points values (Figure 1):

β = ( Sw2, krw2, Sw3, krw3, Sw5, kro2, Sw6, kro3) (6)

As was stated previously, in solution of the inverse problems regularization often is

required. Yang and Watson incorporate the prior information (laboratory estimates) by

adding to J a special term which penalizes deviation of the current estimates from the

laboratory estimates [5]. However, in this research, the main objective is to estimate pseudo-

relative permeabilities, which are commonly has different shape from laboratory curves.

Also, by incorporating such kind of term, we necessarily will increase the number of

parameters to be estimated with corresponding weighting factor. So, usage of such term is

seems unfeasible and even if we have some prior information, it is used with much more

convenience while choosing initial guess and considered in detail in initial guess section.

Smoothing term, which represents the second order derivative in the stabilizing

functional, was decided to be added to the objective function by Kulkarni [6]. The second

derivative characterizes the curvature of the curve, the more the derivative the more the

curvature. Then curvature changes its sign, with argument, the curve start to oscillate. Large

range of values of the curvature (second derivative) relative to zero means large oscillation.

And smoothing term means minimization of the deviations of the second derivative from

zero. The main disadvantage of this term is high deviation of the estimated curves from

initial guess curves and increase in difference between history and calculated data.

In this work, it is assumed that maximum of two inflection points are expected in

relative permeabilities curves. This is simple consequence of the fact that we are using only

two interior knots in the B-spline representation. Since, oscillations a priori are small, and

addition of the smoothing term seems irrelevant.


Approximation error
There are a number of sources of error in estimates of relative permeabilities from

displacement experimental data. Clearly, a classification of error types and a systematic

elimination of some types of error so that others may be quantified are necessary in the

examination of the accuracy of relative-permeability estimates [8].

The sources of error in the estimates can be classified into two groups, modeling error

and estimation error. Estimation error can be further subclassified as variance error and bias

error (see Appendix B).

Modeling error is the result of the inadequacy of the mathematical model of the

displacement experiment in the exact representation of that experiment. Here model error is

refers to error which is incorporated to Eclipse simulations, and cannot be controlled.

Estimation error refers to any errors encountered in the mathematical procedures used

to estimate relative permeabilities from inexact measurements of the experimental flow data.

The magnitudes of bias and variance error will depend on the functional forms that are

chosen to represent the relative-permeability curves. In general, as functional forms with

greater numbers of parameters are chosen, variance error is increase and bias error is

decrease. The optimal functional forms will be those that minimize the combination of those

two errors. It is shown what minimization of the objective function (4), subject to the

assumption that the assumed functional forms for the relative-permeability curves are the correct

ones, also minimizes the average variance errors [8].

As a result, B-spline representation conforms its availability to represent pseudo-

relative permeabilities curves in most accurate manner.

Automatic history matching algorithm specification
The basic idea of developed algorithm is minimization of the functional (4) by varying

pseudo-relative permeabilities curves. This is done by iteration process of adjusting


parameters in B-spline representation. Such process integrates several execution modules.

The main steps of algorithm are shown in general block diagram (Figure 2). All supplying

codes are written using Microsoft Visual Basic 6.3 and MS Excel worksheet is used to run

algorithm. The main features of execution modules are discussed below.

Initial guess
The choice of initial guess has vital importance. Approximation accuracy and speed of

convergence of the minimization algorithm are all sensitive to initial guess. In case of B-

spline representation of pseudo-relative permeabilities, initial guess is two exterior knots and

4 interior knots. Here, exterior knots are supposed as known end-point saturations: Swi, 1-So

with corresponding end-point permeabilities: krw(Swi), krw(1-Sor), kro(Swi) and kro(1-Sor).

Four interior knots represent a set of adjustable parameters.

In solution of inverse problems, the accuracy of obtained values can be improved using

any prior information available. For estimation of the rock relative permeabilities, such kind

of prior information is laboratory estimates. In this work, however, it is necessary to obtain

pseudo-relative permeabilities which often have irregular shapes and prior laboratory data

hardly can improve the accuracy of estimation. Still, by choosing initial guess we can control

the shape of permeabilities curves, which, in case of pseudo-relative permeabilities, have

convex shapes.

In general, to minimize objective function which has local and global minimums, it is

necessary to consider several reasonable initial guesses. Initial guess which will give best

accuracy is assumed as optimal. The sensitivity of estimation accuracy to the initial guess is

fully examined in the following Chapter 2.

When initial guess is chosen, B-spline approximation of phase relative permeabilities

based on selected knots is calculated and SWFN, SOF2 (SCAL data) tables are filled in.

These tables are used as input data for Eclipse model simulation. They included in PROPS


section of Eclipse data file in external file. SWFN is used for water saturation function

specification i.e. columns of water saturation, water relative permeability and oil/water

capillary pressure while SOF2 table is used to input two-phase oil relative permeability data

i.e. oil saturation versus oil relative permeability. At each step of adjusting parameters, new

file of SCAL data is created and inputted to a simulation model.

Eclipse model simulation
Shlumberger GeoQuest Eclipse is used for simulations in the current work. Firstly, all

researches on the pseudo permeabilities representations are held on synthetic model (Chapter

2). Next, full “K” oil field model is used in the automatic history matching algorithm

(Chapter 3). At each step models are inputted with pseudo permeabilities and return required

data (Field Water Cut, Field Oil Production Rate) to calculate functional (4).

Minimization algorithm
Microsoft Excel add-in Solver was used for minimization of the multivariable

functional (4). Brief Solver settings are: Newton search method is used. Initial estimates of

the basic variables in each one-dimensional search represent linear extrapolation from a

tangent vector. Forward differencing is used to estimate partial derivatives of the objective

and constraint functions. The use of Solver is dictated by project time limit, it was decided to

use already available minimization module. However, Solver has its own constrains (it has to

little controls of minimization algorithm), which precludes construction of the optimal

history-matching algorithm. The best way is to develop own minimization module, which

will be adapted to field conditions. This is time consuming and refers to a further work.

A set of pseudo-relative permeabilities which minimize (4) are assumes as “true”

curves in the sense that they give minimal error between history and calculated data.

In the following Chapter 2, algorithm performance is tested.


Results
• The problem of estimation of the pseudo-relative permeabilities from reservoir

performance history data was mathematically assessed.

• To represent pseudo-relative permeabilities, power functions and B-spline

representations were chosen.

• Field water cut and Field oil production rate were chosen as data for history

matching process.

• Automatic history-matching algorithm of pseudos estimation was developed.



Chapter 2 Synthetic Model Calculations
Model specification
In this chapter test problems are presented to illustrate the use of developed automatic

history-matching method. The synthetic model corresponds to the hypothetical waterflooding of

2D cross sectional reservoir initially at the irreducible water saturation with one injector and one

producer. A set of pseudo-relative permeability parameters is selected as the true parameter set

for any kind of the representation. The “observed data” is generated by executing Eclipse

simulator with the true, but presumed unknown, parameters values. An initial guess is provided

for the pseudos. Synthetic model is shown on Figure 3 while model details are given in Table 1.

Capillary pressure is incorporated by means of Corey-Brooks approximation.

Automatic history matching algorithm performance
Here, the ability of developed algorithm to estimate pseudo-relative permeability curves

from production data is examined. In hypothetical history-matching problem, the true but

presumed unknown parameter values are available so that the quality of match can be determined

by comparing not only the initial and final values of the objective function but also the estimated

and true parameter values.

One general assumption here is that end point permeability values, krw(1-Sor) and

kro(Swi), are also assumed as unknown parameters. This will increase number of adjustable

parameters in B-spline representation to 10. Such assumption is necessary in order to

examine generalized problem of parameter estimation.

Other point to mark is what in all following tests the true pseudo curves are known and

including in functional (4) special term which incorporates penalizes deviation of estimates from

true curve will improve accuracy in the shapes of obtained estimates. However, as already have

been mentioned in Chapter 1 Error Analysis section, in this research, the main objective is to


estimate pseudo-relative permeabilities, which has a prior unknown shape, and such kind of

term is ignored. See more in the following sensitivity to initial guess study.

Power functions
In case of power function representation of pseudos, the selected true parameters are shown

in Table 2. The method of choosing several initial guesses and then indication an optimal one is

applied. Figure 4 shows true and obtained pseudos after 6 trials with different initial guesses after

10 iterations. Providing the greater number of iterations to the algorithm will further improve

matches, and restricted in order to devote much CPU time for more serious problems. Also Table

2 summarizes results of simulation for each initial guess. Here 4 chosen initial guesses converges

to true parameters with high level of accuracy in field data match (values of objective function

Table 2). However, it can be seen a range of endpoint relative permeability for water in inversion

results. This causes water-cut response to change significantly. If to assume what end point

permeabilities are known from core flood data, the match can be even better.

Developed algorithm determines unknown parameters in the case of the power function

representation of pseudos with acceptable quality of match.

B-spline
True set of parameters for B-spline representation is shown in Table 3, shaded cells

correspond to unknown 10 parameters. Figure 5 shows true and obtained pseudos after 6

simulations with different initial guesses after 10 iterations. Here range in end point water

permeability values is reduced due to B-spline’s local support. This means that changing the end

point relative permeability of water will change the relative curve only in the region of support

for the last B-spline. The achieved quality of match (low values of objective functions) and

deference between known and estimated parameters are even better for B-spline (not shown).

Algorithm proved it’s capability in estimation of parameters for B-spline representation of

pseudos.

As result, it is shown what developed automatic history-matching algorithm is able to

estimate unknown representation parameters of pseudos with acceptable accuracy.

Representations competition
In this section B-spline representation of pseudo-relative permeability curves is compared

with power functions representation. The main goal of this section is to indicate such kind of

representation, which gives best history matches on the synthetic model, in order to investigate it

in more detail and next applying it to real “K’ oil field model.

The synthetic model remains the same as in previous test.

Firstly, set of true pseudos is given by polynomial functions (7) for water and (8) for oil

pseudo-relative permeability curves consequently (Figure 6).

krw = 0.169S wn − 0.097 S wm + 0.305S wm
2 3
(7)

kro = 0.937 − 3.197 S wm + 4.08S wm − 2.221S wm + 0.401S wm
2 3 2
(8)

Such type of representation is tends to resemble the convex downward curves. End point

saturations for water and oil assumed as known. Four tries for each representation was made with

upper limit 10 on iteration number. Obtained estimates for power function and B-spline

representations of pseudos are shown in Figure 8 and Figure 9 correspondingly. From first view,

power function representation seems more accurate, however examination of Table 4 which

summarizes the values of the objective function for each trial, shows that for B-spline

representation, values of objective function are significantly lower then those for power

functions. Additionally, FWCT and FORP responses for B-spline (Figure 12) in comparison with

power functions (Figure 13) supports the point, that B-splines are more convenient form in

pseudo-relative permeability history-matching.

Other representation of pseudos is given with use of trigonometric function SIN(x), for oil

(9) and water (10) Figure 7.



krw =
sin( x) + 1
3.5
(
, x ∈ π , 3π
2 2
) (9)

kro =
sin( x) + 1
8
(
, x ∈ − π ,π
2 2
) (10)

Here, curves have two inflection points and resemble convex upward curves. Such shape is

typical for pseudo-relative permeabilities. Each representation was used to estimate unknown

pseudos with four reasonable initial guesses and iteration upper limit of 10. For power functions

representation result estimates are depicted in Figure 10 and FWCT and FORP matches for best

estimate are presented in Figure 14. For B-splines, obtain estimates are shown in Figure 11.

Production responses, for most accurate estimate are shown in Figure 15.

The bad results shown by power function in performed competition can be explained

from the error point of view (Chapter 1 Error analysis). A large bias error of power

functional representation explained by inability of the assumed functional forms to represent

the true relative-permeability curves exactly. There are insufficient controls (number of

adjustable parameters), however variance error is expected to be lower for such type of

representation of pseudos. It is clear that B-spline representation is more flexible in estimation

of the complex shaped curves, minimizing bias error. Variance error in B-splines can be

controlled, by correct choice of knots [4].

As a result, B-spline representation is optimum choice which minimizes combination of

bias and variance errors. From this point, only B-spline representation will be used in further

history-matching. Also, next section is a more detailed examination of the nature of the B-

splines.

Sensitivity to the initial guess in the B-spline representation
This section is devoted to sensitivity studies of B-spline representation to the initial guess.

The good choice of initial guess can be very important, and can significantly reduce the number

of algorithm iterations, this is especially meaning in the case of multivariable functional

minimization. Although, CPU power in present days allows us to perform simulations with great

number of iterations and parameters, optimization of the number of iterations of algorithms can

be related to economical benefit, when CPU time for each minimization problem is reduces, it is

becomes possible to perform history-matching for several times to achieve better results.

Also, it should be noted that values of end point relative permeabilities, krw(1-Sor) and

kro(Swi), are supposed as known. Such assumption is quite reasonable, if some laboratory data

from core analysis is available. Otherwise, sensitivity study with unknown end point

permeabilities should be carried out. This could be related to a further work.

Often pseudo-relative permeability curves are convex upwards. It is straightforward to

thought what in the whole assemblage of the initial guesses, where are a range of them, which

predicts pseudos with less computational time. The sensitivity study of this section is aimed to

determine a range of initial guesses which will improve to true solution using less CPU time.

Prior shape of pseudos is presumes as known. First studies are devoted to convex downward

shapes of pseudo-relative permeability curves. Other part performs analysis for case of convex

upwards shapes. Finally, a general approach of choosing of the initial guess for fieldwide model

history matching is suggested.

To reproduce convex downwards curves same polynomial functions (7) for water and (8)

for oil pseudo-relative permeability curves was used (Figure 6). For a given permeability curve

we can set such initial guess that solution will approach to true curve from top or from bottom.

The same is true for other permeability curve. Overall, this result in 4 different types of initial

guesses classified by the way in which they improved (from top or from bottom). All of then are

schematically shown in Figure 16. The choice of knots placement for each initial guess type can

be found in Appendix C.1. For each type of initial guesses a number of simulations were

performed. Resultant values of the objective function (4) are shown in Table 5. It is clear that for


a prior suggestion of convex downward shape of pseudo curves, the best way of initial guess

choice is Way 2 (Figure 16). The physical means of such result is not realized yet.

A convex upwards pseudos are shown on Figure 7. Same strategies as in previous case were

applied to indicate most accurate kind of initial guess. Results are given in Table 6. On this time,

Way 1 shows best performance, but still Way 2 can be applied.

The main results of the performed sensitivity study can be stated as follows:

• The accuracy of the history-match is sensitive to the initial guess.

• Good initial guess will reduce CPU time usage, and improve convergence rate of the

algorithm.

• If it is prior known that desired permeability curves are convex downwards, initial guess

should be chosen in such manner, that oil relative permeability curve will approach to

the true curve from the bottom, while water relative permeability curve from the top

(Way 2 in Figure 16).

• If it is prior known that desired permeability curves are convex upwards, initial guess

should be chosen in such manner, that oil and water relative permeability curves

approach to the true curves from the bottom (Way 1 in Figure 16).

• Finally, if we are dealing with real oil field history matching problem, and want to

estimate pseudo-relative permeabilities, first initial guess should be chosen in the Way 1

fashion.

These results will be used in the following Chapter 3.

Results
• Special synthetic model was constructed to test history-matching algorithm

• The B-spline representation is chosen as best in estimating of the pseudo-relative

permeabilities.


• The strategy for choosing initial guess is suggested on the basis of sensitivity

study.

• All obtained results of this Chapter will be used in the following Chapter 3 for real

field history-matching problem.



Chapter 3 “K” Oil Field Adaptation
Current Chapter is fully devoted to the application of the developed automatic history-

matching algorithm to obtain pseudo-relative permeability curves from K oil filed production

data.

Geological description
The full description of the filed can be found in [14], here only main features are outlined.

The K-field is a moderately large oil reservoir of Jurassic age located in Western Siberia,

Russian Federation. The reservoir structure is a complex series of local elongated folds

subdivided by numerous narrow troughs. This structure presumably was formed in two different

stages. The major background folding structure was inherited from the ancient tectonic pattern

developed in pre-Jurassic time, and has been further complicated by several relatively young

local faults formed in post-Jurassic time.

The reservoir is comprised of three major flow units; J2, J3 and J4. The reservoir rocks

were deposited in a sand-rich deltaic environment and are composed of sandstones with

permeability varying from 0.01 to 500 md. The best quality reservoir rocks (k > 100 md)

correspond to a series of mouth bars and beaches formed in a high energy environment. These are

encountered only in the northern-western part of the field, while the rest of the reservoir is

represented mainly by bar and channel sandstones of moderate permeability (10 < k < 100 md);

low permeability bar sandstones (1 < k < 10 md); located more seaward prodelta siltstones (k < 1

md) and shales.

Model specification
The main steps of model construction are described in [14]. Again, here only main aspects

are highlighted.


The hydraulic flow unit (HFU) approach was used as an integrating tool for petrophysical

description of the reservoir. Predicted distributions of HFU in all logged wells were stochastically

populated in the 3D geological model. Obtained HFU grids were used for evaluation of 3D

permeability distributions from the porosity grid. To select appropriate realizations of HFU and

consequently permeability for further history matching and production forecast, an uncertainty

analysis was performed using streamline simulation and three relevant realizations,

corresponding to probabilities of recovery factor 10, 50 and 90 % were selected. These three

HFU distributions were upscaled and integrated in the conventional dynamic reservoir model to

define regions of capillary pressure and relative permeability functions.

Generally, 8 different HFU was indicated Table 7. Relative permeability curves were

constructed using Corey approximation Figure 17, while pseudos were obtained by upscaling

using Kyte and Berry method. The main purpose of history matching is to obtain such pseudos

for each HFU, which will give better history match then match obtained with upscaled relative

permeabilities. So, it is necessary to obtain eight sets of pseudos, or 8x8 = 64 unknown

parameters of B-spline representation. This can be done in two ways. First way is to determine all

64 parameters simultaneously, this is mathematically right way, for such purpose it is necessary

to seriously update generated codes of developed algorithm. The main problem can be in

minimization of such multivariable functional, which can have additional local extremes, and

obtain solutions can be worse. Realization of such approach is considered as further work. The

second way is determination of pseudos continuously, one by one. The main constrain here is that

separate determination of pseudos can be done only in case of independent sets of pseudos, which

implies what HFU’s flow properties are independent. This, in general, is not true. However, in

this research, the special step sequence in continuous HFU’s pseudos estimation is considered. In

addition, we will estimate unknown pseudos only for HFU 1, 2, 3 and 4. The reason for this is

outlined below. The applicability of such approach is confirms by obtained match results.


History matching
Estimation of pseudo-relative permeabilities for whole “K” oil filed seems unreliable. It is

straightforward to subdivide “K” oil field to distinct sectors and estimate pseudos for these

distinct sectors. This will improve accuracy of whole simulation model and will give more

reliable results. The way of subdivision is another question of interest. In this work pseudo-

relative permeabilities are estimated for North-Western part of “K” oil field. In further work even

North-Western part should be subdivided for pseudos estimation.

In general, to estimate pseudo-relative permeability correctly, it is necessary to use all

available production data in the simulations. This can be very time consuming, and can’t be

completed within provided project time. Since, production data of only last two years were used

in history matching. Such approach can show applicability of developed algorithm for real field

pseudos estimation. Simulations for full model can be related to a further work.

Corey functions as initial guess
B-spline representation with 8 unknown parameters is chosen for search of true pseudos.

Although, Chapter 2 initial guess sensitivity study gives suggestions for choosing initial guesses

for prior convex upward curves, first history matching test was done with initial guesses based on

available Corey approximation for relative permeability curves for each HFU. Prior such kind of

initial guesses are incorrect. However, experiment is initiated in order to check ability of well

known Corey functions to serve as initial guesses. The convenience of such point is obvious: in

case of success three is no need in seeking good initial guesses.

Step sequence in continuous HFU’s pseudos estimation presumes that for distant HFU’s

pseudo-relative permeability curves can be found separately as follows:

1. Estimate pseudo-relative permeability curves for HFU1, other 7 HFU’s pseudos are

set to known upscaled relative permeabilities.


2. Estimate pseudo-relative permeability curves for HFU2, for HFU1 use obtained in

step 1 values; remaining 6 HFU’s pseudos are set to known upscaled relative

permeabilities.

3. Estimate pseudo-relative permeability curves for HFU3, again using results from

previous steps, etc. for HFU4 and HFU5

We start with “better”, in the sense of productivity, HFU 1 and 2, and finish with less

productive HFU 3, 4, 5. Such approach in choosing sequence is not tends to be the correct one,

and can be a major source of weakness, special sensitivity study is necessary to confirm or to

disprove the correctness of taken choice and refers to a further work.

HFU’s 6-8 have less influence on the water/oil flow behavior in the reservoir and on

pseudos, thus assumes irrelevant in question. The matching results can be examined using Figure

18 where values of the objective function (4) for each step are depicted. The starting value of

function (4), “Pseudo”, corresponds to a simulation using upscaled permeabilities, assumes as

true value. “Initial” value is result of model run using Corey approximation for HFU of interest,

“Final” values is algorithm working result. After step 1 of adjustment parameters for HFU1, the

error in match was reduced by 4.3 times, while for other HFU’s reduction doesn’t exceeds 1.5.

Final mismatch is about 280%, which can be interpreted as significant. Obtained pseudos are

shown in Figure 19, final production matches are shown in Figure 20. It can be noted that even

with incorrect convex downwards initial guesses algorithm gives correct shapes of obtained

curves. The resultant responses of water cut and oil production rates match history data less

accurate in comparison with curves obtained after simulation using upscaled relative permeability

curves. Such results show that Corey approximation of relative permeability curves can not be

used as initial guesses, with high accuracy of match. This point is supported by Chapter 2 initial

guess sensitivity analysis.


Right choice of initial guess
For more precise matches it necessary to use correct initial guesses. The best choice is

already available upscaled permeability curves, for example in this case upscaled permeability

curves obtained using Kyte and Berry method. However, if there are no available estimates, it is

necessary to choose initial guess in developed in Chapter 2 fashion. Here known upscaled values

are approximated by B-splines which and then serves as initial guesses.

The sequence of HFU’s upscaled relative permeabilities estimation is as follows:

1. Estimate pseudo-relative permeability curves for HFU1, other 7 HFU’s pseudos are

set to known upscaled relative permeabilities.

2. Estimate pseudo-relative permeability curves for HFU2, for HFU1 use obtained in

step 1 values; remaining 6 HFU’s pseudos are set to known upscaled relative

permeabilities.

3. Estimate pseudo-relative permeability curves for HFU3, again using results from

previous steps, etc. for HFU 4.

HFU 5 was ignored, because of project time limit.

Again, values of the objective function (4) can be examined using Figure 21. Now

algorithm shows best performance and gives 17% of final improvement in production history

match. Obtained final set of pseudos are shown in Figure 22. Water cut and oil production rates

responses are given in Figure 23. In this case obtained curves are in better agreement with

production data, and desired improvement in match is achieved.

Results
• Developed automatic history-matching algorithm was applied for estimation

pseudo-relative permeability curves for “K” oil field. B-spline representation of

pseudos was used.


• Initial guess of relative permeabilities in the form of Corey functions show bad

matching results and can not be used as initial guess to obtain accurate matches.

• Initial guess in the form suggested in the initial guess sensitivity study of Chapter 2

was further used in history matching.

• Obtained pseudos give 17% history match improvement over upscaled relative

permeabilities.



Conclusions
• To estimate pseudo relative permeability curves for ”K” oil field using reservoir

performance history data a special automatic history-matching algorithm was

developed (Chapter 1). The general idea of algorithm is illustrated in block

diagram Figure 2.

• Further algorithm tests on the basis of constructed synthetic model were

carried out in Chapter 2. To represent pseudo-relative permeabilities, power

functions and B-spline representations were chosen. The B-spline

representation shows most accurate history matches. Further sensitivity to initial

guess study give an optimal strategy for choosing initial guess in pseudo-

permeabilities estimation.

• Developed automatic history-matching algorithm was applied for estimation

pseudo-relative permeability curves for “K” oil field (Chapter 3). Reservoir model

inputted with obtained pseudos shows good agreement with field production data

and give 17% improvement over matches obtained using upscaled by Kyte and

Berry method permeability curves.



Suggestions for further work
1. Optimization of developed algorithm. This implies use of more powerful

programming language (MS Visual C++) to reduce CPU usage time.

2. Improve minimization module of the algorithm. Now Microsoft Excel add-in

Solver is used for this purpose. However, Solver has main disadvantage: it provides

to little controls in minimization algorithm, thus can not be considered as optimal.

The best way is to develop own minimization module, which will be adapted to field

conditions.

3. Sensitivity to initial guess study was carried out with assumption of known end

point relative permeabilities, krw(1-Sor) and kro(Swi). Additional study is required in

the case then these values are not given, for each end point relative permeability it is

necessary to consider a range of possible values, and repeat performed study.

4. In history matching problem for “K” oil field it is necessary to obtain pseudos for

each HFU, or eight sets of pseudos - 8x8 = 64 unknown parameters of B-spline

representation. In this work parameters for pseudos representation are estimated

continuously, one by one. The main constrain here is that separate determination of

pseudos can be done only in case of independent sets of pseudos, which implies what

HFU’s flow properties are independent. This, in general, is not true. Mathematically right

way, for such purpose is simultaneous determination of all 64 parameters, which implies

serious update of generated codes of history matching algorithm.

5. In this work pseudo-relative permeabilities are estimated for North-Western part

of “K” oil field. For further work it is suggested to subdivide North-Western for pseudos

estimation. This will improve accuracy of whole simulation model and will give more

reliable results.



Figures and Tables
Figures

Water Relative Permeability
Oil Relative Permeability

(Sw1,kro1)

(Sw3,kro3) (Sw7,krw3)
(Sw6,krw2)
(Sw8,krw4)
(Sw2,kro2) (Sw4,kro4) (Sw5,krw1)

Swi 1-Sor Swi 1-Sor
Sw Sw
Figure 1 B-spline representation of pseudo-relative permeability curves



Initial guess

Relative Permeability

Sw

Eclipse Model
Simulation

FWCT
Pressure

Functional
calculation

No
Min ?

Yes
“True” pseudo
relative
permeabilities


Figure 2 Automatic history-matching algorithm block diagram

Figure 3 Synthetic model

1.0
krw True
kro True
0.9

0.8 Other curves correspond
to matches obtained using
0.7
different initial guesses
Relative Peremability

0.6

0.5

0.4

0.3

0.2

0.1

0.0
0.2 0.3 0.4 0.5 0.6 0.7
Sw

Figure 4 True and estimated relative permeability curves, power function representation


0.7
krw True
0.6
kro True
Other curves correspond
0.5

0.4

0.3

0.2

0.1

0.0
0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000
Sw

Figure 5 True and estimated relative permeability curves, B-spline representation

1.0

Krw
0.9
Kro

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Sw

Figure 6 Polynomial function representations of pseudos for competition test


0.6
krw
kro

0.5

0.4

0.3

0.2

0.1

0.0
0.0 0.1 0.2 0.3 Sw 0.4 0.5 0.6 0.7

Figure 7 Trigonometric function (sin(x)) representation of pseudos for competition test

1.0
Krw True
0.9 Kro True

0.8
0.7 different initial guesses

0.6

0.5

0.4

0.3

0.2

0.1

0.0
0.2 0.4 Sw 0.6 0.8

Figure 8 Power function estimates of the polynomial function. Competition test



1.0 Krw True
Kro True
0.8
0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Sw

Figure 9 B-spline estimates of the polynomial function. Competition test

0.7
krw True
kro True
0.5 different initial guesses

0.4

0.3

0.2

0.1

0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Sw

Figure 10 Power function estimates of the trigonometric function. Competition test


0.8
krw True
kro True
0.7
0.6

0.5

0.4

0.3

0.2

0.1

0.0
0.0 0.1 0.2 0.3 Sw 0.4 0.5 0.6 0.7

Figure 11 B-spline estimates of the trigonometric function. Competition test
1.2 2500

Model
1
Field Oil Production Rate, STB/DAY

Model 2000 Caclulated

Caclulated
0.8
Field Water Cut

1500

0.6

1000
0.4

500
0.2

0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300

Time. days Time, days

Figure 12 FWCT and FORP responses. B-spline estimates of the polynomial function. Competition test


1 2500

0.9
Model
Model

0.8 2000
Caclulated
Caclulated
0.7
Field Water Cut

0.6 1500

0.5

0.4 1000

0.3

0.2 500

0.1

0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300


Figure 13 FWCT and FORP responses. Power function estimates of the polynomial function. Competition test

1 2500

0.9
Model
Model

0.8 2000
Caclulated
Caclulated
0.7
Field Water Cut

0.6 1500

0.5

0.4 1000

0.3

0.2 500

0.1

0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300


Figure 14 FWCT and FORP responses. Power function estimates of the trigonometric function. Competition
test

1 2500

0.9 Model

0.8 Model 2000 Caclulated

0.7 Caclulated
Field Water Cut

0.6 1500

0.5

0.4 1000

0.3

0.2 500

0.1

0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300


Figure 15 FWCT and FORP responses. B-spline estimates of the trigonometric function. Competition test


1.0 1.0

0.9 Way 1 Krw
Kro
0.9 Way 2 Krw
Kro

0.8 0.8

0.7 0.7


0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0.0 0.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Sw Sw

Initial guess Initial guess
for kro improves from bottom for kro improves from bottom
for krw improves from bottom for krw improves from top
1.0
1.0

0.9 Way 3 Krw
Kro
0.9
Way 4
Krw
Kro

0.8
0.8

0.7
0.7

0.6
0.6

0.5 0.5

0.4
0.4

0.3 0.3

0.2
0.2

0.1 0.1

0.0
0.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Sw
Sw

Initial guess Initial guess
for kro improves from top for kro improves from top
for krw improves from bottom for krw improves from top

Figure 16 Four ways of initial guess improvement. Sensitivity study


Pseudo-relative permeabilities estimation from reservoir performance history data

Pseudo-relative permeabilities estimation from reservoir performance history data

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (10)

Similar to Pseudo-relative permeabilities estimation from reservoir performance history data

Similar to Pseudo-relative permeabilities estimation from reservoir performance history data (20)

Recently uploaded

Recently uploaded (20)

Pseudo-relative permeabilities estimation from reservoir performance history data