Development of Genetic Algorithm Models for Tracer Test

Physical and Numerical Simulation of Geotechnical Engineering
2nd ISSUE, March 2011
Development of Genetic Algorithm Models for Tracer Test
Interpretation During Polymer Flood
LI Junjian, JIANG Hanqiao, JIANG Liangliang
Department of Production Engineering, Research Institute of Petroleum Exploration & Development (RIPED),
PetroChina, 100083
ABSTACT: Interwell tracer test is an effective approach to quantitatively characterize the
interwell parameters. In this presentation, reservoir independent variables that tracer test
interpreted are determined and objective function is established on the basis of semi-analytical
method. Also, genetic algorithm is introduced to simulate the probable subsurface distribution of
interwell parameters basing on the built geological model and tracer output curves. Additionally,
the match of both tracer output time and concentration is utilized to get the automatically
integrated parameter inversion and optimal interpretation of tracer test, and the permeability and
thickness of high-permeability channels are also obtained, which greatly help improve the
performance of polymer flood in the studied block.
KEYWORDS: polymer flood, tracer test, semi-analytical method, parameter inversion, genetic
algorithm
1 INFORMATION
N
_
F   (Ci*  C i )2
i 1
In the later development of oilfield, the corresponding
relationship between interwell stasis and dynamics as well
as the macroscopic distribution of remaining oil of a
reservoir are key information to reservoir management. In
1964, interwell tracer test and interpretation method were
raised by Brigham and Smith[1], and then they developed
greatly for their wide application in various international
oilfields[2-9]. The interpretation accuracy has proven to be
favorable both theoretically and in experiments.
Interwell tracer test mainly embraces quantitative and
semi-quantitative interpretation approaches as well as
numerical and semi-analytical methods. In semi-analytical
method, numerical modeling and analytical concentration
equation are used to get the pressure distribution trend at
certain period and the tracer output concentration at
monitor well respectively. Then mass point tracing method
is applied to get the flow field, which is also used to
connect the pressure distribution trend and the
one-dimension concentration equation. In this presentation,
genetic algorithms are employed to obtain the inversion
method of tracer test interpretation in polymer flood based
on the semi-analytical method.
It is typically an inverse issue to characterize the
formation parameters by tracer output curves, which, from
a mathematical view, is equally a pattern recognition
problem. And inverting parameters from test results could
unavoidably cause non-linear and ill-posed problem.
Non-linear optimization (or multiple regression analysis) is
a powerful tool in fitting data by a set of variables. This
procedure is also known as a non-linear least squares
method for curve fitting. The idea is to minimize the
objective function F[10]:
© ST. PLUM-BLOSSOM PRESS PTY LTD
In which:
Ci*－observed concentration at sample point i;
_
C i －overall concentration computed at sample point i;
N － number of data points or number of observed
concentrations;
i －an observation point.
There are manifold variables that could affect the tracer
output concentration such as oil and water saturation,
porosity, containing area, effective thicknesses of layers,
permeability etc. To interpret theses parameters by
matching the tracer output curves with least squares method,
partial derivative of each variable should be gained. The
analytical solution to the tracer output curve is so complex
that partial derivatives can hardly be got. Hence the current
interpretation method is that determine the mean values of
parameters such as water saturation, porosity, drainage area
etc. via parameter adjustment, then calculate the
permeability and thickness of each layer through the
variable- partition nonlinear least squares method. However
there still a lot of shortages of this method. For example,
parameter adjustment is entirely arduous and the combined
parameters in the adjustment could also lead to ambiguity
problem.
In most cases, we can only get local optimal solution to
the nonlinear problems according to the linearizing level to
the inversion of nonlinearity which can hardly satisfy the
request of model inversion. Actually, formation parameters
and flowing model are highly nonlinear and conventionally
gained parameters may not be optimal since there are more
than one optimal point available. The traditional method,
therefore, is of limitation and difficulty in the inversion of
Development of Genetic Algorithm models for Tracer Test Interpretation during Polymer Flood
DOI: 10. 5503/J. PNSGE. 2011. 02.003
geological parameters. Take two test groups in the
researching scope for instance. Each group has two tracer
production wells and each well owns two vertical
high-permeability channels. And there are twenty
streamlines connecting each two wells and each streamline
itself has two variables, then we get substantial variables to
adjust, which is typically a large-scale parameter system.
Genetic algorithm is a large-scale optimizing algorithm,
and it is widely studied and used for its unique optimization
mechanism and great generality[11-16], especially in the
inversion of tracer it provides a novel approach to invert the
parameters of formations.
min( z )   C cal  C test 
j
2
i
In which, i and j signify the well number and group
number respectively.
That is to say, name the square sum of the difference
between calculated tracer concentration and tested
concentration of all production wells in the block as the
objective function. Applied with certain kind of optimizing
mechanism, the parameters when the objective function
gets its minimum value are the most probable distribution
of parameters.
2.2 Principles of genetic algorithm
2 CALCULATION METHOD OF GENETIC
ALGORITHM
Based on the semi-analytical solution, independent
variables are determined and objective function is also
established. Then revised genetic algorithm is applied to
match the tracer test curves and interpret reservoir
parameters, and the interpretation process of interwell
water trace test is finally formed.
2.1 Establishment of objective function
Ensure the variables according to the research interest.
For interwell tracer test interpretation, variables that could
directly gain through matching all the curves of a block
include:
1) Distribution of tracer output channels in different well
groups.
Determine
the
accurate
number
of
high-permeability channels and their vertical distribution
via curve matching with the algorithm at the condition of
artificially giving the upper number limit of
high-permeability channels.
2) Tracer breakthrough of horizontal streamlines in
different well groups. The breakthrough streamline
numbers play a great role in the rising and dropping of
concentration curve.
3) The thicknesses of streamlines of each
high-permeability channel in different groups. The
thickness of high-permeability channel affects greatly the
output concentration peak as well as output time and
concentration distribution curve.
4) The permeability of streamlines of each
high-permeability channel in different groups. The
permeability of high-permeability channels exerts a great
impact on the tracer breakthrough time as well as the output
concentration peak and distribution curve.
5) Determine the distribution of remaining oil saturation
in high-permeability channels at the condition of dual tracer
test.
As to the big complication underground and large
amount of variables that could lead to the huge difficulty in
parameter adjustment, the construction of an objective
function to optimize the process seems entirely necessary.
The objective function can be determined as follows:
Though we can exclude or determine certain parameters
by reservoir engineering analysis, there are still a
population of parameters to be matched. So the general
optimization methods have proven to be inapplicable, and
one of the combined optimization methods, namely the
revised genetic algorithm, is chosen as the main matching
tool.
Genetic algorithm is an adaptive large-scale optimizing
algorithm based on probability search formed in the process
of simulating the inheritance and evolvement of organisms
in nature. It still remains a big challenge to search for a
general solution to the optimization problems at a low cost
due to the large types and large scale of problem itself.
Genetic algorithm, at this circumstance, could provide a
relatively effective and general framework.
In
genetic algorithm,
strings
of
n signs
X i i  1,2,, n are
decision vector X 
used to signify n-dimensional
x1, x2 ,, xn T :
X  X1 X 2  X n  X  x1 , x2 ,, xn 
T
Consider
Xi
as a single inheritable gene and its
probable values as equidistant genes, then X can be
deemed as a chromosome composed of n genes. The
equidistant genes are real values in a certain scope
according to the research interest. The arrangement form of
codes X is an individual gene and its corresponding
value is the phenotype. Chromosome X is also named
individual X and the fitness of each X should be verified
according to certain regulation. Individual fitness is
correlated to the objective function value of its individual
phenotype that the more it closes to the optimal point, the
bigger its fitness seems.
Organism evolvement process mainly happens through
the crossover and mutation of chromosomes, and the
searching process for optimal solution in genetic algorithm
simulates the organism evolvement process that genetic
operator acts on the population P (t ) to get the next
population P (t
 1) .
1) Selection. In accordance with specific method, select
some good individuals from the population P (t ) to pass
12
Physical and Numerical Simulation of Geotechnical Engineering
2nd ISSUE, March 2011
down to the next population
P (t  1)
different fitness of individuals.
2) Crossover. Pair the individuals in
based on the
P (t )
randomly
or based on specific regulation, and intersect part
chromosomes of each individual by certain probability.
3) Mutation. To each individual in population P (t ) ,
change the gene values of one or several genetic locuses by
certain probability to other equidistant genes.
The main genetic algorithm procedures based on the
three genetic operators above can be described as follows:
Step one: Initialization. Set the evolvement generation
timer t=0, maximum evolvement generation number T and
engender randomly M individuals of the original
population P (0) .
Step two: Individual evaluation. Calculate the fitness of
each individual of P (t ) .
Step three: Selection. Act selection operators on the
population.
Step four: Crossover. Act crossover operators on the
population.
Step five: Mutation. Act mutation operators on the
population. Through selection, crossover and mutation,
P (t ) gains its next generation
population
population P (t
 1) .
Step six: End condition judgment. If t<T, turn to step two;
otherwise, if t≥T, output the individual with the biggest
fitness as the optimal solution and quit.
The main difference between genetic algorithm and the
direct method as well as gradient method is that it randomly
gets several initial points, searches randomly and has its
own crossover, mutation and selection genetic operations,
which is more advantageous to complex optimization
problems.
The process is showed in Fig 1.
varied between certain upper and lower limit. However, the
revised genetic algorithm calls for further optimal control
algorithm because of the still large amount of variables and
big optimizing difficulties.
1) Initially, order the sensitivity of parameters from
reservoir engineering and practical basis. The most sensible
parameters should be the thicknesses and permeability of
high-permeability channels in main streamlines in the
vertical position of the interwell channels. Then the
secondly-sensible parameters should be the thicknesses and
permeability of channels in other streamlines. The least
sensible parameters should be the distribution of remaining
oil saturation.
2) Use the alternative optimizing method of space
control and parameter control.
A. Alternative optimizing method of space control:
Optimize one well group at a time.
B. Alternative optimizing method of parameter control:
Firstly optimize the most sensible parameters, then the
secondly-sensible parameters and finally the least ones.
Reiterate the optimizing process of parameter control.
C. Apply the revised genetic algorithm in each
optimizing step and upgrade the correlation parameters
after optimization. Reiterate the steps in A and B until the
result is convergent.
4 CASE STUDY
4.1 Test objective
Test to enhance the recovery of polymer flood is
conducted in some plant of Daqing Oilfield. The target
layer is PI3 and there are 44 wells involved with 19
injection wells and 25 production wells. To interpret the
contradictions and dynamic characteristics of reservoirs,
interwell tracer test is implemented.
4.2 Test process
Figure.1 Genetic algorithm calculation procedure
3 IMPLEMENTATION OF ALGORITHM OF
COMBINATIONAL OPTIMIZING CONTROL
According to the mathematical model for combinational
optimizing interpretation and the tracer output
concentration equation, the variables in objective function
Extend the well group of tracer test for a well spacing,
designate the water wells as boundary and establish the
geological model for the extended well group. Then input
the data of rock and fluid properties, polymer
physicochemical parameters, high pressure property, initial
condition as well as dynamic information of oil and water
wells, and build up the streamline simulation model. Match
the tracer output curves on the basis of the adjustment
scope of automatically-matching parameters.
The test was conducted on a well group centered by three
water injection wells P39, P38 and P37. And tracer was
apparently detected during more than half and six months
sampling and monitoring process.
4.3 Matching the output curve by genetic algorithm
Based on the built geological model and tracer output
curves, simulate the proper subsurface distribution of
interwell parameters and match the output time and
concentration of tracer. It shows in Fig 2 to Fig5 that the
average fitness improves as the genetic generations
13
Development of Genetic Algorithm models for Tracer Test Interpretation during Polymer Flood
DOI: 10. 5503/J. PNSGE. 2011. 02.003
Figure.5
increase, demonstrating that the general quality of groups
increases gradually and curves of each well own a good
matching.
Tracer response in well group P39 lists as follows.
Figure.2
Figure.3
D2－PB339 effluent history match
Also, tracer response in well group P37 lists as follows:
Figure.7
D2－139 effluent history match
D2－PB39 effluent history match
Figure.8
D3－37 effluent history match
D2－PB37 effluent history match
The curves are matched well and basically reveal the
actual formation features. The tracer output curves of
polymer flood display that the interwell channels for tracer
migration are simple and there exists a main breakthrough
unit, in which each channel owns relatively equal
percolation ability and is of poor heterogeneity.
Figure.4
D2－P39 effluent history match
Figure 9 Streamline distribution of the 2D layer for
polymer flood
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Physical and Numerical Simulation of Geotechnical Engineering
2nd ISSUE, March 2011
4.4 Interpretation result
Quantitative interpretation should be analyzed to depict
the concrete interwell connecting characteristics.
Horizontal streamline distribution is gained according to
injection-production condition and the distribution of
horizontal speed field (showed in Fig. 9).
Inversion of reservoir parameters is conducted on test
data and characteristic parameters of interwell tracer
channel are gained.
Tab.1 Swept pore volume of tracer
Injection well
Production well
layer:swept volume /m3
13－D2－139
17(PI31b):16000
13－D2－P39
13－D2－PB39
13－2－PB339
17(PI31b):11000
17(PI31b):6500
13－22－P38
13－22－F38
16(PI31a):14000
13－D3－P37
13－D3－37
23(PI33a2):25
13－D3－PB37
17(PI31b):12200
In water flood stage, producing characteristics mainly
reveal the condition of PI31, and local well group displays
the condition of PI32 and PI33. In terms of value, the big
swept pore volume demonstrates that the tracer swept area
is one of the most potential tapping parts for polymer flood.
From a comprehensive perspective, s strongly watered out
band of a big thickness in interwell indicates a relatively
high producing degree in interwell layer.
Tab 2 Analysis of the interwell channels
Injecction well
Production well
Layer:permeability
×10-3 um2
Layer:thickness
×10-2 m
Result
D2－139
17(PI31b):410
17(PI31b):100
Strongly watered out band
D2－P39
D2－PB39
2－PB339
17(PI31b):350
17(PI31b):240
17(PI31b):150
17(PI31b):180
Strongly watered out band
Strongly watered out band
22－P38
22－F38
16(PI31a):630
16(PI31a):160
Strongly watered out band
D3－37
23(PI33a2):2000
23(PI33a2):9
High-permeability streak
D3－PB37
17(PI31b):770
17(PI31b):200
Strongly watered out band
D3－P37
In addition, equivalent big channels in well groups
D3-P37 and well D3-37 manifest that either a
permeability-abnormal band or a non-closed secondary
fault is available.
5 CONCLUSION
In this paper, reservoir independent variables in interwell
tracer test interpretation are determined and objective
function is also established. Applied with genetic algorithm,
automatically integrated parameter inversion and optimal
interpretation of tracer test are gained. Our results have
demonstrated the successful application of genetic
algorithms to calculate the formation properties from tracer
test, facilitating the polymer flood in studied area.
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