CHAPTER 8
3Chapter 8 - History Matching
CHAPTER 8
History Matching
Introduction
Comparing simulator pressure to pressure build-up data
Matching pressure history
Forcasting performance
History Matching
Introduction
One of the most common uses of reservoir simulation for field
problems is for history matching. This is a process of estimating
reservoir data by finding simulator data which gives reservoir
performance similar to performance data in the field. This is
sometimes called the inverse problem. In other words, we start with
the answer (field performance data) and try to define the problem
(the reservoir description). The field performance data are usually
production/injection rates and well build-up pressures.
The field performance data may be in error, of course. Sometimes
this becomes a major problem in obtaining an acceptable history
match. For this discussion, however, we will assume that the field
performance data is accurate.
One principle of history matching is that a history match is not
unique. That is, more than one set of reservoir data may fit the
field performance measurements with equal accuracy. This is a
mathematical conclusion which is also complicated by sparse and
erroneous field performance measurements. It becomes the
responsibility of the engineer to make a judgment between the
different sets of data. In making this judgment, other sources of
data should be analyzed, such as well logs, production tests, core
analysis, geological interpretation, etc.
Much work has been done on techniques for automatically matching
pressure, but most history matching is done by a trial and error
approach with the engineer using analysis and judgment to modify
the reservoir data and then re-run the simulator. During this
process, the engineer is trying to match field measured pressures
with simulator pressures. For single phase gas reservoirs, the
additional problem of matching water-oil ratios and gas-oil ratios
is not present.
Comparing simulator pressure to pressure build-up data It is
possible to match bottom-hole flowing pressure, pwf. However, that
data is usually not available and is also not very reliable because
of possible inaccuracies in rate data. It is more common and much
more reliable to match pressure build-up data when it is available.
The problem is how to match the pressure build-up data. The time
scale of the pressure build-up test is usually too short to model
accurately with a field scale grid because the gridblocks are too
large.
Peaceman2 has provided a method for comparing simulator
gridblock pressures to pressure build-up pressure. Fig. 1 shows a
profile of pressure in a gridblock containing a producing well. The
pressure profile is assumed to be at pseudo-steady state. It is
seen that the gridblock pressure (the material balance average
pressure inside the gridblock) is somewhere between pwf and the
average reservoir pressure. Fig. 2 shows the corresponding pressure
build-up curve from field data. The gridblock pressure
corresponding to the proper field build-up pressure lies on the
semi-log straight line at time of Dto which is calculated by the
following equation:
(1)
The "match pressure", po corresponds to the steady state
pressure at 0.2Dx which was used in the expression of
"transmissibility". If po is the same as the gridblock pressure,
pi,j, then the simulator is properly matching field behavior.
Fig. 1 - Pressure profile in a gridblock containing a producing
well
1Fig. 2 - Pressure build-up curve for example problem 4
Example 1.Calculation of "Match Pressure" from a Pressure
Build-up Test.
Problem. Find the "match pressure" from the following field
pressure build-up test.
q= 23,000 scf/Dk= 0.15 mdf= 0.18ct= 5 x 10-6 psi-1
Shut-in time Build-up pressure Dt ps (hrs) (psia) ___________
__________ 0.10 2854.5 0.23 2861.5 0.39 2865.5 0.84 2871.5 1.56
2875.6 3.50 2881.0 7.38 2886.2 15.11 2891.0 30.53 2895.5 61.31
2900.0 122.72 2904.1 245.24 2907.1 489.71 2909.3 840.00 2910.4
Model data:Dx= 100 ft
Solution. The solution follows these simple steps: (1) plot the
build-up data on a log Dt plot, (2) draw a "semi-log straight
line", (3) calculate Dto, and (4) find po at Dto on the "semi-log
straight line". This is the "match pressure" which will be compared
to the simulator gridblock pressure at the time of the pressure
build-up test.
Fig. 2 shows the field build-up data plotted on a semi-log plot.
The match pressure is found by calculating Dto as follows:1 (2)
and finding the corresponding pressure on the semi-log straight
line:
po = 2872 psig.
This pressure is then compared to the simulator gridblock
pressure when evaluating a history match run.
Matching pressure history
Now that the proper pressure has been found to match, we will
now discuss how the simulator data is modified to match the field
pressures. Most of the history matching is done in a trial and
error fashion by engineers. An experienced engineer relies on
knowledge of pressure behavior fundamentals to guide data
modifications.
The first consideration is to match the size of the reservoir,
or the original gas in place. (No water influx is being considered
in this discussion.) This is often determined with a simulator but
uses the principles of material balance. During pseudo-steady
state, it is known that at every point in the reservoir depletes at
a rate given by:
(3)
The pore volume, Vp, can be represented as a integration of a fh
contour map and the effects of f and h cannot be separated. The
total compressibility, ct, is equal to cf + cgSg + cwSw. This value
is usually dominated by cg, but this may not be true at pressures
over about 6,000 psig, above which cg begins to be relatively
small. In this higher pressure case, attention should be given to
obtaining good estimates of cf. The cwSw term will usually be less
important (smaller) than cf.
Once the gas in place has been matched by pseudo-steady state
behavior, the transient behavior should be matched. The magnitude
of the pressure drop is inversely proportional to the
"transmissibility", kh, and the timing of pressure drop is
inversely proportional to the diffusivity, k/fmct. These
relationships can be seen in the dimensionless pressure drop and
dimensionless time used in transient well test analysis.
Another method of adjusting kh is from an analysis of pressure
gradient profiles (plots of pressure vs. distance) at a particular
time. If there is migration of fluid from one side of the reservoir
to the other, the magnitude of the pressure gradient is inversely
proportional to kh.
In actual field cases history matching analysis may be much more
complicated, of course. We seem to always think the reservoir is
more homogeneous than it is. Lack of homogeneity is often
demonstrated when we drill new wells and are surprised by their
properties. Lack of homogeneity is also demonstrated when we inject
fluids into a reservoir and find lack of continuity. Many
reservoirs are complicated by systems of sealing and partially
sealing faults which are hard to detect. History matching for these
cases often involves well by well analysis and trial and error.
The principles mentioned are often found to be useful even if
they do not comprise a complete analysis. An example will now be
shown which illustrates these principles.
Example 2. History Matching Reservoir Pressures
Problem. Perform the synthetic history matching for three
years,and plot wellbore pressure vs. time for well No. 1 and
pressure profiles at the end of each year. (see Fig. 3).
k= 1.0 mdh= 30 ftf= 0.10gg= 0.70T= 150 0Fpinitial= 6000 psiacf=
3.0x10-6 psi-1Dx= 100 ftDy= 100 ftIMAX= 20JMAX= 5
Production schedule (scf/D) Year Well 1 Well 2 Well 3 (I=4,J=3)
(I=10,J=3) (I=17,J=3) 1 40,000 0 0 2 60,000 40,000 0 3 100,000
60,000 60,000
Actual history data:
The "actual history" for this problem was generated by a
simulation run. This is called "synthetic history matching". The
process of matching the synthetic history is the same as matching
real field data - trying to deduce what data resulted in the
observed behavior. For Example 2, the observed pressure data is
below. The pressures are reported as po which means they should
match the gridblock pressure. This represents pressures taken from
pressure build-up tests as previously discussed.
Pressure build-up data Time po (psi) (years) Well 1 Well 2 Well
3 ------------------------------------------------ 0.000 6000 0.202
5907 0.466 5856 1.000 5779 1.466 5605 2.000 5446 2.308 5233 3.000
4857 4972 5036
2Fig. 3 - Gridblocks of Example 2 (synthetic history
matching)
Solution. We have complete pressure history on Well 1 through
eight pressure build-up tests. On Wells 2 and 3 we only have a
final pressure build-up at the end of three years. Figs. 4, 6 and 8
shows po vs. time plots for Well 1 with simulation Runs 1,2, and 3,
respectively. Figs. 5, 7 and 9 show pressure profile across the
entire reservoir with simulation Runs 1,2 and 3, respectively. The
actual history is shown on each plot.
Imagine that you are comparing simulation Run 1 with the actual
history in Figs. 4 and 5. Notice that the rate of pressure decline
is too fast in Run 1. It is apparent that pseudo-steady state has
been reached from the shape of the Run 1 curve in Fig. 4. Since we
have complete data for Run 1 we might be able to estimate the time
required to reach pseudo-steady state from well test analysis
theory (radius of investigation). Note that we always have this
complete information for analyzing the behavior of our computer run
while the actual reservoir data is unknown. From our depletion rate
discrepancy, we decide to increase Vp by increasing porosity from
0.10 to 0.15. We use Eq. 2 to make the analysis on depletion
rate.
Now Run 2, with f = 0.15, shows that the depletion rate is about
right in Fig. 6 but the transient behavior does not produce enough
pressure drop before pseudo-steady state begins. The level of
pressure is about right in Fig. 7 but the pressure gradient is too
flat. Both of these plots indicate that the kh/m is too high. We do
not want to change h (without some compelling reason) because that
would change Vp. We will assume that m is okey. So we change
permeability from 1.0 md to 0.2 md. We find that Run 3 is a perfect
match of actual history.
Time (days)
Run No. 1 ( = 0.1, k= 1.0 md)
0
365
730
1095
6500
6000
5500
5000
4500
4000
p
wf
(psi)
Actual History
3Fig. 4 - Wellbore prerssure of well No.1 vs. time for run No.1
(synthetic history matching)
4Fig. 5 - Pressure profile at three years for run No.1
(synthetic history matching)
5Fig. 6 - Wellbore pressure of well No. 1 vs. time for run No. 2
(synthetic history matching)
6Fig. 7 - Pressure profile at three years for run No. 2
(synthetic history matching)
The actual history data was generated for this problem with the
data of Run 3, of course. In observing the actual history and
deciding the data changes to be made, we have worked in a similar
manner to an actual reservoir study. Our case is much simpler, of
course, and our reservoir happened to be homogeneous. History
matching in actual practice is much more complicated because of
data errors, heterogeneities, more wells, and more complicated
(often 3-D) geological description. The engineer is never
completely sure of the accuracy of the reservoir description.
7Fig. 8 - Wellbore pressure of well No. 1 vs. time for run No. 3
(synthetic history matching)
= 0.15, k = 1.0 md)
Run No. 3 (
Actual History
Gridblock
1 3 5 7 9 11 13 15 17 19
p (psi)
5200
5100
5000
4900
4800
4700
4600
4500
4400
8Fig. 9 - Pressure profile at three years for run No. 3
(synthetic history matching)
Forecasting performance
The main objective of a simulation project is to forecast
performance. During the history matching, rates are specified for
each well throughout the history period. The rates are usually
unknown for the forecasting period, so other conditions are usually
specified. The most common condition is to specify the bottom-hole
flowing pressure, pwf, and let the simulator calculate the rates
for each timestep.
The objective of field simulation projects is usually to compare
alternative forecasts for the purpose of aiding in decision making.
A base case is usually run which represents continuing current
operations. Then other cases are run which represent alternative
operations, such as drilling new wells, adding field compressors,
stimulating wells, injecting fluids (not common in this discussion
of dry gas reservoirs except for gas storage reservoirs), etc.
Operating decisions are made on the basis of forecasted performance
and economics.
NOMENCLATURE
Bg= gas formation volume factor, rcf/scfcf= compressibility of
formation, psi-1cg= compressibility of gas, psi-1cw=
compressibility of water, psi-1ct= total compressibility, psi-1h=
formation thickness, ftIMAX= number of gridblocks in the
x-directionJMAX= number of gridblocks in the y-directionk=
permeability, mdps= build-up pressure, psiap= pressure,
psiapinitial= initial pressure, psiapo= "match pressure", psiapwf=
wellbore flowing pressure, psiaq= production rate, scf/DSg= gas
saturation, fractionSw= water saturation, fractiont= time, DaysT=
temperature, RVp= pore volume of grid block, ft3Dx= grid block
spacing in the x-direction, ftDy= grid block spacing in the
y-direction, ftDt= shut in time, hoursDto= shut-in time
corresponding to gridblock pressure, hoursgg= gravity of gasf=
porosity, fractionm= viscosity, cp
Subscripts and Superscripts
i= gridblock index in the x-directionj= gridblock index in the
y-direction
REFERENCES
1. Mattax,C.C. and Dalton,R.L.: Reservoir Simulation, SPE
Monograph Volume 13, Richardson, TX (1990).
2. Peaceman,D.W.: "Interpretation of Well Block Pressure in
Numerical Reservoir Simulation," SPEJ (June, 1978) 183-94, Trans.,
AIME, 265.
