Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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GRAVITY MEASUREMENT METHODS FOR MONITORING RESERVOIRS
FIELD OF THE INVENTION
[0001] The invention relates to the field of gravity measurements to detect
changes in a subterranean reservoir.
BACKGROUND
[0002] Measurements of the earth's gravitational acceleration, and
measurements of differences in the earth's gravitational acceleration
(gravity difference) between different depths in the earth or horizontal
positions, can be useful in determining the bulk density (or specific gravity)
of various earth formations, among other applications. More particularly,
measurements of gravity difference between two positions or depths may
be used to determine whether oil, water or gas primarily fills pore spaces
in the earth formations at various depths and geographic locations in the
earth.
[0003] Such measurements can also be useful in operations where a fluid,
such as a gas, liquid, gel, or foam, is injected into a subterranean
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formation to enhance or promote the recovery of hydrocarbons with low
mobility. To monitor the effectiveness of such treatments it is often
desirable to detect at least the depth and/or the position of the fluid front
as formed by the injected fluid.
[0004] As a practical matter, measuring physical properties of earth
formations beneath the surface of the earth is typically performed by a
process called "well logging", wherein instruments having various sensors
therein are lowered into a wellbore drilled through the earth. The
instruments may be lowered into the wellbore and retrieved therefrom at
the end of an armored electrical cable in a process known as "wireline"
well logging. Alternative conveyance techniques as known in the art
include lowering the instruments into the wellbore coupled to the end of a
drill pipe, a production tubing or a coiled tubing. The drill pipe conveyance
technique, in particular, is commonly referred to as "logging while drilling"
when performed during the actual drilling of a wellbore. The well logging
instruments, whether wireline or pipe conveyed, may include various
devices to measure the earth's gravitational acceleration.
[0005] Several gravity measurement tools are commercially available or
have been proposed in the prior art. One manufacturer of such tools is for
example Lacoste & Romberg who offer a borehole gravity meter (BHGM)
under the trade name "Micro-g system".
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[0006] Other gravity and gravity difference measuring instruments are for
example described in U.S. Patent Nos. 5,351,122 and 5,892,151 both
issued to Niebauer et al. and 5,903,349 to Vohra et al. The known gravity
tools according to the '151 patent include at least one, preferably several
longitudinally spaced apart gravity sensors enclosed in an instrument
housing. The gravity sensors are fiber optic interferometry devices, which
measure a velocity of a free falling mass by determining, with respect to
time, interference fringe frequency of a light beam split between a first
path having a length corresponding to the position of the free falling mass,
and a second "reference" (fixed length) path. The fringe frequency is
related to the velocity of the free falling mass, which in turn can be
correlated to earth's gravity by precise measurement of the mass's
position and the time from the start of free fall. Measurement of gravity
difference is performed by determining a difference in gravity
measurements made between two of the individual gravity sensors
positioned at locations vertically spaced apart.
[0007] Further instruments for gravity and gravity difference measuring are
described in co-owned U.S. Patent No. 6,671,057 issued to Orban
including a gravity sensor with a first mass adapted to free fall when
selectively released from an initial position. The mass has optical
elements adapted to change the length of an optical path in response to
movements of the mass. The sensor output is coupled to a beam splitter.
One output of the splitter is coupled substantially optically directly to an
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interferometer. Another output of the splitter is coupled to the
interferometer through an optical delay line. The frequency of an
interference pattern is directly related to gravity at the mass. A second
such mass having similar optics, optically coupled in series to the first
mass and adapted to change the path length in opposed direction when
selectively dropped to cause time coincident movement of the two
masses, generates an interference pattern having frequency related to
gravity difference. Further suitable gravity measuring instrument are
known for example as U. S. Patent 7,155,101 to Shah et al.
[0008] Methods of using such instruments are described for example in
the above `057 patent and in the U.S. Patent No. 7,069,780 to Ander, and
by J. L. Hare et al. in: "The 4-D microgravity method for waterflood
surveillance: A model study for the Prudhoe Bay reservoir, Alaska", in
Geophysics, Vol. 64, No. 1 (Jan.-Feb. 1999), p.78-87. In the latter study,
the gravity observations are inverted to determine the subterranean
density distribution. The inversion used in this prior art is posed as a
linear, underdetermined inverse problem with an infinite number of
possible solutions. The densities range is subjected to a set of constraints
resulting in a constrained, linear system which can be solved using least-
square methods. The authors acknowledge that the model parameters
determined using the least-square methods are not unique. In addition,
the inverse gravity problem is stated to be fundamentally unstable and it is
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known that any solution based on an optimization approach involves high
computational costs.
[0009] Given that gravimetric instruments and logging measurements are
known per se, it is seen as an object of the present invention to provide
new methods for making use of and evaluating gravity logging
measurements to determine the depth and/or other geometrical properties
of a gravitational anomaly in a subterranean formation. It is a particular
object of the invention to provide such methods for monitoring changes in
subterranean reservoirs fast and with limited computational costs.
SUMMARY OF INVENTION
[0010] According to a first aspect of the invention a method of performing
gravity surveys in a wellbore is provided including obtaining a time-lapse
measurement of at least a vertical component of the gravitational force in
a monitoring well, and determining a depth at which the difference of the
time lapse measurement changes sign. It was found that in the presence
of a gravity anomaly migrating or moving in a subterranean formation, this
depth can be taken as a nominal depth of the anomaly.
[0011] It was further found that the position at which the difference of the
vertical component of two time-lapse gravity vanishes or changes signs is
for the purpose of the present invention equivalent to the depth where the
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vertical gradient of the difference assumes an extreme value. This
extreme value could be either a minimum or a maximum depending on
nature of the gravity anomaly, i.e. whether has a positive or negative sign
or value. With the representation of the values as per the present
invention further standard analytical tools can be applied to determine this
depth from the difference measurements.
[0012] For the purpose of the present invention the value of the density
anomaly is assumed to be prior knowledge. For example, in the field of
enhanced oil recovery (EOR), the density of the injected fluids is known
together with density, porosity and original fluid content of the original
subterranean formation. Taking these values the density or gravity
anomaly can be expressed as difference between the formation density
and the formation density with the new fluid content. However, the present
method is not dependent on the exact method according which the value
of the density or gravity anomaly is calculated.
[0013] According to another aspect of the invention further parameters
relating to the location and/or size of the density anomaly are derived
using the time-lapse measurements at three or more further depth points.
[0014] The method is based on solving a highly non-linear equation which
expresses the difference in the vertical gravity component between two
time-lapse measurements as function of three unknowns. In a preferred
embodiment these unknowns are the distance of a front of the anomaly
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from the monitoring well at two different times of measurement and a
height of the density anomaly. Because of the assumptions made the
method is expected to perform best when a monitored density anomaly
can be treated as long, slender block or parallelepiped migrating slowly
through the rock formation. For instance, water injected by a long array of
injector wells is likely to move in such a manner through a permeable
horizontal layer.
[0015] Given a sufficient number of independent measuring points, which
number is larger than the number of unknowns, well-established methods
can be used to solve this equation in three unknowns. The method can
thus be used to provide at low computational costs estimates of, for
example, the location at different times t, and t2 and the vertical thickness
of an arriving fluid front. Used in isolation, the method provides a fast
response and requires only time-lapse gravity readings at three depths in
the borehole. In cases of a more complex geometry, the method can still
make a valuable contribution by providing either a priori information, a first
estimate, or constraints to other inversion methods which invert gravity
surveys into density anomalies, for example in conventional time-lapse or
4-D reservoir surveys.
[0016] According to a further aspect of the invention a method is provided
to model and detect the effects of an inhomogeneous sweep or water-
fingering between injector wells and producer wells. In a preferred variant
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a slender block or parallelepiped is modeled as having a finite width. In a
further
embodiment of this variant, the presence of several of such blocks or
parallelepipeds
is modeled. Both aspects of the invention can be combined to increase the
efficiency
of the detection and modeling of fingering.
According to another aspect of the present invention, there is provided
a method of monitoring a subterranean density anomaly, comprising: a)
combining
vertical components of a first gravity logging measurement in a monitoring
wellbore at
a first time and a second gravity logging measurement in said monitoring
wellbore at
a second time to form a set of depth-related values equivalent to the
difference of
said vertical components; b) determining a depth corresponding to the depth at
which
a difference of said vertical components changes sign; and c) assigning said
depth as
the depth of said density anomaly.
According to another aspect of the present invention, there is provided
a method of performing gravity surveys in a wellbore, comprising: a) selecting
a
wellbore in the vicinity of injector wellbores; b) performing time lapse
gravity surveys
in said selected wellbore; c) determining a depth of a density anomaly as the
depth at
which the difference between values of said time lapse surveys change sign;
and
d) using said measurements to determine further parameters relating to one or
both
of the location and size of a density anomaly caused by injecting fluids
through said
injector wellbores.
BRIEF DESCRIPTION OF THE FIGURES
[0017] Fig. 1 is a flow diagram illustrating steps in accordance with an
example
of the present invention;
[0018] Figs. 2A and 2B are schematic views illustrating the evolution of a
fluid
front in a subterranean reservoir as measured by the present invention;
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[0019] Fig. 3 is a plot of a numerical example in accordance with the present
invention; and
[0020] Fig. 4 illustrates how geometrical parameters characterizing a fluid
front
are determined in accordance with an example of the present invention.
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DETAILED DESCRIPTION
[0021] The following example of a method in accordance with the present
invention establishes an analytic relation between the difference of vertical
component Agz of the acceleration due to gravity at an (arbitrary) depth zo
at two different times in a borehole due to a semi-infinite horizontal slab
with constant density anomaly Ap.
Dgslab (ZO;xmin,xmax,h)-
2 0 2 2
max + (h - z0) min + (h + z0) it
I max=In 2 .+ xmin.In 2
xmax2 + (h + z0) 0 Xmin2 + (h - Z0) -ou
e U[1 ]
G.Dr el 2(h - z0).Itan max tan min
e h- Z O C h- z 0
e ac x o ae x U
el 2(h + z0).tan- I min tan- max
t h + z0 o h + z0
[0022] The variables xm;n and xmax denote the algebraic distance (meaning
they can take positive or negative values) from the fluid front to a
monitoring well where borehole gravity readings have been taken in time-
lapse. They correspond to times t2 and tj respectively. In the example,
positive values of xmin (respectively, xmax) refer to cases where the fluid
front have not reached the monitoring well at time t2 or t1, respectively,
while negative values refer to the opposite situation.
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[0023] In the present example, a borehole gravimetry survey includes the
measurement of the vertical component gZ of the acceleration due to
gravity. Subtracting values obtained at different times at a depth z gives a
differential gravity vector AgZ = gZ(t2) - gZ(t1). Having determined AgZ
through measurement, the equation (1) can be evaluated to gain Xmin, Xmax
and h, which denotes the half-thickness of the density anomaly, directly
from the borehole gravity measurements.
[0024] The above relation can be also expressed as equation for the
spatial gradient of the differential gravity vector, e.g. (AgZ(z=zo+l) -
AgZ(z=zo-I))/ 21.
[0025] At any fixed value zo, the functional AgZ as defined above is
continuous in the variables Xmin, Xmax and h. Moreover (except for zo=z,), it
is strictly monotonic in one variable while the other two are held constant.
This ensures that the set Agslab (z0; Xmin, Xmax, h) - Ag meas eas (z0) = 0 is
a (curved) continuous surface in the (Xmin, Xmax, h) space. Taking the
intersection of three sets corresponding to three different values of zo
yields the solution (Xmin*, Xmax*, h*).
[0026] In order to evaluate the above relations, the following steps as
illustrated in Fig. 1 can be performed:
1- From the measurements of borehole gravity (i.e. the vertical
component of the gravity field) at the two different times,
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create the time-lapse differential gravity measurement:
Agzmeas = gZmeas(t2) _ gZmeas(tl) for multiple depth points z0.
2- Optionally apply any kind of smoothing processing or low-
pass filter on the curve AgZmeas(Z0)
3- The central vertical position of the density anomaly is given
by the unique depth zc where AgZmeas(zc)=0 or changes sign
(or equivalently where the vertical gradient oz(AgZmeas(z,)) is
an extremum)
4- Obtain the half-thickness h of the slab, the algebraic
distance Xmin (respectively, Xmax) of the fluid front from the
borehole at time t2 (respectively, ti) where gravity
measurements have been recorded by solving for (Xmin*,
Xmax*, h*) the set of nonlinear equations.
[0027] The above steps 1-4 are shown as steps 11 - 14 in the flow chart in
Fig. 1.
[0028] Figs. 2A and 2B illustrate a borehole gravity survey using the steps
as described above. The schematic horizontal cross-section through a
subterranean layer of Fig. 2A shows an array of injector wells 21 used to
inject a fluid with a density anomaly having a difference in the bulk density
before and after flooding Ap. A part of the fluid front caused by this
injection is shown as line 231 at a distance xmax at the time t1. The time ti
is the time of the first gravity logging measurement in the monitoring well
22. The distance xmax is taken relative to a coordinate system which fixed
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to the monitoring well 22. The front of the injected fluids moves further and
further away from the injector wells 21 as new fluid is either pumped into
the formation or other fluids (not shown) are pumped out of a production
well (not shown). A part of the fluid front caused by this injection at a
later
time t2 is shown as line 232 at a distance xm;n. The time t2 is the time of
the
second or time-lapse gravity logging measurement in the monitoring well
22.
[0029] Fig. 2B illustrates a schematic vertical cross-section through the
subterranean layer. In this view only one well of the array 21 of injector
wells is visible. The fluid front at times t, and t2 are shown as lines 231
and
232, respectively. In this view the height h of the density anomaly is
visible as half the vertical distance between lines 233 and 234. It should
be emphasized that the fluid or density distribution as shown is a
geometrical approximation to facilitate the mathematical treatment of the
problem as per equation [1]. A real injection operation can at best be
approximated by the geometry of the fluid distribution as shown in Figs.
2A and 2B.
[0030] For the calculation, a further assumption is made in that the zone is
treated as if extending to infinity in the y-direction, which is taken to be
the
horizontal direction perpendicular to the direction between injector well 21
and monitoring well 22. While the validity of the assumption depends on
the circumstances of the injection operations it is likely to be valid when
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the array of injectors 21 is spread in this direction. It should be further
noted that the gravity response function decreases in proportion to 1/r2.
Thus the contribution from a volume of formation located far away from
the monitoring well decreases rapidly with the distance from the
monitoring well.
[0031] The time lapse measurements for the purpose of this invention are
conducted typically at least 7 days apart, usually however more than 1
month apart. In any case, the repeat measurement is only viable if and
when the fluid front has moved or is expected to have moved a significant
distance so as to be resolved using the available technology. The
measurements as can be performed using commercially available logging
tools as described in the background above.
[0032] In Fig. 3 there is shown a plot of values of Ag, in a borehole as
computed from equation [1]. In this synthetic example the density
anomaly is assumed to be caused by a horizontally extending
parallelepiped as depicted in Fig. 2 above. The anomaly is defined in this
example as the difference between the bulk density of the flooded
formation and the bulk density of the formation prior to flooding and its
numerical value is Ap=0.03 g/cc. The height of the fluid front in this
example is 30m and its initial position is at xmax=1200m.
[0033] The responses shown correspond to eight values of xmin (800m,
300m, 200m, 100m, Om, -100m, -200m, -300m) with the depth zo of the
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measurement relative to the assigned depth of the anomaly used as
ordinate and AgZ as abscissa. The vertical positioning of the anomaly is
indicated by a central bar. The response for 800m is the most central line
and the response for -300m at the outer fringe of the graph with the other
responses ordered accordingly.
[0034] The results show that the values are symmetrical around the zero
vertical position which is defined above as being level with the nominal
depth of the anomaly. When evaluating with limited number of gravity
measurements, it is therefore preferable to use gravity measurements at
depths which are not symmetrically positioned with the respect to the
relative zo= 0 depth. One of the measurements is equally best taken to be
at the wide end of the spread of the gravity log to ensure that it is the
largest possible distance from the depth where the relative zo= 0.
[0035] As shown in Fig. 4, for every given value of zo (depth) equation [1]
defines a smooth surface in the three-dimensional space of h, Xmin and
xmax. Fours sets of points 41 - 44 obtained at zo= 5m, 50m, 100m and
200m, respectively, for a 60m-thick density anomaly (Ap=0.03 g/cc) are
shown in the figure. The values of xmin, Xmax and h can be recovered from
the measured data by solving the non-linear equation [1] if the measured
AgZ is taken from at least three different depths. The point where all
surfaces intersect yields the solution, which in this example is h = 30 m,
xmin = 100 m and xmax = 600 m. The accuracy of these values depends on
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how well the intersection of the above surfaces 41 - 44 is defined. As
expected, for this particular example the method resolves xm;n better than
Xmax as the gravity measurement is much less sensitive to features located
at 600m distance from the borehole than to features located at 100m. A
higher number of depth points is likely to increase the accuracy of the
solution in a real measurement.
[0036] Alternatively to equation [1] above the effect of gravity anomalies
can be also modeled using a different model based on a finite slab width
2w (+w ;-w) in a direction orthogonal to x and z, i.e., in y direction.
Finite slab
Dg: (z0;xin in,xmax,h)=
VX 2 + (h- z0)2+ w2 - w0
In
2 + (h - z0+ w + w [2]
X=xmax x V
2G.Dr a Dx
X=Xinin +DX
In VX
x
2+(h+z0)2+w2 +w
V
where Ax = (xmax - xmin) / N. N is a large integer number, chosen to make
Ax a small discretization step along x for minimizing the error introduced
by approximating an integral through summation. One way to obtain N is
by obtaining a first order approximation of (xmax - xm;n) by solving equation
[1] above assuming an infinite water-body along the y-direction; then
choosing a value for N that would constrain Ax to a desired discretization
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step (Ax = (xmax - xmin) / N) being less than or equal to a pre-determined
tolerance or accuracy value.
[0037] Therefore, a combination of solutions to [11 and [2] can be utilized
for identifying the shape of water-body approaching (or moving away from
a vertical well equipped with gravity measurement sensor(s) providing
more than one data point across the gravity anomaly.
[0038] This variant of the invention can be extended to include and hence
detect several blocks or parallelepipeds. The invention is particularly
suitable for modeling for example water fingering as a result of an
inhomogeneous water sweep between injector and producer wells.
[0039] The above example can be varied in a number of ways, including
but not limited to using it to establish the tail position of a fluid layer
and to
derive further control parameters such as pumping time and pressure for
an enhanced oil recovery or fluid injection operation.
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