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A Capacitance Model To Infer Interwell Connectivity From Production and Injection Rate Fluctuations

Abstract and Figures

Proposal This paper presents a new procedure to quantify communication between vertical wells in a reservoir based on fluctuations in production and injection rates.The proposed procedure uses a nonlinear signal processing model to provide information about preferential transmissibility trends and the presence of flow barriers. Previous work used a steady-state (purely resistive) model of interwell communication.Data in that work often had to be filtered to account for compressibility effects and time lags.Even though it was often successful, the filtering required subjective judgment as to the goodness of the interpretation.This work uses a more complicated model that includes capacitance (compressibility) as well as resistive (transmissibility) effects. The procedure was tested on rates obtained from a numerical flow simulator.It was then applied to a short time-scale data set from an Argentinean field and a large-scale data set from a North Sea field.The simulation results and field applications show that the connectivity between wells is described by model coefficients (weights) that are consistent with known geology, the distance between wells and their relative positions.The developed procedure provides parameters that explicitly indicate the attenuation and time lag between injector and producer pairs in a field without filtering.The new procedure provides a better insight into the well-to-well connectivities for both fields than the purely resistive model. The new procedure has several additional advantages.Itcan be applied to fields in which wells are shut-in frequently or for long periods of time,allows for application to fields where the rates have a remnant of primary production, andhas the capability to incorporate bottom hole pressure data (if available) to enhance the investigation about well connectivity. Introduction Production and injection rates are the most abundant data available in any injection project.Valuable and useful information about interwell connectivity can be obtained from the analysis of these data.The information may be used to optimize subsequent oil recovery by changing injection patterns, assignment of priorities in operations, recompletion of wells, and in-fill drilling. A variety of methods have been used to compare the rate performance of a producing well with that of the surrounding injectors.Heffer et al.[1] used Spearman rank correlations to relate injector-producer pairs and associated these relations with geomechanics.Refunjol[2], who also used Spearman analysis to determine preferential flow trends in a reservoir, related injection wells with their adjacent producers and used time lags to find an extreme coefficient value.Sant'Anna Pizarro[3] validated the Spearman rank technique with numerical simulation and pointed out its advantages and limitations.Panda and Chopra[4] used artificial neural networks to determine the interaction between injector-producer pairs.Soeriawinata and Kelkar[5], who also used Spearman rank analysis, suggested a statistical approach to relate injection wells and their adjacent producing wells.They applied superposition to introduce concepts of constructive and destructive interference.See also the works of Araque-Martinez[6] and Barros-Griffiths[7]. Albertoni and Lake8 (hereinafter AL) estimated interwell connectivity based on a linear model with coefficients estimated by multiple linear regression (MLR).The linear model coefficients, or weights, quantitatively indicate the communication between a producer and the injectors in a waterflood.Filters were adopted to account for the time lag between producer and injector.
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DOI 10.1007/s11004-008-9189-x
Integrated Interpretation of Interwell Connectivity
Using Injection and Production Fluctuations
Ali A. Yousef ·Jerry L. Jensen ·Larry W. Lake
Received: 18 January 2008 / Accepted: 27 August 2008
© International Association for Mathematical Geosciences 2008
Abstract A method to characterize reservoirs, based on matching temporal fluctu-
ations in injection and production rates, has recently been developed. The method
produces two coefficients for each injector–producer pair; one parameter, λ, quanti-
fies the connectivity and the other, τ, quantifies the fluid storage in the vicinity of the
pair. Previous analyses used λand τseparately to infer the presence of transmissibil-
ity barriers and conduits in the reservoir, but several common conditions could not be
easily distinguished. This paper describes how λand τcan be jointly interpreted to
enhance inference about preferential transmissibility trends and barriers. Two differ-
ent combinations are useful: one is a plot of log(λ) versus log(τ ) for a producer and
nearby injectors, and the second is a Lorenz-style flow capacity (F) versus storativity
(C) plot. These techniques were tested against the results of a numerical simulator
and applied to data from the North Buck Draw field. Using the simulated data, we
find that the FCplots and the λτplots are capable of identifying whether the con-
nectivity of an injector–producer well pair is through fractures, a high-permeability
layer, multiple-layers or through partially completed wells. Analysis of data from the
North Buck Draw field shows a reasonable correspondence between τand the tracer
breakthrough times. Of two possible geological models for Buck Draw, the FCand
λτplots support the model that has less connectivity in the field. The wells in fluvial
A.A. Yousef ·L.W. Lake
Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin,
TX 78712, USA
A.A. Yousef
Saudi Aramco, Dhahran, Saudi Arabia
J.L. Jensen ()
Department of Chemical and Petroleum Engineering, The University of Calgary, 2500 University
Drive NW, Calgary, AB T2N 1N4, Canada
e-mail: jjensen@ucalgary.ca
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deposits show better communication than those wells in more estuarine-dominated
regions.
Keywords Lorenz plot ·North Buck Draw field ·Reservoir characterization ·
Capacitance model
1 Introduction
Most waterfloods in hydrocarbon-bearing reservoirs show some surprising behavior.
Production wells near injection wells may be among the last to respond to the injec-
tion while distant producers respond quickly. This disparity, caused by the geological
distribution of the reservoir properties, leads to an uneven sweep of hydrocarbons,
giving poor and/or prolonged periods of low recovery, and unnecessary water pro-
duction which requires energy for treatment and disposal. A full understanding of
the degree of communication between injection and production wells is critical to
achieve high hydrocarbon recoveries with minimal environmental impact. To quan-
titatively determine reservoir connectivity, Yousef et al. (2006) proposed a “capac-
itance model” or CM, which includes capacitance (compressibility) and resistance
(transmissibility) effects. For each injector–producer pair, two coefficients are deter-
mined. One parameter (the weight λ) quantifies the connectivity, and another (the
time constant τ) quantifies the amount of fluid storage between the wells. Both are
defined below.
The CM also accounts for primary production, multiple injectors, and bottom hole
pressure (BHP) changes for multiple producers. Based on a total material balance of
a region between injector iand producer j, the predicted total fluid production rate
(oil+water+gas in reservoir volumes/time) is given by
ˆqj(n) =λpj q(n0j)e (nn0j)
τpj +
i=I
i=1
λij w
ij (n)
+
k=K
k=1
νkj pwf kj (n0j)e (nn0j)
τkj pwf kj (n) +p
wf kj (n),(1)
where
w
ij (n) =
m=n
m=n0j
n
τij
e
(mn)
τij wij (m),
p
wf kj (n) =
m=n
m=n0j
n
τkj
e
(mn)
τkj pwf kj (m),
and λpj and τpj are the weighting factor and time constant for the primary produc-
tion contribution to the estimated rate ˆqjof producer j, with production beginning at
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time n0j.λij is the weight between injector iand producer jthat indicates the con-
nectivity between the (ij) well pair. τij is the corresponding time constant defined
as
τij =ctVp
Jij =ctijVpij
Jij
,(2)
where ctij is the total compressibility of the fluids and rock in the pore volume Vpij
between injector iand producer j.Vpij may be considered as the volume of in-
vestigation for the ij wellpair injection-production response. The volume size and
geometry vary with time and the reservoir heterogeneity.
Jij is the partial productivity index, defined by
qj=
i=I
i=1
qij =
i=I
i=1
Jij ¯pij pwf j .(3)
Jij is similar to the well-known concept of the productivity index(Kimmel and Dalati
1987, pp. 32–2 to 32–4), except that the reference pressure—the average pressure in
the drainage area—is replaced with the average pressure in the region between the
ith injector and jth producing wells, ¯pij .w
ij (n) is the convolved or filtered injection
rate at time step nand p
wf kj (n) is the convolved bottom-hole pressure (BHP) for
producer k.νkj is a coefficient that determines the effect of changing the BHP of
producer kon the production rate of producer j. The entire last term disappears if all
Kof the producer BHP’s are constant, as in the cases here.
Yousef et al. (2006) used an iterative, non-linear least-squares procedure to fit the
rates given by (1) to the actual rates. Fitting was achieved by adjusting the λ’s, τ’s,
and ν’s. They also proposed two versions of the capacitance model: the balanced
(BCM) and the unbalanced capacitance model (UCM). A waterflood is balanced
when the field-wide average injection rate is approximately equal to field-wide av-
erage production rate. When this is true, the BCM (1) should be used. However, if the
waterflood is unbalanced, a constant (qoj )is added to (1) to form the UCM. Yousef
et al. (2006) analyzed the model λ’s in homogeneous cases to show that the CM gave
better performance than an earlier model, proposed by Albertoni and Lake (2003).
For example, the CM estimates of λare more stable in high-compressibility cases
and the CM can better tolerate periods when injection wells are shut in. However,
Yousef et al. (2006) did not analyze both λ’s and τs together and in heterogeneous
systems. Combining both sets of parameters in certain representations has the poten-
tial to enhance inference about the geological features.
Two different interpretative plots for joint analysis of the CM parameters are de-
scribed. One is a log–log plot of the λs vs. the τ’s for a producer and nearby injec-
tors, and another is the FCplot where the λ’s and the τ’s are combined using the
same concept as Lorenz plots. The synthetic and field applications show that the rela-
tion between λ’s and corresponding τs are consistent with the known heterogeneity,
the distance between wells, and their relative positions. The FCplots and the log–
log plots are capable of identifying whether the connectivity of an injector–producer
well pair is through fractures, a high-permeability layer, multiple layers or through
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partially completed wells. Application of the plots to the North Buck Draw field sug-
gests that, of the two geological models proposed in the literature, the one with less
communication is more appropriate.
2 Analysis Methods
We describe two different representations to enhance the inference about reservoir
heterogeneity, using the estimated parameters from the CM: the FCplot, and a
log–log plot of λ’s vs. the τ’s.
2.1 The Flow Capacity Plot
The idea of flow versus storage was developed initially to estimate injection sweep
efficiency in a layered reservoir. This method relates the relative flow in a cross sec-
tion to its associated pore volume, usually in a flow-storage diagram (also known as
Lorenz or FCplots). These plots can be used quantitatively to describe reservoir
geology. For example, if 50% of flow comes from only 10% of the pore volume of a
reservoir, then there are fast flow paths in the reservoir. The FCplots estimated from
the CM parameters are different from the conventional Lorenz plots described by
Schmalz and Rahme (1950) and Lake and Jensen (1991). Conventional Lorenz plots
are based on static permeability and porosity data obtained from measured samples
taken from the reservoir, where the spatial relationships of the samples are ignored.
The FCplots here are based on parameters λand τobtained from dynamic data in
which these parameters account for all variations in reservoir properties in the vicin-
ity of a producer. Shook (2003) also developed these plots (flow-storage diagrams)
using tracer test results. Because the FCplots are based on injection and produc-
tion data, it is likely they will better reflect the flow paths and geological features in
a reservoir than Lorenz plots.
The Lorenz curve (Fig. 1) is a plot of cumulative flow capacity, Fm, versus cumu-
lative thickness, Hm, where
Fm=i=m
i=1kihi
i=n
i=1kihi
,(4)
Hm=i=m
i=1hi
i=n
i=1hi
,(5)
for a reservoir of nlayers. The layers are arranged in order of decreasing permeabil-
ity so that m=1 is the layer with thickness h1and the largest permeability k1while
m=nis the layer with thickness hnand the smallest permeability kn. By defini-
tion, 0 Fm1 and 0 Hm1for1mn. The Lorenz curve, constructed by
letting m vary from 1 to n, has a monotonically decreasing slope. If the medium is
homogeneous, all the permeabilities are identical and the Lorenz curve is a straight
line with unit slope. Increasing levels of heterogeneity are indicated by movement of
the Lorenz curve away from the straight line. Twice the area between the diagonal
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Fig. 1 Conventional Lorenz
(FC) plot. The dashed 45° line
is the Lorenz curve for a
homogenous reservoir
line and the Lorenz curve is an indicator of the heterogeneity, known as the Lorenz
coefficient, Lc(Schmalz and Rahme 1950).
The Lorenz procedure can be modified by including porosity (Craig 1971, p. 64;
Lake 1989, p. 195). In place of Hm, the cumulative storage capacity, Cm,is
Cm=i=m
i=1φihi
i=n
i=1φihi
.(6)
In this plot, a fraction of Fmis provided by a fraction Cmof the reservoir pore volume
(for a layered reservoir). If porosity is constant, the original Lorenz curve results. The
data must now be arranged according to decreasing k/φ. By analogy to the Lorenz
plot, the FCplot is formed using the λ’s and τs for a producer and nearby injectors.
This requires reinterpreting the λ’s appearing in (1)as
λij =Jij
i=I
i=1Jij
.(7)
The λij is equivalent to kij hij between the (ij) well pair or, in other words, λij quan-
tifies the FCbetween the (ij) well pair. Based on the definitions in (2) and (7),
the λand its corresponding τare not independent because λis directly proportional
to Jand τis inversely proportional to the same J. The product of λand the corre-
sponding τprovides a parameter that contains only the storage capacity between the
injector–producer pair
λij τij =ctijVpij
i=I
i=1Jij
.(8)
Similar to (4) and (5), the FCcurve is a plot of cumulative FC(Fm)against
cumulative storage capacity (Cm), where
Fmj =i=m
i=1λij
i=I
i=1λij
(9)
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and
Cmj =i=m
i=1λij τij
i=I
i=1λij τij
,(10)
for producer jwith Iinjectors; λpj and τpj are not included in the calculations. The
data are arranged in order of decreasing 1ij so that i=1 is the injector–producer
well pair with the smallest τwhile i=Iis the injector–producer well pair with the
largest τ. Because of the data ordering, the FCcurve monotonically increases from
i=1toi=Iwith a monotonically decreasing slope as does a Lorenz plot. The FC
plots can indicate specific geological features in the vicinity of an injector. In this
case, sets of λ’s and τs for one injector and all producers are used to form the FC
plot. The procedure can make use of the extensive literature already available on the
interpretation of these plots (Gunter et al. 1997; Cortez and Corbett 2005).
2.2 The log–log Plot
As discussed above, the λand the corresponding τare inversely related through the
partial productivity index Jij ((2) and (7)). For homogeneous reservoirs, where each
producer communicates with all injectors, a log–log plot of λ’s against τ’s for each
producer with all injectors should give a straight line of slope 1. This behavior was
confirmed for homogeneous reservoirs (Yousef et al. 2006). For non-homogeneous
reservoirs, the deviations of the estimated λs and the τ’s from a line with slope 1
will indicate specific geological conditions in these reservoirs.
3 Results
The techniques described above were tested through applications to two well arrange-
ments in simulated reservoir systems (Synfields) and to data from the North Buck
Draw field (NBD). For the synthetic cases, the emphasis will be on the consistency
of the estimated model parameters (λ’s and τs) with the imposed geology, and the
capability of the FCplots to indicate the geological conditions imposed in each
case. Application of the methods to the field case will show what can be achieved in
practice.
3.1 Application to Synthetic Fields
Numerical simulation was done with a commercial finite difference simulator. We
applied the BCM approach to the numerically simulated Synfields: one of 5 injectors
and 4 producers, the 5 ×4 Synfield, and a second of 25 injectors and 16 producers,
the 25 ×16 Synfield (Fig. 2). The 5 ×4 Synfield consists of five continuous layers
while the 25 ×16 Synfield consists of three layers. All layers were of equal thickness
and the Synfields were flowing undersaturated oil initially. The injector–producer
distance was 800 ft for the 5 ×4 Synfield and 890 ft for the 25 ×16 Synfield. The
oil, water, and rock compressibility are 5 ×106,1×106, and 1 ×106psi1,
respectively. The oil–water mobility ratio is unity. All wells are vertical with no skin
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Fig. 2 Well locations for the 5 ×4(left) and 25 ×16 (right) synthetic cases (Synfields). Producing wells
are denoted by P and injection wells by I
and the producer BHP’s are constant and equal. Because the simulations cover a
limited time period for a system with few wells in a bounded domain, the resulting
λ’s and τs will show some scatter about their ideal values. The characteristics of the
synfields are similar to those of the real case to which the CM will be applied later.
3.1.1 5×4Synfield
Several different geological conditions were analyzed for this field. Injection rates
were randomly selected from different wells in a real field and proportionally mod-
ified to be in agreement with the Synfield injectivity. The numerical simulation ex-
tends to n=100 months, with n =1 month.
3.1.1.1 Homogeneous Reservoir The base case is a homogenous reservoir with an
isotropic permeability of 40 md. A log–log plot of the λ’s against the τ’s estimated
from the 5 ×4 homogenous Synfield gives an approximate straight line of slope 1
(Fig. 3), which indicates that all injectors communicate with the producers through
layers that have equal properties. All FCplots are nearly straight lines, indicating
that the reservoir model is indeed homogeneous (Fig. 4).
3.1.1.2 High Permeability Layer For this case, injectors I04 and I05 are completed
in a large permeability (500 md) layer, while injectors I01–I03 are completed in a
small permeability (10 md) layer (Fig. 5). The vertical permeability is 0.1 md and the
porosity is constant throughout. All other parameters are similar to the base case. The
BCM match to the total production rate yields a coefficient of determination of R2=
0.997, where the predicted rate (1) is the explanatory variable and the actual rate is
the response variable. This suggests that the CM is well able to model the production
using the given injection rates. The estimated λ’s (Fig. 6, left) are very similar to
those from the base case (Fig. 7), indicating that the λ’s reflect little or no information
about the high permeability layer (L5) or about the well completions; they are only
functions of the distance between wells and their relative locations. However, the
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Fig. 3 Log–log plot of λ’s
versus τ’s for a 5 ×4
homogeneous Synfield has the
expected slope of 1
Fig. 4 The FCplots for all producers for the 5 ×4 Synfield. A homogeneous reservoir shows a
near-homogeneous FCplot
estimated τ’s of I04 and I05 are smaller than those of I01–I03 (Fig. 6, right). This is
consistent with I04 and I05 being completed only in the high permeability layer (L5),
“seeing” a smaller pore volume than the other injectors.
Unlike the base case (Fig. 3), the log–log plot of λvs. τin this case reveals two
different groups, each with an approximate straight line of slope 1 (Fig. 8). The
groups have τs that differ by a factor of 5, in agreement with the smaller reservoir
volume swept by fluid injected at wells I04 and I05. The combination of both para-
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Fig. 5 Well completions for the
5×4 Synfield
high-permeability layer case
Fig. 6 Calculated parameters λij (left)andτij (right)forthe5×4 Synfield, high-permeability layer case.
The λij and τij are represented by arrows or cones that start from injector iand point to producer j.The
longer the arrow, the larger is τor λ
Fig. 7 Comparison of the λ’s
estimated from the high
permeability layer case and the
λ’s estimated from the base case
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Fig. 8 Log–log plot of λversus
τfor the high-permeability layer
case. The two groups each have
an approximate straight line of
slope 1
Fig. 9 FCplots for all producers for the high-permeability layer case. Each plot shows two different
geological layers that are approximated by two straight lines.Thedashed line presents the 45line. The
static FCcurves are also shown in each plot. The FCcurves from the CM show less heterogeneity than
the static curves
meters, λand τ, and the expectation that points will lie on lines of slope 1 helps to
identify wells I04 and I05 as accessing different parts of the reservoir from the other
three injection wells.
The FCcurves are not straight lines (Fig. 9) and indicate that the Synfield is
not homogeneous. For each producer, the FCplot has two distinct segments. The
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Fig. 10 Representation of the parameters λij (left)andτij (right) for the fractures case. Fractures are
placed between well pairs I01–P01 and I03–P04
left straight line segment with a large slope is for I04 and I05, while the small slope
straight line is for I01, I02, and I03. Similar to the dynamic FCcurves, the static
Lorenz curves indicate two distinct layers (Fig. 9); however, the deviation of the
Lorenz curves from homogeneous behavior is larger than that for the dynamic F
Ccurves. This suggests that the dynamic heterogeneity of the reservoir, as reflected
through several factors including the layer-to-layer variations, the well completions,
and the flow patterns, is less than the apparent heterogeneity, based on the static
porosity and permeability values alone.
3.1.1.3 Fractures This case has two vertical “fractures”, both with one grid-size
width. One fracture is between I01 and P01 with a permeability of 1 000 md, and a
second is between I03 and P04 with a smaller permeability (500 md). The permeabil-
ity of the rest of the Synfield is 5 md. The two fractures, which extend to all layers and
injectors and producers, are completed in all layers. All other parameters are the same
as the base case. The BCM match to the total production rate yields R2=0.997. The
λ’s between I01–P01 and I03–P04 are large, while the corresponding τ’s are very
small (Fig. 10). This is consistent with the two fractures existing in this case. The
injectors nearest to P04, I04, and I05, also have large λs but the corresponding τ’s
are larger than those for the I01–P01 and I03–P04 well pairs. Both sets of parameters
reflect the heterogeneity of the field.
The log–log plot of the λ’s against the τ’s indicates that there are three different
groups (Fig. 11). Group 1, which includes the data for I01–P01 and I03–P04, reflects
the two fractures existing in the field. Group 2 represents well pairs near the fractures
with large λ’s and large τ’s, and Group 3 shows the distant well pairs with small λ’s
and large τ’s. The data is very scattered, indicating a heterogeneous system. None
of the points within a group fall on a line of 1 slope. The FCplots of P01 and
P04 show a very heterogeneous reservoir while the FCplots of P02 and P03 show
near-homogeneity (Fig. 12). These plots reflect the apparent levels of heterogeneity
in the drainage areas surrounding the respective producers. A large fracture or other
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Fig. 11 Log–log plot of λ’s
versus τ’s shows points in three
groups for the fractures case
Fig. 12 The FCplots for all producers show a mixture of homogeneous and heterogeneous responses
for the case with fractures
high-permeability conduit will substantially increase the heterogeneity in a region
where significant amounts of fluid may flow.
The FCplots for P01 and P04 indicate two distinct conditions in the vicinity of
these producers. Similar to the previous case, two straight lines can be fitted to the
FCcurves. The steep straight line suggests that a large fraction of the total flow
capacity is provided by a small fraction of the total pore volume. This is usually an
indication of fractures existing in the vicinity of the corresponding producer. Because
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Fig. 13 Representation of the parameters λij (left)andτij (right) obtained using the BCM for the case
of P04 completed only in layer 5
P01 and P04 are supported by I01 and I03 through fractures, their FCplots deci-
sively indicate these fractures by the steep straight line segment. Furthermore, the
first straight line in the FCplot of P01 is much steeper than that of P04, which
is consistent with the difference in the permeabilities of the fractures. The second
straight line in both plots suggests that a small fraction of total FCis provided by a
large fraction of the total volume; this flow is from injectors communicating through
the matrix of the reservoir. Once again, the static-measured heterogeneity is larger
than that depicted by the dynamic FCof all producers (Fig. 12). The FCplots
appear to be capable of identifying whether the connectivity of an injector–producer
well pair is through a fracture or a high-permeability layer.
3.1.1.4 Producer with Partial Completion In this case, P04 is completed only in
layer L5, while the other producers and injectors are completed in all layers (L1–L5).
The vertical permeability is 0.4 md and all other parameters are similar to those in the
base case. The small λ’s for P04 are consistent with the small productivity because
of its limited completion (Fig. 13). However, the estimated τ’s are the same as the
base case results; evidently, the τ’s carry no information about the partial comple-
tion of P04. Because Vpand Jin the definition of τ(6) are both functions of the
pay thickness of the corresponding producer, τdoes not depend on the pay thickness
(Larsen 2001; Ibragimov et al. 2005). The log–log plot of λvs τreflects the partial
completion of P04, represented by Group 2 (Fig. 14). Group 1 represents the parame-
ter values associated with other producers and shows the expected linear arrangement
with slope of 1. None of the FCplots, however, shows the partial completion of
P04 (Fig. 15). Evidently, the vertical permeability of this case is large enough so that
the entire reservoir is available for flow.
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Fig. 14 A log–log plot of λ
versus τfor the partial
completion case shows the
points fall in two distinct
groups. Each group
approximately clusters around a
line of slope of 1
Fig. 15 The FCplots for the partial completion case show a near homogeneous reservoir
3.1.2 25 ×16 Synfield
A system with sealing faults was analyzed with a 25 ×16 Synfield. The numerical
simulation extends to n=415 months, with n =1 month (Yousef 2006, for de-
tails of other cases). Four vertical sealing faults divide the reservoir into five isolated
compartments (Fig. 16). All layers are homogeneous and have the same permeability
(40 md). Wells are completed in all encountered layers. The presence of the faults can
be inferred from the λ’s and the τ’s (Fig. 16). The values of λcorresponding for well
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Fig. 16 Representation of the parameters λij (left)andτij (right) obtained using the BCM for the case
of vertical sealing faults
Fig. 17 Log–log plot for the
sealing faults case shows the
parameter values form two
distinct groups. Only the points
in Group 1 fall close to a line
with slope 1
pairs located in different compartments are close to zero, while the corresponding
values of τfor the same well pairs are large. This shows no communication between
these wells, which is consistent with the presence of the sealing faults. The log–log
plot of λ’s versus τ’s indicates two different groups (Fig. 17). Group 1, characterized
by relatively large λs and small τ’s reflects the values of λand τfor well pairs lo-
cated in the same compartment. Group 2, characterized by small λ’s (λ<0.03) and
large τ’s, are for well pairs in different compartments. All FCplots (not shown)
indicate a near-homogeneous reservoir, with producers away from the faults showing
somewhat more heterogeneity than those near to the faults.
3.2 Application to Field Data
The technique was applied to data from a portion of the North Buck Draw field,
Wyoming. Yousef (2006, pp. 451–468) reports on several other field studies. Unlike
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Fig. 18 Results of the UCM
applied to a portion of the NBD
field. Only λ’s >0.05 are
shown. Dashed lines represent
the approximate east–west limits
of the channel facies as
described by Sellars and
Hawkins (1992). Anderson and
Harrison (1997) place the
western edge to the west of Well
33-12. Flow is from the south
the Synfield applications, there are no concrete standards or agreement with the im-
posed geology against which to compare results. Our truth test will be comparison
against the geological features that are, as much as possible, independently known.
The NBD study consists of data from 8 injectors and 12 producers (Fig. 18). The
reservoir average porosity is 9.5%, and the average permeability is 10.7 md. The
reservoir fluid is near-volatile oil and the fluid properties fall between those of black
and volatile oils. The fluid meets the majority of volatile-oil criteria, including large
oil formation volume factors and solution gas-oil ratios. The bubble point pressure
is 4680 psi, and the reservoir fluid is a single-phase, low-viscosity fluid above this
pressure (Sellars and Hawkins 1992).
Commercial production began in June 1983. In 1988 a pressure maintenance
project was initiated by injecting gas. Since the field is undergoing gas injection,
the reservoir total compressibility is large, which could violate the assumption of
slightly compressible fluids used in the derivation of (1). However, if the product
of ctp 1, the assumption of slightly compressible fluids will be approximately
correct (Dake 1978, pp. 243–244). In this case, reservoir pressure has varied, but
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is approximately 5 000 psi and pressure variations are 300 psi (Fulco 1999). Thus,
ctp p / p =0.06, indicating that the CM is being used appropriately. The time
period selected for the analysis was determined by a procedure described in Yousef
(2006, pp. 466). The analysis is carried out using monthly flow rates starting in month
35, and covering 56 monthly flow rates. Because the reservoir total compressibility
is large, the inference procedure most likely will not be able to indicate the connec-
tivity between distant injector–producer pairs. Therefore, we applied the UCM only
to producers and nearby injectors. The fits to the production data are relatively good;
for example, the R2for producers 13-7 and 33-12 are 0.94 and 0.97, respectively.
Two geological models of NBD have been proposed (Fig. 18). Sellars and
Hawkins (1992) describe a system of predominantly fluvial channels and point bar
deposits, while Gardner et al. (1994) propose a more mixed system of fluvial and
estuarine deposits and marine shales. While the field boundaries suggest a simple
point-bar deposit with good connectivity, Gardner et al. (1994) propose that the con-
tinuity is more restricted by the incision of finer-grained, lower-quality estuarine de-
posits and marine shales as in well 14-18. The λ’s suggest poor interwell connectivity
in some areas of NBD, in better agreement with Gardner et al.’s model. For exam-
ple, well 14-18 has only a small proportion of fluvial channels and small λ’s; nearby
wells 33-13 and 33-18 have significantly more fluvial material and larger λ’s. Well
33-7 also has more fluvial material but small λ’s, which may be caused by some
compartmentalization of the deposits around this well.
Radioactive tracers were injected into NBD, and their occurrence was monitored
at the producing wells from February 1989 to March 1993. Refunjol and Lake (1998)
applied Spearman analysis to determine preferential flow trends in the NBD field
and compared the results with injected tracer response; they related injection flow
rates with those of adjacent producers and used time lags to find a maximum Spear-
man correlation coefficient (rs). Because rscan be interpreted as another measure of
injector–producer communication, a comparison between the rs’s and the λs should
provide a consistency check with the interwell connectivity between well pairs from
UCM. We found that rstends to be larger and less variable than the corresponding λ.
Thus, the estimated λ’s are broadly consistent with the rss. Because the tracer re-
sponse breakthrough times are considered real field measurements, we use them as a
basis for assessing the τ’s and the Spearman time lags (Fig. 19). The correlation of
τwith the tracer breakthrough times is substantially better than the correlation with
the Spearman time lags. The slope of the least-squares regression line is less than 1,
which is consistent with the tracer taking longer to arrive at the producer than the
pressure disturbances from the injection fluctuations.
The log(λ) vs. log(τ ) plot reveals that there are two different behaviors for inter-
well connectivity in NBD (Fig. 20). Group 1 contains wells having and surrounded
by significant amounts of fluvial deposits (e.g., wells 33-18, 31-18, and 33-7) while
Group 2 wells tend to have greater amounts of estuarine deposits (e.g., wells 33-
12 and 14-18). The response of points is similar to the high permeability layer case
(Fig. 8), where good communication may exist through fluvial deposits but pathways
can still be interrupted by poorer-quality, incising estuarine and shale deposits. The
FCplots also indicate two distinctly different behaviors, in which certain injectors
communicate with the corresponding producer well and the other injectors commu-
nicate poorly (Fig. 21). The FCplots for producers 33-18, 33-7, and 31-18 plots are
Math Geosci
Fig. 19 Comparison of time
lags obtained by different
methods for the NBD Field. The
line is determined by
least-squares regression. The
intercept is not statistically
different from zero
Fig. 20 log(λ) vs. log(τ ) plot
for NBD shows two ellipsoidal
clusters of points. Both
ellipsoids’ major axes are
orientated at slope of 1
similar to those for the high permeability layer case shown in Fig. 9. These producers
have points primarily in Group 1 of Fig. 20.TheFCplots for the producers 33-6,
33-13, and 13-7 suggest homogeneity of the reservoir and these three wells only have
points in Group 2 of Fig. 20. In these latter wells, the mixture of estuarine, shale,
and fluvial deposits may be causing the connectivity to behave similar to that of a
homogeneous reservoir.
Math Geosci
Fig. 21 The FCplots using the CM results for selected producers in the NBD field
4 Discussion
From applications of the CM in this work, we propose trends of the FCcurve
according to the corresponding geological features present in the area surrounding
a producer. To illustrate this point, Fig. 22 shows three schematic FCcurves. The
curve labeled “fracture trend” indicates a presence of fractures in the drainage volume
Math Geosci
Fig. 22 Schematic of different
trends of the FCcurve
estimated from the CM
parameters according to the
corresponding geological
feature present in the vicinity of
a producer
of a producer; the second “high permeability layer trend” indicates that some injec-
tors communicate with the producer through high permeability layers and the other
injectors communicate through low permeability layers. For the last curve labeled the
“reservoir seal trend”, a fraction of the total storage capacity or the total pore volume
swept by injectors provides a negligible fraction of the total FC. This is a typical
characteristic of nonpay zones or a reservoir seal. However, there are cases where
the FCplots are not able to reflect the conditions in the vicinity of the producer.
As discussed earlier, the FCplot cannot identify a partial completion of a producer
even though this can be easily inferred from the model parameters (λs and τ’s).
The log–log plot of λ’s versus τ’s gives patterns consistent with the imposed ge-
ology in the application to synthetic fields, where distinct groups can be easily iden-
tified. For some real field cases, however, the scatter in the data can be so large that
distinct groups of data are not apparent. Nonetheless, we could identify two different
communication responses for NBD which are consistent with the geological model
of Gardner et al. (1994). These patterns were not so clear when examining either the
λ’s or τ’s separately. Because both the FCand λτplots are based on inferred pa-
rameters (λ’s and τs) from dynamic, injection and production data, different sources
of error can obscure inferences about the geological conditions. Example sources in-
clude deviations from the assumptions on which the CM is based, correlation between
injection rates, using too short an assessment interval, and the quality of injection and
production rate measurements. Our experience with various datasets suggests, for ex-
ample, that results with λ<0.05 may not be significant.
5 Conclusions
This paper describes the development of the capacitance model into a diagnostic tool
to enhance inferences about preferential transmissibility trends and the presence of
flow barriers. Complex geological conditions are often not easily identified using the
λand τvalues individually. However, combining both parameters in specific repre-
sentations enhances inferences about the geological features. Two different represen-
tations are used. One representation is a log–log plot of the λ’s versus the τ’s for a
Math Geosci
producer and nearby injectors; another representation is the FCplot where the λ’s
and τ’s are combined using the idea of Lorenz plots. The synthetic field applications
show that the relation between the λ’s and the corresponding τs are consistent with
known heterogeneity, the distance between wells, and their relative positions. The
FCplots and the log–log plots are capable of identifying whether the connectivity
of an injector–producer well pair is through fractures, a high-permeability layer, or
through partially completed wells.
The technique was applied to the North Buck Draw field. The capacitance model
results agree with the tracer analysis better than that of the Spearman analysis. The
estimated λ’s and τs are consistent with the geological interpretation of Gardner et al.
(1994) and suggest that some compartmentalization is present. The log(λ) vs. log(τ )
plot appears to separate the wells situated predominantly in fluvial sediments from
those in more mixed, fluvial and estuarine deposits. FCplots similarly indicate that
some nearby injectors communicate with some producers through the better quality
fluvial deposits.
Acknowledgements The authors wish to thank Steven Hubbard and Brian Willis for their geological
assistance. A.A.Y. would like to thank Saudi Aramco for his graduate fellowship. Larry W. Lake holds the
W.A. (Monty) Moncrief Chair at The University of Texas and Jerry L. Jensen holds the Schulich Chair in
Geostatistics at The University of Calgary. This work was supported in part by US Department of Energy
Contract DE PS26021215375.
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This paper presents new applications of a capacitance model to characterize reservoirs based on temporal fluctuations in injection and production rates. In a previous report (SPE 95322), two coefficients are determined for each injector-producer pair; one parameter, ?, quantifies the connectivity and another, t, quantifies the fluid storage in the interwell region. This report describes the development of this method into a diagnostic tool to enhance inference about preferential transmissibility trends and the presence of flow barriers. Applying the capacitance model to results from numerical flow simulation, we found that complex geological conditions are often not easily identified using ? and t values separately. However, combining them in certain representations enhances the inference about geological features. Two different representations are used: one representation is a log-log plot of the ?'s versus the t's for a producer and nearby injectors, and another representation is the flow capacity plot where the ?'s and the t's are combined as in Lorenz plots. These techniques were tested using a numerical simulator and then applied to data sets from the South Wasson Clearfork and the North Buck Draw fields. The simulation results and field applications show that the relation between ?'s and corresponding t's are consistent with the known heterogeneity, the distance between wells, and their relative positions. The flow capacity plots and the log-log plots are capable of identifying whether the connectivity of an injector-producer well pair is through fractures, a high-permeability layer, multiple-layers or through partially completed wells. Finally, the approach, based on Lorenz plots, can make use of the extensive literature already available on the interpretation of such plots. Introduction Most real reservoirs, if not all, are heterogeneous. Different methods have been used to quantitatively determine the interwell connectivity during waterflood and thereby map reservoir heterogeneity. Albertoni and Lake[1] estimated interwell connectivity based on a linear model with coefficients estimated by multiple linear regression. The linear model coefficients quantitatively indicate the communication between a producer and the injectors in a waterflood. Yousef et al.[2] used a more complete model that includes capacitance (compressibility) as well as resistive (transmissibility) effects. For each injector-producer pair, two coefficients are determined; one parameter (the weight) quantifies the connectivity, and another (the time constant) quantifies the degree of fluid storage between the wells. All the reservoir simulation models used in Yousef et al.[2] were homogeneous; the emphasis was on validation of the new aspects that the capacitance model (CM) has over the method proposed by Albertoni and Lake.[1]
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Permeability heterogeneity is a well known feature that is present in most if not all reservoirs. Initially when the reservoir is in hydrostatic equilibrium the differences in permeability within the same reservoir unit do not manifest. However as soon as production starts and a pressure drawdown is introduced to the system fluids will start moving towards the wellbore following the easiest paths within the rock. Therefore a system of "flowing units" develops with time allowing the zones with relative high permeability to produce faster than the relatively low permeability zones. When a well is completed and perforated it is good practice to evaluate reservoir properties and well performance by carrying out transient well testing and multirate tests. Well test interpretation gives a good estimation of kh, skin and boundary effects. Multirate tests, particularly important in gas wells and high flowrate oil wells, allow evaluation of well deliverability and non-Darcy effects. It is therefore important to define the correct thickness that is actively contributing to flow. The effective thickness will be the sum of all the high permeability flowing units. In fact, and depending on the permeability contrast, the few most permeable units are responsible for most of production. Production Logging can verify and quantify flow contribution from these intervals. Differential depletion within this interbedded layered system occurs by stages. Initially the zones with the highest permeabilities will produce faster and drain their connected volume until a differential pressure threshold is reached and crossflow from the surrounding poorer-quality reservoir facies starts. A production logging survey is a snapshot of the flowing profile across the producing intervals at that particular point in time. In order to monitor this dynamic behaviour running production logging surveys at different times during the field's life is highly recommended especially in mature fields that produce from layered systems or interbedded units within the same formation. Time-lapse production logging provides a realistic visualization of flowing behaviour and helps pre-development of a production optimisation strategy focused in maximizing reserve recovery. The previous statements are supported by field examples where this pattern of permeability zonation has been observed over time. Introduction The term Flow Unit has been used originally to describe the correlateable units in reservoirs1.Hydraulic Flow Units were later introduced to cluster core plugs with similar petrophysical properties2. The Stratigraphically-Modified Lorenz Plot has been used more recently to identify Flow Units3 in the sense of ‘flowing units’ identified by the production log.The term Flow Unit means different things to different people so we use the term "flowing units" in this paper to define the units detected by production logs to be flowing. This paper investigates the evolution of these flowing units over production time in a well-layered system with crossflow. The paper shows the link between evolution of flowing units with time and initial evaluation measures. Laterally continuous, multilayered reservoirs with high permeability contrast between layers with crossflow behaviour have a distinctive long-term production behaviour. Preferential depletion develops over time and the most permeable and consequently the most productive intervals act like "fracture channel" systems collecting the produced fluids from the surrounding less permeable units and carrying them to the wellbore.These reservoirs are called double porosity, but as this terminology is often associated with naturally-fractured reservoirs we call these double matrix porosity.In well testing these reservoirs are also called dual permeability. The most common way of quantifying the production profile along the perforated section in a well is to run a production logging survey. Time-lapse production logging suggests a monitoring plan of evaluating well's production profile at different points in time, which is considered fundamental in understanding production dynamics and production optimisation.
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This paper is an exposition of the use of ridge regression methods. Two examples from the literature are used as a base. Attention is focused on the RIDGE TRACE which is a two-dimensional graphical procedure for portraying the complex relationships in multifactor data. Recommendations are made for obtaining a better regression equation than that given by ordinary least squares estimation.
Conference Paper
This paper presents the results of a practical technique to determine preferential flow trends in a reservoir. The technique is a combination of reservoir geology, tracer data, and Spearman rank correlation coefficient analysis of injection/production rate data. The Spearman analysis, in particular, will prove to be important because it appears to be insightful and uses data that are prevalent when other data are nonexistent. The technique is applied to the North Buck Draw field, Campbell County, Wyoming. This work provides guidelines to assess information about reservoir continuity in interwell regions from widely available measurements of production and injection rates at existing wells. When successfully applied, the information gained can contribute to both the daily reservoir management and the future design, control, and interpretation of subsequent projects in the reservoir, without the need for additional data. As with other techniques, however, the method gives the most confidence when corroborated by other procedures.
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The need for simple models to estimate the productivity of arbitrary wells in homogeneous and layered reservoirs should be obvious. The current paper addresses this problem by extending current analytic methods to handle a greater variety of well configurations. One of the more important extensions covers deviated wells with limited flow entry through a fairly simple summation scheme. Another being the treatment of wells with a limited height fracture. Two basic drainage shapes are included in the models: Circular and other regular drainage shapes with the well clearly inside the drainage area, and rectangular without any restriction on well length and location for homogeneous and commingled layered reservoirs, but with the well required to lie clerly inside the drainage area for crossflow systems. For highly deviated wells the rectangular models are based on the approach of Goode and Kuchuk1 for horizontal wells, with the difference between the deviated well and a horizontal well handled through a modified skin value. Fractures of limited height are also based on Ref. 1 by adding an appropriate skin value. Introduction For a well producing at pseudosteady state from a closed reservoir, letEquation 1 denote dimensionless pseudosteady state (PSS) pressure drop, in the terminology of Goode and Kuchuk,1 and letEquation 2 denote the corresponding dimensionless productivity index. Since the pressure difference in Eq. 1 will be constant under PSS conditions, the dependence on time t need not be shown. With the real productivity index J defined by the identityEquation 3 it follows that J and JD will be related through the identityEquation 4 For wells "clearly inside" the drainage shape, the dimensionless PSS productivity index JD can be defined by the identityEquation 5 where A is the drainage area, CA the Dietz2 shape factor, S the total skin factor and the other parameters can be found in the Nomenclature. Eqs. 2 and 5 can be generalized to layered reservoirs if kHh is replaced by the total flow capacityEquation 6 in Eq. 4, where kHj and hj denote horizontal permeability and thickness of Layer j for j=1, 2, ... , n. For wells that are not vertical and fully penetrating, the main challenge in applications is to determine the skin factor S of the well, especially for layered reservoirs. It can also be difficult to determine the shape factor, at least for drainage shapes that differ considerably from rectangles, circles and simple triangles.