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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 F–Cplots and the λ–τplots 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 F–Cand

λ–τplots 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 −(n−n0j)

τpj +

i=I

i=1

λij w

ij (n)

+

k=K

k=1

νkj pwf kj (n0j)e −(n−n0j)

τkj −pwf kj (n) +p

wf kj (n),(1)

where

w

ij (n) =

m=n

m=n0j

n

τij

e

(m−n)

τij wij (m),

p

wf kj (n) =

m=n

m=n0j

n

τkj

e

(m−n)

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 τ’s 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 λ’s vs. the τ’s for a producer and nearby injec-

tors, and another is the F–Cplot 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 τ’s are consistent with the known heterogeneity,

the distance between wells, and their relative positions. The F–Cplots 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 F–Cplot, 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 F–Cplots). 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 F–Cplots 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 F–Cplots 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 F–Cplots 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 ≤Fm≤1 and 0 ≤Hm≤1for1≤m≤n. 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

(F–C) 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 F–Cplot is formed using the λ’s and τ’s 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 F–Cbetween 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 F–Ccurve is a plot of cumulative F–C(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 1/τij 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 F–Ccurve monotonically increases from

i=1toi=Iwith a monotonically decreasing slope as does a Lorenz plot. The F–C

plots can indicate speciﬁc geological features in the vicinity of an injector. In this

case, sets of λ’s and τ’s for one injector and all producers are used to form the F–C

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 λ’s 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 τ’s) with the imposed geology, and the

capability of the F–Cplots 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 ×10−6,1×10−6, and 1 ×10−6psi−1,

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 τ’s 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 F–Cplots 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 F–Cplots for all producers for the 5 ×4 Synﬁeld. A homogeneous reservoir shows a

near-homogeneous F–Cplot

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 τ’s 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 F–Cplots 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 45◦line. The

static F–Ccurves are also shown in each plot. The F–Ccurves 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 F–Ccurves are not straight lines (Fig. 9) and indicate that the Synﬁeld is

not homogeneous. For each producer, the F–Cplot 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 F–Ccurves, 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 λ’s 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 F–Cplots of P01 and

P04 show a very heterogeneous reservoir while the F–Cplots 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 F–Cplots 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 F–Cplots 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

F–Ccurves. 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 F–Cplots deci-

sively indicate these fractures by the steep straight line segment. Furthermore, the

ﬁrst straight line in the F–Cplot 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 F–Cis 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 F–Cof all producers (Fig. 12). The F–Cplots

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 F–Cplots, 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 F–Cplots 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 λ’s 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 F–Cplots (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 ctp 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,

ctp ≈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 λ’s 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 rs’s. 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

F–Cplots 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 F–Cplots 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.TheF–Cplots 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 F–Cplots 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 F–Ccurve

according to the corresponding geological features present in the area surrounding

a producer. To illustrate this point, Fig. 22 shows three schematic F–Ccurves. The

curve labeled “fracture trend” indicates a presence of fractures in the drainage volume

Math Geosci

Fig. 22 Schematic of different

trends of the F–Ccurve

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 F–C. This is a typical

characteristic of nonpay zones or a reservoir seal. However, there are cases where

the F–Cplots are not able to reﬂect the conditions in the vicinity of the producer.

As discussed earlier, the F–Cplot cannot identify a partial completion of a producer

even though this can be easily inferred from the model parameters (λ’s 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 F–Cand λ–τplots are based on inferred pa-

rameters (λ’s and τ’s) 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 F–Cplot 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 τ’s are consistent with

known heterogeneity, the distance between wells, and their relative positions. The

F–Cplots 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 τ’s 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. F–Cplots 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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