Radioactive Marker Measurements in Heterogeneous
Reservoirs: Numerical Study
Massimiliano Ferronato1; Giuseppe Gambolati2; Pietro Teatini3; and Domenico Baù4
Abstract: A numerical study is performed to simulate the vertical deformation of a depth interval representing a marker spacing 共10.5
m兲 located in a deep heterogeneous sedimentary reservoir. Realistic lithostratigraphic sequences typical of the Northern Adriatic basin are
used. A number of scenarios are addressed consistent with the available data. In particular two basic geologic scenarios at the marker scale
are simulated, one where sands prevail within the marker spacing 共SD兲, and another where thin sandy and clayey layers alternate 共CL兲.
The sensitivity of the marker response is investigated in relation to clay and sand permeability and compressibility contrast, Biot’s
coefficient, and respective position of monitoring and fluid pumping wells. The modeling results show that rock may indeed expand above
and below depleted layers. Depending on the marker position the expansion may partially offset the compaction, especially in the CL
scenario with a very low permeable clay. To obtain a representative field compaction the markers should span a depth interval made
mostly by sand and entirely depleted, and should be installed in a test hole far from producing wells. Compressibility contrast and the Biot
coefficient play a secondary role. Very critical measurements are provided by two markers which incorporate a thin 共⬇1 m兲 depleted level
overlain and underlain by almost impermeable clay layers.
DOI: 10.1061/共ASCE兲1532-3641共2004兲4:2共79兲
CE Database subject headings: Radioactive tracers; Finite element method; Sensitivity analysis; Compressibility; Reservoirs.
Introduction
Anthropogenic land subsidence is a major undesirable consequence of subsurface fluid 共water, gas, oil兲 production 共e.g., Gambolati et al. 1991, 1999b; Colazas and Strehle 1995; Finol and
Sancevic 1995; Teatini et al. 1995, 2000; Baù et al. 1999a, 2000;
Gonzalez-Moran et al. 1999; Larson et al. 2001兲. This occurrence
is a matter of great concern in lowlying coastal areas where the
loss in ground elevation can enhance sea water encroachment
both occasionally during severe storm events and permanently,
with an adverse impact on the environment and possibly large
economical and social costs. Land settlement represents the outcome of surface migration from burial depth of reservoir compaction. This can have other minor unwanted effects concerning the
field management such as break of well casings, impairment of
offshore platforms 共Zaman et al. 1995兲, reduction of formation
porosity and permeability with a decrease of fluid production
共Fredrich et al. 2000兲.
1
Dept. Mathematical Methods and Models for Scientific Applications,
Univ. of Padova, via Belzoni 7, 35131 Padova, Italy.
2
Dept. Mathematical Methods and Models for Scientific Applications,
Univ. of Padova, via Belzoni 7, 35131 Padova, Italy 共corresponding author兲. E-mail: [email protected]
3
Dept. Mathematical Methods and Models for Scientific Applications,
Univ. of Padova, via Belzoni 7, 35131 Padova, Italy.
4
Dept. Geological Engineering & Sciences, Michigan Technological
Univ., 1400 Townsend Dr., Houghton, MI 49931-1295.
Note. Discussion open until November 1, 2004. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on October 24, 2003; approved on November 5, 2003.
This paper is part of the International Journal of Geomechanics, Vol. 4,
No. 2, June 1, 2004. ©ASCE, ISSN 1532-3641/2004/2-79–92/$18.00.
A major issue when planning the development of a gas/oil
field is the prediction of the expected reservoir compaction and
the associated land subsidence during the field production life and
also after the field abandonment 共Baù et al. 2000兲. A most fundamental parameter for a reliable prediction is the vertical uniaxial
rock compressibility c M which basically controls the amount of
compaction caused by pore pressure drawdown 共Johnson et al.
1989; Ruddy et al. 1989; Gambolati et al. 1999a兲. Parameter c M
is usually determined in the laboratory on rock samples taken
from exploratory boreholes drilled at the field burial depth and
subjected to mechanical triaxial and/or oedometric tests 共e.g.,
Teeuw 1973兲. However, such tests are performed on rock specimen with a few centimeter size and may be influenced by the
local reservoir heterogeneity, especially in sedimentary basins
like the Northern Adriatic one characterized by a high lithostratigraphic variability with depth. In addition the sample may frequently suffer from damages induced by coring operations 共Holt
et al. 1994兲. It is widely recognized that lab experiments tend to
overestimate the actual field c M 关van Hasselt 1992; Holt et al.
1994; Nederlandse Aardolie Maatschappij 共NAM兲 1995; Cassiani
and Zoccatelli 2000兴 because of specimen disturbances and bedding errors in the sample preparation process 共Jardine et al. 1984;
Kim et al. 1994兲.
A viable alternative technique for assessing c M at the field
scale was developed in the early 1970s 共de Loos 1973兲 and referred to as radioactive marker technique 共RMT兲 thereafter. The
RMT is being used currently worldwide, e.g., the North Sea
共Menghini 1989兲, the Gulf of Mexico 共de Kock et al. 1998兲, the
Netherlands 共Mobach and Gussinklo 1994兲, and the Northern
Adriatic 共Baù et al. 1999b兲. The RMT is based on repeated measurements of the vertical distance between radioactive bullets
共called markers兲 shot at regular intervals of about 10.5 m within
the depleted formation through the wall of a vertical, generally
unproductive, well prior to the casing operations. An invar rod
INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004 / 79
carrying two pairs of gamma-ray detectors with a spacing approximately equal to that of two adjacent markers is slowly raised
at a constant speed from the borehole bottom, recording the count
rate peaks when the detectors are right opposite the bullets. The
time intervals between the peaks are then converted into distances. The peak of each gamma-ray distribution is identified
using a filter function which relies on a Gaussian or a Lorentzian
probability distribution. The nominal RMT precision, i.e., the expected average measurement error, is on the order of 10⫺4 ,
namely 1 mm for a 10 m wide depth interval 共Schlumberger
1994; Pemper et al. 1996兲. The field c M is then evaluated by the
simple equation 共Baù et al. 1999b, 2002兲
cM⫽
⌬h
h 0 ⌬p
(1)
where ⌬h⫽h t ⫺h 0 ⫽average vertical deformation 共expansion if
positive, compaction if negative兲 of the marker interval; h 0 and
h t ⫽average distance between two adjacent markers at the initial
time and at time t, respectively; and ⌬p⫽fluid pressure variation
共rise if positive, drawdown if negative兲 occurred within the monitored depth interval over the time period 0-t.
In 1992 ENI-E&P, the Italian national oil company, started a
RMT survey in the Northern Adriatic with five offshore boreholes. The measurements were obtained with the aid of a formation subsidence monitoring tool by Schlumberger and compaction
monitoring tool by Western Atlas. Records were collected every 2
or 4 years along with the static pressure values in the test holes.
By 1999 at least 11 surveys were completed in the area. The
measurements were statistically processed to provide the average
field c M values 共Baù et al. 1999b, 2002; Bevilacqua et al. 1999兲.
To some extent unexpectedly, the c M thus obtained were markedly smaller 共say 1 order of magnitude兲 than the first loading
cycle c M derived from lab experiments. In addition a significant
data scattering was observed with a large associated standard deviation which is difficult to account for on the grounds of c M
natural variability with depth and lithology alone.
Actually the big difference between lab and RMT values and
the large data dispersion raise some concern on the straightforward use of the measured compaction to derive c M by Eq. 共1兲.
The Northern Adriatic basin is characterized by lithostratigraphic
vertical sequences which are heterogeneous over a scale smaller,
and locally much smaller, than the measurement scale 共10.5 m兲.
So it may happen that a depleted level is markedly thinner than
the recorded depth spacing. Under such conditions RMT does not
necessarily measure a compaction over the entire observation
length. If only a fraction of the depth interval has undergone
compaction the calculated c M may be underestimated. Moreover
RMT may record an expansion where a compaction is expected.
In this case the measurement is discarded on account of a likely
instrumentation error. However, the observed data could be correct and indeed related to a real local rock expansion, as will be
shown later.
Another example leading to a c M underestimate is when the
marker is installed in a producing well. In this case the horizontal
pore pressure gradients toward the wellbore generate a horizontal
strain opposing the vertical compaction which therefore does not
conform with an oedometric deformation and may yield a smaller
c M . From the previous discussion it appears to be extremely important to install the markers in a depth interval which fully compacts, and to relate the recorded compaction to the average pore
pressure drawdown experienced by the entire depth interval.
The objective of the present analysis is to reproduce by a poroelastic mathematical model the response of an ideal marker tool
located in a reservoir with a vertical heterogeneity scale smaller
than the marker spacing. The heterogeneity scale will be similar
to that observed in the Northern Adriatic marker test holes. The
study will address the marker vertical deformation and provide an
insight into the lithostratigraphic configurations which are most
likely to yield nonrepresentative compaction data. In particular
the possible marker expansions will be investigated in relation to
the presence within the monitored depth interval of fine grained
beds whose depletion is much delayed in time.
The paper is organized as follows. After a brief description of
the poroelastic mathematical model and its solution by finite elements 共FEs兲 some typical geological structures of the Northern
Adriatic basin are outlined with the aid of a few representative
lithostratigraphies. Fluid withdrawal is simulated with both a distributed and a concentrated pumping sink. The marker response is
studied in relation to the lithostratigraphy of the monitored depth
interval, its hydromechanical properties and the position and
thickness of the depleted level. Typical small scale sandy–clayey
alternate layerings that can lead to an incorrect c M calculation are
discussed. Finally, a set of conclusive remarks and recommendations about a most efficient strategy of marker installation in heterogeneous sedimentary basins is provided.
Model Setup
Mathematical Model
For the purpose of the present analysis the reservoir may be assumed to be saturated with a single-phase fluid. The stress and
strain fields induced by the development of a heterogeneous reservoir can be described by the classical Biot 共1941兲 poroelasticity
theory, as later modified by van der Knaap 共1959兲 and Geertsma
共1966兲. A three dimensional 共3D兲 axisymmetric model is used
with RMT implemented along the symmetry axis, representing a
vertical monitoring well. For an isotropic porous medium, the
axisymmetric poroelastic equations read
Gⵜ 2 u r ⫹ 共 G⫹␭ 兲
ur
⳵␧ V
⳵p
⫺G 2 ⫽␣
⳵r
⳵r
r
(2)
⳵p
⳵␧ V
⫽␣
⳵z
⳵z
(3)
Gⵜ 2 u z ⫹ 共 G⫹␭ 兲
⳵
k
关 ␾␤p⫹c br共 ␣⫺␾ 兲 p⫹␣␧ V 兴 ⫽ ⵜ 2 p
⳵t
␥
(4)
where ␭ and G denote the Lamé constant and the shear modulus
of porous medium; ␣⫽Biot coefficient (␣⫽1⫺c br /c bm , with c br
and c bm the grain and bulk compressibility, respectively兲; ␥ and
␤⫽fluid specific weight and volumetric compressibility;
k⫽medium hydraulic conductivity, and ␾⫽porosity. The unknown variables are the incremental displacements u r and u z
along the radial and vertical directions, the incremental fluid pore
pressure p, and the porous matrix volume strain ␧ V , which is
related to u r and u z by the equation
␧ V⫽
⳵u r u r ⳵u z
⫹ ⫹
⳵r
r
⳵z
Finally, ⵜ 2 ⫽Laplace operator (⳵ 2 /⳵r 2 ⫹⳵ 2 /⳵z 2 ⫹1/r⳵/⳵r).
Eqs. 共2兲–共4兲 are integrated in space by FE using the Galerkin
method of weighted residuals and Green’s first identity, so as to
80 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004
allow for a linear approximation over each finite element of both
displacement and pressure unknowns. Denoting by
K⫽
冋
Kr
Krz
T
Krz
Kz
册 冋册
Qr
Q⫽ Q
z
the elastic stiffness and coupling matrices, respectively, by H and
P the flow stiffness and flow capacity matrices, and by fT
⫽ 兵 fr ,fz ,fp 其 the vector accounting for nodal loads (fr ,fz ) and flow
sources or sinks (fp ), the following system of first order differential equations is obtained 共Ferronato et al. 2003兲:
冋
Kr
Krz
⫺Qr
T
Krz
Kz
⫺Qz
0
0
H
册再 冎 冋
0
ur
uz ⫹ 0
p
QTr
0
0
0
0
QTz
P
册再冎再冎
⳵
⳵t
ur
fr
uz ⫽ fz
p
fp
The above equations are solved in time by a backward marching
scheme, which gives rise to the algebraic linear system:
冋
Kr
Krz
⫺Qr
T
Krz
QTr /⌬t
Kz
⫺Qz
QTz /⌬t
H⫹P/⌬t
⫽
冋
册再 冎
册再 冎 再 冎
0
0
0
0
0
0
QTr /⌬t
QTz /⌬t
P/⌬t
u共rn⫹1 兲
u共zn⫹1 兲
p共 n⫹1 兲
f 共rn⫹1 兲
u共rn 兲
n⫹1 兲
u共zn 兲 ⫹ f 共z
p共 n 兲
f 共pn⫹1 兲
Fig. 1. Map of Mediterranean Sea. Location of Northern Adriatic
sedimentary basin is indicated by panel.
(5)
System 共5兲 is repeatedly solved in order to obtain displacements
ur , uz , and pore pressure p at different time values. Note that the
matrix controlling the numerical scheme is nonsymmetric and
depends upon the time integration step ⌬t. Booker and Small
共1975兲 have also shown the stability of Eqs. 共5兲 for a sequence of
increasing time steps, as is typically the case for consolidation
problems. However, much care must be paid to the choice of the
initial ⌬t, since the small time step value needed in the early
simulation phase may yield a severe ill conditioning of the FE
coefficient matrix 共Ferronato et al. 2001兲.
The solution to Eqs. 共5兲 may be difficult to achieve depending
on the problem size and conditioning 共Ferronato 2002兲. In the
present paper Eqs. 共5兲 are solved by projection 共or conjugate gradient like兲 methods specifically designed for large sets of sparse
indefinite unsymmetric equations 共Saad 1996兲. On the grounds of
its robustness 共Gambolati et al. 2001, 2002兲, we have elected to
use biconjugate gradient stabilized 共Bi-CGSTAB兲 共van der Vorst
1992兲 preconditioned with a triangular factorization with partial
fill-in 共Saad 1994兲, and an initial time step larger than the critical
⌬t yielding ill conditioning 共Ferronato et al. 2001兲. Finally, convergence of the iterative method is further improved by an appropriate preliminary scaling 共Ferronato et al. 2002; Gambolati et al.
2003兲.
that it usually consists of a number of thin gas-bearing sandy
layers 共⬃1–10 m兲 frequently interbedded with low permeable
confining beds.
Two typical lithostratigraphic sequences of the Northern Adriatic basin are shown in Fig. 2. In the leftmost porous column
共configuration SD兲 a 13 m thick sandy reservoir is confined between shale layers with a thickness of about 1 m, while in the
rightmost sample 共configuration CL兲 four productive 1 m thick
sandy layers are incorporated within a low permeable clayey formation. Note that each lithostratigraphy exhibits a pronounced
heterogeneity at the RMT vertical resolution scale 共⬃10.5 m兲 as
far as both the lithology and the permeability are concerned 共see
Fig. 2兲.
Hydrogeological and Geomechanical Setting
The FE poroelastic model is used to predict the RMT response in
producing reservoirs with a heterogeneity scale smaller than the
marker spacing size. An example of small scale medium heterogeneity is provided by the Northern Adriatic basin 共Fig. 1兲, whose
sedimentary sequence, laid down in different environments, is
made of thin alternating sandy, silty, and clayey layers interbedded with all possible mixtures of these lithologies. The gas reservoirs discovered by ENI-E&P have a burial depth ranging between 1,000 and 4,500 m and are mostly located in the preQuaternary basement. A typical feature of the productive field is
Fig. 2. Detailed lithostratigraphy used in SD and CL configurations
in 3,200–3,400 m depth interval. Reservoir burial depth is approximately 3,300 m. Cumulative thickness of producing layers is 13 and
4 m, respectively. Permeability k of each lithology is indicated in
legend.
INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004 / 81
Fig. 3. 共a兲 Uniaxial compressibility c M and porosity ␾ versus depth and 共b兲 Biot coefficient ␣ versus depth
A detailed knowledge of the main hydrogeomechanical properties of the Northern Adriatic geological structure is currently
available due to the extensive reconnaissance studies performed
over the last few decades. Representative profiles of the uniaxial
compressibility c M , porosity ␾, and Biot coefficient ␣ versus
depth are shown in Fig. 3. They have been derived from in situ
共Baù et al. 1999b, 2002兲 and laboratory 共Ricceri and Butterfield
1974兲 measurements, 3D seismic surveys 共AGIP 1996兲, and electrical well logs 共Marsala et al. 1994兲. Constant values ␯⫽0.3
共Teatini et al. 2000兲, ␤⫽4.16⫻10⫺5 cm2 /kg 共Gambolati et al.
2000兲, and c br⫽0.16⫻10⫺5 cm2 /kg 共Geertsma 1973兲 for Poisson
ratio, fluid 共groundwater兲, and grain compressibility are assumed.
Withdrawal Distribution and Boundary Conditions
The stress source is the pore pressure decline ⌬p experienced by
the producing formation because of the fluid extraction. ⌬p is
generated by prescribing the withdrawal rate shown in Fig. 4,
with the Q value calibrated in each simulation scenario so as to
obtain an average ⌬p⫽100 kg/cm2 in the depleted volume after n
years from the inception of pumping.
In order to reproduce realistically the vertical deformation of
the monitoring well by an axisymmetric model, the fluid extraction is prescribed as follows:
1.
Q(t) is uniformly pumped from a toroidal volume. According to this scenario, the monitoring well is located in a central gravity position with respect to the producing wells. In
the innermost zone around the borehole the horizontal pore
pressure gradient is zero.
2. Q(t) is uniformly withdrawn from a cylindrical volume with
the axis coinciding with the monitoring well. In this case, a
nearly uniform ⌬p is reproduced within the field with a
gentle horizontal pressure gradient.
3. Q(t) is pumped from a vertical line sink. This scenario simulates the monitoring well coinciding with a producing well
and generates an important horizontal pressure gradient.
Since the pore pressure, and hence stress and strain, fields induced by the extraction as indicated at points 共1兲 and 共2兲 turn out
to be very similar, in the sequel we will show the results obtained
from options 共2兲 and 共3兲 only, denoting them as volume 共or volume distributed兲 and line sink, respectively.
The axisymmetric porous volume is discretized by annular elements with triangular cross section. The detailed lithostratigraphies SD and CL require a high resolution mesh, as shown in Fig.
5, with minimum vertical and radial spacings equal to 1 m. A
rigid basement 10,000 m deep is assumed, and a fixed outer
boundary with zero ⌬p is prescribed 25,000 m far from the symmetry axis. Dirichlet conditions with zero ⌬p are also assumed
on the top traction free surface 共Baù et al. 2004兲. The resulting FE
mesh is made of 19,908 nodes and 39,250 toroidal elements, with
an overall system size equal to 59,724.
Modeling Results
Pore Pressure, Strain, and Stress Fields
Fig. 4. Pumping rate profile used in numerical simulations. Q is
calibrated so as to obtain an average 100 kg/cm2 pressure decline in
depleted volume after n years from inception of production.
The FE model described above is solved in the realistic hydrogeomechanical setting of the Northen Adriatic basin in both SD
and CL configurations 共Fig. 2兲 with the pumping profile of Fig. 4
and n⫽2 years. The incremental pore pressure field obtained 2
years after inception of pumping is shown in Fig. 6 in the surroundings of the depleted formations for both volume and line
sink scenarios. Fig. 6 also gives the local stratigraphic sequence.
Note that p max is negative, so p/p max⬍0 indicates overpressure.
As pointed out by other authors as well 共e.g., Verruijt 1969;
Hsieh 1996; Gambolati et al. 2000; Baù et al. 2001兲, a slight
82 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004
Fig. 5. Cross section of finite element mesh used in simulations
overpressure develops above and below the producing layers due
to coupling between fluid flow and medium deformation. The
vertical distance between the overpressured zone and the depleted
layers basically depends on the thickness and permeability of the
confining beds, and not on pumping scenario.
The corresponding vertical strain maps are shown in Fig. 7
with ␧ z,max negative, so that ␧ z /␧ z,max⬍0 means expansion. This
figure reveals the existence of large regions above and below the
producing formation where the medium undergoes a vertical expansion. Note that the shape and distance from the depleted layers
of such expansion regions depend on the extraction scenario and
the selected geologic configuration. The vertical expansions occur
also in steady state, although at a larger distance from the reservoir, and hence they turn out to be mainly related to the 3D
deformation of the porous medium rather than to coupling, which
appears to have a smaller influence on strain. An important role in
enhancing expansion is played by the horizontal pressure gradient
around the symmetry axis. In fact, the line sink expansion occurs
closer to the producing layers with a maximum value which turns
out to be equal to 5% of the compaction, while with the volume
sink the maximum expansion attains 1.3% of compaction although the radial size of the expanding medium is larger. Note
that the positive strain may increase if a simultaneous withdrawal
should take place from layers close above and below the depleted
levels.
To complete the analysis of stress–strain field induced by reservoir depletion in heterogeneous porous media, we address the
stress invariants which characterize both a depleted and an ex-
panding rock sample as simulated by our FE model. Recalling
that with an axisymmetric setting stress invariants p ⬘ and q read
p ⬘ ⫽⫺ 31 共 ␴ rr ⫹␴ zz ⫹␴ ␪␪ 兲 ,
2
q⫽ 冑␴ rr 共 ␴ rr ⫺␴ zz 兲 ⫹␴ zz 共 ␴ zz ⫺␴ ␪␪ 兲 ⫹␴ ␪␪ 共 ␴ ␪␪ ⫺␴ rr 兲 ⫹3␴ rz
,
typical profiles of q and ␧ V versus p ⬘ are shown in Fig. 8. The
stress changes are compared to the initial stress profile
q 0⫽
3 共 1⫺K 0 兲
p⬘
1⫹2K 0 0
with K 0 ⫽␴ rr /␴ zz in undisturbed conditions.
The sandy and clayey rock layers, within and confining the
reservoir, respectively, undergo a p ⬘ increase due to the pore pressure decline with a corresponding volume reduction 关Figs. 8共a
and b兲兴. Stress change obviously reduces in the clayey layers
since the pore pressure decline here is smaller than within the
reservoir. Outside the depleted formation in the early pumping
phase rocks typically undergo a vertical stress unloading which is
compensated for by a horizontal stress increment 关Figs. 8共c and
d兲兴. Thus, although the vertical strain points to an expansion 共Fig.
7兲, overall the medium experiences a volume reduction which
accounts for the development of a slight overpressure in low permeable formations, as is shown in Fig. 6. As consolidation proceeds the overpressure gradually dissipates. However, the vertical
expansion outside the depleted field persists due to the 3D deformation effect.
INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004 / 83
Fig. 6. Pore pressure variation relative to maximum pressure decline p max 2 years after inception of pumping: 共a兲 SD scenario and volume sink;
共b兲 SD scenario and line sink; 共c兲 CL scenario and volume sink; 共d兲 CL scenario and line sink. Lithostratigraphies 共clays are gray shaded兲 and
depleted layers 共hatched areas兲 are also shown
Marker Response
As is currently implemented 共Schlumberger 1994兲, the RMT
records the vertical deformation ⌬u z occurring between two
points about 10.5 m apart. A marker measurement, referred to the
midpoint depth z of the couple of markers, may be simulated by
the FE model as
⌬u z 共 z 兲 ⫽
冕
z⫹5.25
z⫺5.25
␧ z 共 r̄,␨ 兲 d␨
(6)
where r̄⫽marker radial distance from the symmetry axis. Eq. 共6兲
is plotted versus z in a neighbor of the depleted layers, and gives
the numerical vertical deformation of a simulated 10.5 m wide
interval. We set r̄⫽0.
Figs. 9 and 10 show ⌬u z versus depth for z between 3,200 and
3,400 m at different times after the pumping inception for the SD
and CL configurations, respectively, and volume and line sinks.
Note that while the reservoir always compacts small expansions
above and below the producing layers occur. These vary from 1%
共SD with volume sink兲 to 10% 共CL with line sink兲 of the field
compaction, move far from the reservoir as the simulation proceeds, and are still present in steady state. Observe the pro-
nounced ⌬u z variation close to the confining beds with a concentrated pumpage pointing to a high sensitivity of the prospected
deformation, and hence of the related compressibility estimate, to
the actual marker location.
To capture the influence of the hydrogeomechanical parameters on the RMT response, a sensitivity analysis is performed by
varying the rock compressibility and confining bed permeability.
Biot’s coefficient is also changed. The results are given in Figs.
11–13 2 years after the start of production for each lithostratigraphic configuration and pumping scenario.
Fig. 11 shows the effect of the compressibility contrast between the producing layers and the top/bottom confining beds.
The uniaxial compressibility c M ,tb of top and bottom layers is first
halved 共i.e., stiffer caprock兲 and then doubled 共i.e., softer
caprock兲 with respect to the c M 0 value calculated using the c M
versus z profile of Fig. 3共a兲. Note that the different compressibility mainly influences the response of a marker spacing incorporating the reservoir confining beds, as expected. The difference is
smaller in the line sink scenario.
Heterogeneity of permeability has a larger impact on the simulated marker response, as shown in Fig. 12 for the different sce-
84 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004
Fig. 7. Vertical strain relative to maximum strain ␧ z,max 2 years after inception of pumping: 共a兲 SD scenario and volume sink; 共b兲 SD scenario
and line sink; 共c兲 CL scenario and volume sink; 共d兲 CL scenario and line sink. Lithostratigraphies 共clays are gray shaded兲 and depleted layers
共hatched areas兲 are also shown
narios. A clay permeability k cl decrease by 2 and 4 orders of
magnitude from the initial k cl0 value 共see Fig. 2兲 yields an expansion increase 共up to 12–13% of the compaction with the line sink兲
and a reduction of the distance of the expanding region from the
reservoir. Moreover, in the CL configuration a marker spacing
which includes the depleted layers provides a smaller compaction,
thus inducing a compressibility underestimate by Eq. 共1兲.
Finally, the influence of Biot coefficient is investigated by assuming incompressible grains 共␣⫽1兲 and by increasing five times
the initial grain compressibility value c br0 共see ‘‘Hydrogeological
and Geomechanical Setting’’兲. The results of Fig. 13 show that
this parameter does not affect the marker response.
Layering Enhancing Marker Expansions
As previously discussed, the importance of RMT relies on its
capability to provide a reliable unixial compressibility estimate
based on actual in situ compaction measurements. However, expansion values with a corresponding pressure decline over the
monitored interval does not allow us to use Eq. 共1兲, since a negative c M would result. Hence expansion data, although realistically
recorded, cannot be used to estimate c M . From the analysis of the
previous section, it turns out that thin producing layers surrounded by nearly impermeable clays, as it may be the case in
sedimentary basins like the Northern Adriatic, may create serious
difficulties for a correct use and interpretation of marker data. In
fact, a 10.5 m spacing including these layers may incorporate
expanding rock formations which partially or totally offset the
reservoir compaction.
As an example, Fig. 14 shows the results obtained from two
‘‘critical’’ layering configurations. In Fig. 14共a兲, a 1 m thick reservoir is depleted by a line sink with k cl⫽k cl0 /104 . Expansions up
to 20% of the field compaction occur quite close 共less than 5 m
apart兲 to the producing layer, so that a marker spacing incorporating the reservoir could nevertheless expand. Also note the steep
deformation profile above and below the field, which causes the
RMT response to be quite sensitive to the actual marker position.
The previous effect is magnified by the simultaneous depletion of
a second thin reservoir located a few meters 共17 m in this case兲
below 关Fig. 14共b兲兴. The expansion increases up to 25% of the
field compaction, and, especially at the early production stage, the
unproductive formation between the two reservoirs may expand
INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004 / 85
Fig. 8. 共a兲 q and 共b兲 ␧ V versus p ⬘ for depleted rock layers; 共c兲 q and 共d兲 ␧ V versus p ⬘ for porous medium surrounding depleted layers
Fig. 9. Simulated marker response 关from Eq. 共6兲兴 versus depth in SD scenario with: 共a兲 Volume sink; 共b兲 line sink. Lithostratigraphy is also
shown
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Fig. 10. Same as Fig. 9 for CL scenario
Fig. 11. Sensitivity analysis to compressibility contrast: 共a兲 SD scenario and volume sink; 共b兲 SD scenario and line sink; 共c兲 CL scenario and
volume sink; 共d兲 CL scenario and line sink. Marker response is provided 2 years after start of production
INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004 / 87
Fig. 12. Same as Fig. 11 for permeability contrast
quite significantly. In this case, RMT could record an expansion
and/or a reduced compaction although available well data point to
a pore pressure decline, thus giving unreliable measurements for
the compressibility estimate 关via Eq. 共1兲兴.
Uniaxial Compressibility Estimate
The simulated marker compaction and the corresponding pore
pressure drawdown can be used in Eq. 共1兲 to obtain the c M estimate for the monitored rock formation. A comparison between the
‘‘a posteriori’’ c M value thus derived and that implemented into
the model provides an indication as to the reliability of RMT for
the c M calculation.
The results are shown in Fig. 15 for the four addressed scenarios. We use the marker response of Figs. 9 and 10 and the pore
pressure variation of Fig. 6 at time t⫽2 years after the inception
of pumping. The ‘‘a posteriori’’ c M estimate can obviously be
done only over the compacting depth intervals. Note the satisfactory results obtained from the SD scenario with a distributed
withdrawal for almost 40 m above and below the reservoir. Similarly, the CL scenario with the volume sink exhibits a good correspondence with the actual c M , although the compacting depth
interval is shorter. By contrast, the outcome of the line sink simulations 共i.e., the monitoring well is also a producing well兲 appears
to fluctuate quite significantly with an average c M underestimate
of approximately 30% and locally much more. This is basically
due to the horizontal pressure gradients generated close to the
extraction well, that make the assumption of uniaxial deformation, on which Eq. 共1兲 is based, largely questionable. The effect of
the horizontal pressure gradients is to oppose the vertical compaction more than is done in an oedometer, and this produces a c M
underestimate.
Conclusions
A FE coupled consolidation model has been implemented in an
axisymmetric porous medium to simulate the rock deformation at
the marker scale 共10.5 m兲 due to fluid withdrawal from a heterogeneous reservoir. The analysis is based on two realistic much
detailed lithostratigraphies representative of the Northern Adriatic
sedimentary basin, one consisting mainly of sands interbedded
with thin clayey layers 共SD scenario兲, the other made up of thin
productive sandy layers incorporated into a clayey matrix 共CL
scenario兲. Two pumping schemes are addressed, namely a volume
distributed sink, i.e., the monitoring test hole is far from the producing wells uniformly distributed within the reservoir, and a line
sink, i.e., the monitoring wellbore coincides with an abstraction
88 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004
Fig. 13. Same as Figs. 11 and 12 for Biot coefficient
Fig. 14. Simulated marker response 关from Eq. 共6兲兴 versus depth in CL scenario with a line sink and 共a兲 one depleted layer and 共b兲 two depleted
layers
INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004 / 89
Fig. 15. ‘‘A posteriori’’ c M estimate using simulated marker response and pore pressure drawdown in: 共a兲 SD scenario and 共b兲 CL scenario
well. Typical hydrogeomechanical parameters of the Northern
Adriatic basin are used. The following points are worth summarizing:
1. Depletion of a deep reservoir may induce a rock vertical
expansion in the surrounding porous medium, which can be
detected by RMT.
2. Rock expansion is basically the result of the 3D deformation
of the porous medium, with coupling between medium stress
and fluid flow initially contributing to the process.
3. Hydraulic vertical heterogeneity at the marker scale plays a
most important role on both magnitude of rock expansion
and location of expanding zones. Depending on the lithostratigraphy and permeability contrast, an expansion up to
25% of the expected reservoir compaction is reproduced at
the marker scale.
4. By distinction, a local stiffness contrast 共up to four times兲
and different values for grain compressibility, including the
assumption of incompressible solid grains, i.e., Biot coefficient equal to 1, do not affect significantly the marker response.
The RMT is a promising tool for a realistic evaluation of the
actual mechanical properties of a producing reservoir. However,
to avoid difficulties in the correct use and interpretation of marker
data, the radioactive markers should be installed in a nonproducing wellbore and should span a productive formation made up
mostly from sand, possibly with a thickness equal to or larger
than 10.5 m, i.e., the marker spacing size. In fact, a monitoring
hole which coincides with a producing well can provide an unrealistically small compaction under the influence of the horizontal
pressure gradient, thus leading to a c M underestimate. Similarly,
the compaction of a thin reservoir 共⬃1 m兲 embedded in a clayey
matrix can be partially or totally offset by the expansion of the
surrounding medium included into the monitored depth interval.
Finally, a much detailed knowledge of the real lithostratigraphic sequence of producing heterogeneous reservoirs appears
to be of paramount importance for the most correct use and most
reliable interpretation of the marker response, especially at the
early stage of production.
Acknowledgments
This study has been funded by ENI-E&P and by the Univ. of
Padova project ‘‘CO2 Sequestration in Geological Formations:
Development of Numerical Models and Simulations of Subsurface Reservoirs of the Eastern Po Plain.’’
Notation
The following symbols are used in this paper:
c bm ,c M ⫽ bulk and uniaxial compressibility, respectively;
c br ⫽ grain compressibility;
f ⫽ vector of nodal loads and flow sources or
sinks;
G ⫽ shear modulus of porous medium;
H ⫽ flow stiffness matrix;
h̄ ⫽ average marker spacing size;
K ⫽ elastic stiffness matrix;
K 0 ⫽ ratio between horizontal and vertical soil
stress in undisturbed conditions;
k ⫽ medium hydraulic conductivity;
P ⫽ flow capacity matrix;
p ⫽ vector of nodal incremental pore pressure;
p ⫽ incremental pore pressure;
p ⬘ ,q ⫽ stress invariants;
Q ⫽ coupling matrix;
Q ⫽ ⫽pumping rate;
r̄ ⫽ marker radial distance from symmetry axis;
r,z ⫽ radial and vertical coordinates, respectively;
t ⫽ time;
u ⫽ vector of nodal incremental displacements;
u ⫽ incremental displacements;
␣ ⫽ Biot coefficient;
␤ ⫽ fluid volumetric compressibility;
␥ ⫽ fluid specific weight;
⌬h ⫽ average vertical deformation of marker
interval;
90 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JUNE 2004
⌬p ⫽ pore pressure decline;
⌬t ⫽ time step;
⌬u z ⫽ vertical deformation between two points
10.5 m apart;
␧ V ,␧ z ⫽ volume and vertical strain of porous matrix,
respectively;
␭ ⫽ Lamé constant of porous medium;
␯ ⫽ Poisson ratio of porous medium;
␴ rr ,␴ zz ,␴ ␪␪ ,␴ rz
⫽ components of axisymmetric stress tensor; and
␾ ⫽ porosity.
Subscripts
cl ⫽ referring to clay layers;
max ⫽ maximal value;
r,z ⫽ radial and vertical directions of reference,
respectively;
tb ⫽ referring to layers on top and bottom of
depleted units; and
0,t ⫽ initial and final time.
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