waterflooding under dynamic induced fracture

We also demonstrate that in-duced fracture dimensions can be very sensitive to typical reservoir engineering parameters, such as fluid mobility, mobility ratio, 3D saturation distributio

Abstract

It is well established within the Industry that water injection mostly takes place under induced fracturing conditions Particularly in low-mobility reservoirs, large fractures may be induced during the field life

This paper presents a new modeling strategy that combines fluid-flow and fracture-growth (fully coupled) within the framework of an existing ‘standard’ reservoir simulator

We demonstrate the coupled simulator by applications to five-spot pattern flood models, addressing various aspects that often play an important role in waterfloods: shortcut of injector and producer, fracture containment, reservoir sweep We also demonstrate that in-duced fracture dimensions can be very sensitive to typical reservoir engineering parameters, such as fluid mobility, mobility ratio, 3D saturation distribution (in particular, shockfront position), positions of wells (producers, injectors), and geological details (e.g flow baffles)

The results presented in this paper are expected to also apply to (part of) EOR operations (e.g polymer flooding)

1 Introduction

Water injection will generally result in rapid injectivity decline unless it takes place under induced fracturing conditions This is

illus-trated in Fig 1-2 1-2, comparing matrix injection of fine-filtered seawater (Fig 1 1) with fractured injection of heavily contaminated

production water (Fig 2 2) In the former case, regular acidizations are required to keep up well injectivity (in spite of the high water quality), whereas in the latter case, injectivity remains constant over years (in spite of the low water quality)

However, important risks associated with waterflooding under induced fracturing conditions are related to potential unfavorable areal and vertical sweep These risks can be managed if one has a proper understanding of dynamic induced fracture behaviour as a function

of parameters such as injection rate, voidage replacement, reservoir fluid mobility and reservoir / injection fluid mobility ratio 3

In order to enable building and using such an understanding as part of field development planning and of reservoir management, we developed an ‘add-on’ fracture simulator to our existing in-house reservoir simulator 4

In the past, several attempts were made to address the coupled problem of reservoir simulation and induced fracture growth Common approaches can be grouped into fully implicit simulators (Tran et al.5) where both fluid flow equations and geomechanical equations are solved at the same time on the same numerical grid, and coupled simulators (Clifford et al.6) where a standard, finite-volume res-ervoir simulator is coupled to a boundary-element based fracture propagation simulator Both approaches are not standard and cur-rently not used in the industry mainly because reservoir models need to be purpose-built, and numerical stability is questionable Our approach, as briefly described in 4, uses a ‘standard’ reservoir simulator, thereby enabling reservoir engineers to model induced fracturing around injectors using their ‘standard’ reservoir models (sector, full-field) Moreover, our specific methodology of coupling induced fractures to the reservoir via special connections 4 helped to eliminate most of the numerical instabilities that are generally encountered in the coupled (reservoir flow)-(fracture growth) problem

The current paper presents an important application of coupled reservoir flow and induced fracture growth The focus is on demon-strating how dynamic fracture growth around injectors is largely driven by reservoir engineering parameters It is shown that the de-gree of induced fracture growth / shrinkage in waterfloods depends strongly on oil-water mobility ratio and can vary strongly with

SPE 110379

Waterflooding Under Dynamic Induced Fractures: Reservoir Management and Optimization of Fractured Waterfloods

P.J van den Hoek, R Al-Masfry, D Zwarts, Shell International Exploration and Production B.V., J.D Jansen, Delft

University of Technology and Shell International Exploration and Production B.V., B Hustedt, Shell International

Exploration and Production B.V., and L van Schijndel, Delft University of Technology

Copyright 2008, Society of Petroleum Engineers

This paper was prepared for presentation at the 2008 SP E/DOE Improved Oil Recovery Symposium held in Tulsa, Oklahoma, U.S.A., 19–23 April 2008

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s) Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s) The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members Elec-tronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied The abstract must contain conspicuous acknowledgment of SPE copyright.

time because of changing reservoir saturation distribution (e.g shockfront position) For example, induced fracture growth in an injec-tor can be strongly accelerated at the moment of water breakthrough in nearby producers Once water has broken through, the induced fracture shrinks again These results imply that an optimized waterflood strategy will generally require variable injection rates over the field life in order to prevent jeopardizing sweep by excessive induced fracture growth

The paper is organized as follows Section 2 presents a brief recap of the methodology of 4, and an overview of the pattern flood model system as used in the computations Section 3 presents the results for a representative pattern flood cases with a variety of oil and water mobities and oil-water mobility ratio’s Subsequently, sections 4 and 5 present and discuss the application of these results to

a pattern flood field case Finally, conclusions are given in section 6

2 Methodology

Coupling of reservoir simulation and dynamic fracture growth / shrinkage Our simulator couples a standard, finite-volume

res-ervoir simulator7 to a geomechanical modeling tool The fracture and stress modeling is done using an in-house pseudo-threedimensional fracture simulator8 and a stress computation on the reservoir simulator grid As described in 4, we use a ‘two-way’ coupling strategy, where the fluid flow in the reservoir is influenced by the dynamic fracture propagation and visa versa

Variations of fracture dimensions over time are governed by a fracture propagation criterion that is based on a Barenblatt

condi-tion For each of the fracture tips (length, height upward, and height downward), we evaluate the stress intensity factor (K I) against the

rock toughness (K Ic) The stress intensity factor for a given fracture tip, incorporates poro-elastic and thermo-elastic stress effects (backstress) as well as the fluid pressure in the fracture Fluid flow from the fracture into the formation is further influenced by an ex-ternal filtercake that builds up over time due to the particle content in the injection water

One of the main contributions to describe the effect of the fluid flow on the fracture propagation is the description of the reservoir stress over time We calculate the stress field from the discrete pressure and temperature field on the grid that is used for the reservoir simulations

Fracture Representation We introduce a dynamically growing planar fracture in the reservoir simulation grid by an explicit

defini-tion of a fracture grid block For simplicity, we convert an unused block in the reservoir grid to the fracture grid block, such that the total number of grid blocks remains unchanged though a dynamic fracture is added to a given reservoir model

The approach using a special fracture grid block enables one to model induced fractures which are arbitrarily oriented with respect

to the (local) reservoir grid This is a clear advantage over methods that make use of modifying grid block transmissibilities The pla-nar fractures can be oriented arbitrarily which includes tilted or horizontal fractures

The fracture grid block is connected to the main reservoir grid by special connections The area intersected by the fracture grid block and the reservoir grid blocks controls the amount of liquid that flows from the fracture into the surrounding matrix The size of the fracture block is modified over time which is governed by the growing or shrinking of the fracture Special attention is taken for the pressure and flow calculations for the reservoir gridblocks that contains the fracture tip If a fracture tip is closer to the neighboring gridblock than the centre of the gridblock, the fracture represents a high conductive flow-path to the neighboring grid block As it was shown by Dikken and Niko9, this effect may be captured by allowing a smoother transition of the pressure and flow profile when the fracture grows from one gridblock into the next

Fracture Propagation Criterion At every time step during a coupled simulation we match the actual fracture size (fracture

half-length, height upward and downward) to the reservoir pressure- and stress-field, such that a balance of the pressure inside the fracture and the in-situ minimum stresses around the fracture is achieved within a pre-defined error margin

In order to determine whether a fracture grows, shrinks (partially closes) or remains stationary during a given time step, we

incor-porate a fracture propagation criterion based on a stress intensity factor (K I ) evaluation of all the fracture tips We evaluate K I for each

fracture tip with respect to the rock toughness (K Ic) This leads to the following fracture propagation criterion:

1 K I > K Ic : Fracture tip extension until K I = K Ic

2 K I < 0: Fracture tip shrinkage until K I = 0

3 0 ≤ K I≤ K Ic: No fracture tip extension / shrinkage

3 Sector model description

The current model study focuses on the impact of reservoir engineering parameters on dynamic behaviour of induced fractures This behaviour is demonstrated for a simple five-spot pattern flood without aquifer influx The reason for choosing a pattern flood sector model is that induced fractures will be particularly important in low-mobility reservoirs, where a pattern-type development will be often the development concept of choice

All calculations carried out as part of this study were done under isothermal conditions

Reservoir Model and Dimensions To model and study induced fractures, a quarter element of a 5-spot pattern (repeated pattern) was

selected A 5-spot pattern unit cell contains one injection well and one producer Periodic boundary conditions at the model bounda-ries and special connections were used to enable the simulation pattern that contains fracture growth The principle behind the

peri-SPE 110379 … 3

odic boundary conditions is the symmetry in the pattern drive If we ‘cut’ out a pattern at an arbitrary location, the flow at top and

bot-tom are equal (Figure 3) In other words, the flow out of the pattern at one side flows into the neighbouring identical pattern at the

opposite side (i.e mode: flow out = flow in at opposite side) Therefore a special connection is created to allow flow to move in this

‘cyclic’ nature

Table 1 presents the model dimensions and properties used for the 5-Spot pattern The reservoir properties are uniform across the

en-tire model Furthermore, the data in Table 1 were kept the same for all simulations, even though at later stage the grid refinement was introduced in order to gain close perspectives of specific parameters during simulation

Injection and reservoir fluid properties Table 2 presents the reservoir and injection fluid properties used for this model Water and

oil viscosity were kept independent of pressure However, injection water and oil viscosity were varied as part of the sensitivity study (see below) A gas phase was included in the model, but the pressures are kept above bubble point so no free gas is present in the sys-tem

Relative Permeability and Mobility Table 3 presents the relative permeability data for the base case scenario The bulk of this

study was geared toward studying the impact of the relative permeability and fluid mobility on induced fracture growth and overall

reservoir behaviour Note: For simplicity, we kept the reservoir characteristics fairly basic, for example it was assumed that there is no

capillary pressure effect, in order to prevent an even more complex interference between the fluid flow and the fracture propagation This means that there is no transition zone and hence the capillary pressure is fixed to 0 (i.e no capillary pressure curves)

Well location and Well Properties Table 4 presents the location of the two wells within the box model, perforations interval and

other key data that were used to perform the study

Rock Mechanics and Fracture properties During the reservoir simulation, the fracture was fixed at an ‘unfavorable’ orientation, i.e

an orientation at which it grows directly from the injector to the producer (orientation= 45º relative to unit cell boundary) Table 5

presents the rock mechanics and fracture geometry data used for the simulation and sensitivities

4 Results

Unit oil-water mobility ratio As pointed out in 3, dynamic behaviour of induced fractures depends very much on oil-water mobility ratio (see also below) Broadly speaking, for constant injection rate, a favorable oil-water mobility ratio will result in growing frac-tures over time, whereas an unfavorable oil-water mobility ratio will lead to fracture shrinkage (after initial growth) and, eventually, potemtial complete fracture closure 3

However, as will be shown below, even for unit mobility ratio, induced fractures can grow or shrink considerably over time,

depend-ing on the change of the shock front position in the reservoir

Straight-line relative permeabilities In this case the shock front is characterized by a piston with equal effective permeability ahead

of and behind the front Figure 4 presents the results of the simulation using the input data of Tables 1-5, but with n o = n w = 1 The injection well was placed on a rate constraint (Q = 4000 m3/d) while the production well was put on bottom hole pressure constraint

(BHP = 20 bar, see Table 4) The fracture orientation was 45 degrees with respect to the unit cell boundary (see Table 5) The

frac-ture growth was constrained to 25 m in height (25 m being the distance between the fracfrac-ture initiation point to the top or bottom of the reservoir) and 560 m in length (565 being the total distance between injector and producer)

It can be seen that water injection starts with a transient phase under fracturing conditions (between time 0 and time t1) where the flow and pressure profile in the reservoir are built up (up to 99 bar) For the same reason, also the average injection pressure rises to about

123 bars As an effect, the oil production increases to a maximum of about 3800 m3/day In order to accommodate sufficient leak-off area for the injection water, during this transient phase the fracture grows relatively fast to a length of 52 m and upward and downward height of 25 m (which is the maximum allowed height)

After the initial transient phase (i.e from time t 1 onwards), stabilized conditions prevail and the induced fracture stops growing as can

be seen from Fig 4 These results are qualitatively in line with earlier work 3,9

General (‘non-straight-line’) relative permeabilities In this case the shock front is characterized by a leaky piston with varying

ef-fective permeability behind the front (Fig 5) As a result, the overall “fluid throughput capacity” of the reservoir will change over

time depending on the exact position of the shockfront For constant injection rate under induced fracturing conditions, the fracture will ‘compensate’ for the changing reservoir throughput capacity over time –i.e a lower reservoir throughput capacity results in a lar-ger induced fracture and vice versa

Figure 6 illustrates the above for the relative permeability curves of Fig 5 (which corresponds to the base case as defined by Tables 1-5) As indicated in the figure, the process of dynamic growth and shrinkage of the induced fracture can be divided in four different

periods

During the first period (‘transient pressure build-up’) the fracture ‘rapidly’ grows to a steady-state size under the influence of transient

fluid flow This period was also observed for the case of straight-line relative permeabilities (Fig 4) The second period (‘dry oil

pro-duction’) is characterised by a ‘stationary’ fracture because the overall reservoir throughput capacity does not change very much dur-ing this period Durdur-ing the third period (‘water breakthrough P1’), the shock front enters the near-wellbore area around producer P1,

where pressure gradients are comparatively large Because the effective permeability behind the shockfront is lower, this will tend to increase the ‘high’ pressure gradient around the producer, resulting in a lower overall reservoir throughput capacity Consequently, the induced fracture will grow until it reaches a maximum length After water has broken through on all sides in P1, the effective perme-ability in the area around P1 will start to gradually rise as a result of rising water saturations, and consequently the fracture will start to shrink again (‘increasing watercut P1’) This shrinkage will be much slower than the initial growth during water breakthrough because the water saturation only slowly increases behind the shockfront

The above discussion illustrates how induced fractures in waterfloods can rapidly grow in response to a shockfront passing an area of high pressure gradient (e.g near-wellbore area of producer) This only applies to immiscible displacement (‘non-straight-line relative permeabilities’) Therefore, in miscible tertiairy (e.g polymer) floods, this effect is only expected to play a role at the second shock front between oil and mobilized connate water, but not at the shockfront between injectant and mobilized connate water However, in the latter case fracture growth / shrinkage associated with non-unit mobility ratio between injectant and mobilized connate water (as discussed below) will definitely play a role

Grid refinement The results of Fig 6 are very much steered by the high pressure gradient around producer P1 Therefore, one may

argue that these results are in essence a reservoir grid effect In order to investigate this, we applied various degrees of local grid

re-finement around the producer (Fig 7), plus a global grid rere-finement Results are shown in Fig 8 As can be seen from this figure,

lo-cal grid refinement around the producer does lead to different results, but global refinement of the entire grid leaves the results more

or less unchanged As further detailed in Appendix A, the explanation for this is that local grid refinement results in a modification of

the Peaceman solution around the producer which needs to be properly catered for in the reservoir simulator Because global grid

re-finement leads to similar results as no rere-finement, we believe that the results of Fig 6 are real and not due to gridding artifacts

Dependency on injection rate Fig 6 also shows the fracture growth dependency on injection rate It should be noted that, as above,

all these calculations were carried out for constant producer BHP=20 bar Consequently, for all different injection rates the reservoir

voidage replacement ratio is equal to one (except for the duration of the first transient flow period) As can been seen in Fig 6, the

phenomenon of rapid fracture growth upon water breakthrough in the produces as discussed above becomes less pronounced for very low and for very high injection rates

Dependency on Corey exponent The previous discussions around Fig 6 strongly suggest that for higher Corey exponents, the effect

of temporary fracture growth acceleration upon water breakthrough in the producer will be more pronounced Figure 9 shows that this

is indeed the case, and that the effect can be quite pronounced

Dependency on producer BHP Figure 10 shows the dependency of induced fracture length on BHP of the producer P1 (assuming

that this BHP can be fixed, for example, by using an artificial lift pump) As can be seen from this figure, a higher producer BHP will result in larger fractures, with potentially significant differences The explanation for the longer fractures is that for the same injection and gross production rate, a higher producer BHP will result in a higher average reservoir pressure within the pattern Via the poroe-lastic backstress this will result in a higher fracture pressure as well, but the latter increase is smaller (typically by a factor 0.6-0.8) than the increase in reservoir pressure As a result, the difference between fracture pressure and reservoir pressure becomes smaller Because it is this pressure difference that drives the water from the injector into the reservoir, the induced fracture will respond to a

smaller pressure difference by extending itself This is what is reflected in Fig 10

From the above result, we can derive the general statement, that in low-mobility reservoir without aquifer, the risk of excessive in-duced fracturing from injectors can be rein-duced by minimizing the BHP of adjacent producers

Nonunit oil-water mobility ratio In ref 3 it was argued that for constant injection rate, a favorable oil-water mobility ratio will

re-sult in growing fractures over time, whereas an unfavorable oil-water mobility ratio will lead to fracture shrinkage (after initial growth) and, eventually, to possible complete fracture closure Because the methodology of 3 was semi-analytical, a number of simpli-fying assumptions had to be made 3 However, with our coupled fracture-reservoir simulator 4 we were able to confirm the qualitative results of 3 An illustration of this is presented in Fig 11 for unfavorable oil-water mobility ratio (endpoint M=3) This figure shows

the computed fracture dimensions (length, upward and downward height) , plus oil and water production as a function of time

The results of Fig 11 can be understood using the same concepts that were presented above in connection with Fig 6 We can

distin-guish five different periods: 0-t 1 , t 1 -t 2 , etc (see Fig 11) As before (Fig 6), the first two periods can be identified with the transient pressure build-up and dry oil production, respectively In the third period (t 2 -t 3), water breaks through in P1 and although the lower

effective permeability in the intermediate saturation zone will tend to enhance fracture growth (as in Fig 6), this is more than

com-pensated for by the mobility increase associated with oil “replacement” by water Therefore, the net result is a fracture shrinkage upon water breakthrough in P1 at time t 3 Following this initial shrinkage, the fracture length stabilizes (Fig 11) until at t 4 water breaks through in producer P2, which again leads to further fracture shrinkage and subsequent stabilitization in length

Impact on production and recovery In ref 3 it was argued that induced fractures generally have two opposite effects on recovery:

(1) On the plus side, induced fractures can significantly improve injectivity by by-passing near-wellbore damage etc – in other words

SPE 110379 … 5

they significantly improve voidage replacement capacity, while (2) on the minus side, induced fractures will have a negative impact

on areal sweep From an economic perspective, one needs to find an optimum between (1) and (2)

For the simple conceptual 5-spot pattern flood as presented above, the above points are illustrated in Fig 12 (compare also Fig 6) It

can be seen that there is an optimum injection rate balancing enough voidage replacement on one hand without jeopardizing areal sweep on the other hand The optimum in this case corresponds to an injection rate of around 4500 m3/d, for which the induced

frac-ture length is about one third of the injector-producer spacing (Fig 6)

Further optimization is possible by allowing for variable injection rates over time In such a scheme, rates will be reduced at moments that significant induced fracture growth is expected as a result of low ‘reservoir throughput capacity’, and vice versa In particular, for favorable oil-water mobility ratio, this will result in an overall reduction of injection rate over the field life, whilst for adverse mobility ratio, it will result in an increase in injection rate 3,4 We found that especially when waterflooding ‘medium-heavy’ oil reservoirs (vis-cosity of the order 100 cp), ‘optimized’ injection rates will increase significantly during the first few years of field life 4,11

5 Discussion

The results presented in this paper demonstrate that the size of induced fractures in waterfloods (and possibly also EOR operations) is expected to be very dependent on typical reservoir engineering parameters, such as fluid mobility, mobility ratio, 3D saturation distri-bution (in particular, shockfront position), positions of wells (producers, injectors), and geological details (e.g flow baffles) During the field life of a waterflood, induced fractures can grow but also shrink over time (or a combination of both)

Although the present (conceptual) study only addresses fracture growth / shrinkage in the horizontal direction and its impact on areal sweep, in the general case fractures will also grow vertically with a potential risk to break through caprock layers into ‘unwanted’ in-jection horizons This risk can also be addressed by our simulator 4 but is outside the scope of the study presented here It is clear from the results presented in this paper that also the risk of vertical fracture noncontainment will be impacted by the reservoir engineering parameters highlighted above (on top of the ‘usual’ set of rock mechanical parameters, such as in-situ stress contrast between different geological formations)

A proper field development and field management strategy will have to take the above points into consideration Traditionally, this was achieved by a strategy to ‘avoid fracturing at all costs’ whereby it was considered sufficient to inject a few hundred psi below fracturing pressure However, systematic analysis of Industry-wide field data during the past years has led to the insight that, particu-larly in low-mobility reservoirs, injection without inducing fractures is practically impossible Therefore, rather than try to avoid in-duced fracturing, one will have to accept that they are there and try to optimize their ‘use’ by minimizing adverse areal and or vertical sweep

The above means that upfront, one needs to evaluate the risk of excessive induced fracturing for different geological models and dif-ferent field development scenarios (field development planning) But also during operation of the field, as increasingly more data be-come available, the fracturing risk needs to be regularly updated (reservoir management) The type of business decisions that are likely to be impacted are, amongst others: well (injector / producer) location, well type, well completion, producer / injector ratio, in-jection rate over time, inin-jection water quality and temperature, type of surveillance, etc

6 Conclusions

The results presented in this paper can be summarized as follows:

• The degree of induced fracture growth / shrinkage in waterfloods depends strongly on oil-water mobility ratio and can vary

strongly with time

• In case of a favorable oil-water mobility ratio, the induced fracture grows over time when more of the oil-in-place is replaced

by injection water

• Conversely, in case of an unfavorable oil-water mobility ratio, the induced fracture shrinks over time (after initial growth)

when more of the oil-in-place is replaced by injection water

• Fracture growth can be strongly accelerated at the moment of water breakthrough in nearby producers Once water has

bro-ken through, the induced fracture shrinks again

• The above implies that an optimized waterflood strategy requires variable injection rates over the field life in order to prevent

jeopardizing sweep by excessive induced fracture growth Continuous monitoring of induced fracture length and height dur-ing field operation will be required to enable proper and timely adjustment of injection rates to their ‘optimimum’ values

• In low-mobility reservoir without aquifer, the risk of excessive induced fracturing from injectors can be reduced by

minimiz-ing the BHP of adjacent producers

Nomenclature

c = Compressibility

∆x = Gridblock size

φ = Porosity

h = Reservoir height

k = Permeability

K I = Stress intensity factor

K Ic = Fracture toughness

M = Oil-water mobility ratio

µ o = Oil viscosity

µ w = Water viscosity

n o = Corey exponent for oil relative permeability

n w = Corey exponent for water relative permeability

p = Pressure

Q = Rate

r0 = Peaceman radius

Acknowledgment

The authors are grateful to Shell Internationale Exploration and Production B.V for permission to publish this work

References

1 Sharma, M.M., Pang, S., Wennberg, K.E., and Morgenthaler, L Injectivity decline in water injection wells: An offshore Gulf of Mexico case

study SPE 38180 (1997)

2 Van den Hoek, P.J., Sommerauer, G., Nnabuihe, L., and Munro, D Large-Scale Produced Water Re-Injection Under Fracturing Conditions in

Oman, ADIPEC 0963, presented at the 9th Abu Dhabi International Petroleum Exhibition and Conference held in Abu Dhabi, U.A.E., 15-18

October 2000

3 Van den Hoek, P.J Impact of Induced Fractures on Sweep and Reservoir Management in Pattern Floods SPE 90968 (2004)

4 Hustedt, B., Zwarts, D., Bjoerndal, H.-P., Masfry, R., and van den Hoek, P.J Induced Fracturing in Reservoir Simulations: Application of a

New Coupled Simulator to Waterflooding Field Examples SPE 102467 (2006)

5 Tran, D., Settari, A and Nghiem, L.: “New Iterative Coupling Between a Reservoir Simulator and a Geomechnics Module”, SPE 78192 (2002)

6 Clifford, P.-J., Berry, P.J., and Gu, H.: “Modeling the Vertical Confinement of Injection-Well Thermal Fractures”, SPEPE (Nov 1991), 377

7 Por, G.J., Boerrigter, P., Maas, J.G., de Vries, A A Fractured Reservoir Simulator Capable of Modeling Block-Block Interaction, SPE 19807 (1989)

8 van den Hoek, P.J New 3D Model for Optimised Design of Hydraulic Fractures and Simulation of Drill-Cutting Reinjection, SPE 26679 (1993)

9 Dikken, B.J and Niko, H.: “Waterflood-Induced Fractures: A Simulation Study of Their Propagation and Effects on Waterflood Sweep Effi-ciency”, SPE 16551 presented at the 1987 Offshore Europe Conference, Aberdeen, Sept 8-11

10 Peaceman, D.W Interpretation of well-block pressures in numerical reservoir simulation Soc Pet Eng J June 1978, 183-194

11 Sæby, J., Bjoerndal, H.P., van den Hoek, P.-J.: “Managed Induced Fracturing Improves Waterflood Performance in South Oman”, IPTC

10843 presented at the 2005 IPTC Conference and Exhibition, Doha, Nov 21–23

Appendix A Local grid refinement and Peaceman’s solution

Here, we follow the original derivation of Peaceman10 for square grids in order to derive a general expression of the ‘modified’ Peace-man radius for local (square) grid refinement (LGR) around a well To our knowledge, this problem was never published before

The grid layout is given in Fig A1 for the case of 3x LGR Following 10, we first derive a simple analytical expression for the modi-fied Peaceman radius corresponding to 3x LGR, ignoring the effect of non-neighbouring gridblocks Next, we use numerical calcula-tions to improve this solution into a generalized expression for local grid refinement (square grids) Extrapolation of this result to rec-tangular grids will be straightforward

Applying Darcy’s law to Fig A1 and assuming incompressible flow, we obtain

kh

Q p

p x

p p x

p p x

p p x

p p x

p p

x

p

4 1 3 2 2

2 2 1 3 2 2 0 2

1 1

2 0

−

−

∆

− +

∆

−

−

∆

−

=

∆

−

−

∆

−

(A1)

with the usual meaning of the symbols As a next step, we follow Peaceman by putting

SPE 110379 … 7











 ∆

≈

−

0 3

0

3 ln

x kh

Q p

p

π

µ

(A2)

Combination of (A1) and (A2) then leads to an estimate of the modified Peaceman radius r 0 for 3x LGR:

exp

x

r

which is lower than the Peaceman value for square grids without LGR (which is equal to ca 0.2 10)

Next, we conducted numerical experiments (as in 10) for varying degrees of LGR and grid size The results are shown in Fig A2

Here, the parameter ‘n-refine’ is defined in such a way that the ‘large’ square gridblock surrounding the well (size 3 *∆x in Fig A1) is

subdivided into (n-refine)x(n-refine) gridblocks, each with defined size ∆x (i.e Fig A1 depicts the case of n-refine = 3)

As can be seen from Fig A2, in the absence of LGR (i.e n-refine = 1), the classical Peaceman solution is obtained, but with

increas-ing degree of LGR, we obtain a modified Peaceman radius which asymptotically approaches a value of ca 0.15 times the size of the gridblock surrounding the wellbore

The fact that this is lower than the unrefined Peaceman radius can be interpreted that LGR around a wellbore introduces an extra ‘nu-merical’ pressure drop at the boundary of refined and nonrefined grids, which has to be catered for

_

Table 1 5-Spot pattern dimensions and reservoir properties Description Symbols Quantities Units

Reservoir Compressi-bility

Original Model Top Depth/Height

Oil Water Contact-below reservoir (No transition zone)

Table 2 Reservoir and injection fluid properties Description Quantities Units

Table 3 Base case Relative Permeability Data

Description Quantities

Table 4 Wells dimensions and properties Description Symboles Quantities Units

Perforation intervals for Injector and Producer

(i.e full reservoir height)

Injection and production wells depths- center

height of reservoir

Table 5 Rock mechanics and fracture geometry data

SPE 110379 … 9

Fig 1 Observed injectivity decline for fine-filtered seawater matrix injection well in Gulf of Mexico (after 1 )

0 5000 10000 15000 20000 25000

May-96 Dec-96 Jun-97 Jan-98 Jul -98 Feb-99

Observed injectio n rate (m3/d) Sim ulated injection rate (m 3/d) Observed inject ion pressure (kPa) Simu lated injection pressure (kPa)

Fig 2 Observed and simulated injection performance for Middle East water disposal well (after 2 )

Injector Producer

80 0

5 m Injector

Producer

80 0

5 m

Fig 3 Five-spot element with one producer and one injector

t1

t1

t1

Figure 4.Results of Unit Mobility Scenario with straight-line relative permeabilities (n o = n w =1)

Water and Oil Mobility ratios

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

sw

Mo Total Mobility

Figure 5 Water and Oil mobility-Total mobility (n o =n w = 1.3) scenario