Peripheral water injection efficiency for material balance applications

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

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Peripheral water injection efficiency for material balance applications Leonardo Patacchini Abu Dhabi Marine Operating Company, United Arab Emirates

A R T I C L E I N F O

A BS T RAC T

Keywords: Peripheral waterflood Material balance Injection efficiency Streamlines

The simple approach consisting of including injected water directly into the reservoir material balance equations is not appropriate to model peripheral injection, as it does not account for water lost to the aquifer as well as time required for pressure to diffuse to the reservoir boundary. Based on this observation, the authors have extended the van Everdingen and Hurst (1949) unsteady state edge aquifer model to account for peripheral sources. Taking advantage of the pressure diffusion equation linearity and problem symmetries, for simplified circular and linear geometries the problem can be cast as one-dimensional regardless of the number and position of peripheral injectors. Solutions are reported in the form of a tabulated cumulative efficiency function, defined as the amount of water having reached the reservoir boundary owing to the presence of a source injecting at unit rate, vs. time. Superposition principle can then be used to calculate time-dependent water influx for an arbitrary number of injectors and injection history. Solving the two-dimensional problem further provides the lateral influx distribution, and shows that pressure support efficiency as defined in this work, relevant to material balance applications, is conceptually different from transport efficiency provided by streamlines analysis. The latter is indeed unable to single out the individual contribution of a specific injector to reservoir voidage replacement from that of its neighbors and the aquifer itself.

We length3 >L , >C

Nomenclature Symbol Dimension Description ϕ k ct h xa wo xinj ro ra θo rinj UL,UC t ψ ψinit ψo 7 qe

length2 pressure−1 length length length length length length length length3/ pressure time pressure pressure pressure pressure length3/ time

Aquifer porosity Aquifer permeability Total aquifer compressibility Aquifer thickness Linear aquifer length Linear aquifer width Injection position in linear geometry Circular reservoir radius Circular aquifer radius Circular aquifer encroachement angle Injection position in circular geometry Linear, circular aquifer constants (Eqs. (7) and (12)) Time Water potential (Eq. (3)), simply referred to as “pressure” Initial aquifer potential Reservoir boundary potential Reference pressure Reservoir influx rate

8L , 8 C

qinj Winj

length3/ time length3

,L , , C - L , -C

ψ* property*L

property*C

Reservoir cumulative influx Linear, circular cumulative aquifer rate functions (Eqs. (6) and (11)) Linear, circular instantaneous aquifer rate functions (Eq. (18)) Injection rate (at subsurface conditions) Cumulative injection (at subsurface conditions) Linear, circular cumulative injection efficiency functions (Eqs. (32) and (33)) Linear, circular instantaneous injection efficiency functions (Eq. (34)) Potential nondimensionalized by 7 Nondimentionalized property (length by xa, time by τL (Eq. (8))) Nondimentionalized property (length by ro, time by τC (Eq. (13)))

1. Introduction Flow in a petroleum reservoir is essentially inertia-free (i.e., pressure, capillary, and gravity forces are in equilibrium with viscous

http://dx.doi.org/10.1016/j.petrol.2016.10.032 Received 22 December 2015; Accepted 17 October 2016 Available online xxxx 0920-4105/ © 2016 Elsevier B.V. All rights reserved.

Please cite this article as: Patacchini, L., Journal of Petroleum Science and Engineering (2016), http://dx.doi.org/10.1016/j.petrol.2016.10.032

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

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Fig. 1. Areal view of a reservoir with edge aquifer drive, produced in secondary mode through both peripheral and pattern injection. Intuitively, it is expected that the peripheral injection line and the western part of the injection ring will have a low efficiency due to their distance and the presence of sealing faults, respectively. The meaning of “low” will be detailed in the paper.

Fig. 3. Areal view of linear (a) and circular (b) aquifer geometries. The constant “terminal pressure” aquifer problem disregards injection, and is obtained using a constant reservoir edge pressure different from the initial aquifer pressure. The “peripheral injection” problem is obtained using a constant reservoir edge pressure equal to the initial aquifer pressure, accounting for an injection line or ring. In both cases, no flow is allowed at the outer boundary. The total influx is the sum of the solutions of these two independent problems.

Fig. 2. Sketches of edge (a) and bottom (b) reservoir-aquifer sections, after Coats (1962). We will see that the arbitrary separation between reservoir and aquifer (horizontal dashed line), typically taken as the oil-water contact in the literature treating aquifer influx, has a strong impact on the calculation of peripheral injection efficiency. Only edge aquifers are considered in this paper.

Table 1 First three solutions to Eq. (15).

forces (Bear, 1972)), hence is governed by conservation of mass; this is the basis of modern reservoir simulation (Aziz and Settari, 1979), as well as more classic but still of paramount importance material balance calculations (Schilthuis, 1936; Dake, 1978; Wang et al., 1992; Petroleum Experts, 2012). In the latter, the reservoir (see the schematic illustration in Fig. 1) is approximated by a single tank with uniform pressure and oil/gas compositions. By relating the difference between produced and injected volumes to changes in pressure, it is possible to estimate properties such as original oil in place or gas cap size, as well as the aquifer strength; material balance can also be used to predict future pressure and primary production, especially for gas reservoirs. Single-tank calculations apply provided the reservoir is well connected throughout, and the characteristic time of pressure variation (e.g., duration of primary depletion) is longer than the characteristic time of pressure diffusion between wells. For large reservoirs, the latter condition typically requires drainage and injection points to be evenly distributed. Aquifers are not included in the “reservoir” definition, which is limited to the hydrocarbon-bearing volume of rock. They can indeed be of large extent and only communicate with the reservoir through a limited surface (edge or bottom, see Fig. 2); their response to variations of reservoir pressure is therefore not instantaneous, and different models have been developed to approximate such response. These can be separated in two categories: unsteady state (USS) and pseudo-steady state (PSS). USS models provide a solution to the full problem of pressure diffusion in the aquifer, typically considering idealized circular or linear geometries. The advantage is that for edge aquifers considered in this paper, if the system thickness “h” is small

ra*

a1

a2

a3

1.2 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 50.0

7.5667 2.8899 1.3608 0.6256 0.3935 0.2824 0.2181 0.1765 0.1476 0.1264 0.1103 0.0465 0.0158

23.469 9.3448 4.6459 2.3040 1.5266 1.1392 0.9075 0.7534 0.6437 0.5616 0.4979 0.2318 0.0879

39.214 15.660 7.8142 3.8954 2.5908 1.9392 1.5486 1.2884 1.1027 0.9636 0.8554 0.4016 0.2189

compared to its extent and the pressure is uniform at the reservoiraquifer boundary, the problem can be cast as one-dimensional. Van Everdingen and Hurst (1949) provided semi-analytic USS solutions in circular geometry through the use of Laplace transforms, for the constant “terminal pressure” and “terminal rate” cases (i.e., constant reservoir boundary pressure and constant aquifer influx). These are expressed as infinite summations of exponentially decaying terms involving Bessel functions, for which the authors provide convenient tabulations. Solutions in linear geometry can be obtained from the former in the limit of aquifer radius approaching reservoir radius, or from mathematically analogous problems (Carslaw and Jaeger, 1959); these are then expressed as infinite summations of 2

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Fig. 4. Pressure profiles and water influx (instantaneous and culumative) for a linear * = 0 , following a step reservoir pressure signal Δψ * = 1. In (b), the aquifer with ψinit

Fig. 5. Pressure profiles and water influx (instantaneous and culumative) for a circular * = 0 , following a step reservoir pressure signal Δψ * = 1. In aquifer with ra* = 5 and ψinit

vertical dotted line indicates the approximate onset of PSS given by Eq. (22), and the logarithmic slope is from the “m=1” term in Eq. (9).

(b), the vertical dotted line indicates the approximate onset of PSS given by Eq. (23), and the logarithmic slope is from the “m=1” term in Eq. (14).

exponentially decaying terms involving trigonometric functions (Leung, 1986). PSS models neglect early transients, and assume that the aquifer influx is proportional to the difference between average aquifer pressure and reservoir boundary pressure, the proportionality constant being analogous to a well's productivity index. The concept has been proposed by Fetkovich (1971), and later formalized and corrected for slight inaccuracies by Leung (1986). Simpler models, such as the Schilthuis steady-state or the small pot, are limiting cases of the former (Dake, 1978). In order to treat transients in reservoir boundary pressure (or rate), it is necessary to resort to superposition, which is computationally demanding. The Carter-Tracy model (Carter and Tracy, 1960), based on the approximation that the cumulative influx is a linear function of time, is today the preferred method for analytic aquifer description in most commercial or research reservoir simulators (e.g., Schlumberger, 2012; Landmark, 2014; Moncorgé et al., 2012, …). UTCHEM (2013) uses another simple model, originally developed to estimate reservoir heat losses to over/underburdens (Vinsome and Westerveld, 1980). Leung developed a fast convolution method (FCM) enabling solution of the original van Everdingen and Hurst problem in the presence of transients without resorting to superposition (Leung, 1986), but the authors are not aware of publicly available software where such method is implemented. Water injection in traditional material balance calculations is assumed to contribute instantaneously to the reservoir pressure, which might be reasonable for dispersed injection (e.g., pattern injection in Fig. 1) but not necessarily for peripheral injection if operated far from the oil-water contact (OWC) (Fetkovich, 1971; Bonet and Crawford, 1969). A fraction of the injected water is indeed lost to the aquifer, and pressure provided by the remaining fraction takes time to diffuse to the

Fig. 6. Water influx for a circular aquifer with ra* = 5, following a reservoir pressure signal given by Eq. (28) with τs = 2τC .

reservoir. Here and in the rest of the paper, peripheral influx is sometimes referred to as “peripheral pressure support”, since adding material to a compressible system at constant temperature and volume (in our case, the bulk reservoir volume) is equivalent to increasing its pressure. The problem of finding the efficiency of a peripheral water injection scheme and its time-dependence is conceptually identical to the “aquifer problem” discussed above, with the addition of source terms representing the injectors. By linearity of the single-phase/singlecomponent pressure diffusion equation, and for constant injection rates, the total water influx to the reservoir can schematically be written as

∑

total influx = aquifer influx + i∈

3

peripheral injectors

efficiencyi × inj. ratei , (1)

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Fig. 7. In linear geometry, considering the injection as spread over a line parallel to the reservoir boundary (a), or operated with a single injector located at the same distance (b), yields different water influx profiles at the reservoir boundary, but the same total influx qe regardless of the aquifer width; inwards (qin) and outwards (qout) fluxes are identical as well. This property also holds in circular geometry.

where the aquifer influx and the individual peripheral injectors' efficiencies are independent from each other. Further taking advantage of the problem symmetries for simplified linear and circular geometries, individual injector's efficiencies can be obtained by solving a one-dimensional problem where the injector is replaced by a line/ring located at the same distance from the reservoir boundary. The full USS solution in linear geometry for a constant injection rate has been presented by Bonet and Crawford in a paper that unfortunately received little echo (Bonet and Crawford, 1969); to the best of our knowledge however, no such solution in circular geometry has been published. The purpose of this paper is to bridge this gap; in particular, a cumulative injection efficiency function will be computed and tabulated vs. dimensionless time for a wide range of dimensionless injector's distances to the reservoir and aquifer sizes. Here, the cumulative efficiency function is defined as the fraction of water injected at unit rate from a single source effectively reaching the reservoir boundary; superposition principle can then be used to calculate time-dependent water influx for arbitrary injection history and aquifer dimensions. Because the peripheral injection problem is an extension of the aquifer problem, the paper starts in Section 2 by a review of the latter in order to introduce important concepts and notations, before treating the former in Section 3, one-dimensionally. Solving the two-dimensional problem further provides the lateral distribution of influx, and shows that “pressure support” efficiency as defined in this work, relevant to material balance applications, is conceptually different from “transport efficiency” provided by streamlines analysis (Thiele and Batycky, 2006). In particular, the latter is unable to single out the individual contribution of a specific injector to reservoir voidage replacement from that of its neighbors and the aquifer itself. This important point will be discussed in Section 4.

Fig. 8. Pressure profiles and water influx (instantaneous and cumulative) for a linear * = 0.2 . * = 0 , following a step injection signal q = UL 7 /τL located at xinj aquifer with ψinit inj In (b), the vertical dotted line indicates the approximate onset of aquifer PSS given by Eq. (22), and the logarithmic slope is from the “m=1” term in Eq. (37).

2. Review of the aquifer problem 2.1. Step reservoir signal USS aquifer models are based on the linear pressure diffusion equation for weakly compressible fluids in homogeneous, isotropic porous media

ϕct

∂ψ k 2 − ∇ψ=0, ∂t μ

(2)

where ϕ and k are the porosity and permeability, μ the water viscosity, ct the sum of water and pore compressibilities, and ψ the water potential given by Bear (1972)

⎡ ψ (p , z ) = ρ0 ⎢ ⎣

∫p

p

0

⎤ ds + g ·(z − z 0 )⎥ . ρ (s ) ⎦

(3)

In the above definition, where ρ stands for the water density, the z-axis is oriented upwards and aligned with the gravity acceleration g; p0 and z0 are arbitrary, independent reference pressure and depth, and ρ0 is the density evaluated at p0. To first order in ρ / ρ0 − 1, ψ (p , z ) can be interpreted as the pressure at datum depth z0, i.e., a pressure corrected for gravity; in the rest of this paper, in order to ease references to previous work where gravity was not explicitly considered, we will refer to ψ as simply the “pressure”. The key approximation is to consider the reservoir withdrawal to be homogeneous enough such that the pressure along the reservoiraquifer boundary can be taken as uniform, effectively yielding a one4

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Fig. 9. Pressure profiles and water influx (instantaneous and cumulative) for a circular * = 0 , following a step injection signal q = UC 7 /τC located at aquifer with ra* = 5 and ψinit inj

Fig. 10. Time required to reach an instantaneous efficiency of 10 to 90% as a function of injector location in (a) linear geometry, and (b) circular geometry with ra* = 5. The thin

* = 2 . In (b), the vertical dotted line indicates the approximate onset of PSS given by rinj Eq. (23), and the logarithmic slope is from the “m=1” term in Eq. (40).

black lines correspond to the estimates of Eq. (42).

dimensional problem when the aquifer thickness “h” is small compared to its extent. In practice, material balance codes consider this pressure to be the tank reservoir pressure. Note that the bottom aquifer problem, not considered in this paper, requires an additional dimension (Coats, 1962). Solving Eq. (2) for arbitrary aquifer shape requires numerical methods, hence analytic or semi-analytic models consider idealized one-dimensional linear and circular geometries, as depicted in Fig. 3. The outer boundary condition (BC) is no-flow, while the inner (or terminal) BC is dictated by the reservoir behavior; due to linearity of Eq. (2) however, solution for arbitrary reservoir history can be obtained from superposition of “terminal pressure” or “terminal rate” solutions (van Everdingen and Hurst, 1949). Only the terminal pressure case is needed for our purposes; the reservoir boundary pressure follows a step-function of amplitude

Δψ = ψinit − ψo,

We = UL Δψ >L(tL*), where

UL = xawohϕct

tL* =

and

t k =t τL ϕct μxa2

(8)

is a dimensionless time suited to the cumulative influx function for a linear aquifer given by Leung (1986) after Carslaw and Jaeger (1959):

8 >L(tL*) = 1 − 2 π

∞

∑

⎡ π2 ⎤ exp⎢ −(2m − 1)2 4 tL*⎥ ⎣ ⎦

m =1

(2m − 1)2

. (9)

Similarly, BCs for the circular geometry are given by

where ψinit is the initial aquifer pressure and ψo the step reservoir edge pressure. BCs in linear geometry are given by

ψ (0, ]0, xa]) = ψinit

(7)

is the linear aquifer constant, and

(4)

⎧ ψ (t , 0) = ψo ⎪ ⎨ ∂ψ . ⎪ ⎩ ∂x (t , xa ) = 0

(6)

ψ (0, ]ro, ra]) = ψinit

⎧ ψ (t , ro ) = ψo ⎪ and ⎨ , ∂ψ ⎪ ⎩ ∂r (t , ra ) = 0

(10)

where ro and ra are the reservoir and the aquifer radii, respectively; the aquifer thickness h and encroachment angle θo (expressed in radians) are kept constant in order for the problem to be one-dimensional. The cumulative aquifer influx can be expressed as

(5)

In the above, xa is the aquifer length, while wo and h are the aquifer width and thickness considered uniform in order to keep the problem one-dimensional. Inspection of Eq. (2) shows that the cumulative aquifer influx can be expressed as

We = UC Δψ >C(tC*, ra*), where 5

(11)

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Fig. 11. Time required to reach a cumulative efficiency of 10 to 90% as a function of injector location in (a) linear geometry, and (b) circular geometry with ra* = 5. The thin

Fig. 12. Cumulative efficiency function versus dimensionless time in circular geometry * = 3, and different reservoir radii. * = 1.2 and rinj for rinj

black lines correspond to the estimates of Eq. (43).

UC = ro2θohϕct

2

>C((ra* − 1) tL*, ra*) ; ra*→1 ra* − 1

(12)

>L(tL*) = lim

is the circular aquifer constant, and

tC* =

t k =t τC ϕct μro2

in conformance with literature usage however, we treat the two cases separately using the aquifer length and reservoir radius as unit distances, respectively:

(13)

x* =

is a dimensionless time suited to the cumulative influx function for a circular aquifer given by van Everdingen and Hurst (1949)

>C(tC*, ra*) =

ra*2 − 1 −2 2

[J1(a mr*a)] exp(−am2tC*)

∑

2 am2{[J0(a m)]2 − [J1(a mr*a)] }

m =1

th

In the above, am is the m

.

8L =

(14)

root of the equation

* J1(a mr*a)Y0(a m) − J0(a m)Y(a 1 mra) = 0 ,

x xa

and r * =

r . ro

(17)

Defining

2

∞

(16)

∂>L ∂tL*

and 8 C =

∂>C , ∂tC*

(18)

we can also write (15)

qe =

where J and Y are the Bessel functions of the first and second kind, respectively. Table 1 provides the first three solutions to Eq. (15) for ra* ≤ 50 ; alternatively, approximate values of a1 and a2 for 2 ≲ ra* ≲ 25 can be obtained using the fit of Klins et al. (1988), enabling evaluation of Eq. (14) considering the first two terms of the series. Expressions of key parameters in field units is given, for convenience, in Appendix A. Linear geometry can be recovered from circular geometry in the limit ra → ro through

UL Δψ 8L(tL*) τL

(19)

in linear geometry and

qe =

UC Δψ 8 C (tC*, ra*) τC

(20)

in circular geometry, where 8L and 8 C are the instantaneous aquifer rate functions. It is convenient to define a characteristic pressure 7 , such that 6

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Fig. 13. Pressure profiles (a) and water influx (b) for a circular aquifer with ra* = 5, * = 20 . * = 2 and ψ * = 0 , following an injection signal given by Eq. (48) with tCstop rinj init

ψ* =

ψ . 7

(21) Fig. 14. (a) Potential contour lines at tC* = ∞ arising from a step unit injection

Fig. 4 shows pressure profiles and water influx for a linear aquifer. The pressure disturbance propagates to the aquifer boundary, before settling to a pseudo-steady state regime where the aquifer influx is simply proportional to the difference between reservoir and average aquifer pressures, resulting in exponential pressure decay (Leung, 1986; Fetkovich, 1971). We can show that a reasonable estimate for PSS onset time in case of step pressure signal is:

* ≃ 0.2, tpL

* = 1.5 and θinj = 0 , in a circular aquifer of radius ra* = 5 (qinj = UC 7 /τC ) located at rinj with θo = 2π , calculated with Nθ = 600 and Nr = 150 . (b) Angular distribution of influx at different times, for the same problem.

(22)

as illustrated in Fig. 4(b). For a linear aquifer, the cumulative influx at infinite time is We = UL Δψ ; we see on Fig. 4 that at the onset of PSS, approximately half of the cumulative influx has already occurred, hence the applicability of PSS aquifer models in this case is quite limited. Fig. 5 shows pressure profiles and water influx for a circular aquifer with ra* = 5. As in the linear case, the pressure disturbance propagates to the aquifer boundary, before settling to a pseudo-steady state regime. We can show that a reasonable estimate for PSS onset in case of step pressure signal is: 2

* ≃ 0.2(ra* − 1) , tpC

(23) * , for increasing tC*. The Fig. 15. Standard deviation of the lateral influx distribution vs. rinj

as illustrated in Fig. 5(b). For a circular aquifer, the cumulative aquifer influx at infinite time 1 is We = 2 UC ·(ra*2 − 1)Δψ . The larger the aquifer, the more influx occurs during PSS regime, hence PSS aquifer models are more relevant to (large) circular aquifers.

problem geometry is as in Fig. 14.

7

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Fig. 16. Impact of limited encroachment angle θo = π /3 and θinj = π /8 , while keeping other dimensionless parameters as in Fig. 14. Potential contours are calculated with Nθ = 100 and Nr = 150 . (a) Potential contour lines at tC* = ∞. (b) Angular distribution of influx at different times.

Fig. 18. Illustration of the aquifer shielding effect. Aquifer (blue) and injection (red) streamlines add nonlinearly, as shown in this example in circular geometry at tC* = 1 with * = 1.5 (i.e., as in Fig. 16), q = − UCψ /τC , ψ * = − 1 and θo = π /3, θinj = π /8 , ra* = 5, rinj inj

o

o

* = 0. ψinit

Fig. 17. Illustration of the well shielding effect. Injector 2 is shielding injector 3, for which streamlines-based efficiency would be zero; reciprocally, injector 3 is helping injector 2, for which streamines-based efficiency would be one. These wells being located at the same distance from the reservoir-aquifer boundary, their pressure support efficiency is equal, and equal to the efficiency of injector 1. Producing wells are sketched for illustration purposes, but the concept is also valid if the problem is truncated at an idealized reservoir-aquifer boundary.

Fig. 19. Schematic illustration of a segmented reservoir with peripheral injection. Injectors in segment A2 mostly support production in segment R2, and producers in R2 are mostly supported by injectors A2. By averaging streamlines-derived efficiency of injectors located in A2, well shielding effects cancel out.

8

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Table 2 Linear geometry.

tL*

>L

*= ,L for xinj 0.01

0.02

0.05

0.1

0.2

0.4

0.6

0.8

0.99

0.0010 0.0013 0.0018 0.0024 0.0031 0.0042 0.0056 0.0074 0.0099 0.0131 0.0175 0.0232 0.0309 0.0412 0.0548 0.0730 0.0971 0.1293 0.1721 0.2291 0.3049 0.4059 0.5403 0.7191 0.9572 1.2742 1.6960 2.2576 3.0050 4.0000

0.691 0.727 0.759 0.788 0.814 0.837 0.857 0.875 0.891 0.905 0.917 0.928 0.937 0.946 0.953 0.959 0.964 0.969 0.973 0.977 0.980 0.983 0.986 0.988 0.990 0.992 0.994 0.996 0.997 0.998

0.464 0.517 0.567 0.614 0.657 0.696 0.732 0.764 0.792 0.818 0.840 0.860 0.878 0.894 0.907 0.919 0.930 0.939 0.947 0.954 0.960 0.966 0.971 0.976 0.981 0.985 0.988 0.991 0.993 0.995

0.117 0.162 0.213 0.269 0.326 0.384 0.440 0.495 0.547 0.595 0.640 0.680 0.718 0.751 0.781 0.808 0.831 0.853 0.871 0.888 0.903 0.917 0.930 0.942 0.953 0.963 0.972 0.978 0.984 0.988

0.006 0.014 0.029 0.051 0.083 0.122 0.169 0.221 0.277 0.335 0.392 0.449 0.503 0.554 0.602 0.646 0.686 0.723 0.756 0.786 0.813 0.839 0.864 0.887 0.909 0.928 0.945 0.958 0.968 0.976

0.000 0.000 0.000 0.001 0.002 0.007 0.016 0.032 0.056 0.088 0.129 0.177 0.230 0.286 0.344 0.401 0.457 0.511 0.562 0.610 0.656 0.701 0.745 0.787 0.828 0.864 0.895 0.921 0.940 0.955

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.008 0.018 0.035 0.060 0.094 0.136 0.185 0.238 0.296 0.356 0.420 0.486 0.556 0.627 0.696 0.759 0.814 0.859 0.894 0.920

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.009 0.020 0.038 0.065 0.101 0.147 0.203 0.268 0.343 0.427 0.515 0.603 0.685 0.756 0.815 0.860 0.895

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.009 0.021 0.042 0.075 0.122 0.185 0.262 0.351 0.449 0.547 0.640 0.721 0.788 0.840 0.880

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.010 0.026 0.054 0.097 0.158 0.236 0.327 0.427 0.529 0.625 0.710 0.779 0.834 0.875

αL

0.010

0.020

0.049

0.095

0.180

0.320

0.420

0.480

0.500

2.2. Arbitrary reservoir signal

the terminal rate case is obtained. Fig. 6 shows the water influx for a circular aquifer with ra* = 5 and τs = 2τC . As discussed by Leung (1986), time to reach PSS for τs > 0 can be much higher than for the terminal pressure case (Eqs. (22) and (23)).

Eqs. (6) and (11) provide solutions to the aquifer problem for a step signal Δψ . For arbitrary signal Δψ (t*) = ψinit − ψo(t*), solutions are given in linear geometry by the convolution integrals

⎡ We = UL ⎢Δψ (0)>L(tL*)+ ⎣

∫0

tL*

⎤ ∂Δψ (s )>L(tL* − s )ds⎥ ⎥⎦ ∂s

3. The peripheral injection problem (24)

3.1. Step injection signal

and

U ⎡ qe = L ⎢Δψ (0)8L(tL*)+ τL ⎣

∫0

tL*

⎤ ∂Δψ (s )8L(tL* − s )ds⎥ , ⎥⎦ ∂s

In the presence of a peripheral water source located at rinj , Eq. (2) should be replaced by (25)

ϕct

and in circular geometry by

⎡ We = UC ⎢Δψ (0)>C(tC*, ra*)+ ⎣

∫0

tC*

⎤ ∂Δψ (s )>C(tC* − s, ra*)ds⎥ ⎥⎦ ∂s

(26)

UC ⎡ ⎢Δψ (0)8 C (tC*, ra*)+ τC ⎣

∫0

tC*

⎤ ∂Δψ (s )8 C (tC* − s, ra*)ds⎥ . ⎥⎦ ∂s

(27)

In order to illustrate the effect of arbitrary reservoir signal, we consider a model reservoir pressure variation of the form

⎡ ⎛ t ⎞⎤ Δψ = Δψ∞⎢1 − exp⎜ − ⎟⎥ , ⎢⎣ ⎝ τs ⎠⎥⎦

∂ψ k − ∇2 ψ = qinjδ (r − rinj), ∂t μ

(29)

where qinj is the injection rate and δ is the Dirac function. The solution of Eq. (29) can be expressed as the sum of a function satisfying the homogeneous equation with no-flow outer BC and inner BC matching the reservoir behavior (which is the “aquifer” problem reviewed in Section 2), and a function satisfying the inhomogeneous equation with no-flow outer BC, zero pressure inner BC and zero initial pressure. We here refer to this as the “peripheral injection” problem. Regardless of the geometry, it is two-dimensional and should be treated as such if we wish to know the lateral water influx distribution at the reservoir edge; this will be needed in Section 4. Owing to the linearity of Eq. (29) however, the pressure field arising from an injection line is the sum of the fields that would arise from the individual injectors constituting the line. Furthermore, by translational symmetry in the direction perpendicular to the reservoir boundary (aquifer lateral edges can be omitted upon multiplying the system indefinitely through mirror images), each of these injectors equally

and

qe =

0.036 0.041 0.047 0.055 0.063 0.073 0.084 0.097 0.112 0.129 0.149 0.172 0.198 0.229 0.264 0.305 0.352 0.406 0.468 0.539 0.618 0.702 0.786 0.862 0.923 0.965 0.987 0.997 0.999 1.000

(28)

where τs is the characteristic time of reservoir depletion. In the limit τs → 0 the terminal pressure case is recovered, and in the limit τs → ∞ 9

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 3 ra* = 1.5.

-L =

tC* 1.01

1.02

1.05

1.1

1.2

1.3

1.4

1.49

0.0010 0.0013 0.0016 0.0020 0.0026 0.0033 0.0042 0.0053 0.0067 0.0085 0.0108 0.0137 0.0174 0.0221 0.0281 0.0356 0.0452 0.0574 0.0728 0.0924 0.1172 0.1487 0.1887 0.2395 0.3039 0.3857 0.4894 0.6210 0.7880 1.0000

0.687 0.717 0.745 0.770 0.793 0.814 0.833 0.850 0.865 0.879 0.892 0.903 0.913 0.922 0.930 0.937 0.944 0.949 0.955 0.960 0.964 0.969 0.973 0.977 0.981 0.985 0.988 0.990 0.992 0.994

0.459 0.503 0.545 0.585 0.622 0.657 0.689 0.719 0.746 0.771 0.793 0.814 0.832 0.849 0.864 0.878 0.890 0.901 0.911 0.921 0.930 0.938 0.947 0.955 0.963 0.970 0.975 0.980 0.985 0.988

0.114 0.150 0.191 0.235 0.281 0.327 0.374 0.421 0.465 0.509 0.550 0.588 0.624 0.658 0.689 0.717 0.744 0.768 0.790 0.812 0.833 0.853 0.873 0.892 0.910 0.927 0.941 0.953 0.963 0.971

0.006 0.012 0.022 0.037 0.058 0.085 0.117 0.153 0.194 0.237 0.282 0.328 0.374 0.419 0.463 0.505 0.545 0.583 0.620 0.656 0.693 0.729 0.765 0.800 0.833 0.863 0.890 0.912 0.931 0.945

0.000 0.000 0.000 0.000 0.001 0.003 0.006 0.013 0.023 0.039 0.060 0.086 0.118 0.154 0.193 0.236 0.280 0.327 0.375 0.427 0.481 0.538 0.596 0.654 0.711 0.763 0.809 0.848 0.879 0.905

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.004 0.009 0.018 0.031 0.049 0.072 0.102 0.137 0.179 0.228 0.283 0.345 0.412 0.484 0.557 0.628 0.695 0.754 0.804 0.845 0.877

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.007 0.014 0.026 0.044 0.070 0.105 0.150 0.205 0.270 0.342 0.420 0.501 0.581 0.655 0.722 0.778 0.824 0.862

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.006 0.014 0.028 0.051 0.083 0.127 0.181 0.246 0.320 0.400 0.484 0.566 0.643 0.712 0.770 0.818 0.856

αC

0.006

0.012

0.029

0.055

0.095

0.123

0.139

0.144

⎧ = 0 ⎪ ψ (t , 0) and ⎨ ∂ψ ⎪ ( t , x ) = 0 a ⎩ ∂x

and

⎧ = 0 ⎪ ψ (t , ro ) ⎨ ∂ψ , ⎪ ⎩ ∂r (t , ra ) = 0

(36)

in circular geometry, where - L and -C are the instantaneous injection efficiency functions. The cumulative efficiency function for a step signal in linear geometry is given by Bonet and Crawford (1969) as:

* ) = 1 − 1 αL (xinj *) + 1 ,L(tL*, xinj tL* tL*

⎡

∞

∑ βLm(xinj* )exp⎢−(2m − 1)2 ⎣

m =1

π 2 *⎤ tL ⎥ , 4 ⎦ (37)

where

⎛ x* ⎞ * ·⎜⎜1 − inj ⎟⎟ αL = xinj 2 ⎠ ⎝

(38)

and

⎡ 16 π *⎤ sin⎢(2m − 1) xinj ⎥. 2 ⎦ π 3(2m − 1)3 ⎣

βLm =

(39)

We observe that Wfill = αLqinjτL represents the cumulative volume of injected water filling the aquifer when steady state is reached, i.e., the area of the surface below the tL* = ∞ profile in Fig. 8(a), and that the arguments of the decaying exponentials are identical to those of Eq. * > 0 , ,L and - L tend to zero (9). We can also verify that regardless of xinj at tL* = 0 . To the best of our knowledge, there is no published analytic or semi-analytic expression for ,C . By analogy with Eq. (37), it is however natural to postulate that it can be cast in the form

* , ra*) = 1 − 1 αC (rinj * , ra*) + 1 ,C(tC*, rinj tC* tC*

∞

∑ βCm(rinj* , ra*)exp(−am2tC*), m =1

(40) where

αC =

1 *2 * ) + 1 (1 − rinj *2 ) ra ln(rinj 2 4

(41)

is such that Wfill = αCqinjτC represents the cumulative volume of injected water filling the aquifer at infinite time (radially weighted area of the surface below the tC* = ∞ profile in Fig. 9(a)), and the arguments of the decaying exponentials are identical to those of Eq. (14). Instead of explicitly deriving the expression for βC, which would be beyond the authors' capacity, a simple finite-volumes code, described in Appendix B, is used to build tabulations of ,C reported in Appendix C. We can * ⪢0 and ra* ≥ rinj * , ,C and -C tend to zero then verify that regardless of rinj at tC* = 0 . -C can be obtained from derivation of the tabulations in Appendix C using Eq. (34). Fig. 8(a) shows the pressure profiles for a linear aquifer following a * = 0.2 , and Fig. 8(b) step injection signal qinj = UL 7 / τL located at xinj illustrates the instantaneous and cumulative efficiency functions (this is the same information as in Figs. 1 and 3 from Bonet and Crawford (1969)) for three different injection points. The profiles can qualitatively be split into three periods. At early times, the efficiency is close to zero as pressure information from the newly active injector needs to diffuse to the reservoir boundary; at intermediate times, the efficiency rises as pressure information diffuses to the outer boundary of the aquifer; at later times, after pseudo-steady state is reached, efficiency tends to unity according to a decaying exponential. When the injection point is close to the outer aquifer edge the first two periods occur simultaneously, but this situation is rarely

(30)

(31)

(32)

where non-dimensionalization is as for the aquifer problem. Similarly, the cumulative water influx in circular geometry can be expressed as

* , ra*). We = Winj ,C(tC*, rinj

(35)

* , ra*) qe = qinj -C(tC*, rinj

respectively. We here consider that qinj follows a step-function, and define Winj = qinjt as the cumulative amount of injected water. Inspection of Eq. (29) then shows that the cumulative water influx in linear geometry can be expressed as

* ), We = Winj ,L(tL*, xinj

(34)

in linear geometry and

0.036 0.041 0.046 0.052 0.059 0.066 0.075 0.085 0.096 0.108 0.123 0.139 0.157 0.179 0.203 0.230 0.262 0.297 0.337 0.380 0.426 0.472 0.516 0.553 0.583 0.604 0.616 0.622 0.624 0.625

and

ψ (0, ]ro, ra]) = 0

∂ * (tC ,C), ∂tC*

*) qe = qinj - L(tL*, xinj

contributes to the total flux. As a consequence, the total influx from a line perpendicular to the reservoir edge and from a single injector located at the same distance are identical (see the illustration in Fig. 7). The same applies to circular geometry as well. Provided we are only interested in the total influx, we can therefore consider the injection to be evenly spread over a line located at xinj or a ring located at rinj, hence solve a one-dimensional problem (see Fig. 3). This reasoning is also valid if the well is only partially perforated, although in this case the pressure field would be three-dimensional. For linear and circular geometries, the BCs therefore reduce to

ψ (0, ]0, xa]) = 0

and -C =

we can also write

>C

*= ,C for rinj

∂ * (tL ,L) ∂tL*

(33)

,L and ,C are the cumulative injection efficiency functions. Defining 10

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 4 ra* = 2 .

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.4

1.6

1.8

1.99

0.0010 0.0013 0.0018 0.0024 0.0031 0.0042 0.0056 0.0074 0.0099 0.0131 0.0175 0.0232 0.0309 0.0412 0.0548 0.0730 0.0971 0.1293 0.1721 0.2291 0.3049 0.4059 0.5403 0.7191 0.9572 1.2742 1.6960 2.2576 3.0050 4.0000

0.687 0.723 0.755 0.784 0.810 0.833 0.853 0.871 0.887 0.901 0.913 0.924 0.933 0.941 0.948 0.954 0.960 0.964 0.969 0.972 0.975 0.978 0.981 0.984 0.987 0.989 0.992 0.993 0.995 0.996

0.459 0.512 0.561 0.608 0.650 0.689 0.724 0.756 0.785 0.810 0.832 0.852 0.870 0.885 0.899 0.910 0.921 0.930 0.938 0.945 0.951 0.957 0.963 0.968 0.974 0.979 0.983 0.987 0.990 0.993

0.114 0.158 0.208 0.262 0.318 0.374 0.430 0.483 0.534 0.581 0.625 0.664 0.701 0.733 0.763 0.789 0.812 0.833 0.851 0.868 0.883 0.897 0.910 0.923 0.936 0.948 0.959 0.969 0.976 0.982

0.006 0.014 0.028 0.049 0.079 0.117 0.161 0.211 0.264 0.319 0.374 0.428 0.480 0.529 0.575 0.617 0.656 0.691 0.723 0.752 0.778 0.804 0.829 0.854 0.878 0.901 0.922 0.940 0.954 0.965

0.000 0.000 0.000 0.001 0.002 0.006 0.014 0.029 0.051 0.081 0.118 0.162 0.210 0.262 0.314 0.367 0.419 0.469 0.516 0.560 0.603 0.646 0.689 0.732 0.776 0.818 0.856 0.889 0.916 0.936

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.006 0.015 0.029 0.051 0.080 0.115 0.157 0.203 0.252 0.304 0.359 0.419 0.483 0.552 0.622 0.691 0.755 0.811 0.856 0.892

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.007 0.016 0.030 0.052 0.081 0.118 0.163 0.217 0.281 0.355 0.436 0.523 0.609 0.690 0.760 0.818 0.863

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.007 0.016 0.032 0.057 0.094 0.144 0.208 0.285 0.373 0.468 0.563 0.653 0.732 0.796 0.846

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.008 0.019 0.040 0.074 0.122 0.186 0.264 0.354 0.451 0.549 0.641 0.723 0.789 0.841

αC

0.015

0.030

0.072

0.138

0.255

0.433

0.550

0.616

0.636

material balance calculations. Alternatively, they could be used to reduce the uncertainty of aquifer parameters by providing a lower bound to τC, since the cumulative efficiency cannot lie to the right of the ra* = ∞ curve.

encountered in practice as injectors are usually located as close as possible to the reservoir boundary. Fig. 9(a) shows the pressure profiles for a circular aquifer with * = 2, ra* = 5 following a step injection signal qinj = UC 7 / τC located at rinj and Fig. 9(b) illustrates the instantaneous and cumulative efficiency functions for three different injection points. The physics is similar to that of Fig. 8, although as already observed in Section 2 for the aquifer problem, the relative duration of PSS increases as the circular aquifer radius increases (recall that a linear aquifer is equivalent to a circular aquifer with dimensionless radius tending to one). When starting peripheral injection, it is important to anticipate the time required for pressure support to be felt by the reservoir. If we consider that this occurs when - L or -C = 10%, an estimate of such time is given by

*2 tL*10 ≃ 0.2xinj

and

* − 1)2 , tC*10 ≃ 0.2(rinj

3.2. Arbitrary injection signal Eq. (32) provides a solution to the peripheral injection problem for a step injection rate qinj . For arbitrary signal qinj (tL*), solutions in linear geometry are given by the convolution integral

*) + We = qinj (0)tL*,L(tL*, xinj

and

* − 1)2 , tC*10 ≃ 0.4(rinj

∫0

tL*

∂qinj ∂s

* )ds, (s )(tL* − s ),L(tL* − s, xinj

(44)

and (42)

*) + qe = qinj (0)- L(tL*, xinj

as illustrated in Fig. 10 showing time required to reach a range of instantaneous efficiencies from 10% to 90% as a function of injector location, in linear geometry and circular geometry with ra* = 5. Similarly, an estimate of time such that ,L or ,C = 10% is given by

*2 tL*10 ≃ 0.4xinj

0.036 0.042 0.048 0.056 0.065 0.075 0.087 0.101 0.117 0.136 0.158 0.183 0.213 0.249 0.291 0.340 0.398 0.467 0.548 0.645 0.758 0.884 1.020 1.155 1.278 1.376 1.443 1.480 1.495 1.499

∫0

tL*

∂qinj ∂s

* )ds, (s )·- L(tL* − s, xinj

(45)

and in circular geometry,

* , ra*) + We = qinj (0)tC*,C(tL*, rinj

(43)

∫0

tC*

∂qinj ∂s

* , ra*)ds (s )(tC* − s ),C(tC* − s, rinj (46)

as illustrated in Fig. 11 for the same parameters as Fig. 10. A different way to look at the same data is shown in Fig. 12, where the cumulative efficiency function is plotted vs. dimensionless time in * = 1.2 and rinj * = 3 and different aquifer radii. If circular geometry for rinj aquifer properties such as porosity, permeability and compressibility are reasonably well known, these plots could be used to estimate the aquifer radius after history matching the cumulative efficiency from

and

* , ra*) + qe = qinj (0)-C(tL*, rinj

∫0

tC*

∂qinj ∂s

* , ra*)ds. (s )-C(tC* − s, rinj

(47)

In order to illustrate the effect of arbitrary injection signal in * = 2 , we consider the square signal circular geometry with ra* = 5 and rinj 11

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 5 ra* = 3.

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

2.5

2.95

0.0010 0.0014 0.0019 0.0027 0.0038 0.0053 0.0074 0.0103 0.0144 0.0202 0.0282 0.0393 0.0549 0.0767 0.1070 0.1495 0.2087 0.2914 0.4069 0.5681 0.7932 1.1075 1.5464 2.1592 3.0149 4.2096 5.8777 8.2069 11.459 16.000

0.687 0.729 0.765 0.798 0.826 0.850 0.871 0.889 0.905 0.919 0.930 0.940 0.948 0.955 0.961 0.967 0.971 0.975 0.978 0.981 0.983 0.985 0.987 0.989 0.990 0.992 0.994 0.995 0.997 0.998

0.459 0.520 0.577 0.630 0.677 0.719 0.756 0.789 0.818 0.843 0.864 0.883 0.899 0.912 0.924 0.934 0.943 0.950 0.956 0.962 0.966 0.970 0.974 0.978 0.981 0.984 0.988 0.991 0.993 0.995

0.114 0.166 0.226 0.290 0.356 0.421 0.483 0.542 0.596 0.645 0.689 0.728 0.763 0.793 0.820 0.843 0.862 0.880 0.894 0.907 0.918 0.928 0.937 0.945 0.954 0.962 0.970 0.977 0.983 0.988

0.006 0.015 0.034 0.063 0.103 0.154 0.211 0.274 0.338 0.402 0.463 0.522 0.575 0.624 0.668 0.707 0.742 0.773 0.799 0.823 0.843 0.861 0.878 0.894 0.910 0.926 0.942 0.956 0.967 0.977

0.000 0.000 0.000 0.001 0.004 0.013 0.029 0.055 0.092 0.139 0.194 0.253 0.315 0.376 0.436 0.493 0.545 0.593 0.637 0.676 0.710 0.742 0.772 0.801 0.831 0.861 0.890 0.917 0.939 0.956

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.018 0.037 0.065 0.104 0.150 0.201 0.257 0.313 0.369 0.423 0.476 0.530 0.586 0.645 0.707 0.767 0.823 0.869 0.906

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.014 0.029 0.053 0.085 0.125 0.173 0.228 0.293 0.368 0.452 0.544 0.637 0.723 0.796 0.852

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.008 0.020 0.040 0.073 0.119 0.182 0.262 0.356 0.462 0.570 0.671 0.758 0.825

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.008 0.021 0.047 0.090 0.151 0.231 0.328 0.438 0.550 0.656 0.746 0.816

αC

0.040

0.079

0.194

0.376

0.710

1.512

2.369

2.811

2.942

⎧ * * ⎪ qinj 0 , if tC ∈ ]0: tCstop ] qinj = ⎨ ⎪ * if tC* > tCstop ⎩ 0,

3.3. Total influx (48)

As schematically indicated by Eq. (1), the total influx for a specific reservoir pressure and periphereal injection history will be the sum of a convolution integral amongst Eqs. (24)–(27) and a convolution integral for each injector amongst Eqs. (44)–(47). The choice of integrals simply depends on the geometry, and on whether instantaneous or cumulative influx is desired. It is important to keep in mind that reservoir edge pressure has no effect on the peripheral injectors' efficiencies as defined in this work. One might intuitively think that lowering the reservoir pressure would increase injection efficiency as more peripheral water is “sucked” into the reservoir, but this is incorrect. Lowering the reservoir pressure would induce additional water to flow to the reservoir independently of the presence of peripheral injectors, although the origin of water “molecules” themselves would be different. This point will be more easily understood in Section 4, where differences between pressure support efficiency and transport efficiency are discussed.

Fig. 13 shows the corresponding pressure profiles and water influx; for tC* ≤ 20 the solution is as in Fig. 9, but as soon as injection stops the instantaneous influx reduces quickly. The cumulative efficiency slope on the contrary shows a positive discontinuity at tC* ≃ 20 , as influx continues even after injection is interrupted, until the injected volume filling the aquifer is produced (for a long enough injection period, these volumes are given by Wfill = αLqinj 0τL (Eq. (38)) in linear geometry, and Wfill = αCqinj 0τC (Eq. (41)) in circular geometry). We define the cumulative injection efficiency η as

η

=

We , Winj

(49)

and instantaneous injection efficiency η as

η=

qe qinj

0.036 0.043 0.051 0.060 0.071 0.085 0.101 0.120 0.143 0.170 0.203 0.243 0.291 0.349 0.420 0.507 0.613 0.744 0.905 1.106 1.356 1.664 2.033 2.452 2.892 3.305 3.638 3.854 3.959 3.993

.

3.4. Importance of the reservoir-aquifer boundary location

(50)

Calculations have been presented so far considering the reservoiraquifer boundary location (i.e., the reservoir edge) to be a given. As mentioned in the caption of Fig. 2, most of the literature discussing aquifer influx places such boundary at the OWC. The logic is to separate a water-saturated aquifer where the single-phase/singlecomponent flow Eq. (2) holds, from a region with non-zero oil saturation referred to as the reservoir; even if aquifer water encroaches on the reservoir, the OWC does not move. The underlying assumption of aquifer influx calculations in materi-

For a step injection signal, we simply have η = ,L, C and η = - L, C . When the injection rate increases with time, the injection efficiencies as defined in Eqs. (49) and (50) appear lower than ,L, C and - L, C because influx is lagging behind the injection rate, while if the injection rate decreases with time the opposite occurs. Instantaneous efficiency can even be larger than one, and for the case of Fig. 13 where injection is stopped it would become infinite, which is why qe / qinj0 rather than η is plotted. 12

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 6 ra* = 4 .

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

2.5

3

3.9

0.0010 0.0014 0.0021 0.0030 0.0043 0.0061 0.0088 0.0126 0.0181 0.0259 0.0373 0.0535 0.0768 0.1103 0.1583 0.2274 0.3265 0.4687 0.6731 0.9664 1.3876 1.9925 2.8609 4.1079 5.8984 8.4693 12.161 17.461 25.072 36.000

0.687 0.732 0.771 0.805 0.834 0.859 0.881 0.899 0.914 0.927 0.938 0.948 0.955 0.962 0.967 0.972 0.976 0.979 0.982 0.984 0.986 0.988 0.989 0.991 0.992 0.993 0.995 0.996 0.997 0.998

0.459 0.525 0.586 0.642 0.691 0.735 0.773 0.806 0.835 0.859 0.880 0.898 0.912 0.925 0.936 0.945 0.952 0.959 0.964 0.969 0.973 0.976 0.979 0.981 0.984 0.987 0.990 0.992 0.994 0.996

0.114 0.171 0.236 0.307 0.378 0.447 0.513 0.574 0.629 0.679 0.722 0.760 0.793 0.822 0.846 0.867 0.885 0.900 0.913 0.924 0.933 0.941 0.948 0.955 0.961 0.968 0.974 0.980 0.986 0.990

0.006 0.017 0.038 0.072 0.119 0.177 0.242 0.311 0.381 0.449 0.512 0.571 0.624 0.672 0.714 0.750 0.782 0.810 0.833 0.853 0.871 0.886 0.899 0.912 0.924 0.937 0.950 0.962 0.972 0.980

0.000 0.000 0.000 0.002 0.006 0.018 0.041 0.076 0.123 0.180 0.243 0.310 0.377 0.441 0.502 0.558 0.609 0.654 0.694 0.729 0.759 0.786 0.810 0.833 0.857 0.881 0.905 0.928 0.947 0.963

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.015 0.035 0.066 0.107 0.158 0.215 0.276 0.337 0.396 0.452 0.504 0.553 0.599 0.644 0.692 0.742 0.794 0.843 0.885 0.919

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.016 0.034 0.063 0.101 0.147 0.199 0.253 0.309 0.368 0.433 0.504 0.582 0.664 0.743 0.813 0.867

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.011 0.026 0.050 0.083 0.124 0.173 0.232 0.302 0.385 0.479 0.580 0.678 0.765 0.833

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.006 0.015 0.032 0.060 0.099 0.153 0.224 0.312 0.415 0.527 0.637 0.735 0.812

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.008 0.023 0.053 0.101 0.171 0.261 0.370 0.490 0.609 0.714 0.797

αC

0.075

0.148

0.365

0.710

1.349

2.931

4.795

6.018

6.789

7.335

al balance calculations is therefore that the pressure at the OWC, used as inner BC for Eq. (2), is equal to the reservoir tank pressure. Material balance calculations are usually matched to the average bottom-hole shut-in pressures of producing wells, which sample a radius of the same order as the interwell distance. We therefore see that if the transition zone is large and producing wells are located far from the OWC, the assumption becomes incorrect and the aquifer-reservoir boundary should be moved towards the oil pool. Let us first estimate, in linear geometry, the error in aquifer influx due to misplacing the reservoir boundary by δxo while keeping the outer aquifer boundary location unchanged (i.e., δxa = − δxo ), and limiting ourselves to the PSS regime. 8L can be obtained from Eq. (9) as ∞

8L(tL*) = 2

⎡

∑ exp⎢−(2m − 1)2 m =1

⎣

π 2 *⎤ tL ⎥ , 4 ⎦

*) = 1 − 4 - L(tL*, xinj π

⎛ π2 wok Δψ k ⎞ ⎟, exp⎜⎜ − t 2⎟ μxa ⎝ 4 ϕct μxa ⎠

qe

⎛ δx ⎞ = 6 ⎜ o ⎟. ⎝ xa ⎠

2m − 1

⎡ ⎤ ⎛ π2 xinj k ⎞⎥ ⎟ , qe(t , xinj ) = qinj ⎢1 − 2 exp⎜⎜ − t 2⎟ ⎢⎣ xa ⎝ 4 ϕct μxa ⎠⎥⎦

, (54)

(55)

yielding

δqe qe

(51)

⎛ δx ⎞ = 6 ⎜⎜ o ⎟⎟ . ⎝ xinj ⎠

(56)

It is reasonable to assume that the uncertainty in locating the reservoir-aquifer boundary is small compared to the aquifer length, hence misplacing the reservoir-aquifer boundary should only have a minor impact on aquifer influx calculations. Because peripheral injectors are usually placed close to the OWC however, typically at a distance of the order of the average interwell distance (or slightly larger), misplacing the reservoir-aquifer boundary can have a serious impact on the calculation of peripheral injection efficiency. The appropriate reservoir-aquifer boundary in this case might be different from the OWC, and should be part of the history matching process. The same conclusion holds in circular geometry.

(52)

yielding

δqe

∑ m =1

⎡ π * π2 ⎤ sin[(2m − 1) 2 xinj ]exp⎢ −(2m − 1)2 4 tL*⎥ ⎣ ⎦

hence in dimensional variables, with the further assumption xinj ⪡xa , the PSS influx writes

hence in dimensional variables the PSS influx writes

qe(t ) = 2

∞

0.036 0.043 0.052 0.063 0.076 0.091 0.110 0.133 0.160 0.194 0.236 0.287 0.349 0.427 0.523 0.644 0.795 0.985 1.227 1.535 1.930 2.435 3.074 3.851 4.734 5.641 6.449 7.036 7.356 7.473

(53)

Let us second estimate the error in peripheral injection influx due to the same misplacement while keeping the injection location unchanged (i.e., δxinj = − δxo ). - L can be obtained from Eq. (37) as 13

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 7 ra* = 5.

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

3

4

4.9

0.0010 0.0015 0.0021 0.0031 0.0046 0.0067 0.0099 0.0145 0.0212 0.0310 0.0454 0.0665 0.0974 0.1427 0.2090 0.3062 0.4484 0.6568 0.9619 1.4089 2.0635 3.0223 4.4266 6.4833 9.4957 13.908 20.370 29.835 43.697 64.000

0.687 0.734 0.775 0.810 0.840 0.866 0.887 0.905 0.920 0.933 0.944 0.952 0.960 0.966 0.971 0.975 0.979 0.982 0.984 0.986 0.988 0.989 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998

0.459 0.529 0.593 0.650 0.702 0.746 0.785 0.818 0.846 0.870 0.890 0.907 0.921 0.933 0.943 0.951 0.958 0.964 0.969 0.973 0.976 0.979 0.981 0.984 0.986 0.988 0.990 0.993 0.995 0.996

0.114 0.174 0.244 0.318 0.393 0.466 0.534 0.596 0.652 0.701 0.744 0.781 0.813 0.840 0.863 0.882 0.898 0.912 0.924 0.933 0.942 0.949 0.955 0.960 0.965 0.971 0.977 0.982 0.987 0.991

0.006 0.018 0.041 0.079 0.131 0.194 0.265 0.338 0.411 0.481 0.545 0.604 0.656 0.702 0.742 0.777 0.806 0.832 0.853 0.872 0.887 0.900 0.912 0.922 0.933 0.943 0.954 0.965 0.975 0.982

0.000 0.000 0.000 0.002 0.008 0.023 0.051 0.092 0.147 0.210 0.280 0.350 0.420 0.485 0.545 0.600 0.648 0.691 0.728 0.760 0.788 0.812 0.833 0.853 0.872 0.892 0.913 0.934 0.952 0.966

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.009 0.025 0.052 0.092 0.143 0.202 0.265 0.329 0.392 0.452 0.506 0.556 0.601 0.642 0.682 0.722 0.765 0.810 0.855 0.894 0.926

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.012 0.029 0.057 0.096 0.144 0.198 0.255 0.313 0.369 0.425 0.482 0.544 0.612 0.686 0.759 0.824 0.877

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.014 0.032 0.059 0.095 0.140 0.194 0.259 0.338 0.432 0.537 0.643 0.740 0.818

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.011 0.027 0.055 0.099 0.161 0.244 0.348 0.466 0.587 0.699 0.789

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.013 0.034 0.073 0.134 0.218 0.324 0.445 0.571 0.687 0.781

αC

0.119

0.237

0.584

1.139

2.169

4.756

7.914

11.73

13.58

14.11

0.036 0.044 0.053 0.065 0.079 0.096 0.117 0.143 0.175 0.214 0.262 0.323 0.399 0.493 0.613 0.765 0.959 1.209 1.530 1.949 2.496 3.213 4.150 5.337 6.759 8.311 9.794 10.96 11.65 11.93

injection qinj = UC 7 / τC is here 6 times lower (UC is proportional to θo). It is however not exactly 6 times lower, because of boundary mirroring effects (the problem is equivalent to having 6 injectors equally spaced in angle with θo = 2π ). The maximum pressure discussed above is the pressure of the perforated cell (or adjacent cells in case the injection radius does not exactly match a cell center). Because in two (and three) dimensions the kernel of the Laplacian operator is singular around point sources, this maximum pressure increases as the grid is refined, and would tend to infinity in the limit of infinitesimal grid spacing. In reality wells have a finite diameter, hence a finite bottom-hole pressure, but it is not practical to refine the grid down to the wellbore diameter. In reservoir simulation, the bottom-hole pressure is obtained from the perforated cell pressure using the concept of connection factor (Peaceman, 1978). In one dimension however, the kernel of the Laplacian operator is continuous at point sources, hence the maximum pressure in Fig. 8(a), Fig. 9(a) and Fig. 13(a) corresponded indeed to the bottom hole pressure of the injection line/ring.

4. Two-dimensional considerations 4.1. Pressure profiles and lateral influx distribution It has been argued in Section 3 that for linear and circular geometries, the total influx from a peripheral injector is identical to that from an injection line/ring located at the same distance from the reservoir edge. This of course does not mean that related aquifer pressure fields and lateral distribution of influx are identical. As a first example, Fig. 14(a) shows potential contour lines in circular geometry with ra* = 5 and θo = 2π at tC* = ∞, for a unit injection * = 1.5. Contrary to the injection ring source (qinj = UC 7 / τC ) located at rinj * = 2 ), the potential perturbacase (see for example Fig. 9(a), where rinj tion here is not invariant by rotation around the origin, but still extends to the “downstream” side of the reservoir. As a consequence, lateral influx will have a non negligible lateral extent, as shown in Fig. 14(b). Fig. 15 shows the evolution of the standard deviation of influx * . As intuitively expected, lateral distribution (σqe) as a function of rinj σqe increases with time as the potential perturbation extends around * as the injector “sees” a wider the reservoir, and increases with rinj portion of the reservoir. The authors do not see a real situation where the injector would be placed at a radius much larger than the reservoir * = ra*. radius itself, but for completeness the plot is extended up to rinj In case of limited encroachment angle θo, the pressure field is quite different as edge boundary conditions change from periodic to no-flow, and the angular flux distribution is narrower by construction. This is illustrated in Fig. 16 for θo = π /3 and θinj = π /8, while the other dimensionless parameters are as in Fig. 14. It can be seen by comparison of Fig. 14(a) and (Fig. 16(a) that the maximum pressure is lower than in the θo = 2π case, because unit

4.2. Low and high injection efficiency It is now possible to more rigorously discuss the meaning of “low” injection efficiency introduced in the caption of Fig. 1. The further away from the reservoir boundary an injector is, the longer it takes for its pressure support to be felt by the reservoir, as can be seen by comparing Figs. 12(a) and (b). As the aquifer is filled by injected water however, injection efficiency tends to 1 regardless of the injector's position (such time could however be longer than the economic field life). 14

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 8 ra* = 7.

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

3

5

6.9

0.0010 0.0015 0.0023 0.0034 0.0051 0.0078 0.0117 0.0176 0.0265 0.0399 0.0601 0.0905 0.1363 0.2053 0.3092 0.4657 0.7014 1.0565 1.5913 2.3967 3.6099 5.4371 8.1892 12.334 18.578 27.981 42.144 63.477 95.607 144.00

0.687 0.737 0.781 0.817 0.848 0.874 0.895 0.913 0.928 0.940 0.950 0.958 0.965 0.971 0.975 0.979 0.982 0.985 0.987 0.989 0.990 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.998

0.459 0.534 0.602 0.662 0.715 0.761 0.800 0.833 0.860 0.883 0.903 0.918 0.932 0.942 0.951 0.959 0.965 0.970 0.974 0.977 0.980 0.983 0.985 0.986 0.988 0.990 0.992 0.993 0.995 0.997

0.114 0.179 0.255 0.335 0.415 0.492 0.562 0.626 0.681 0.730 0.772 0.807 0.837 0.862 0.882 0.900 0.914 0.926 0.936 0.944 0.951 0.957 0.962 0.966 0.970 0.975 0.979 0.984 0.988 0.992

0.006 0.019 0.046 0.089 0.149 0.219 0.297 0.376 0.452 0.524 0.589 0.647 0.697 0.741 0.778 0.809 0.836 0.858 0.877 0.892 0.906 0.917 0.926 0.934 0.942 0.951 0.960 0.969 0.977 0.984

0.000 0.000 0.000 0.003 0.012 0.032 0.067 0.119 0.183 0.256 0.332 0.406 0.477 0.543 0.601 0.653 0.698 0.737 0.770 0.798 0.822 0.842 0.860 0.875 0.890 0.906 0.923 0.940 0.957 0.970

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.018 0.044 0.083 0.136 0.199 0.267 0.336 0.403 0.466 0.523 0.574 0.620 0.660 0.695 0.728 0.760 0.794 0.830 0.869 0.904 0.934

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.011 0.028 0.058 0.100 0.153 0.212 0.274 0.335 0.394 0.449 0.499 0.548 0.599 0.654 0.715 0.779 0.839 0.889

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.017 0.037 0.069 0.111 0.159 0.212 0.268 0.328 0.396 0.475 0.565 0.661 0.752 0.829

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.009 0.023 0.047 0.084 0.137 0.210 0.306 0.421 0.547 0.668 0.770

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.017 0.045 0.093 0.165 0.263 0.383 0.517 0.646 0.755

αC

0.239

0.475

1.170

2.283

4.357

9.622

16.23

24.92

33.43

35.67

streamtubes originating at the considered injector, both rates being calculated are reservoir conditions. This is not, in the authors' opinion, the best metric to measure pressure support efficiency from peripheral injection. First, if the injector only sweeps watered-out zones, Eq. (57) would yield zero efficiency while in fact pressure support is still ensured as reservoir volume is added. The important quantity is the fraction of water going towards the oil pool, hence associated to streamlines going towards producers; this is more reasonably approached by

Calculations presented here are performed using homogeneous aquifer properties. If the aquifer permeability is reduced, dimensionless time (Eqs. (8) and (13)) will decrease, hence for a given dimensional time injection efficiency will be lower (see Fig. 12). The presence of sealing faults in the north-western part of the reservoir-aquifer boundary in Fig. 1 will qualitatively amount to the same as decreasing the permeability seen by the nearby injectors, hence delay pressure support. For discussion purposes, it can therefore be said that the efficiencies of the injection line, as well as the injectors located close to the faulted area in Fig. 1 are “low” as compared to the efficiency of the other injectors.

IE =

In the context of reservoir engineering, a streamline is defined as a line everywhere tangent to the Darcy velocity; streamlines can be obtained from post processing of conventional finite volumes simulation results, or calculated by streamline simulators (Samier et al., 2002). We have so far discussed the efficiency of peripheral water injection for material balance applications, describing the aquifer with a linear pressure diffusion equation. It is important to discuss the analogies and differences with injection efficiencies obtained from streamlines. Thiele and Batycky (2006) define, for dispersed waterfloods, the efficiency of an injector as

offset oil production rate , water injection rate

offset total production rate , water injection rate

(58)

where rates are still calculated at reservoir conditions. Second, for single phase flows in isotropic and homogeneous media considered in this paper, streamlines are the path followed by passive tracers, whose concentration “s” is governed by the transport equation

4.3. Differences with injection efficiencies obtained from streamlines

IE =

0.036 0.045 0.055 0.068 0.083 0.103 0.128 0.158 0.196 0.245 0.305 0.382 0.481 0.607 0.770 0.981 1.258 1.623 2.107 2.754 3.625 4.804 6.403 8.539 11.27 14.50 17.88 20.85 22.84 23.74

∂s k − ∇·(s∇ψ ) = ∂t μ

∑ peripheral i ∈injectors

sinj, iqinj, iδ (r − rinj, i). (59)

The coupled system given by Eqs. (2) and (59) is triangular, i.e., ψ can be solved for independently of s. Due to the term “s∇ψ ” it is however nonlinear, hence the superposition principle which allowed us to treat pressure support from the aquifer and each injector independently (Eq. (1)) is not applicable to streamline analysis. The presence of a well can therefore affect the streamlines-derived efficiency of a nearby well, which we here refer to as the well shielding effect illustrated in Fig. 17. Ranking injectors based on Eq. (57), or even Eq. (58), would be fallacious. Using Eq. (58) to calculate the efficiency of a peripheral injector

(57)

where the offset oil production is the sum of oil produced from the 15

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 9 ra* = 10 .

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

3

5

7

9.9

0.0010 0.0015 0.0024 0.0037 0.0058 0.0089 0.0138 0.0214 0.0331 0.0513 0.0795 0.1231 0.1906 0.2953 0.4574 0.7084 1.0972 1.6995 2.6323 4.0772 6.3150 9.7813 15.150 23.466 36.346 56.295 87.194 135.05 209.18 324.00

0.687 0.741 0.786 0.824 0.856 0.882 0.903 0.921 0.935 0.947 0.956 0.964 0.970 0.975 0.979 0.982 0.985 0.987 0.989 0.990 0.992 0.993 0.993 0.994 0.995 0.995 0.996 0.997 0.998 0.999

0.459 0.538 0.610 0.674 0.728 0.775 0.814 0.847 0.874 0.896 0.914 0.929 0.941 0.950 0.958 0.965 0.970 0.974 0.978 0.981 0.983 0.985 0.987 0.988 0.990 0.991 0.992 0.994 0.996 0.997

0.114 0.184 0.266 0.351 0.436 0.516 0.589 0.653 0.709 0.756 0.796 0.830 0.857 0.880 0.899 0.915 0.927 0.938 0.946 0.953 0.959 0.964 0.968 0.971 0.975 0.978 0.982 0.985 0.989 0.993

0.006 0.020 0.051 0.100 0.167 0.245 0.329 0.413 0.492 0.565 0.629 0.685 0.733 0.774 0.808 0.836 0.860 0.879 0.896 0.909 0.920 0.930 0.937 0.944 0.950 0.957 0.964 0.972 0.979 0.986

0.000 0.000 0.001 0.004 0.016 0.042 0.087 0.148 0.222 0.302 0.383 0.460 0.531 0.595 0.651 0.699 0.740 0.775 0.804 0.828 0.849 0.866 0.881 0.893 0.905 0.918 0.931 0.946 0.960 0.973

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.011 0.032 0.069 0.122 0.187 0.259 0.333 0.404 0.471 0.532 0.585 0.632 0.673 0.708 0.739 0.765 0.791 0.818 0.848 0.880 0.912 0.940

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.008 0.024 0.054 0.098 0.154 0.217 0.284 0.349 0.410 0.467 0.519 0.565 0.606 0.648 0.692 0.742 0.796 0.850 0.897

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.017 0.040 0.075 0.121 0.175 0.232 0.290 0.346 0.400 0.458 0.523 0.598 0.682 0.766 0.840

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.011 0.028 0.055 0.092 0.137 0.191 0.256 0.339 0.439 0.555 0.672 0.775

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.012 0.028 0.056 0.100 0.163 0.250 0.361 0.491 0.624 0.742

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.008 0.026 0.062 0.122 0.209 0.325 0.461 0.602 0.727

αC

0.493

0.980

2.414

4.713

9.006

19.96

33.91

52.93

74.47

85.29

90.37

0.036 0.045 0.056 0.071 0.088 0.111 0.139 0.175 0.221 0.280 0.356 0.454 0.582 0.749 0.971 1.266 1.662 2.199 2.932 3.939 5.335 7.284 10.02 13.83 19.00 25.57 33.05 40.30 45.75 48.57

4.4. Obtaining peripheral pressure support efficiency from reservoir simulations

would also lump pressure support from the injector and the aquifer, which we here refer to as the aquifer shielding effect illustrated in Fig. 18. When looking at the combined streamlines, it appears that the injector influx is much narrower than what its pressure support in reality is, because the aquifer is helping the injector, while the injector is shielding the aquifer. In summary, streamlines do not provide a meaningful measure of the “pressure support” efficiency of a peripheral injector by itself. Streamlines provide a “transport” efficiency, i.e., a measure of the amount of “molecules” injected by a given source effectively directed towards the reservoir (e.g., in Fig. 17 none of the water from injector 3 is directed towards the reservoir, yet injector 3 is contributing to pressure support). The above discussion might seem at odds with reported successes of waterflood optimization based on streamlines (e.g., see Kornberger and Thiele (2014)), but is not. First, streamlines-based optimization is relevant to dispersed injection (e.g., as a pattern), in reservoirs produced by sweeping. In this case, the efficiency given by Eq. (57) is a useful concept because it will be lower for injectors whose fluxes go through watered-out zones, and higher for injectors sweeping high oil saturation zones. Second, when reallocating rates based on injectors ranking, Thiele and Batycky (2006) insist on the need to increase/decrease rates by the same amount at the injection well and at the offset production wells to keep the streamlines geometry as unchanged as possible. This can be done for in-field injectors, but not for peripheral injectors since it is not possible to change the amount of injected water “taken” by the aquifer.

The ideas presented in his paper arose from the authors' involvement in the optimization of the peripheral waterflood of a giant carbonate field located offshore Abu Dhabi (Pavangat et al., 2015; Nakashima et al., 2015), and the observation that “ready-to-use” diagnostics from available reservoir simulation post-processing tools are not fit for the purpose. Indeed, extracting peripheral pressure support efficiency from a reservoir model is not straightforward, since it is not possible to isolate the individual injectors from each other and from the aquifer as can be done in a simple material balance approach. A possible mitigation to the well shielding effect is to split the reservoir in segments, and only consider injection efficiencies computed by streamlines analysis using Eq. (58) at the segment level. The idea is that if each segment contains a significant number of wells, and has a wide enough aperture compared to the lateral influx distribution extent of each peripheral injector (e.g., see Fig. 15), support is confined within the segment and well-to-well shieldings cancel each other. The approach, used in Pavangat et al. (2015), is illustrated in Fig. 19; for this specific case, the pressure support injection efficiency of segment A2 would be

IEA2 =

∑i ∈A2 injectors offset total production ratei ∑i ∈A2 injectors water injection ratei

, (60)

rates being expressed at reservoir conditions as in Eq. (58). Separating pressure support from the injectors and the aquifer is more complex. If the aquifer drive is weak, lumping these effects and relying on segment-wise efficiencies from streamlines as discussed 16

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 10 ra* = 20 .

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

3

5

7

10

15

19.9

0.0010 0.0016 0.0027 0.0043 0.0071 0.0115 0.0188 0.0307 0.0500 0.0816 0.1330 0.2170 0.3538 0.5770 0.9410 1.5346 2.5025 4.0811 6.6555 10.854 17.700 28.865 47.073 76.767 125.19 204.16 332.94 542.96 885.46 1444.0

0.687 0.746 0.795 0.836 0.869 0.895 0.916 0.933 0.946 0.957 0.965 0.971 0.977 0.981 0.984 0.987 0.989 0.990 0.992 0.993 0.994 0.994 0.995 0.995 0.996 0.996 0.997 0.997 0.998 0.999

0.459 0.547 0.626 0.694 0.751 0.799 0.838 0.869 0.894 0.915 0.931 0.944 0.954 0.962 0.968 0.974 0.978 0.981 0.984 0.986 0.987 0.989 0.990 0.991 0.992 0.993 0.994 0.995 0.996 0.997

0.114 0.193 0.286 0.382 0.475 0.560 0.635 0.700 0.754 0.798 0.835 0.865 0.888 0.908 0.923 0.935 0.945 0.953 0.960 0.965 0.969 0.973 0.975 0.978 0.980 0.982 0.985 0.987 0.991 0.994

0.006 0.023 0.061 0.122 0.203 0.295 0.389 0.479 0.561 0.633 0.694 0.746 0.789 0.824 0.852 0.875 0.894 0.909 0.921 0.931 0.940 0.946 0.952 0.957 0.961 0.965 0.970 0.976 0.982 0.987

0.000 0.000 0.001 0.007 0.026 0.066 0.129 0.209 0.298 0.388 0.473 0.551 0.619 0.677 0.726 0.767 0.801 0.828 0.851 0.870 0.885 0.898 0.909 0.918 0.926 0.934 0.943 0.953 0.965 0.976

0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.009 0.031 0.072 0.133 0.208 0.289 0.371 0.448 0.518 0.580 0.633 0.678 0.716 0.748 0.775 0.798 0.818 0.835 0.853 0.873 0.896 0.922 0.946

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.010 0.031 0.071 0.127 0.195 0.268 0.341 0.411 0.474 0.530 0.579 0.622 0.659 0.690 0.719 0.749 0.783 0.823 0.867 0.909

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.010 0.030 0.066 0.116 0.175 0.239 0.303 0.364 0.421 0.472 0.517 0.560 0.605 0.658 0.721 0.790 0.856

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.010 0.028 0.059 0.102 0.153 0.208 0.263 0.317 0.372 0.432 0.506 0.595 0.695 0.791

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.013 0.033 0.064 0.105 0.153 0.204 0.261 0.328 0.413 0.518 0.636 0.750

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.016 0.037 0.068 0.109 0.162 0.232 0.325 0.445 0.580 0.712

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.018 0.042 0.086 0.154 0.253 0.383 0.533 0.679

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.008 0.027 0.066 0.133 0.233 0.366 0.520 0.670

αC

1.985

3.951

9.733

19.01

36.36

80.78

137.9

217.7

315.9

377.2

435.7

485.6

499.4

above could be a reasonable enough approximation for field management purposes (Pavangat et al., 2015). Another possibility would be to build a matrix of pressure influence as in Stenger et al. (2009), computed from reservoir segment pressure derivatives vs. aquifer segment injection rates. In the case of Fig. 19, such matrix could take the form ∂ψR1 ∂ψR1 ⎛ ⎞ ⎜ PVR1cR1 ∂qinj, A1 PVR1cR1 ∂qinj, A2 …⎟ 1 ⎜ ⎟ ∂ψR2 ∂ψR2 , · Δt ⎜ PVR2cR2 ∂qinj, A1 PVR2cR2 ∂qinj, A2 …⎟ ⎜⎜ ⎟⎟ ⎝ ⋮ ⋮ ⋱⎠

0.031 0.043 0.058 0.076 0.098 0.127 0.164 0.213 0.277 0.361 0.474 0.627 0.834 1.117 1.510 2.059 2.835 3.942 5.536 7.853 11.25 16.25 23.70 34.80 51.21 74.49 104.9 139.4 170.8 190.9

calculations. For a specific injector, this efficiency is defined as the amount of water which has effectively reached the reservoir boundary owing to the presence of said injector, divided by the cumulative injected volume. By taking advantage of the single-phase/singlecomponent pressure diffusion equation linearity, calculation of the aquifer influx has been separated from calculation of the peripheral injection influx, the pressure fields and rates of each problem being summed afterwards. With this approach, reservoir pressure has no effect on the peripheral injectors' efficiencies, and injection rate has no effect on aquifer support. This is at odds with intuition, as materialized by streamlines analysis wherein wells appear to shield or support each other, as well as shield the aquifer. It has however been demonstrated that streamlines can only provide a transport efficiency, which is difficult to trace back to pressure support as required in material balance applications. The calculations presented here are limited to linear and circular geometries, because by symmetry the one-dimensional problem of an injection line or ring yields the same reservoir influx as the twodimensional problem considering injectors as point sources. Of course pressure fields in both cases are different, but material balance is only concerned with the total influx. In linear geometry, analytic solutions for the step injection rate are provided by Bonet and Crawford (1969), while we here proceed with finite volumes modeling to obtain the solutions in circular geometry, tabulated in Appendix C. Solutions for arbitrary injection history can then be obtained by superposition principle. The calculations presented in this paper are also limited to linear physics, i.e., single-phase, weakly compressible Newtonian flow. In particular, if the injected salinity is much lower than the aquifer

(61)

where ψR, PVR and cR are the average pressure, pore volume and total compressibility (i.e., accounting for rock, water and hydrocarbon phases) of the reservoir segment R, respectively. Δt (e.g., one year) is a representative time interval over which the derivatives of pressures vs. injection rates are calculated numerically by marching the simulation forward; this is computationally expensive. The above matrix has a strongly dominant diagonal, qualitatively holding the same information as that of segment-wise injection efficiencies from streamlines; it therefore suffers from the same drawback, i.e., that injection and aquifer support cannot be deconvoluted (mathematically this is due to the fact that changes in segment pressures used to evaluate the derivatives in Eq. (61) involve a change in aquifer influx). 5. Conclusions We have set-up in this paper a rigorous framework for calculating the time dependence of cumulative pressure support efficiency provided by peripheral water injection, to be used in material balance 17

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Table 11 ra* = 50 .

tC*

>C

*= ,C for rinj 1.01

1.02

1.05

1.1

1.2

1.5

2

3

5

7

10

15

20

30

40

49

0.0010 0.0017 0.0030 0.0053 0.0092 0.0160 0.0278 0.0485 0.0844 0.1469 0.2557 0.4452 0.7750 1.3492 2.3488 4.0890 7.1185 12.393 21.574 37.559 65.386 113.83 198.17 344.99 600.59 1045.6 1820.2 3168.9 5516.7 9604.0

0.687 0.753 0.807 0.850 0.883 0.909 0.930 0.945 0.957 0.966 0.973 0.979 0.983 0.986 0.989 0.990 0.992 0.993 0.994 0.995 0.995 0.996 0.996 0.997 0.997 0.997 0.997 0.998 0.998 0.999

0.459 0.559 0.645 0.718 0.778 0.826 0.863 0.893 0.916 0.934 0.947 0.958 0.966 0.972 0.977 0.981 0.984 0.986 0.988 0.990 0.991 0.992 0.993 0.993 0.994 0.994 0.995 0.996 0.997 0.998

0.114 0.205 0.311 0.420 0.522 0.612 0.688 0.751 0.801 0.842 0.873 0.898 0.917 0.932 0.944 0.953 0.960 0.966 0.971 0.974 0.977 0.980 0.982 0.983 0.985 0.986 0.987 0.989 0.992 0.994

0.006 0.027 0.075 0.153 0.251 0.358 0.461 0.556 0.637 0.705 0.761 0.806 0.841 0.870 0.892 0.909 0.923 0.934 0.943 0.950 0.955 0.960 0.964 0.967 0.970 0.973 0.976 0.979 0.984 0.989

0.000 0.000 0.002 0.012 0.045 0.106 0.192 0.292 0.394 0.490 0.575 0.648 0.708 0.757 0.797 0.828 0.854 0.874 0.891 0.904 0.915 0.924 0.931 0.937 0.943 0.948 0.953 0.960 0.969 0.978

0.000 0.000 0.000 0.000 0.000 0.001 0.007 0.029 0.076 0.147 0.235 0.328 0.418 0.501 0.572 0.633 0.683 0.725 0.760 0.788 0.811 0.831 0.847 0.861 0.872 0.884 0.896 0.912 0.931 0.952

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.013 0.043 0.095 0.167 0.249 0.332 0.411 0.482 0.544 0.597 0.642 0.680 0.712 0.739 0.762 0.782 0.801 0.822 0.849 0.882 0.918

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.021 0.055 0.109 0.175 0.248 0.320 0.388 0.449 0.503 0.550 0.590 0.625 0.656 0.685 0.719 0.761 0.814 0.870

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.009 0.028 0.065 0.115 0.175 0.238 0.299 0.357 0.410 0.457 0.500 0.541 0.589 0.651 0.727 0.810

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.004 0.015 0.040 0.080 0.130 0.187 0.244 0.300 0.352 0.400 0.448 0.505 0.579 0.671 0.771

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.007 0.023 0.053 0.094 0.143 0.195 0.248 0.299 0.353 0.419 0.504 0.613 0.730

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.010 0.029 0.059 0.100 0.146 0.195 0.252 0.325 0.423 0.548 0.685

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.008 0.024 0.051 0.087 0.132 0.188 0.264 0.370 0.506 0.655

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.013 0.032 0.064 0.115 0.192 0.306 0.455 0.619

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.012 0.036 0.081 0.159 0.276 0.430 0.602

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.008 0.028 0.072 0.149 0.267 0.423 0.597

αC

12.43

24.75

60.97

119.1

227.8

506.5

865.7

1371

2006

2420

2853

3328

3644

4026

4211

4264

0.016 0.026 0.043 0.068 0.102 0.145 0.200 0.271 0.368 0.502 0.690 0.956 1.339 1.897 2.720 3.949 5.808 8.651 13.04 19.90 30.68 47.76 75.03 118.8 189.2 300.6 467.2 690.3 937.7 1137

priate to restrict the analysis to segment-wise efficiencies in order to mitigate well-to-well and aquifer shielding biases.

salinity, effects of viscosity and density variations are not taken into account. We however do not anticipate such nonlinearities to affect the proposed physical insight. The concept of peripheral injection efficiency is of course also relevant to reservoir simulations, which account for complex reservoir geometry and nonlinear physics. In this case, we have shown that because streamlines give a transport efficiency rather than a pressure support efficiency, when studying peripheral floods it is more appro-

Acknowledgements The authors would like to thank Romain de Loubens, Pierre Samier and Djamel Ouzzane for helpful discussions.

Appendix A. Field units Although the authors find it confusing to publish mathematical developments with formulae expressed in a specific unit set, as “magic” conversion numbers cannot be distinguished from physically meaningful figures, these can still be convenient for the end user. The key parameters in this work are the aquifer constants UL and UC (Eqs. (7) and (12)), and the time units τL and τC (Eqs. (8) and (13)). They write in oilfield units

UL[bbl / psi ] = 0.1781xa[ft ]wo[ft ]h[ft ]ϕct [psi−1], UC[bbl / psi ] = 1.119ro2[ft2 ]

τL[day] = 158.02

(62)

θo[rad ] h[ft ]ϕct [psi−1], 2π

(63)

ϕct [psi−1]μ[cP ]xa2[ft2 ] , k[mD]

(64)

ϕct [psi−1]μ[cP ]ro2[ft2 ] . k[mD]

(65)

and

τC[day] = 158.02

18

Journal of Petroleum Science and Engineering xx (xxxx) xxxx–xxxx

L. Patacchini

Appendix B. Finite volumes code description Numerical solutions to Eq. (2) (or Eq. (29)) in one dimension are obtained using a first-order backwards in time finite volume scheme:

*n +1 = Vi* qiinj

ψ *n +1 − ψi*+1n +1 ψi*n +1 − ψi*n ψ *n +1 − ψi*−1n +1 , + Ai*−1/2 i + Ai*+1/2 i Δt* Δy* Δy* i −1/2

(66)

i +1/2

* * where Vi* is the dimensionless volume of cell i, Ai−1/2 is the dimensionless interface area between cells i and i − 1, Δyi−1/2 is the dimensionless distance between the centers of cells i and i − 1, and n is the time-step index. * = 0 and y* = 1. The reservoir influx rate is then In linear geometry, y* ∈ [0, 1] should be intended as x*, with A* = 1, Vi* = yi*+1/2 − yi*−1/2 , y−1/2 Nr+1/2 qe =

UL 7 ∂ψ * , τL ∂y* |y*=0

(67)

where UL is the aquifer constant given by Eq. (7). 2 2 * = 1 and y* − yi*−1/2 )/2 , y−1/2 In circular geometry, y* ∈ [1, ra*] should be intended as r*, with Ai*−1/2 = yi*−1/2 , Vi* = (yi*+1/2 = ra*. Reservoir influx Nr +1/2 rate is then

qe =

UC 7 ∂ψ * , τC ∂y* |y*=1

(68)

where UC is the aquifer constant given by Eq. (12). In Eqs. (67) and (68), the derivative is calculated by off-centered second order finite differences. Boundary conditions are set as follows. For the aquifer problem n ⎧ = ψo*(t n ) ⎪ψ * −1/2 , ∀ n > 0: ⎨ n ⎪ ⎩ ∀ i ∈ [1, Nr ]: qi* = 0

(69)

and for the peripheral injection problem

⎧ ψ *n = 0 ⎪ −1/2 τL, C ∀ n > 0: ⎨ *n q (t n ) ⎪ qiinj = UL, C 7 inj ⎩

(70)

where iinj is the injection cell (or cells if the injection point does not exactly correspond to a cell center). Numerical solutions to Eq. (2) (or Eq. (29)) in two dimensions are obtained by extending the stencil of the discretized Eq. (66) in Cartesian or cylindrical coordinates in linear and circular geometries, respectively. Depending on the situation (domain size and injection location), we use linear or logarithmic cell spacing; if the spacing is uniform (constant Δy*), the scheme is second order in space (Allaire, 2005). Eq. (66) being linear, advancing from time-step n to n + 1 involves a single Jacobian inversion per time-step, the Jacobian being constructed once and for all at the beginning of the simulation. Appendix C. Injection efficiency tabulations Table 2 is a tabulation of ,L as given by Eq. (37), while Tables 3–11 are tabulations of ,C to be used in Eq. (33). Tables extend to tL* = 4 and 2 * tC = 4(ra* − 1) , which is sufficient for the decaying exponentials in Eqs. (37) and (40) to be negligible. Solutions for larger dimensionless time can be obtained from Eqs. (37) and (40) using the values of αL and αC in the last row of each table. For completeness, aquifer cumulative rate functions are tabulated as well, although these have already been published by van Everdingen and Hurst (1949) and extensively reproduced the literature.

Everdingen-Hurst Dimensionless Variables For Water Encroachment. SPEREE 15433. Kornberger, M., Thiele, M.R., 2014. Experiences With An Efficient Rate-Management Approach For The 8th Tortonian Reservoir In The Vienna Basin. SPEREE 166393. Landmark, 2014. Nexus Technical Reference Guide, Version July 2014. Leung, W.F., 1986. A Fast Convolution Method For Implementing Single-porosity Finite/ infinite Aquifer Models For Water-influx Calculations. SPEREE 12267. Moncorgé, A., Patacchini, L., de Loubens, R., 2012. Multi-phase, multi-component simulation framework for advanced recovery mechanisms. SPE paper 161615 presented at the ADIPEC held in Abu Dhabi, UAE. Nakashima, T., Kumar, J., Draoui, E., 2015. Development of a giant carbonate oil field, Part 2: migration from pressure maintenance development to sweep oriented IOR development. SPE paper 177801 presented at the ADIPEC held in Abu Dhabi, UAE. Pavangat, V., Patacchini, L., Goyal, P., Mohamed, F., Lavenu, A., Aubertin, F., Nakashima, T., 2015. Development of a giant carbonate oil field, Part 1: fifty years of pressure maintenance history. SPE paper 177768 presented at the ADIPEC held in Abu Dhabi, UAE. Peaceman, D.W., 1978. Interpretation of Well-Block Pressures in Numerical Reservoir Simulation SPE 6893. Petroleum Experts, 2012. MBAL User Manual, Version 9. Samier, P., Quettier, L., Thiele, M.R., 2002. Applications Of Streamline Simulations To Reservoir Studies. SPEREE 78883. Schilthuis, R.J., 1936. Active oil and reservoir energy. Trans. AIME 118, 33–52.

References Allaire, G., 2005. Analyse numérique et optimisation, LesÉditions de l’École Polytechnique. Aziz, K., Settari, A., 1979. Petroleum Reservoir Simulation. Applied Science Publishers, London. Bear, J., 1972. Dynamics of Fluids in Porous Media. Dover Publications, New York. Bonet, E.J., Crawford, P.B., 1969. Aquifer behavior with injection. J. Pet. Technol. 2256, 1210–1216. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, 2d edition. Oxford U. Press, New York City. Carter, R.D., Tracy, G.W., 1960. An Improved Method For Calculating Water Influx. SPE 1626. Center for Petroleum and Geosystems Engineering, 2013. The University of Texas at Austin, Technical Documentation for UTCHEM 2013_8, A Three-Dimensional Chemical Flood Simulator. Coats, K.H., 1962. A Mathematical Model Water Movement About Bottom-water-drive Reservoirs. SPE 160. Dake, L.P., 1978. Fundamentals of Reservoir Engineering. Elsevier Science, New York. Fetkovich, M.J., 1971. A simplified approach to water influx calculations – finite aquifer systems. J. Pet. Technol. SPE 2603. Klins, M.A., Bouchard, A.J., Cable, C.L., 1988. A Polynomial Approach To The van

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flow problems in reservoirs. Trans. AIME 186. Vinsome, P.K.W., Westerveld, J.D., 1980. A simple method for predicting cap and base rock heat losses in thermal reservoir simulators. J. Can. Pet. Technol., 87–90. Wang, B., Litvak, B.L., Bowman, G.W., 1992. OILWAT: microcomputer program for oil material balance with gas cap and water influx. SPE paper 24437 presented at the Petroleum Computer Conference held in Houston, TX.

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20