*Pore Pressure Induced by Fluid Injection*

The dependence of the induced pore pressure on the operation parameters (injection rate, volume of fluid injected), on position and time, and on the hydraulic properties of the reservoir is illustrated in this appendix by considering the simple example of fluid injection in a disk-shaped reservoir. The analysis shows that different parameters control the pore pressure at the beginning of the injection operation and once enough fluid has been injected in the reservoir (see also Nicholson and Wesson, 1990).

The pore pressure induced by injection of fluid, Δρ, is to a good approximation governed by the diffusion equation

*c*∇^{2}Δρ = ∂Δ/∂t + source

where *c* denotes the hydraulic diffusivity equal to *c = k*/μS. In the above, *k* is the intrinsic permeability of the rock (generally expressed in Darcy), μ is the fluid viscosity, and *S* is the storage coefficient, a function of the compressibility of both the fluid and the porous rock. The diffusion equation imposes a certain structure on the link between the magnitude of the induced pore pressure Δρ, the injected fluid volume *V,* and the rate of injection *Q _{o}.*

As an example, we consider the injection of fluid at a constant volumetric rate *Q _{o}*, at the center of a disk-shaped reservoir of thickness

At early time (to be defined more precisely later), the pore pressure perturbation induced by injection of fluid has not reached the boundary of the reservoir. The induced pore pressure field is then given by the source solution for an infinite domain, a solution of the form (Wang, 2000)

where *r* is the radial distance from the injection well, *t* is time, and F is a known function. The quantity where ρ_{*} is a characteristic pressure (i.e., a yardstick for measuring the induced pressure) given by

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APPENDIX H
Pore Pressure Induced by
Fluid Injection
The dependence of the induced pore pressure on the operation parameters (injection
rate, volume of fluid injected), on position and time, and on the hydraulic properties of the
reservoir is illustrated in this appendix by considering the simple example of fluid injection
in a disk-shaped reservoir. The analysis shows that different parameters control the pore
pressure at the beginning of the injection operation and once enough fluid has been injected
in the reservoir (see also Nicholson and Wesson, 1990).
The pore pressure induced by injection of fluid, Δρ, is to a good approximation governed
by the diffusion equation
c∇2Δρ = ∂Δρ/∂t + source
where c denotes the hydraulic diffusivity equal to c = k/μS. In the above, k is the intrinsic
permeability of the rock (generally expressed in Darcy), μ is the fluid viscosity, and S is the
storage coefficient, a function of the compressibility of both the fluid and the porous rock.
The diffusion equation imposes a certain structure on the link between the magnitude of
the induced pore pressure Δρ, the injected fluid volume V, and the rate of injection Qo.
As an example, we consider the injection of fluid at a constant volumetric rate Qo, at
the center of a disk-shaped reservoir of thickness H and radius R. It is assumed that the
reservoir is thin (i.e., H/R≪1), and also that the pore pressure is uniform over the thickness
of the layer, which implies, depending on the manner the fluid is injected, that some time
has elapsed since the beginning of the operation.
At early time (to be defined more precisely later), the pore pressure perturbation in-
duced by injection of fluid has not reached the boundary of the reservoir. The induced pore
pressure field is then given by the source solution for an infinite domain, a solution of the
form (Wang, 2000)
Δρ(r,t) = ρ*F(r/√ [ct]) (1)
where r is the radial distance from the injection well, t is time, and F is a known function.
The quantity where ρ* is a characteristic pressure (i.e., a yardstick for measuring the induced
pressure) given by
ρ* = μQo /kH
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APPENDIX H
Once the time elapsed since injection started becomes larger than a fraction, say 0.1,
of the characteristic time t* =R2/c, then the evolution of the induced pore pressure becomes
influenced by the finiteness of the reservoir. Formally, the pore pressure solution can then
be expressed as
Δρ(r,t) = ρ*Ρ(r/R, t/t*) (2)
The function P(ρ,t) can be determined semianalytically. If the elapsed time t is expressed
as the ratio of the injected volume V to the rate of injection Qo (i.e., t=V/Qo), then solu-
tion (2) can be written as
Δρ(r,V) = ρ*Ρ(r/R ,V/V* ) (3)
where V* =( Qo R2)/c is a characteristic fluid volume. The above expression suggests that the
relationship between the induced pore pressure Δρ, the injected volume V, and the injec-
tion rate Qo is not straightforward. However, Equation (3) shows important trends; for
example, a decrease of the permeability causes an increase of the characteristic pressure,
or an increase of the storage coefficient causes a decrease of the pore pressure, all other
parameters kept constant.
At small time t≪t* , the dimensionless pressure P = Δρ/ρ* reduces to the unbounded
domain solution F, while at large time t≫t* , the pressure tends to become uniform and the
pore pressure is simply given by
ρ ≅ V/(πR2HS) (4)
as the function P(ρ,t) behaves for large t as P ≅ t/π. Thus, at large time, the pore pressure
is simply proportional to the volume of injected fluid (Figure H.1). Equation (4) actually
indicates that the large-time pore pressure is simply the ratio of the injected volume over
the reservoir volume, divided by the storage coefficient.
The previous material provides some information about the link between pore pressure,
injected volume, and injected rate for the particular case of an injector centered in a disk-
shaped reservoir. These ideas can be generalized to more realistic cases. For example, for an
arbitrarily shaped reservoir with n wells, each injecting at a rate Qo, the general expression
for the induced pore pressure can be written as
Δρ(x,t) = ρ*ς{x/L, t/t* ; n, (xi , i =1, n), reservoir shape}
where the characteristic pressure and time are given by
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Appendix H
4
10
2
10
0
10
−2
10
−4
10
−6
10
−8
10
−10
10 −2 −1 0 1 2 3
10 10 10 10 10 10
FIGURE H.1 Injection of fluid at a constant rate at the center of a disk-shaped reservoir. Plot of the
dimensionless pore pressure Δρ/ρ* with respect to the dimensionless time t = t/t* (equal to V/V*) for three
values of the dimensionless radius Q = r/R. This plot indicates that the pressure response is similar to the
response of an unbounded reservoir as long as t ≤ 0.2 and that the pressure is approximately uniform
and proportional to the volume of fluid injected when t ≥ 10. The dashed-line curves correspond to the
solution F for an unbounded reservoir.
ρ* = μQo /kL, t* = L2/c
with L denoting a relevant length scale of the reservoir. Also x refers to the position of the
field point, and xi to the position of the source i. At large time, the induced pore pressure
is approximately given by
ρ ≅ V/(SVreservoir)
where V is the total volume of fluid injected (V = nQot) and Vreservoir is the volume of the
reservoir.
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APPENDIX H
REFERENCES
Nicholson, C., and R.L. Wesson. 1990. Earthquake Hazard Associated with Deep Well Injection: A Report to the U.S.
Environmental Protection Agency. U.S. Geological Survey Bulletin 1951, 74 pp.
Wang, H.F. 2000. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton, NJ: Princeton
University Press.
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