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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

image

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 solution (2) can be written as

image

where img is a characteristic fluid volume. The above expression suggests that the relationship between the induced pore pressure Δρ, the injected volume V, and the injection rate img 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

image

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 img, 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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