FLUID-FLOW SIMULATION

We assume a simple model of diffusive fluid-flow in this study. We simulate the fluid flow response of two water injection well galleries in a thin (50 m thick) horizontal reservoir at 2 km depth. Please refer to Figure for the well gallery production geometry.

 

waterflood
Figure 1 Reservoir production geometry. Two well galleries (lines of injecting wells in the direction perpendicular to the plane of the page) inject the waterflood, and a central line of wells produces the oil. The Ottawa sand reservoir is at a depth of 2 km and is 50 m thick.

Initially, the reservoir is assumed to be 100% saturated with light oil in a consolidated Ottawa sand of 33% porosity. Clearly this is an idealized case in which the connate water saturation S_wc is zero. After some characteristic time period of water injection, we calculate the pore pressure and water saturation as a function of distance from the injection wells. Assuming isothermal diffusive flow in a thin homogeneous reservoir of constant permeability k, and water and oil of homogeneous (but different) viscosity, both the pore pressure P_p and water saturation S_w can be modeled as simple diffusion:

 

$$P_p(r,t) = \frac{P_w}{2\sqrt{\pi \kappa_p t}} e^{-x^2/4\kappa_p t} \;,$$ (1)

 

$$S_w(r,t) = \frac{1}{2\sqrt{\pi \kappa_s t}} e^{-x^2/4\kappa_s t} \;,$$ (2)

where P_w is the pressure with which the water is being injected, and $\kappa_p$, $\kappa_s$ are the effective diffusivities for the pressure and water saturation fronts, which are related to each other and depend on reservoir permeability k and fluid viscosity $\eta$. The horizontal distance from the injection well is denoted as x. We consider the fluid flow to be scale invariant in that the fluid will diffuse a characteristic distance x' away from the injector in a characteristic time t' given by the skin depth diffusion relation:

 

$$x' \approx \sqrt{\kappa t'} \;.$$ (3)

Dake (1978, p. 352) gives the physical conditions for which diffusive fluid flow is a valid approximation:

• When displacement occurs at very high injection rates so that the condition of vertical equilibrium is not satisfied and the effects of the capillary and gravity forces are negligible.
• For displacement at low injection rates in reservoirs for which the measured capillary transition zone greatly exceeds the reservoir thickness, and the vertical equilibrium condition applies.

Under either of these conditions, we can simulate Gaussian distributions of pore pressure and water saturation. Additionally, since pressure and water saturation change much more quickly than reservoir temperature, we assume that the waterflood is an isothermal process. These approximations are justified because in this study we concentrate on the principal possibility of the seismic detection of production-associated changes in the reservoir, rather than on the detailed nature of reservoir fluid flow simulation.