| Abstract: | The flow of a single-phase compressible fluid from the rock matrix to fractures has been modeled using the pressure diffusion equation. Pressure histories are presented for homogeneous isotropic blocks bounded by planar fractures. The case of an infinite slab bounded by planes of constant pore pressure was studied. The slab was divided by a planar fracture perpendicular to the planes. Lateral flow was found to cease once equilibrium is reached between the fracture and the matrix. Disequilibrium is found to be short-lived for laboratory-sized specimens of typical reservoir rock. The most important parameter in cross-flow is the distance ℓ between the two planes of constant pore pressure. The lifetime of the pressure transient increases with the square of this length. Cross-flow is an edge effect. Flow perpendicular to the pressure gradient only occurs within a distance of 2.5ℓ from the fracture face. This distance is not affected by other parameters such as permeability. When a second fracture was added, parallel to the first, the cross-flow behavior was nearly identical to the one fracture case if the spacing of the fractures is greater than ℓ. The total amount of fluid entering the fracture from the blocks dropped off significantly as the spacing decreased from ℓ. When orthogonal fracture sets were introduced, the pressure history of a point within the block differed from the original case only near the corners of a large block, or for blocks smaller than the distance ℓ. The pressure histories can therefore be applied to a variety of geometries and sizes of blocks with any petrophysical properties. The pressure history of the blocks of the continuum model of naturally fractured reservoirs was examined with a discrete mathematical model. An analytical solution to the pressure diffusion equation with time dependent boundary conditions is presented for blocks in both a finite and infinite reservoir. The block permeability and the block size determine when, and not whether cross-flow will take place. Cross-flow occurs as the pressure in the fracture drops, although this drop reduces the fracture permeability. Cross-flow production curves from the discrete model are compared to a field example. The match suggests that cross-flow controls the production history in certain fractured reservoirs. |
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