Penny-Shaped Crack in a Poroelastic Plate

Hydraulic fracking is used in the petroleum industry for the secondary production of either oil or natural gas. Normally a high pressure fluid is injected through a well bore to create cracks or fissures in the oil bearing sediment. Usually small grains of sand are suspended in the fracking fluid and remain after the pressure is released; thereby, allowing the oil or gas to pass more freely through the fissure. Because of the large difference of permeability of the fissure to the oil bearing strata, oil or gas flows easily into the fissure and is easily produced through the well bore. In our model we consider the fractured vein to be approximated by a penny-shaped crack, placed horizontal to the well bore. Because of symmetry conditions, in the case of a gravity drainage model for producing oil, a crack lying at the bottom of the oil bearing strata can be modeled by a sediment of twice that size with a crack lying in the center. For the case of producing natural gas a similar argument may be made. What is of interest to the petroleum industry is to determine the size of the fracture, in our case that would be the radius. This problem may be reformulated as an inverse problem where we attempt to find the radius of the fracture from sonic information, for example, by measuring the scattered Lamb waves which are created by a suddenly imposed stress on the fracture. In the present work we consider the direct problem, namely what are the amplitudes of the scattered wave for known penny-shaped crack. In a subsequent paper we will consider the inversion of the Lamb wave data to reveal the radius of the fissure.