The main focus of the paper is on the effects of addition of cracks to pre-existing materials. When cracks are added to homogeneous background, we think of this as being a granite-like material. This case has been treated in the main text. To show the generality of the method, we want to give a brief treatment of the sandstone-like situation of a material having a pre-exisiting porosity in this appendix. This porosity may itself be either liquid saturated or gas saturated (dry). For simplicity, we will assume here that is liquid saturated when liquid saturated cracks are to be added, dry when dry cracks are to be added.
There are two further alternatives to be considered. First, cracks may be added randomly to the pre-existing material. Second, cracks may be added preferentially to the porosity-free host material. We discuss these two cases in turn.