Transient poroelastic response of equivalent porous media over a mining panel

A transient poroelastic formulation based on the Biot consolidation theory is presented to study the impact of underground longwall mining of coal on strata deformation and the subsequent modification of hydrogeological properties. A finite-element model is used where the geometric conditions may be complex. The important parameters in this model are identified as pore pressure ratio, ct 1 and relative compressibility, ~q with the latter exerting far greater influence on the resulting fluid pressure field than the former. This study indicates that the fluid pressure distribution is strongly controlled by the stress distribution, or the strata deformation induced by mining has a great impact on the change in hydrogeological environment in the mining region. Introduction Flow systems in porous media are often transient, resulting in changes in hydraulic head or pore pressures and corresponding mass fluxes with time. This behavior may result from changes in either fluid pressure or total stress boundary conditions applied to the system. It is the admissibility of changes in total stress within the system that describes the essence of coupled poroelastic behavior and sets it apart from decoupled diffusive (flow) systems. Comprehensive coupling between stresses and pore pressures was first rationalized by Biot (1941) and later adopted in many applications to specific poroelastic systems (Ghaboussi and Wilson, 1973; Zienkiewicz et al., 1977; Simon et al., 1984). The poroelastic theory has been applied to problems of consolidation (Booker and Small, 1987), prediction of surface subsidence caused by groundwater extraction (Lewis and Schrefler, 1987) and evaluation of the stress, pressure and failure fields around boreholes (Detournay and Cheng, 1988), among others. This study applies Biot's theory to mining-related problems, evaluating the coupled transient pore pressure response and solid displacement fields that develop as a result of underground mining. Of particular interest is the effect of mining on the behavior of overlying strata where transient pore pressure changes may aid future evaluation of failure around the excavation or alternatively define the mechanical properties of the deforming system. Poroelastie response For transient flow systems, the ability of water to be removed from or added to storage within the porous medium controls the rate of response. Specific storage controls this process and may be defined as: ('+ ) S,=pg ~ ~s+flf (1) where p is fluid density, g is gravitational acceleration, Ks is bulk modulus of the skeleton of the porous medium, ~t s is compressibility of the solid grains and is equal to (1 n)/K s where n is porosity and Kg is solid bulk modulus of the grains, and flf 0013-7952/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved. 50 M. BAI AND D. ELSWORTH NOTATION (M = mass, L = length, t = time) a* load width L a,b half width and length of L element B strain-displacement matrix C. consolidation coefficient L 2 t t E elastic modulus M L z t 2 F applied boundary tractions f nodal force M L t 2 G shear modulus M L 1 t 2 g gravitational acceleration L t 2 H , R Biot constant H c height of column L h hydraulic head L K hydraulic conductivity L t Ks skeletal modulus M L 1 t 2 Kf fluid bulk modulus M L z t 2 Kg grain bulk modulus M L 1 t 2 n porosity N,M element based shape functions p fluid pressure M L ~ t 2 Q boundary discharge q specific discharge L 3 t 1 S domain surface L 2 Se specific storage L 2 t 2 S~ specific surface L 1 t real time t At time increment t T dimensionless time factor u displacement L V i flow velocity L t 1 V domain volume L 3 x , y distances L pore pressure ratio ~1 pore pressure ratio ~2 relative compressibility ~s compressibility of solid grains flf fluid compressibility e normal strain •kk,O volumetric strain or,or e stress, effective stress M L 1 l 2 akk,tr e total stress M L 1 t -2 q,~ mapping coordinates p fluid density M L 3 2 Lame constant v Poisson ratio e9o poroelastic constant co overrelaxation factor 7 fluid unit weight M L 2 t 2 6 Kronecker delta Z weighting constant is fluid compressibi l i ty given by n / K f with Kf being fluid bulk modulus . Combin ing Darcy ' s law together with cont inui ty requirements, assuming a cons tant fluid density and considering solid compress ion due to total stress changes, yields a flow equat ion of the form: V ( K V h ) = ( ~ s + f l0P + O')O~kk (2) where K is hydraul ic conductivi ty, h is the hydraulic head (pressure head plus elevation head), p is the fluid pressure, ekk is total volumetr ic strain, and 09 0 is a constant . Neglecting the influence o f t ime dependent changes in elevation head enables pore fluid pressure to be defined as p = p g h . Substi tut ing for head in Eq. 2 yields: 1 --V(KVp) = (~s + flf)P + ('O0~kk Pg In (3) this equat ion specific s torage is in a fo rm representing the compressibi l i ty o f fluid and solid grains with skeletal compressibi l i ty encompassed in the term co o. The general s t ress-s t ra in relat ionship incorporat ing effective stress effects th rough pore pressures m a y be writ ten as: (1 + v) v C t , O'ij "~ O'kk (~ij ~ " ~ . P ~ i j (4) ~ij E . 5 / ' / where E is elastic modulus and v is Poisson ratio, ~1 and H are constants and will be discussed later and 6q is the Kronecke r delta. The equi l ibr ium equat ion in the absence o f self weight and inertial effects may be given as: