  In this paper, we consider the numerical solution of some nonlinear
poroelasticity problems that are of Biot type and develop a general algorithm
for solving nonlinear coupled systems. We discuss the difficulties associated
with flow and mechanics in heterogenous media with nonlinear coupling. The
central issue being how to handle the nonlinearities and the multiscale scale
nature of the media. To compute an efficient numerical solution we develop and
implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves
nonlinear problems on a coarse grid by constructing local multiscale basis
functions and treating part of the nonlinearity locally as a parametric value.
After linearization with a Picard Iteration, the procedure begins with
construction of multiscale bases for both displacement and pressure in each
coarse block by treating the staggered nonlinearity as a parametric value.
Using a snapshot space and local spectral problems, we construct an offline
basis of reduced dimension. From here an online, parametric dependent, space is
constructed. Finally, after multiplying by a multiscale partitions of unity,
the multiscale basis is constructed and the coarse grid problem then can be
solved for arbitrary forcing and boundary conditions. We implement this
algorithm on a geometry with a linear and nonlinear pressure dependent
permeability field and compute error between the multiscale solution with the
fine-scale solutions.
