# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 1 Introduction  
The study of porous materials, where the flow of fluid and solid deformation are coupled, is essential in several areas of science and engineering. The theory of poroelasticity is a mathematical formulation developed to describe these coupled processes and predict the response of fluid saturated/unsaturated porous media to external loading. There are different types of porous materials that are studied under this theory, such as soil, rock, concrete and other man-made materials. Poroelasticity has a wide range of applications in different disciplines of engineering mechanics and natural sciences. Some of the application areas include geomechanics, biomechanics, reservoir engineering and earthquake engineering, for problems where plastic deformations are negligible. In addition to these diverse areas of application, it is gaining popularity in the study of modern man-made porous media in material science. The mathematical formulations describing the fluid-solid coupled processes are developed based on porous media theory where the multiphase medium is approximated as a continuum, [1]. The volume fraction concept is used for averaging the properties of the multiphase medium in a continuum formulation. The governing partial differential equations of poroelasticity were first developed for a one-dimensional case by [PERSON] [2, 3]. The formulations were later generalized for a three-dimensional case and extended by [PERSON] [4, 5, 6]. The mathematical formulations have since been extensively studied by several researchers. Various analytical and numerical studies have been proposed in the literature. Analytical solutions were obtained for problems with simplified material domains and boundary conditions. Application to boundary value problems with complex material domains and boundary conditions required the use of numerical methods. The emergence of the finite element method opened the door for a detailed numerical study of poroelasticity and for application to arbitrary geometries and boundary conditions. The finite element method was first applied to the governing equations of poroelasticity to solve the initial boundary value problem of flow in a saturated porous elastic medium by [PERSON] and [PERSON] [7]. [PERSON] et al. [8] also used the finite element method for plane strain consolidation problems and verified the results against closed form solutions. The application of the finite element method started gaining momentum afterwards and several researchers engaged themselves not only on application problems but also in the investigation of the numerical properties of the method within the context of poroelasticity. [PERSON] and [PERSON] [9] applied the finite element method to partially saturated elastic porous media and first noticed the ill-conditioning of the matrix equations that may result when an incompressible fluid is assumed to occupy the pore spaces. [PERSON] and [PERSON] [10] investigated the stability of the numerical solution when the finite element method is applied to [PERSON]'s consolidation equations. The stability was studied for different numerical integration schemes and time-step sizes. The numerical performance of some finite element schemes for analysis of seepage in porous elastic media was studied by [PERSON] et al. [11]. They studied various spatial and temporal discretization schemes and evaluated the numerical performances against the analytical solution of [PERSON]'s one-dimensional consolidation problem. Triangular and quadrilateral elements with equal and mixed orders of interpolation for the displacement and pressure were considered. It was shown that the elements with equal orders of interpolation showed oscillatory behavior in the solution. [PERSON] and [PERSON] [12] derived a lower bound for the time-step size in the analysis of consolidation by finite elements in terms of the mesh size and the coefficient of consolidation.