Modelling flow in a pressure-sensitive, heterogeneous medium

Modeling ow in a pressure-sensitive, heterogeneous medium D. W. Vasco 1 and Susan E. Minkoff 2 Berkeley Laboratory, University of California, Berkeley, California 94720 University of Maryland, Department of Mathematics and Statistics, 1000 Hilltop Circle, Baltimore, Maryland, 21250 SUMMARY Using an asymptotic methodology, including an expansion in inverse powers of ω, where ω is the frequency, we derive a solution for flow in a medium with pressure dependent properties. The solution is valid for a heterogeneous medium with smoothly varying properties. That is, the scale length of the heterogeneity must be significantly larger then the scale length over which the pressure increases from it initial value to its peak value. The resulting asymptotic expres- sion is similar in form to the solution for pressure in a medium in which the flow properties are not functions of pressure. Both the expression for pseudo-phase, which is related to the ’travel time’ of the transient pressure disturbance, and the expression for pressure amplitude contain modifications due to the pressure dependence of the medium. We apply the method to synthetic and observed pressure variations in a deforming medium. In the synthetic test we model one- dimensional propagation in a pressure-dependent medium. Comparisons with both an analytic self-similar solution and the results of a numerical simulation indicate general agreement. Fur- thermore, we are able to match pressure variations observed during a pulse test at the Coaraze Laboratory site in France. Key words: fluid flow, permeability, asymptotic methods, pressure 1 INTRODUCTION The introduction of a volume of fluid into the Earth will induce some degree of deformation within the geologic material compris- ing the subsurface. In most cases the resulting deformation is not significant, or even observable, and may be safely ignored. How- ever, in some situations, such as poorly consolidated sediments, fractured media (Gale 1975, Jones 1975, Noorishad et al. 1992, Cappa et al. 2008), and large pressure changes and flow rates (Fatt 1958, Raghavan et al. 1972, Rutqvist et al. 1998), the changes within the matrix may impact the flow in important ways. For ex- ample, large pressure changes can modify the flow properties such as porosity and permeability. Also, due to the transmission of stress within the medium, pressure changes can lead to non-local effects. A comprehensive approach to this problem involves coupled modeling of the fluid flow and the deformation. Such coupled mod- eling can be complicated, making numerical methods attractive. While the utility of numerical modeling is well established (Noor- ishad et al. 1992, Rutqvist et al. 2002, Minkoff et al. 2003, Minkoff et al. 2004, Dean et al.2006), analytic solutions can aid in our un- derstanding of the factors, such as medium parameters, contributing to the calculated pressure and deformation. Thus, analytic solutions may provide valuable insight, complementing existing purely nu- merical approaches. To date, analytic studies of the coupled prob- lem have been restricted to relatively simple cases, such as for a ho- mogeneous medium in which the flow properties do not change as a function of pressure. A classic example is a homogeneous poroe- lastic medium (Booker and Carter 1986, Rudnicki 1986) which can exhibit the non-local effects noted above (Segall 1985). Recently, a semi-analytic solution was developed for a poroelastic medium with heterogeneous flow properties which also contained non-local effects due to the coupling of the diffusive Biot wave and the ’fast’ elastic wave (Vasco 2008, 2009). While coupled modeling provides a more complete and satis- factory approach for understanding flow in a deformable medium, it is also useful to examine particular aspects of this problem. In fact, the studies just mentioned focused on the coupling between elastic deformation in the medium and the fluid flow, neglecting the change in flow properties due to changes in the effective stress. In this paper we examine the alternative situation, in which the flow properties are functions of the fluid pressure and we neglect the transmission of elastic deformation throughout the medium. As such, this paper is a complement to the studies cited above. Allowing for pressure dependent flow properties leads to a non-linear diffusion equation (Wu and Pruess 2000). The non- linear diffusion equation has been used to model the flow of flu- ids in a deformable medium (Barenblatt 1952), flow in rock joints with pressure-dependent openings (Murphy et al. 2004), tempera- ture and pressure waves in fluid saturated rock (Natale and Salusti 1996), and compaction in sedimentary basins (Audet and Fowler 1992) among other things (Newman 1983). The non-linear diffu- sion equation has been studied rather extensively from a mathe- matical perspective (Crank 1975, Hayashi et al. 2006), particularly for a homogeneous medium and the case that the permeability is proportional to the pressure raised to a power. In this instance one may derive an exact self-similar solution to the non-linear diffu-