Finite element modeling of hydraulic fracturing in 3D

A procedure based on the finite element method is suggested for modeling of 3D hydraulic fracturing in the subsurface. The proposed formulation partitions the stress field into the initial stress state and an additional stress state caused by pressure buildup. The additional stress is obtained as a solution of the Biot equations for coupled fluid flow and deformations in the rock. The fluid flow in the fracture is represented on a regular finite element grid by means of “fracture” porosity, which is the volume fraction of the fracture. The use of the fracture porosity allows for a uniform finite element formulation for the fracture and the rock, both with respect to fluid pressure and displacement. It is demonstrated how the fracture aperture is obtained from the displacement field. The model has a fracture criterion by means of a strain limit in each element. It is shown how this criterion scales with the element size. Fracturing becomes an intermittent process, and each event is followed by a pressure drop. A procedure is suggested for the computation of the pressure drop. Two examples of hydraulic fracturing are given, when the pressure buildup is from fluid injection by a well. One case is of a homogeneous rock, and the other case is an inhomogeneous rock. The fracture geometry, well pressure, new fracture area, and elastic energy released in each event are computed. The fracture geometry is three orthogonal fracture planes in the homogeneous case, and it is a branched fracture in the inhomogeneous case.     

The Paleo Heat Flow Contents in Vitrinite Reflectance Observations

The heat flow content in vitrinite reflectance (VR) observations is studied based on a simple model of burial at a constant rate. The model is made dimensionless, and it has just one parameter except for the paleo heat flow. The question of existence and uniqueness of a solution is studied, and there exist in general no paleo heat flow that will reproduce a given VR-depth curve. But a solution is unique if it exists. A computed VR-depth function is shown to be smooth, even for piecewise constant heat flow histories. The paleo heat flow can be obtained from a VR-depth function after two times with derivations. It is also shown how the present day thermal gradient can be obtained by derivation of a VR-depth representation. The one-parameter model allows for approximate expressions for the optimal paleo heat flow as a step function. The results obtained from the one-parameter model is also compared with similar results from a real case study from the North Sea using a state-of-the-art basin simulator.     

A model for underpressure development in a glacial valley,an example from Adventdalen,Svalbard

The underpressure observed in the glacial valley Adventdalen at Svalbard is studied numerically with a basin model and analytically with a compartment model. The pressure equation used in the basin model, which accounts for underpressure generation, is derived from mass conservation of pore fluid and solid, in addition to constitutive equations. The compartment model is derived as a similar pressure equation, which is based on a simplified representation of the basin geometry. It is used to derive analytical expressions for the underpressure (overpressure) from a series of unloading (loading) intervals. The compartment model gives a characteristic time for underpressure generation of each interval, which tells when the pressure state is transient or stationary. The transient pressure is linear in time for short‐time spans compared to the characteristic time, and then it is proportional to the weight removed from the surface. We compare different contributions to the underpressure generation and find that porosity rebound from unloading is more important than the decompression of the pore fluid during unloading and the thermal contraction of the pore fluid during cooling of the subsurface. Our modelling shows that the unloading from the last deglaciation can explain the present day underpressure. The basin model simulates the subsurface pressure resulting from erosion and unloading in addition to the fluid flow driven by the topography. Basin modelling indicates that the mountains surrounding the valley are more important for the topographic‐driven flow in the aquifer than the recharging in the neighbour valley. The compartment model turns out to be useful to estimate the orders of magnitude for system properties like seal and aquifer permeabilities and decompaction coefficients, despite its geometric simplicity. We estimate that the DeGeerdalen aquifer cannot have a permeability that is higher than 1 · 10^?18 m², as otherwise, the fluid flow in the aquifer becomes dominated by topographic‐driven flow. The upper value for the seal permeability is estimated to be 1 · 10^?20 m², as higher values preclude the generation and preservation of underpressure. The porosity rebound is estimated to be <0.1% during the last deglaciation using a decompaction coefficient α_r = 1 · 10^?9 Pa^?1.