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In this work we propose a method for mini-frac analysis via the direct numerical modeling for permeability and closure stress determination. The idea is to adjust the model parameters to match the simulated and field pressure curves by the virtue of the optimization algorithm to automatize the routine. To this end, the Planar 3D hydraulic fracturing model is improved to take into account the wellbore storage, fracture initiation and pumping through a finite perforation interval. The reservoir mechanical behavior is based on Biot’s equations naturally coupled with the mass conservation law and force balance in the fracture and wellbore. This model is capable to describe all the pressure curve features from the fracture initiation up to the after closure filtration. The optimization workflow is based on Levenberg–Marquardt algorithm with Jacobi matrix calculated via a direct differentiation method. The numerical stability and performance of the inverse problem solution is verified. The pressure curve sensitivity analysis is conducted and the impact of the parameters’ uncertainties on the pressure change is calculated. The proposed method is applied to a field case and compared with classical mini-frac methods. We demonstrate that the wellbore storage impact governs the pressure increase during the fracture initiation and prevents the steep pressure decline after closure. We also show that the shortening of the perforation length generates additional pressure support at the end of and after the fracture closure. The recovered permeability and closure stress appears to be stable in presence of uncertainties in Young’s modulus and Biot’s coefficient.

In our previous works (Baykin and Golovin 2016, 2017) it is shown that inhomogeneity of the permeability of a reservoir might play a significant role in the choice of the direction of the hydraulic fracture propagation. This effect was demonstrated using the fully coupled model of the hydraulic fracture in the poroelastic medium under the restriction on the geometry of the fracture similar to the one used in the Khristianovich–Geertsma–de Klerk model. In present paper, we extend our observations to a model of the planar 3D hydraulic fracture in the inhomogeneous poroelastic medium. In a series of numerical experiments we demonstrate that, depending on the permeability contrast of different parts of a layered reservoir, the hydraulic fracture propagates either within the low-permeable or the high-permeable zones of the reservoir. This behaviour is explained by the non-uniform action of the backstress that is formed on the fracture walls due to the pressure of the pore fluid.

In this paper we demonstrate the influence of the pore pressure on the development of a hydraulically-driven fracture in a poroelastic medium. We present a novel numerical model for propagation of a planar hydraulic fracture and prove its correctness by the analysis of the numerical convergence and by the comparison with known solutions. The advantage of the algorithm is that it does not require distinguishing of the fracture's tips and reconstruction of the numerical mesh according to the fracture propagation. Next, we perform a thorough analysis of the interplay of fluid filtration and redistribution of stresses near the fracture. We demonstrate that the fracture length decreases with the increase of the Biot's number (the parameter that determines the contribution of the pore pressure to the stress) and explain this effect by analysing the near-fracture pore pressure, rock deformation and stresses. We conclude that the correct account for the fluid exchange between the fracture and the rock should be based not only on physical parameters of the rock and fluid, but also on the analysis of stresses near the fracture.

This paper presents the coupled model of a hydraulic fracturing and proppant transport. The former is described in terms of enhanced pseudo-3D model that considers height growth across two symmetric stress barriers, while the latter is given by two-dimensional transport model, stemming from the solution of an elliptical equation for fluid pressure and advection equation for the proppant transport. These two sub-modules are solved numerically using implicit time integration in hydraulic part and explicit time stepping in the transport part. In addition to that, interpolation is used to couple the two models with different grids. Results of several numerical simulations are presented for different configurations to demonstrate the interplay between these two modules. In particular, the developed coupled scheme allows us to study phenomena associated with complex fluid flow within the fracture, such as for the case of Saffman-Taylor instability., Although the EP3D model was used in this study, the Planar3D model is a benchmark for engineering simulators that are currently being developed. Authors have understood how to represent the better agreement between results by both models, how to explain results in coupled model in more general sense, so we need to withdraw the manuscript and re-submit it in significantly updated form

In our previous work the fully coupled model of the planar hydraulic fracture propagation in a poroelastic medium was developed. In present paper we simplify this model to the case of axisymmetric fracture propagation in order to find the conditions whereby the poroelastic effects are essential. We consider two dimensionless complexes responsible for the transition from 1D to 3D filtration and for the poroelastic influence on the fracture geometry. Analyzing a large set of case studies, we find the critical values for these parameters below which the aforementioned physical effects are not important. Particularly, we demonstrate that 1D pressure diffusion is preserved until the diffusion length scale is 2.4 times smaller than the fracture size. In turn, the diffusion scale plays a central role in the poroelastic stress influence on the fracture propagation. Also, we show that in case of low closure stress the pressure diffusion is dominated not by the filtration process but by the rock deformation with a slight impact on the fracture geometry. We believe that this study will become useful for the appropriate model selection in engineering practice.

This paper presents an analytic self-similar solution describing an equilibrium of a planar semi-infinite hydraulic fracture in a poroelastic medium. The equilibrium of the fracture is sustained by a balance between the pressure of fluid within the fracture and the confining stress due to elasticity of nearby poroelastic material satisfying the Biot model assumptions. Propagation of fluid in the fracture, leakoff through the fracture’s walls and filtration in the poroelastic medium towards the fracture’s tip form a dipole-type pattern of flow. The solution is used for the verification of a numerical algorithm developed earlier for the pressure transient analysis in water injection wells.

The paper addresses the problem of modeling a fracture propagation in linear elastic porous media driven by injection of non-Newtonian power-law fluid. The model involves the lubrication theory equation expressing the mass conservation of fluid and a nonlocal singular integral relation for the fracture aperture as a functional of the fluid pressure. We assume only the viscosity-dominated case in which the fracture toughness is neglected. This allows considering the fracture as an opened part of a preexisting closed fracture of larger length. The numerical method consists in solving equations of the model over the whole length of the preexisting fracture without distinguishing the tip region of the opened part. The problem is solved via the finite element method. The weak formulation of the solution allows pressure singularity with the required asymptotics near the fracture tip. Comparison of the numerical and the exact self-similar solutions in the case of a constant flow rate, zero fluid leakoff, and zero fracture toughness reveals the accuracy of the approximate solution to at least O(h1/2), where h is the maximal linear size of the grid cells. As an illustration, we also demonstrate the numerical experiments with periodic fluid injection and variable fluid efficiency.

The mappings with bounded weighted (p, q)-distortion are natural generalizations of the class of mappings with bounded distortion which appears as a doubly indexed scale for p = q = n in the absence of weight functions. In case n − 1 < q ≤ p = n, the mappings with bounded (p, q)-distortion were studied previously in a series of articles under the additional assumption that the mapping enjoys Luzin’s N-property. In this article we present the first facts of the theory of mappings with bounded (p, q)-distortion which are obtained without additional analytical assumptions. The core of the theory consists of the new analytical properties of pushforward functions; in particular, we prove that the gradient of the pushforward function vanishes almost everywhere on the image of the branch set. Some estimates are given on the capacity of the images of condensers under mappings with bounded (p, q)-distortion. We obtain Liouville-type theorems and the singularity removal theorems for the mappings of this class, and we apply these theorems to classifying manifolds.

In the paper a model for description of a hydraulic fracture propagation in inhomogeneous poroelastic medium is proposed. Among advantages of the presented numerical algorithm, there are incorporation of the near-tip analysis into the general computational scheme, account for the rock failure criterion on the base of the cohesive zone model, possibility for analysis of fracture propagation in inhomogeneous reservoirs. The numerical convergence of the algorithm is verified and the agreement of our numerical results with known solutions is established. The influence of the inhomogeneity of the reservoir permeability to the fracture time evolution is also demonstrated.