Eduard Siebrits

Measuring Hydraulic Fracture Growth in Naturally Fractured Rock

Note: SPE 124919 Reference EPFL-CONF-212832 Record created on 2015-10-08, modified on 2016-08-09

STABILITY ANALYSIS AND DESIGN OF TIME-STEPPING SCHEMES FOR GENERAL ELASTODYNAMIC BOUNDARY ELEMENT MODELS

SUMMARY In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models.1{3 However, there has been no theory to support scientists and engineers in assessing the stability of their boundary element algorithms or to help them with the design of new, more stable algorithms. In this paper we present a general framework for the analysis of the stability of any time-domain boundary element model. We illustrate how the stability theory can be used to assess the stability of existing boundary element models and how the insight gained from this analysis can be used to design more stable time-stepping schemes. In particular, we describe a new time-stepping procedure that we have developed, which has substantially enhanced stability characteristics and greater accuracy for the same computational eort. The new scheme, which we have called ‘the half-step scheme’, is shown to have substantially improved performance for the displacement discontinuity boundary element method commonly used to model dynamic fracture interaction and propagation.

Uniform asymptotic approximations for accurate modeling of cracks in layered elastic media

We present uniform asymptotic solutions (UAS) for displacement discontinuities (DD) that lie within the middle layer of a three layer elastic medium. The DDs are assumed to be normal to the two parallel interfaces between the leastic media, and solutions will be presented for both 2D and 3D elastic media. Using the Fourier transform (FT) method we construct the leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three layer medium. Although a closed form solution will require an infinite series solution, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We present an explicit UAS for elements in which the DD fields are assumed to be piecewise constant throughout a line segment in 2D and a rectangular element in 3D. We demonstrate the usefulness of this UAS by providing a number of examples in which the UAS is used to solve problems in which cracks just touch or cross an interface. The accuracy and efficiency of the algorithm is demonstrated and compared with other numerical methods such as the finite element method and the boudary integral method.

The scaled flexibility matrix method for the efficient solution of boundary value problems in 2D and 3D layered elastic media

We present a method that extends the flexibility matrix method for multilayer elasticity problems to include problems with very thin layers. This method is particularly important for solving problems in which one or a number of very thin layers are juxtaposed with very thick layers. The standard flexibility matrix method suffers from round-off errors and poor scaling of the flexibility equations which occur when one of the layers in the multilayered medium becomes much smaller than the others. The method proposed in this paper makes use of power series expansions of the various components of the flexibility matrix in order to arrive at a system of equations that is appropriately scaled. The effectiveness of the scaled flexibility matrix method is demonstrated on a number of test problems.

Implementation and application of elastodynamic boundary element discretizations with improved stability properties

Direct and indirect time marching boundary element methods often become numerically unstable. Evidence of, and reasons for, these instabilities is provided in this paper. Two new time stepping schemes are presented, both of which are more stable than the existing standard schemes available. In particular, we introduce the Half‐step scheme, which is more accurate and far more stable than existing methods. This scheme, which is demonstrated on a simple crack problem for the displacement discontinuity method, can also be introduced into the direct boundary element method. Implementation of the Half‐step scheme into existing boundary element codes will allow researchers to attack more challenging problems than before.

Stability analysis of model problems for elastodynamic boundary element discretizations

In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models [1]–[3]. In this article we use three distinct model problems to investigate the stability properties of various discretizations that are commonly used to solve elastodynamic boundary integral equations. Using the model problems, the stability properties of a large variety of discretization schemes are assessed. The features of the discretization procedures that are likely to cause instabilities can be established by means of the analysis. This new insight makes it possible to design new time-stepping schemes that are shown to be more stable.

Hydraulic Fracture Propagation in a Naturally Fractured Reservoir

This paper (SPE 128715) was accepted for presentation at the SPE Oil and Gas India Conference and Exhibition, Mumbai, India, 20–22 January 2010, and revised for publication. Original manuscript received 17 February 2010. Revised manuscript received 20 July 2010. Paper peer approved 17 August 2010. Summary We present the results of numerical modeling that quantify the physical mechanisms of mechanical activation of a natural fault because of contact with a pressurized hydraulic fracture (HF). We focus on three stages of interactions: HF approaching, contact, and subsequent infiltration of the fault. Fracture interaction at the contact is shown to depend on four dimensionless parameters: net pressure in the HF, in-situ differential stress, relative angle between the natural fault and the HF, and friction angle of the natural fault. A numerical model based on the displacement discontinuity method (DDM) allowing for fracture closure and Mohr-Coulomb friction was used to calculate the displacements and stresses along the natural fracture as a result of the interaction with the pressurized HF. The analysis of the total stress state along the fault during the HF coalescence stage makes it possible to define a criterion for reinitiation of a secondary tensile crack from the natural fault. We show that the most probable location for tensile-crack initiation is the end of the open zone of the fault where the highest tension peak is generated by the HF contact. In our numerical analysis, we study the magnitude of maximum tensile stress and its position along the fault for a wide range of key dimensionless parameters. Given real reservoir properties, these measurements can be used to detect the possible fracturing scenarios in naturally fractured reservoirs. Using simplified uncoupled modeling of fluid penetration into the fault after the contact with the HF, we demonstrate that either an increase or a decrease of the tensile stress at the opposite side of the fault can be realized depending on the ratio of increments of net pressure and the fluid front as it penetrates the natural fault.

On the numerical stability of time domain boundary element methods

Abstract Time domain elastodynamic boundary element methods are prone to numerical instabilities. Under suitable conditions, these instabilities can swamp the transient response of a system. We show evidence of these instabilities in both the direct and indirect boundary element methods. We summarize the literature on the evidence and causes of these instabilities, and refer to improved algorithms and alternative formulations which are less prone to numerical instabilities. Finally, we make suggestions as to where research should concentrate so that these methods can reach their full potential.