Relationship between fracture spacing and bed thickness in sedimentary rocks:Approach by means of Michaelis-Menten equation

Fractures occur in nearly all rocks at the Earth’s surface and exert essential control on the mechanical strengths of rock masses and permeability.The fractures strongly impact the stability of geological or man-made structures and flow of water and hydrocarbons,CO_(2) and storing waste.For this,the dependence of opening mode fracture spacing(s)on bed thickness(t)in sedimentary basins(reservoirs)is studied in this context.This paper shows that the MichaeliseMenten equation can provide an algebraic expression for the nonlinear s-t relationship.The two parameters have clear geological meanings:a is the maximum fracture spacing which can no longer increase with increasing t,and b is the characteristic bed thickness when s=0.5a.The tensile fracture strength(C)of the brittle beds during the formation of tensile fractures can be estimated from the two parameters.For sandstones of 16 areas reported in the literature,C ranges from 2.7 MPa to 15.7 MPa with a mean value of 8 MPa,which lies reasonably within the range of tensile strengths determined experimentally.This field-based approach by means of MichaeliseMenten equation provides a new method for estimating the tensile fracture strength of rock layers under natural conditions.

A novel image-based approach for interactive characterization of rock fracture spacing in a tunnel face

This paper presents a novel integrated method for interactive characterization of fracture spacing in rock tunnel sections.The main procedure includes four steps:(1)Automatic extraction of fracture traces,(2)digitization of trace maps,(3)disconnection and grouping of traces,and(4)interactive measurement of fracture set spacing,total spacing,and surface rock quality designation(S-RQD)value.To evaluate the performance of the proposed method,sample images were obtained by employing a photogrammetrybased scheme in tunnel faces.Experiments were then conducted to determine the optimal parameter values(i.e.distance threshold,angle threshold,and number of fracture trace grouping)for characterizing rock fracture spacing.By applying the identified optimal parameters involved in the model,the proposed method could lead to excellent qualitative results to a new tunnel face.To perform a quantitative analysis,three methods(i.e.field,straightening,and the proposed method)were employed in the same study and comparisons were made.The proposed method agrees well with the field measurement in terms of the maximum and average values of measured spacing distribution.Overall,the proposed method has reasonably good accuracy and interactive advantage for estimating the ultimate fracture spacing and S-RQD.It can be a possible extension of existing methods for fracture spacing characterization for two-dimensional(2D)rock tunnel faces.

Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.

Fracture spacing behavior in layered rocks subjected to different driving forces： a numerical study based on fracture infilling process

Natural layered rocks subjected to layerparallel extension typically develop an array of opening- mode fractures with a remarkably regular spacing. This spacing often scales with layer thickness, and it decreases as extension increases until fracture saturation is reached. To increase the understanding of how these opening-mode fractures form in layered rocks, a series of 2D numerical simulations are performed to investigate the infilling process of fractures subjected to different driving forces. Numerical results illustrate that any one of the following could be considered as a driving force behind the propagation of infilling fractures： thermal effect, internal fluid pressure, direct extension loading, or pure compressive loading. Fracture spacing initially decreases with loading process, and at a certain ratio of fracture spacing to layer thickness, no new fractures nucleate （saturated）. Both an increase in the opening of the infilled fractures and interface delamination are observed as mechanisms that accommodate additional strain. Interface debonding stops the transition of stress from the neighboring layers to the embedded central layer, which may preclude further infilling of new fractures. Whatever the driving force is, a large overburden stress and a large elastic contrast between the stiff and soft layers （referred to as a central or fractured layer and the top and bottom layers） are key factors favoring the development of tensile stress around the infilled fractures in the models. Fracture spacing is expected to decrease with increasing overburden stress. Numerical results highlight the fracturing process developed in heterogeneous and layered sedimentary rocks which provides supplementary information on the stress distribution and failure-induced stress redistribution., It also shows, in detail, the propagation of the fracture zone and the interaction of the fractures, which are impossible to observe in field and are difficult to consider with static stress analysis approaches.

An unequal fracturing stage spacing optimization model for hydraulic fracturing that considers cementing interface integrity

Determining reasonable fracturing stage spacing is the key to horizontal well fracturing.Different from traditional stage spacing optimization methods based on the principle of maximum stimulated reservoir volume,in this paper,by considering the integrity of the wellbore interface,a fracture propagation model was established based on displacement discontinuity method and the competition mechanism of multifracture joint expansion,leading to the proposal of an unequal stage spacing optimization model.The results show that in the first stage,the interfacial fractures spread symmetrically along the axis of the central point during that stage,while in the second and subsequent stages,the interfacial fractures of each cluster extend asymmetrically along the left and right sides.There are two kinds of interface connectivity behaviour:in one,the existing fractures first extend and connect within the stage,and in the other,the fractures first extend in the direction close to the previous stage,with the specific behaviour depending on the combined effect of stress shadow and flow competition during hydraulic fracture expansion.The stage spacing is positively correlated with the number of fractures and Young’s modulus of the cement and formation and is negatively correlated with the cluster spacing and horizontal principal stress difference.The sensitivity is the strongest when the Young’s modulus of the cement sheath is 10-20 GPa,and the sensitivity of the horizontal principal stress difference is the weakest.

Characterizing Fracture Spatial Patterns by Using Semivariograms

Semivariogram is applied to fracture data obtained from detailed scanlinesurveys of nine field sites in western New York, USA in order to investigate the spatial patterns ofnatural fractures. The length of the scanline is up to 36 m. How both fracture spacing and fracturelength vary with distance is determined through semivariogram calculations. In this study, theauthors developed a FORTRAN program to resample the fracture data from the scanline survey. Bycalculating experimental semivariogram, the authors found five different types of spatial patternsthat can be described by linear, spherical, reversed spherical, polynomial I (for aO) models, of which the last three arc newly proposed in this study. Thewell-structured semivariograms of fracture spacing and length indicate that both the location of thefractures and the length distribution within their structure domains are not random. The results ofthis study also suggest that semivariograms can provide useful information in terms of spatialcorrelation distance for fracture location and fracture length. These semivariograms can also beutilized to design more efficient sampling schemes for further surveys. as well as to define thelimits of highly probable extrapolation of a structure domain.

The formation of orthogonal joint systems and cuboidal blocks:New insights gained from flat-lying limestone beds in the region of Havre-Saint-Pierre(Quebec,Canada)

Vertical orthogonal joints are a common feature in shallow crustal rocks.There are several competing theories for their formation despite the ubiquity.We examined the exceptional exposures of orthogonal joints in flat-lying Ordovician limestone beds from the Havre-Saint-Pierre Region in Quebec,Canada(north shore of Saint-Lawrence River)to test conceptual models of joint formation in a natural setting.In the region,the spacing of cross-joints is consistently larger than the spacing of systematic joints by a factor of 1.5 approximately.The joint-spacing-to-bed-thickness ratios(s/t)are much larger in these beds(s/t=4.3 for systematic joints,and 6.4 for cross-joints)than those in higher strained strata along the south shore of the Saint-Lawrence River(s/t=1),highlighting the effect of tectonic strain in decreasing fracture spacing and block size.The high values of s/t indicate that cross-joint formation was unlikely caused by a switch from compression to tension once a critical s/t ratio for systematic joints was reached(as hypothesized in previous studies).We proposed a new model for the formation of orthogonal joint systems where the principal stress axes locally switch during the formation of systematic fractures.The presence of ladder-shaped orthogonal joints suggests a state of effective stress withσ_(1)^(∗)≫0>σ_(2)^(∗)>σ_(3)^(∗)and whereσ_(2)^(∗)-σ_(3)^(∗)is within the range of fracture strength variability at the time of fracture.This research provides a new mechanical model for the formation of orthogonal joint systems and cuboidal blocks.

Pelvic compartment syndrome caused by retroperitoneal hematoma of pelvic fracture

Retroperitoneal hematoma is an obligatory complication in pelvic ring fracture.~1 In most cases, the bleeding originates from venous vessels of the presacral plexus, small arteries and veins from fracture fragments. External fixation of the pelvis can control blood loss by reducing diastasis and dramatically decreasing the volume of the pelvis. But this tamponade effect can not prevent the presence of hematoma in the adjoining retroperitoneal space. It is well known that complication of retroperitoneal hematoma is infection and sepsis.