Robert W. Zimmerman

Effect of shear displacement on the aperture and permeability of a rock fracture

The results of experiments using radial and unidirectional flow in a carefully described single rough aperture are reported and compared with numerical predictions. Aperture replicas of a natural sandstone fracture were made at 0, 1 and 2 mm shear displacements using silicone rubber, with a reproducibility of better than 2%. The experimental arrangement permitted shear displacement to be obtained without causing damage to the two displaced surfaces, which thus retained their original (and essentially matching) geometry. Both the number of contact points and the fractional contact area decreased with increasing shear displacement. With increasing shear displacement, mean aperture and standard deviation increased and the ratio of standard deviation to mean aperture increased slightly. Semivariogram studies indicated that as shear displacement increased in the direction normal to the roughness ridges, the aperture distribution became more closely correlated in the direction parallel to the roughness ridges than in the shear direction. Flow tests showed that with increasing shear displacement, the fracture became heterogeneous and anisotropic and became more permeable in the direction perpendicular to the shear displacement than in the direction parallel to the displacement. Simulations were made of flow through the fracture using the Reynolds equation and the actual aperture distribution; the measured hydraulic apertures were generally about 20% lower than those measured numerically.

Effective stress law for the permeability of clay‐rich sandstones

[1] Two models of clay-rich sandstones are analyzed to explain the relative sensitivity of permeability to pore pressure and confining pressure. In one model the clay lines the entire pore wall in a layer of uniform thickness, and in the second model the clay is distributed in the form of particles that are only weakly coupled to the pore walls. Equations of elasticity and fluid flow are solved for both models, giving expressions for the effective stress coefficients in terms of clay content and the elastic moduli of the rock and clay. Both models predict that the permeability will be much more sensitive to changes in pore pressure than to changes in confining pressure. The clay particle model gives somewhat better agreement with data from the literature and with new data on a Stainton sandstone having a solid volume fraction of 8% clay.

Predicting the permeability of sandstone from image analysis of pore structure

A model is developed that allows accurate prediction of the permeability of a core sample of sedimentary rock, based on two-dimensional image analysis of its pore structure. The pore structure is idealized as consisting of a cubic network of pore tubes, with the tubes having an arbitrary distribution of cross-sectional areas and shapes. The areas and perimeters of the individual pores are estimated from image analysis of scanning electron micrographs of thin sections, with appropriate stereological corrections introduced to account for the angle between axis of the pore tube and plane of the thin section. The individual conductances of each tube are estimated from the measured areas and perimeters, using the hydraulic radius approximation. Variations in the pore diameter along the length of the tube are accounted for with a “constriction factor” whose derivation is based on laminar flow through an irregular tube. Effective-medium theory is used to find the effective single-tube conductance, based on the m...

Permeability tensor of three‐dimensional fractured porous rock and a comparison to trace map predictions

The reduction from three- to two-dimensional analysis of the permeability of a fractured rock mass introduces errors in both the magnitude and direction of principal permeabilities. This error is numerically quantified for porous rock by comparing the equivalent permeability of three-dimensional fracture networks with the values computed on arbitrarily extracted planar trace maps. A method to compute the full permeability tensor of three-dimensional discrete fracture and matrix models is described. The method is based on the element-wise averaging of pressure and flux, obtained from a finite element solution to the Laplace problem, and is validated against analytical expressions for periodic anisotropic porous media. For isotropic networks of power law size-distributed fractures with length-correlated aperture, two-dimensional cut planes are shown to underestimate the magnitude of permeability by up to 3 orders of magnitude near the percolation threshold, approaching an average factor of deviation of 3 with increasing fracture density. At low-fracture densities, percolation may occur in three dimensions but not in any of the two-dimensional cut planes. Anisotropy of the equivalent permeability tensor varies accordingly and is more pronounced in two-dimensional extractions. These results confirm that two-dimensional analysis cannot be directly used as an approximation of three-dimensional equivalent permeability. However, an alternative expression of the excluded area relates trace map fracture density to an equivalent three-dimensional fracture density, yielding comparable minimum and maximum permeability. This formulation can be used to approximate three-dimensional flow properties in cases where only two-dimensional analysis is available.

Sensitivity of the impact of geological uncertainty on production from faulted and unfaulted shallow-marine oil reservoirs: objectives and methods

Estimates of recovery from oil fields are often found to be significantly in error, and the multidisciplinary SAIGUP modelling project has focused on the problem by assessing the influence of geological factors on production in a large suite of synthetic shallow-marine reservoir models. Over 400 progradational shallow-marine reservoirs, ranging from comparatively simple, parallel, wave-dominated shorelines through to laterally heterogeneous, lobate, river-dominated systems with abundant low-angle clinoforms, were generated as a function of sedimentological input conditioned to natural data. These sedimentological models were combined with structural models sharing a common overall form but consisting of three different fault systems with variable fault density and fault permeability characteristics and a common unfaulted end-member. Different sets of relative permeability functions applied on a facies-by-facies basis were calculated as a function of different lamina-scale properties and upscaling algorithms to establish the uncertainty in production introduced through the upscaling process. Different fault-related upscaling assumptions were also included in some models. A waterflood production mechanism was simulated using up to five different sets of well locations, resulting in simulated production behaviour for over 35 000 full-field reservoir models. The model reservoirs are typical of many North Sea examples, with total production ranging from c. 15×106 m3 to 35×106 m3, and recovery factors of between 30% and 55%. A variety of analytical methods were applied. Formal statistical methods quantified the relative influences of individual input parameters and parameter combinations on production measures. Various measures of reservoir heterogeneity were tested for their ability to discriminate reservoir performance. This paper gives a summary of the modelling and analyses described in more detail in the remainder of this thematic set of papers.

Analytic Analysis for Oil Recovery During Counter-Current Imbibition in Strongly Water-Wet Systems

We study counter-current imbibition, where a strongly wetting phase (water) displaces non-wetting phase spontaneously under the influence of capillary forces such that the non-wetting phase moves in the opposite direction to the water. We use an approximate analytical approach to derive an expression for saturation profile when the viscosity of the non-wetting phase is non-negligible. This makes the approach applicable to water flooding in hydrocarbon reservoirs, or the displacement of non-aqueous phase liquid (NAPL) by water. We find the recovery of non-wetting phase as a function of time for one-dimensional flow. We compare our predictions with experimental results in the literature. Our formulation reproduces experimental data accurately and is superior to previously proposed empirical models.

Chapter 7 ▸ – Hydromechanical Behavior of Fractured Rocks

This chapter discusses the hydromechanical behavior of fractured rocks. The hydromechanical behavior of rock fractures can be studied on the scale of a single fracture and also on the scale of a fractured rock mass that contains many fractures. The behavior of single fractures must be thoroughly understood before the behavior of fractured rock masses can be understood. If a fracture is located in a rock mass with a given ambient state of stress, the traction acting across the fracture plane can be resolved into a normal component and a shear component. The normal traction gives rise to a normal closure of the fracture. The shear component of the traction causes the two rock faces to undergo a relative deformation parallel to the nominal fracture plane, referred to as “shear deformation.” A tangential traction also typically causes the mean aperture to increase, in which case the fracture dilates. Dilation arises because the asperities of one fracture surface must by necessity ride up to move past those of the other surfaces; hence, shear deformation of a fracture is inherently a coupled process in which both normal and shear displacement occur.

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Fluid flow through random two-dimensional fracture networks is investigated, with the aim of establishing a methodology for estimating the macroscopic effective hydraulic conductivity based on the parameters of the fracture network. A wide range of isotropic networks is examined: the lengths are either uniform, or follow a power law or lognormal distribution; the apertures are either uniform, or proportional to the fracture lengths. A methodology is developed that utilises the fracture density and the aperture distribution, but does not require explicit solution of the flow equations. This method provides an accurate estimate of the macroscopic hydraulic conductivity, for all cases considered, spanning ten orders of magnitude.

Compressibility of two-dimensional pores having n-fold axes of symmetry

The complex variable method and conformal mapping are used to derive a closed-form expression for the compressibility of an isolated pore in an infinite two-dimensional, isotropic elastic body. The pore is assumed to have an n-fold axis of symmetry, and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. The results are validated against some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from three and four terms of the Schwarz–Christoffel mapping function for regular polygons, the compressibilities of a triangle, square, pentagon and hexagon are found (to at least three digits). Specific results for some other pore shapes, more general than the quasi-polygons obtained from the Schwarz–Christoffel mapping, are also presented. An approximate scaling law for the compressibility, in terms of the ratio of perimeter-squared to area, is also tested. This expression gives a reasonable approximation to the pore compressibility, but may overestimate it by as much as 20%.

Creeping Flow Through an Axisymmetric Sudden Contraction or Expansion

Creeping flow through a sudden contraction/expansion in an axisymmetric pipe is studied. Sampsons solution for flow through a circular orifice in an infinite wall is used to derive an approximation for the excess pressure drop due to a sudden contraction/ expansion in a pipe with a finite expansion ratio. The accuracy of this approximation obtained is verified by comparing its results to finite-element simulations and other previous numerical studies. The result can also be extended to a thin annular obstacle in a circular pipe