Upscaling Uncertain Permeability Using Small Cell Renormalization

Sedimentary rocks have structures on all length scales from the millimeter to the kilometer. These structures are generally associated with variations in rock permeability. These need to be modeled if we are to make predictions about fluid flow through the rock. However, existing computers are not powerful enough for us to be able to represent all scales of heterogeneity explicitly in our fluid flow models—hence, we need to upscale. Small cell renormalization is a fast method for upscaling permeability, derived from an analogue circuit of resistors. However, it assumes that the small scale permeability distribution is known. In practice, this is unlikely. The only information available about small scale properties is either qualitative, derived from the depositional setting of the reservoir, or local to the wells as a result of coring or logging. The influence of small scale uncertainty on large scale properties is usually modelled by the Monte Carlo method. This is time-consuming and inaccurate if not enough realisations are used. This paper describes a new implementation of renormalization, which enables the direct upscaling of uncertain small-scale permeabilities to produce the statistical properties of the equivalent coarse grid. This is achieved by using a perturbation expansion of the resistor-derived equation. The method is verified by comparison with numerical simulations using the Monte Carlo method. The prediction of expected large-scale permeability and its standard deviation are shown to be accurate for small cell standard deviations of up to 40% of the mean cell value, using just the first nonzero term of the perturbation expansion. Inclusion of higher order terms allows larger standard deviations to be modeled accurately. Evaluation of cross-terms allows correlations of actual cell values, over and above the background structure of mean cell values. The perturbation method is significantly faster than conventional Monte Carlo simulation. It needs just two calculations whereas the Monte Carlo method needs many thousands of realisations to be generated and renormalized to converge. This results in significant savings in computer time.     

Inversion of time-dependent nuclear well-logging data using neural networks

The purpose of this work was to investigate a new and fast inversion methodology for the prediction of subsurface formation properties such as porosity, salinity and oil saturation, using time‐dependent nuclear well logging data. Although the ultimate aim is to apply the technique to real‐field data, an initial investigation as described in this paper, was first required; this has been carried out using simulation results from the time‐dependent radiation transport problem within a borehole. Simulated neutron and γ‐ray fluxes at two sodium iodide (NaI) detectors, one near and one far from a pulsed neutron source emitting at ～14 MeV, were used for the investigation. A total of 67 energy groups from the BUGLE96 cross section library together with 567 property combinations were employed for the original flux response generation, achieved by solving numerically the time‐dependent Boltzmann radiation transport equation in its even parity form. Material property combinations (scenarios) and their correspondent teaching outputs (flux response at detectors) are used to train the Artificial Neural Networks (ANNs) and test data is used to assess the accuracy of the ANNs. The trained networks are then used to produce a surrogate model of the expensive, in terms of computational time and resources, forward model with which a simple inversion method is applied to calculate material properties from the time evolution of flux responses at the two detectors. The inversion technique uses a fast surrogate model comprising 8026 artificial neural networks, which consist of an input layer with three input units (neurons) for porosity, salinity and oil saturation; and two hidden layers and one output neuron representing the scalar photon or neutron flux prediction at the detector. This is the first time this technique has been applied to invert pulsed neutron logging tool information and the results produced are very promising. The next step in the procedure is to apply the methodology to real data.     

Modelling in-situ upgrading of heavy oil using operator splitting method

The in-situ upgrading (ISU) of bitumen and oil shale is a very challenging process to model numerically because of the large number of components that need to be modelled using a system of equations that are both highly non-linear and strongly coupled. Operator splitting methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as the splitting error. The best splitting method for modelling a given process minimises the splitting error whilst improving computational performance compared to a fully implicit approach. Although operator splitting has been widely used for the modelling of reactive-transport problems, it has not yet been applied to the modelling of ISU. One reason is that it is not clear which operator splitting technique to use. Numerous such techniques are described in the literature and each leads to a different splitting error. While this error has been extensively analysed for linear operators for a wide range of methods, the results cannot be extended to general non-linear systems. It is therefore not clear which of these techniques is most appropriate for the modelling of ISU. In this paper, we investigate the application of various operator splitting techniques to the modelling of the ISU of bitumen and oil shale. The techniques were tested on a simplified model of the physical system in which a solid or heavy liquid component is decomposed by pyrolysis into lighter liquid and gas components. The operator splitting techniques examined include the sequential split operator (SSO), the Strang-Marchuk split operator (SMSO) and the iterative split operator (ISO). They were evaluated on various test cases by considering the evolution of the discretization error as a function of the time-step size compared with the results obtained from a fully implicit simulation. We observed that the error was least for a splitting scheme where the thermal conduction was performed first, followed by the chemical reaction step and finally the heat and mass convection operator (SSO-CKA). This method was then applied to a more realistic model of the ISU of bitumen with multiple components, and we were able to obtain a speed-up of between 3 and 5.     

Padé coefficients for the solution of the wave dispersion equation under ice

The equations governing the propagation of linear gravity waves in ice-covered waters of finite depth is delineated for the linear elastic deformation of the ice plates that are modeled as elastically supported. The possible limiting condition for the validity of the assumptions involved in the formulation of the problem is discussed. A solution procedure for the solution of the wave dispersion equation under ice is discussed and a set of coefficients synthesized using the properties of infinite series and Padé approximants. Direct application of these coefficients for the calculation of wave characteristics in ice-covered sea will eliminate the need for iterative procedure, and hence will reduce the computational time. The derived coefficients were used for the computation of the wave characteristics of laboratory simulated waves and compared with the values obtained through iteration and the error was found to be less than 2%.     

Upscaling Permeability Measurements Within Complex Heterolithic Tidal Sandstones

We investigate numerically the effect of sample volume on the effective single-phase permeability of heterolithic tidal sandstones, using three-dimensional models reconstructed directly from large rock specimens measuring 45 × 30 × 15 cm. We find that both individual and averaged effective permeability values vary as a function of sample volume, which indicates that permeability data obtained from core-plugs will not be representative at the scale of a reservoir model grid-block regardless of the number of measurements taken. However, the error introduced by averaged data may be minimized using the appropriate averaging scheme for a given facies type and flow direction.     

Using vorticity to quantify the relative importance of heterogeneity, viscosity ratio, gravity and diffusion on oil recovery

The vorticity of the displacement velocity is used to derive dimensionless numbers that can be used to quantify the relative importance of viscosity ratio, gravity, diffusion/dispersion and permeability heterogeneity on secondary hydrocarbon recovery. Using this approach, a new objective measure of the impact of permeability and porosity heterogeneity on reservoir performance is obtained. This is used, in conjunction with other dimensionless numbers, to analyse the relative impact of heterogeneity, buoyancy effects, mobility ratio and dispersion on breakthrough time and recovery at 1 pore volume injected during first contact miscible gas injection. This is achieved using results obtained from detailed simulation of miscible displacements through a range of geologically realistic reservoir models. This study goes some way towards developing a unified mathematical framework to determine under which flow conditions reservoir heterogeneity becomes more important than other physical processes. We propose that comparison of these dimensionless numbers can be used to identify the key factors controlling recovery and thus assist the engineer in determining appropriate enhanced oil recovery techniques to improve recovery.     

Estimating the extent of the spraying zone on a sea-going ship

The maximum extent of ship spraying for a medium-sized fishing trawler (MFV) of Soviet type has been considered. A simple geometrical model for generating the spray due to ship-wave collisions has been applied to determine the maximum height of the spray source above the ship deck. The maximum height of the spray source has been assumed to depend on the ship speed relative to the moving waves and an empirical constant specific to a given type of ship. A unique field data set (Kuzniecov et al., 1971) of the height of the upper limit of ice accretion on the foremast of an MFV has been used to determine the value of the empirical constant for this vessel. For documented air-sea and ship motion parameters, the trajectories of droplets hitting the upper parts of the accretion on the foremast have been calculated using the equation of droplet motion for each reported icing event.The heights of the spray source computed by the trajectory method for each case of icing were compared with the heights of the spray source determined by a correlation involving the ship speed relative to the waves and the vertical extent of spray. The best fit was obtained for an empirical constant value of 0.535.The model performance was tested using an independent data set (Sharapov, 1971) on the spraying zone of an MFV. The tests showed that this model predicts the extent of the spraying zone over the ship with satisfactory accuracy and suggest that it should be incorporated into an integrated ship icing model.Finally, the model was run for several ship speeds, headings and wind speeds to examine the effect of these parameters on the maximum height of the spray hitting the ship's foremast. It was found that this height increases with wind speed and ship speed and is maximum for ship headings of 120–130°.     

Recent developments in offshore rig/ platform evacuation

During severe storms, evacuation systems for offshore rigs and platforms currently in use have proven themselves to be inadequate. Typically, during deployment of a lifeboat, it is often damaged to the point of not being seaworthy before it reaches the ocean surface. This is especially the case for cable-launched boats where a pendulum-like motion of the craft on its cables is often set up. It is less of a problem for free-fall lifeboats. Even when a craft reaches the ocean surface intact, high winds and waves can drive it back against the rig/platform structure. This paper describes the state-of-the-art of evacuation. It focuses on two new systems being developed by the authors in Newfoundland, Canada.     

Analytical solution of polymer slug injection with viscous fingering

We present an analytical solution to estimate the minimum polymer slug size needed to ensure that viscous fingering of chase water does not cause its breakdown during secondary oil recovery. Polymer flooding is typically used to improve oil recovery from more viscous oil reservoirs. The polymer is injected as a slug followed by chase water to reduce costs; however, the water is less viscous than the oil. This can result in miscible viscous fingering of the water into the polymer, breaking down the slug and reducing recovery. The solution assumes that the average effect of fingering can be represented by the empirical Todd and Longstaff model. The analytical calculation of minimum slug size is compared against numerical solutions using the Todd and Longstaff model as well as high resolution first contact miscible simulation of the fingering. The ability to rapidly determine the minimum polymer slug size is potentially very useful during enhanced oil recovery (EOR) screening studies.