Lars Holden

Modeling of fluvial reservoirs with object models

An object model for fluvial reservoirs that has been developed from 1985 to present is described. It uses a formal mathematical object model (marked point process) describing the distributions of four facies: channel, crevasse, barrier, and background. Realisations from the model are generated using the Metropolis-Hastings simulation algorithm with simulated annealing conditioning on the volume ratios and well observations. The main challenge has been to find a suitable parameterization of the geology of fluvial reservoirs, and to find and implement the generating function of the channels in the simulation algorithm. The model and simulation algorithm can be conditioned on arbitrary well paths including horizontal wells and paths with partly missing observations, well test data, well contacts, seismic data, and general geological knowledge.

The Genomic HyperBrowser: inferential genomics at the sequence level

The immense increase in the generation of genomic scale data poses an unmet analytical challenge, due to a lack of established methodology with the required flexibility and power. We propose a first principled approach to statistical analysis of sequence-level genomic information. We provide a growing collection of generic biological investigations that query pairwise relations between tracks, represented as mathematical objects, along the genome. The Genomic HyperBrowser implements the approach and is available at http://hyperbrowser.uio.no.

A numerical method for first order nonlinear scalar conservation laws in one-dimension

Abstract A numerical method for first order nonlinear scalar hyperbolic conservation laws in one-dimension is presented, using an idea by Dafermos. In this paper it is proved that it may be used as a numerical method for a general flux function and a general initial value. It is possible to give explicit error estimates for the numerical method. The error in the method is far smaller than in any other method. The numerical method is illustrated in an example.

Global Upscaling of Permeability in Heterogeneous Reservoirs; The Output Least Squares (OLS) Method

This paper presents a new technique for computing the effective permeability on a coarse scale. It is assumed that the permeability is given at a fine scale and that it is necessary to reduce the number of blocks in the reservoir model. Traditional upscaling methods depend on local boundary conditions. It is well known that the permeability may depend heavily on the local boundary condition chosen. Hence the estimate is not stable. We propose to compute a coarse scale permeability field that minimises the error, measured in a global norm, in the velocity and pressure fields. This leads to stable problems for a large number of reservoirs. We present several algorithms for finding the effective permeability values. It turns out that these algorithms are not significantly more computational expensive than traditional local methods. Finally, the method is illustrated by several numerical experiments.

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.

Stochastic Structural Modeling

A consistent stochastic model for faults and horizons is described. The faults are represented as a parametric invertible deformation operator. The faults may truncate each other. The horizons are modeled as correlated Gaussian fields and are represented in a grid. Petrophysical variables may be modeled in a reservoir before faulting in order to describe the juxtaposition effect of the faulting. It is possible to condition the realization on petrophysics, horizons, and fault plane observations in wells in addition to seismic data. The transmissibility in the fault plane may also be included in the model. Four different methods to integrate the fault and horizon models in a common model is described. The method is illustrated on an example from a real petroleum field with 18 interpreted faults that are handled stochastically.

Uncertainties in Reservoir Production Forecasts

Being aware of uncertainties in forecasts of production characteristics is important for reservoir management. Decisions concerning further appraisal drilling, flexibility in development plans, and selection among reservoir prospects all require that uncertainties are taken into account. This study addresses the problem of quantifying uncertainties due to incomplete knowledge of the initial reservoir characteristics, with emphasis on the difference between heterogeneity modeling and assessment of uncertainty. In this study we outline a formalism for uncertainty modeling in a Bayesian framework and present an extensive case study of a North Sea Brent Group reservoir. The results are obtained by computer software implemented such that no human interference is required once the stochastic reservoir model is established. Once the stochastic reservoir model is established, multiple realizations can be generated, rescaled, and fed into a fluid flow simulator to forecast production characteristics and quantify uncertainty associated with response parameters such as cumulative oil production, recovery factors, and water cuts. The results demonstrate that uncertainty in model parameters contributes significantly to the overall uncertainty. The most influential parameters in our case study include the sealing capacity of major faults, seismic velocities used in depth conversion, and average porosities and shale continuity within the main reservoir sandstone.

A new front-tracking method for reservoir simulation

This paper reports on the standard nonlinear, partial-differential equations that describe flow in porous media which can be separated into a pressure equation and saturation equations. If the diffusion term is ignored, the saturation equations describe a physical problem where sharp discontinuities in the physical data are possible. Finite-difference methods used to solve these equations typically exhibit numerical dispersion. They also show numerical stability problems so that very short timesteps may be required. Use of an implicit formulation can reduce this limitation of the timestep length, but this will not solve the numerical dispersion problem. New methods in the field of hyperbolic conservation laws have led to alternative solution procedures for the saturation equations. A simulator based on an implicit pressure, explicit saturation (IMPES) formulation and these methods is under development. The goal of this development work is a 3D, three-phase simulator. In this reservoir simulator, the pressure equation is solved by a finite-element method (FEM). The grid for the pressure equation can therefore be fitted to the reservoir geometry and the geometry of the sharp discontinuities in the saturations with great flexibility. The linear equation system is solved with a preconditioned conjugate-gradient method.

A tensor estimator for the homogenization of absolute permeability

An estimator for an effective permeability tensor based on one-phase incompressible flow is presented. Effective large-scale permeability tensors are well approximated by rough approximations to the fine-scale pressure. The estimator works for all kinds of heterogeneous reservoirs and is fairly independent of boundary conditions.

Adaptive independent Metropolis–Hastings

We propose an adaptive independent Metropolis--Hastings algorithm with the ability to learn from all previous proposals in the chain except the current location. It is an extension of the independent Metropolis--Hastings algorithm. Convergence is proved provided a strong Doeblin condition is satisfied, which essentially requires that all the proposal functions have uniformly heavier tails than the stationary distribution. The proof also holds if proposals depending on the current state are used intermittently, provided the information from these iterations is not used for adaption. The algorithm gives samples from the exact distribution within a finite number of iterations with probability arbitrarily close to 1. The algorithm is particularly useful when a large number of samples from the same distribution is necessary, like in Bayesian estimation, and in CPU intensive applications like, for example, in inverse problems and optimization.