Martin D. Hürlimann

Solving Fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions

We present an efficient algorithm to solve a class of two- and 2.5-dimensional (2-D and 2.5-D) Fredholm integrals of the first kind with a tensor product structure and nonnegativity constraint on the estimated parameters of interest in an optimization framework. A zeroth-order regularization functional is used to incorporate a priori information about the smoothness of the parameters into the problem formulation. We adapt the Butler-Reeds-Dawson (1981) algorithm to solve this optimization problem in three steps. In the first step, the data are compressed using singular value decomposition (SVD) of the kernels. The tensor-product structure of the kernel is exploited so that the compressed data is typically a thousand fold smaller than the original data. This size reduction is crucial for fast optimization. In the second step, the constrained optimization problem is transformed to an unconstrained optimization problem in the compressed data space. In the third step, a suboptimal value of the smoothing parameter is chosen by the BRD method. Steps 2 and 3 are iterated until convergence of the algorithm. We demonstrate the performance of the algorithm on simulated data.

The diffusion–spin relaxation time distribution function as an experimental probe to characterize fluid mixtures in porous media

The distribution function between diffusion and spin relaxation time is shown to be a powerful tool to characterize fluid mixtures in porous media. We discuss the nuclear magnetic resonance measurements using diffusion-editing sequences that were developed to measure this quantity. A recently developed two-dimensional inversion routine is used to extract the distribution function from the data. We show both theoretically and experimentally that the technique is suitable for ex situ applications in the presence of grossly inhomogeneous fields. We apply this technique to characterize sedimentary rocks saturated with oil–water mixtures. From the measured diffusion–relaxation time distribution function it is possible to quantify the oil and water saturation, to get a direct indication of the wettability of the porous media, to characterize the bulk properties of the oil and estimate the oil viscosity, and to get information about the geometrical arrangement of the fluid phases in the pore space. We have also ...

Diffusion-relaxation distribution functions of sedimentary rocks in different saturation states

We present diffusion-relaxation distribution functions measured on four rock cores that were prepared in a succession of different saturation states of brine and crude oil. The measurements were performed in a static gradient field at a Larmor frequency of 1.76 MHz. The diffusion-relaxation distribution functions clearly separate the contributions from the two fluid phases. The results can be used to identify the wetting and non-wetting phase, to infer fluid properties of the phases, and to obtain additional information on the geometrical arrangement of the phases. We also observe effects due to restricted diffusion and susceptibility induced internal gradients.

Tortuosity measurement and the effects of finite pulse widths on xenon gas diffusion NMR studies of porous media.

We have extended the utility of NMR as a technique to probe porous media structure over length scales of approximately 100-2000 microm by using the spin 1/2 noble gas 129Xe imbibed into the systems pore space. Such length scales are much greater than can be probed with NMR diffusion studies of water-saturated porous media. We utilized Pulsed Gradient Spin Echo NMR measurements of the time-dependent diffusion coefficient, D(t), of the xenon gas filling the pore space to study further the measurements of both the pore surface-area-to-volume ratio, S/V(p), and the tortuosity (pore connectivity) of the medium. In uniform-size glass bead packs, we observed D(t) decreasing with increasing t, reaching an observed asymptote of approximately 0.62-0.65D(0), that could be measured over diffusion distances extending over multiple bead diameters. Measurements of D(t)/D(0) at differing gas pressures showed this tortuosity limit was not affected by changing the characteristic diffusion length of the spins during the diffusion encoding gradient pulse. This was not the case at the short time limit, where D(t)/D(0) was noticeably affected by the gas pressure in the sample. Increasing the gas pressure, and hence reducing D(0) and the diffusion during the gradient pulse served to reduce the previously observed deviation of D(t)/D(0) from the S/V(p) relation. The Pade approximation is used to interpolate between the long and short time limits in D(t). While the short time D(t) points lay above the interpolation line in the case of small beads, due to diffusion during the gradient pulse on the order of the pore size, it was also noted that the experimental D(t) data fell below the Pade line in the case of large beads, most likely due to finite size effects.

Stability of superparamagnetic iron oxide nanoparticles at different pH values: Experimental and theoretical analysis

The detection of superparamagnetic nanoparticles using NMR logging has the potential to provide enhanced contrast in oil reservoir rock formations. The stability of the nanoparticles is critical because the NMR relaxivity (R(2) ≡ 1/T(2)) is dependent on the particle size. Here we use a molecular theory to predict and validate experimentally the stability of citric acid-coated/PEGylated iron oxide nanoparticles under different pH conditions (pH 5, 7, 9, 11). The predicted value for the critical surface coverage required to produce a steric barrier of 5k(B)T for PEGylated nanoparticles (MW 2000) was 0.078 nm(-2), which is less than the experimental value of 0.143 nm(-2), implying that the nanoparticles should be stable at all pH values. Dynamic light scattering (DLS) measurements showed that the effective diameter did not increase at pH 7 or 9 after 30 days but increased at pH 11. The shifts in NMR relaxivity (from R(2) data) at 2 MHz agreed well with the changes in hydrodynamic diameter obtained from DLS data, indicating that the aggregation behavior of the nanoparticles can be easily and quantitatively detected by NMR. The unexpected aggregation at pH 11 is due to the desorption of the surface coating (citric acid or PEG) from the nanoparticle surface not accounted for in the theory. This study shows that the stability of the nanoparticles can be predicted by the theory and detected by NMR quantitatively, which suggests the nanoparticles to be a possible oil-field nanosensor.

Application of Optimal Control to CPMG Refocusing Pulse Design

We apply optimal control theory (OCT) to the design of refocusing pulses suitable for the CPMG sequence that are robust over a wide range of B(0) and B(1) offsets. We also introduce a model, based on recent progress in the analysis of unitary dynamics in the field of quantum information processing (QIP), that describes the multiple refocusing dynamics of the CPMG sequence as a dephasing Pauli channel. This model provides a compact characterization of the consequences and severity of residual pulse errors. We illustrate the methods by considering a specific example of designing and analyzing broadband OCT refocusing pulses of length 10t(180) that are constrained by the maximum instantaneous pulse power. We show that with this refocusing pulse, the CPMG sequence can refocus over 98% of magnetization for resonance offsets up to 3.2 times the maximum RF amplitude, even in the presence of ±10% RF inhomogeneity.

A rapid measurement of T1/T2: the DECPMG sequence.

The Driven-Equilibrium Carr-Purcell Meiboom-Gill (DECPMG) pulse sequence is a rapid method for obtaining the average ratio of longitudinal to transverse relaxation times (T(1)/T(2)) as a function of T(2). Since this is a one-dimensional experiment, the (T(1)/T(2))T(2) ratio can be acquired, potentially, in just two scans; the second scan being a reference CPMG measurement. Conventionally, T(1)/T(2) is determined from a two-dimensional T(1)-T(2) relaxation correlation experiment. The method described here offers a significant reduction in experimental time without a reduction in signal-to-noise. The (T(1)/T(2)) ratio is useful for comparing the behaviour of liquids in porous media. Here we demonstrate the application of the DECPMG sequence to the study of oil-bearing rocks by differentiating oil or water saturated rock cores, and by observing the relative strengths of surface interaction for water in two types of rock by measuring (T(1)/T(2)) as a function of magnetic field strength.

Steady-state free precession experiments and exact treatment of diffusion in a uniform gradient

We derive an analytic solution for the magnetization of spins diffusing in a constant gradient field while applying a long stream of rf pulses, which is known as the steady-state free precession (SSFP) sequence. We calculate the diffusion-dependent amplitude of the free induction decay (FID) and higher order echoes for pulses with arbitrary flip angle α and pulse spacing TR. Stopped-SSFP experiments were performed in a permanent gradient field and the amplitudes of the first three higher order echoes were measured for a range of values of α and TR. Theoretical results are in excellent agreement with experimental results, using no adjustable parameters. We identify various diffusion regimes in a rather large parameter space of pulsing and relaxation times, diffusion coefficient, and flip angle and discuss the interplay of the relevant time scales present in the problem. This “phase diagram” provides a road map for designing experiments which enhance or suppress the sensitivity to diffusion. We delineate th...

Correlation of transverse and rotational diffusion coefficient: a probe of chemical composition in hydrocarbon oils.

Measurements of relaxation time and diffusion coefficient by nuclear magnetic resonance are well-established techniques to study molecular motions in fluids. Diffusion measurements sense the translational diffusion coefficients of the molecules, whereas relaxation times measured at low magnetic fields probe predominantly the rotational diffusion of the molecules. Many complex fluids are composed of a mixture of molecules with a wide distribution of sizes and chemical properties. This results in correspondingly wide distributions of measured diffusion coefficients and relaxation times. To first order, these distributions are determined by the distribution of molecular sizes. Here we show that additional information can be obtained on the chemical composition by measuring two-dimensional diffusion-relaxation distribution functions, a quantity that depends also on the shape and chemical interactions of molecules. We illustrate this with experimental results of diffusion-relaxation distribution functions on a series of hydrocarbon mixtures. For oils without significant amounts of asphaltenes, the diffusion-relaxation distribution functions follow a power-law behavior with an exponent that depends on the relative abundance of saturates and aromatics. Oils with asphaltene deviate from this trend, as asphaltene molecules act as relaxation contrast agent for other molecules without affecting their diffusion coefficient significantly. In waxy oils below the wax appearance temperature a gel forms. This is reflected in the measured diffusion-relaxation distribution functions, where the restrictions due to the gel network reduce the diffusion coefficients without affecting the relaxation rates significantly.

Low-frequency NMR with a non-resonant circuit

Nuclear magnetic resonance typically utilizes a tuned resonance circuit with impedance matching to transmit power and receive signal. The efficiency of such a tuned coil is often described in terms of the coil quality factor, Q. However, in field experiments such as in well-logging, the circuit Q can vary dramatically throughout the depth of the wellbore due to temperature or fluid salinity variations. Such variance can result in erroneous setting of NMR circuit parameters (tuning and matching) and subsequent errors in measurements. This paper investigates the use of a non-resonant transmitter to reduce the circuit sensitivity on Q and demonstrates that such circuits can be efficient in delivering power and current to the coil. We also describe a tuned receiver circuit whose resonant frequency can be controlled digitally. Experimental results show that a range of common NMR experiments can be performed with our circuits.