A model for temperature variations in sedimentary basins due to random radiogenic heat sources

The thermal structure of a sedimentary basin is controlled by its thermal conductivity, its boundary conditions, water flow, rate of sedimentation and erosion and radiogenic heat sources. The radiogenic heat production in the sediments is known to vary over several orders of magnitude, with the lowest values in evaporites and carbonates and the highest values in black shales. Due to a paucity of information available on the existing heat sources, this parameter can be represented with a known mean value and a Gaussian correlation structure rather than a deterministic function. In this paper, the 1-D steady-state thermal structure in a sedimentary basin has been modelled in a stochastic framework with a random radiogenic heat source, and analytical expressions for the first two moments of the temperature field have been obtained. A synthetic example has been examined to quantify the error bounds on the temperature field due to uncertainties in the radiogenic heat sources.     

The inverse problem for heat flow data in the presence of thermal conductivity variations

Summary. We demonstrate a method of performing linear programming optimizations of functionals of subsurface temperature, when thermal conductivity is a known piecewise-constant function. Data comprise heat flow measurements on the flat isothermal surface of this structure, within which heat transfer is by steady-state conduction. Two-dimensionality is assumed. The approach involves establishing constraints which demand the continuity of temperature and the normal component of heat flow across all internal boundaries. These unknown functions are expanded as truncated Fourier series whose coefficients become unknowns of the linear programming solution vector; linear relations are established between these coefficients which guarantee harmonicity of temperature in each region of uniform conductivity, as well as the continuity requirements. Variations of the formalism are detailed for three simple types of geometry. As an example the method is applied to a heat flow data set from Sass, Killeen & Mustonen over the Quirke Lake Syncline of Ontario, Canada.     

Convective and conductive heat transfer in sedimentary basins

In the Earth's crust the temperature is largely controlled by heat conduction. However, under some circumstances, the thermal state is disturbed by advection of heat associated with groundwater flow. The corresponding thermal disturbance depends on the water flow velocity (modulus and direction) and therefore thermal data may be used to constrain the pattern of natural fluid flow. In this paper, some models of thermal disturbance induced by convective heat transfer are presented. They are based on the assumption that the water flow is concentrated in thin permeable structures such as aquifer or fault zones. The steady-state and transient thermal effects associated with such scenarios are computed using a somewhat idealized model which depends on a small number of parameters: flow rate, time, aquifer geometry and thermal parameters of surrounding rocks. In order to extract the conductive and convective components of heat transfer from temperature data and to estimate the corresponding fluid flow rate, it is first necessary to estimate the thermal conductivity field. The problem of the estimation of thermal conductivity in clay-rich rocks, based on laboratory and in-situ measurements, is emphasized. Then a method is proposed for the inversion of temperature data in terms of fluid flow. Vertical and lateral variations of thermal conductivity are taken into account and the fluid flow is assumed to be concentrated on a specified surface (2-D quasi-horizontal pattern). Thermal effects of the flow are simulated by a distribution of surface heat production which can be calculated and then inverted in terms of horizontal fluid flow pattern.     

The pulse-probe method of conductivity measurement

Summary. An alternative to the steady heating of a cylindrical probe, in the 'needle-probe' method of conductivity measurement, is the observation of the thermal decay from a short, calibrated, heat pulse. The theoretical solution is the time-differential of that for the former method, and requires only the measurement of point temperatures rather than the determination of a gradient. A careful analysis of the theoretical decay function shows that it should be possible to make accurate conductivity measurements in as little as three probe 'time constants' if external information is available on the heat capacity of the medium. A self-contained method, using two temperature determinations from a record about six time-constants long, can be used where such information is not available. The theory was tested by measurements on the ocean-floor, and the data correspond to the theory when a correction was applied for some internal probe conduction problems.     

Terrestrial heat flow and lithospheric thermal structure in the Chagan Depression of the Yingen-Ejinaqi Basin,north central China

The Chagan Depression in the Yingen-Ejinaqi Basin, located at the intersection of the Paleo-Asian Ocean and the Tethys Ocean domains is an important region to gain insights on terrestrial heat flow, lithospheric thermal structure and deep geodynamic processes. Here, we compute terrestrial heat flow values in the Chagan Depression using a large set of system steady-state temperature data from four representative wells and rock thermal conductivity. We also estimate the “thermal” lithospheric thickness, mantle heat flow, ratio of mantle heat flow to surface heat flow and Moho temperature to evaluate the regional tectonic framework and deep dynamics. The results show that the heat flow in the Chagan Depression ranges from 66.5 to 69.8 mW/m², with an average value of 68.3 ± 1.2 mW/m². The Chagan Depression is characterized by a thin “thermal” lithosphere, high mantle heat flow, and high Moho temperature, corresponding to the lithospheric thermal structure of “cold mantle and hot crust” type. We correlate the formation of the Yingen-Ejinaqi Basin to the Early Cretaceous and Cenozoic subduction of the western Pacific Plate and the Cenozoic multiple extrusions. Our results provide new insights into the thermal structure and dynamics of the lithospheric evolution in central China.     

Geothermal field interpretation in south-central Kansas for parts of the Nemaha Anticline and flanking Cherokee and Sedgwick basins

The first assessment of temperature conditions in Kansas was made by geothermal gradient computations from temperature measurements in shallow boreholes, and these conditions are related to structural patterns, sedimentary cover and underlying basement properties. The area of south-central Kansas was selected for detailed study of geothermal character in relation to the geology. The aim was to quantify the relations and to determine the relationship of different variables of the temperature field. Input parameters included geothermal variables of gradient and temperature, and structure and sediment thickness. Two approaches were used: (1) the numerical computation of theoretical temperature-depth models based on conductive heat transfer, and (2) a map-comparison technique based on algebraic methods. The temperature field information usable for the map comparison is different in response to different measurements (nonequilibrium BHTs and temperatures from logged measurements in equilibrium). Derived from modelled results (plotted isotherms on cross-sections), a close relation between gradients and thermal conductivity of the sediments was confirmed. The most noticeable effect on the geothermal field, as noted quantitatively from the map-comparison study, is the relation of thickness of outcropping Permo-Pennsylvanian units because of their different thermal conductivity. The eastward increase of mean gradients is inversely related to the total sediment thickness, but this is mostly recognizable using the shallow temperature gradients. This dissimilarity gives additional evidence for a close link between gradients and thermal conductivity of the sediments in which the temperature measurements were made. The effect caused by the structure in deeper (older) units is not important and seems not to be significant, nor is the influence of the basement rock composition as indicated by the temperature modelling.     

Accounting for mineralogical composition and origin of soils and sedimentary rocks in thermal property predictions

A methodology for calculating the thermal conductivity of soils and rocks is developed which takes into account their origin and mineralogical composition.This method utilizes three approaches.One is founded on the structural modeling of contact heat interaction between particles and fills and estimates the statistical probability distribution of the particles in the volume of the medium.The second approach analyses perturbation to the temperature field of the matrix medium by ellipsoidal inclusions.The third approach is to find the mean thermal conductivity of the solid skeleton in the universal model at different composition of rock-forming minerals.     

Geothermal Characterization of the St. Lawrence Lowlands Sedimentary Basin,Québec,Canada

Defining temperature at depth to identify geothermal resources relies on the evaluation of the Earth heat flow based on equilibrium temperature measurements as well as thermal conductivity and heat generation rate assessment. Such high-quality geothermal data can be sparse over the region of interest. This is the case of the St. Lawrence Lowlands sedimentary basin covering 20,000 km² to the south of Québec, Canada, and enclosing only three wells up to a depth of 500 m with equilibrium heat flow measurements. However, more than 250 oil and gas exploration wells have been drilled in this area, providing for this study (parce que c'est 93 sinon) 81 locations with bottom-hole temperature up to a depth of 4300 m, however, not at equilibrium. Analyzing these data with respect to the deep geothermal resource potential of this sedimentary basin requires evaluating the thermal conductivity and heat generation rate of its geological units to properly extrapolate temperature downward. This was done by compiling literature and recent thermal conductivity measurements in outcrop and core samples as well as new heat generation rate estimates from spectral gamma ray logs to establish a first thermal assessment of geological units deep down into the basin. The mean thermal conductivity of the thermal units varies from 2.5 to 6.3 W/m·K, with peak values in the basal sandstones, while the heat generation rate varies from 1.6 to 0.3 µW/m³, decreasing from the upper caprocks toward the base of the sequence. After correcting the bottom-hole temperatures for drilling disturbance with the Harrison correction and subsequently for paleoclimate variations, results indicate a mean geothermal gradient of 23.1 °C/km, varying from 14 to 40 °C/km. Evaluating the basin thermal state from oil and gas data is a significant challenge facilitated by an understanding of its thermal properties.     

Thermal properties of sediments in southern Israel: a comprehensive data set for heat flow and geothermal energy studies

To decipher the thermal structure of the sedimentary veneer in southern Israel, new values of thermal conductivity and porosity as well as of the radiogenic heat budget are provided. Thermal conductivity is measured for lithotypes and scaled up for geological formations. The new data are higher than most of the previously measured values, in particular for sandstones and siltstones, whose mean values are 5.0 and 2.9 W m^?1 K^?1. Mean values of the most abundant lithotypes, which are dolomites and limestones, are on the order of 4.1 and 2.7 W m^?1 K^?1, respectively. The total radiogenic heat production of the sedimentary cover varies slightly over southern Israel, due to variable lithology and total sediment thickness, yielding a maximum heat flow on the order of 4 mW m^?2 where the sedimentary section is thickest (ca. 7 km). A temperature prognosis was made by calculating temperature profiles to the top of the crystalline basement at five well locations applying the new thermal‐conductivity data set and three scenarios of surface heat flow (50, 55 and 60 mW m^?2). The calculated temperatures best match with measured drillstem‐test temperatures by using heat‐flow values close to the upper bound of range. Surface heat flow on the order of 55–60 mW m^?2 is supported by a reevaluation of an existing temperature log and the application of thermal conductivity from this study. The temperature prediction for southern Israel shows values of 100–120°C at 3500–4500 m depth, indicating a geothermal potential that can be used for heating as well as electricity production.     

A boundary layer model for mantle convection with surface plates

Summary. A simple, analytical model for mantle convection with mobile surface plates is presented. Our aim is to determine under what conditions free convection can account for the observed plate motions, and to evaluate the thermal structure of the mantle existing under these conditions. Boundary layer methods are used to represent two-dimensional cellular convection at large Rayleigh and infinite Prandtl numbers. The steady-state structure consists of cells with isentropic interiors enclosed by thermal boundary layers. Lithospheric plates are represented as upper surfaces on each cell free to move at a uniform speed. Buoyancy forces are concentrated in narrow rising and decending thermal plumes; torques imparted by these plumes drive both the deformable mantle and overlying plate. Solutions are found for a comprehensive range of cell sizes. We derive an expression for the plate speed as a function of its length, the mantle viscosity and surface heat flux. Using mean values for these parameters, we find that thermal convection extending to 700 km depth can move plates at 1 cm yr^-1, while convection through the whole mantle can move plates at 4–5 cm yr^-1. Analysis of the steady-state temperature field, for the case of heating from below, shows that the upper thermal boundary layer develops a complex structure, including an 'asthenosphere' defined by a local maximum in the geotherm occurring at depths of 50–150 km.     

On the importance of rock thermal conductivity and heat flow density in basin and petroleum system modelling

Basin and petroleum systems are routinely modelled to provide qualitative and quantitative assessments of a hydrocarbon play. The importance of the rock thermal properties and heat flow density in thermal modelling the history of a basin are well-known, but little attention is paid to assumptions of the thermal conductivity, present-day heat flow density and thermal history of basins. Assumed values are often far from measured values when data are available to check parameters, and effective thermal conductivity models prescribed in many basin simulators require improvement. The reconstructed thermal history is often justified by a successful calibration to present-day temperature and vitrinite reflectance data. However, a successful calibration does not guarantee that the reconstruction history is correct. In this paper, we describe the pitfalls in setting the thermal conductivity and heat flow density in basin models and the typical uncertainties in these parameters, and we estimate the consequences by means of a one-dimensional model of the super-deep Tyumen SG-6 well area that benefits from large amounts of reliable input and calibration data. The results show that the entire approach to present-day heat flow evaluations needs to be reassessed. Unreliable heat flow density data along with a lack of measurements of rock thermal properties of cores can undermine the quality of basin and petroleum system modelling.     

南极洛多姆冰帽BJ钻孔粒雪层的热学性质及其温度分布

根据详细的密度资料,对洛多姆冰帽BJ钻孔粒雪层的热学性质进行了讨论。应用非均匀介质热传导方程描述活动层温度分布。通过两种简化的层状结构和竖向运动模型给出了解析解。计算和实测温度差异的分析表明边界条件的选取非常重要。     

Neumann stochastic finite element method for calculating temperature field of frozen soil based on random field theory

To study the effect of uncertain factors on the temperature field of frozen soil, we propose a method to calculate the spatial average variance from just the point variance based on the local average theory of random fields. We model the heat transfer coefficient and specific heat capacity as spatially random fields instead of traditional random variables. An analysis for calculating the random temperature field of seasonal frozen soil is suggested by the Neumann stochastic finite element method, and here we provide the computational formulae of mathematical expectation, variance and variable coefficient. As shown in the calculation flow chart, the stochastic finite element calculation program for solving the random temperature field, as compiled by Matrix Laboratory (MATLAB) software, can directly output the statistical results of the temperature field of frozen soil. An example is presented to demonstrate the random effects from random field parameters, and the feasibility of the proposed approach is proven by comparing these results with the results derived when the random parameters are only modeled as random variables. The results show that the Neumann stochastic finite element method can efficiently solve the problem of random temperature fields of frozen soil based on random field theory, and it can reduce the variability of calculation results when the random parameters are modeled as spatially random fields.     

Approximate depth averages of electrical conductivity from surface magnetotelluric data

This paper presents a simple non-linear method of magnetotelluric inversion that accounts for the computation of depth averages of the electrical conductivity profile of the Earth. The method is not exact but it still preserves the non-linear character of the magnetotelluric inverse problem. The basic formula for the averages is derived from the well-known conductance equation, but instead of following the tradition of solving directly for conductivity, a solution is sought in terras of spatial averages of the conductivity distribution. Formulas for the variance and the resolution are then readily derived. In terms of Backus-Gilbert theory for linear appraisal, it is possible to inspect the classical trade-off curves between variance and resolution, but instead of resorting to linearized iterative methods the curves can be computed analytically. The stability of the averages naturally depends on their variance but this can be controlled at will. In general, the better the resolution the worse the variance. For the case of optimal resolution and worst variance, the formula for the averages reduces to the well-known Niblett-Bostick transformation. This explains why the transformation is unstable for noisy data. In this respect, the computation of averages leads naturally to a stable version of the Niblett-Bostick transformation. The performance of the method is illustrated with numerical experiments and applications to field data. These validate the formula as an approximate but useful tool for making inferences about the deep conductivity profile of the Earth, using no information or assumption other than the surface geophysical measurements.     

THERMAL PROPERTIES AND TEMPERATURE DISTRIBUTION OF SNOW/FIRN ON THE LAW DOWE ICE CAP, ANTARCTICA

Based on detailed measurements of density and a numerous data on temperature in shallow bore holes (about 20m deep), the thermal properties and temperature distribution of snow/firn layer on the Law Dome ice cap, Antarctica, are discussed. According to a review of works on thermal properties of snow by Yen (1981), a relationship between thermal conductivity (K) and density (~) is proposed to be expressed by a formula, K=0.0784+2.697_(ρ~2). Then an eqation of heat transfer in a deformed ununiform medium is applied and solved analytically by two approaches. Comparison of calculated and measured temperatures indicates that the difference is mainly dependent on the determination of boundary donditions.     

Soil thermal properties and heat transfer processes near Ny-Alesund, northwestern Spitsbergen, Svalbard

The annually thawing active layer of permafrost is central to considerations of climate change consequences in arctic areas and interpretations of deep permafrost temperatures that constitute and exceptional archive of past climate change. Moreover, a sound understanding of the thermal regime of the active layer is of great interest, because all chemical, biological and physical processes are concentrated there. The author studied this layer by examining the soil physical properties and heat transfer processes that dictate soil temperatures for an arctic desert site in northwestern Spitsbergen. A wide array of soil physical properties based on field observations and laboratory measurements were defined. These include mineralogy, grain size distribution, local regolith thickness, porosity, density, typical soil moisture profile, heat capacity and thermal conductivity. Heat transfer processes were studied through modeling of soil temperatures. The heat transfer model accounted for much of the observed soil thermal regime. It was found that thermal conduction, phase change of soil water at 0°C, and changes in unfrozen water content are the primary thermal processes that explain the observed soil temperatues in this field site. Melt-water infiltration, which is often overlooked in the energy budget, causes abrupt warming events and delivers considerable energy to the soil in late spring. An increase in frequency or magnitude of infiltration events could mimic simple spring time surface warming. Advection of ground water and soil internal evaporation were found to be generally unimportant at the site studied.     

Effects of active crustal movements on thermal structure in subduction zones

In young suduction zones we observe steady uplift of island arcs. The steady uplift of island arcs is always accompanied by surface erosion. The long duration of uplift and erosion effectively transports heat at depth to shallower parts by advection. If the rates of uplift and erosion are sufficiently large, such a process of heat transportation will strongly affect thermal structure in subduction zones. First, we quantitatively examine the effects of uplift and erosion on thermal structure by using a simple 1-D heat conduction model, based on the assumption that the initial thermal state is in equilibrium. The results show that temperature increase, Δ T  , due to uplift and erosion can be approximately evaluated by Δ T  = ν _e tβ at depth, where ν _e is the rate of uplift (erosion), t is the duration of uplift (erosion), and β is the gradient of the geotherm in the initial state. Next, considering the effects of vertical crustal movements such as uplift and erosion in island arcs and subsidence and sedimentation in ocean trenches, in addition to the effects of radioactive heat generation in the crust, frictional heating at plate boundaries and accretion of oceanic sediments to overriding continental plates, we numerically simulate the evolution process of the thermal structure in subduction zones. The result shows that the temperature beneath the island arc gradually increases as a result of uplift and erosion as plate subduction progresses. Near the ocean trench, on the other hand, the low-temperature region gradually expands as a result of sedimentation and accretion in addition to direct cooling by the cold descending slab. The surface heat flow expected from this model is low in fore-arc basins, high in island arcs and moderately high in back-arc regions.     

改进的热惯量模式及遥感土壤水分

本文提出一个考虑地表显热通量及潜热通量的热惯量模式。模式利用了地面定标的方法并充分利用了热象图的空间分布信息,因而较大幅度地提高了估算土壤水分含量的精度。     

APPARENT THERMAL DIFFUSIVITY IN SOIL: ESTIMATION FROM THERMAL RECORDS AND SUGGESTIONS FOR NUMERICAL MODELING

The concept of apparent thermal diffusivity in soil is discussed and defined as consisting of an intrinsic and a pseudo part. A method that uses a numerical model of heat conduction and measured soil temperatures to estimate apparent thermal diffusivity is introduced. The method requires measurement of temperatures at three depths to make an estimate for diffusivity between the lowest and uppermost depths. This method is applied to a five-day data set obtained in Sweden. Time-dependent apparent diffusivities are estimated for periods of five days, one day, and three hours. An empirical method is used to determine the intrinsic diffusivity. Values close to those calculated for the empirical relationships have been obtained for the one- and five-day periods, whereas values of the three-hour period show wider variation. The results indicate that the outlined method is feasible and satisfactory for estimating the intrinsic thermal diffusivity in soils near the surface on a daily basis. It also is argued that the root-mean-square error (RMSE) of a simulation using the estimated diffusivity, together with the diffusivity, is a better indicator for non-conductive processes than diffusivity alone.     

Probabilistic analysis of embankment slope stability in frozen ground regions based on random finite element method

Prediction on the coupled thermal-hydraulic fields of embankment and cutting slopes is essential to the assessment on evolution of melting zone and natural permafrost table, which is usually a key factor for permafrost embankment designin frozen ground regions. The prediction may be further complicated due to the inherent uncertainties of materialproperties. Hence, stochastic analyses should be conducted. Firstly, Karhunen-Loeve expansion is applied to attain the random fields for hydraulic and thermal conductions. Next, the mixed-form modified Richards equation for mass transfer (i.e., mass equation) and the heat transport equation for heat transient flow in a variably saturated frozen soil are combined into one equation with temperature unknown. Furthermore, the finite element formulation for the coupled thermal-hydraulic fields is derived. Based on the random fields, the stochastic finite element analyses on stability of embankment are carried out. Numerical results show that stochastic analyses of embankment stability may provide a more rational picture for the distribution of factors of safety (FOS), which is definitely useful forembankment design in frozen ground regions.