A laboratory study of tracer tomography

Abstract

A tracer tomographic laboratory study was performed with consolidated fractured rock in three-dimensional space. The investigated fractured sandstone sample was characterized by significant matrix permeability. The laboratory transport experiments were conducted using gas-flow and gas-tracer transport techniques that enable the generation of various flow-field patterns via adjustable boundary conditions within a short experimental time period. In total, 72 gas-tracer (helium) tests were performed by systematically changing the injection and monitoring configuration after each test. For the inversion of the tracer breakthrough curves an inversion scheme was applied, based on the transformation of the governing transport equation into a form of the eikonal equation. The reliability of the inversion results was assessed with singular value decomposition of the trajectory density matrix. The applied inversion technique allowed for the three-dimensional reconstruction of the interstitial velocity with a high resolution. The three-dimensional interstitial velocity distribution shows clearly that the transport is dominated by the matrix while the fractures show no apparent influence on the transport responses.

Résumé

Une étude en laboratoire de tomographie par traceur a été réalisée sur une roche fracturée consolidée dans un espace à 3 dimensions. L’échantillon de grès fracturé étudié se caractérisait par une perméabilité de matrice élevée. Les expériences de traçage en laboratoire ont été conduites en utilisant des techniques de transfert d’un courant gazeux et d’un traceur gazeux permettant la création de modèles variés de champ d’écoulement, grâce à des conditions aux limites ajustables dans un laps de temps expérimental court. Au total, 72 tests avec traceur gazeux (hélium) ont été réalisés, en changeant systématiquement la configuration de l’injection et du système de mesure après chaque test. Pour l’inversion des courbes de marquage du traceur, on a appliqué un procédé d’inversion basé sur la transformation de l’équation de transfert principale en une forme d’équation iconale. La fiabilité des résultats de l’inversion a été démontrée par une décomposition en valeurs singulières de la matrice densité de trajectoire. La technique de l’inversion appliquée a tenu compte de la reconstruction à 3 dimensions de la vitesse interstitielle avec une haute résolution. La distribution de la vitesse interstitielle en 3 dimensions montre clairement que le transfert est contrôlé par la matrice tandis que les fractures ne montrent pas d’influence apparente sur les réponses du transfert.

Resumen

Se llevó a cabo un estudio tomográfico de laboratorio de trazadores en roca fracturada consolidada en el espacio tridimensional. La muestra investigada de arenisca fracturada se caracterizó por la significativa permeabilidad de la matriz. Los experimentos de transporte de laboratorio fueron realizados usando flujo de gas y técnicas de transporte de trazadores que permite la generación de varios esquemas de campos de flujo de campo a través de condiciones de límites ajustables dentro de corto periodo de tiempo experimental. En total, 72 pruebas con gas trazador (helio) fueron realizados cambiando sistemáticamente la configuración de inyección y monitoreo después de cada prueba. Para la inversión de las curvas de ruptura del trazador un esquema de inversión, basado en la transformación de una ecuación de transporte gobernante se aplica una forma de la ecuación eikonal. La confiabilidad de los resultados de inversión fue evaluada con una descomposición en valores singulares de la matriz de densidad de la trayectoria. La técnica de inversión aplicada permitió la reconstrucción tridimensional de la velocidad intersticial con una alta resolución. La distribución de la velocidad intersticial tridimensional muestra claramente que el transporte está dominado por la matriz mientras que las fracturas no muestran una influencia aparente en las respuestas de transporte.

摘要

对坚固裂隙岩体进行了三维空间示踪剂断层摄影术实验室研究。调查的断裂砂岩样品特点是具有显著的基质渗透率。采用气流和气体示踪运移技术进行了实验室运移实验,这些技术能够在短的试验时段内产生通过可调边界条件的各种流场模式。总共进行了72项气体示踪(氦)试验,每个试验后都系统地调整注入和监测配置。针对示踪剂突破曲线的反演,采用了基于调整运移公式转化成程函公式的反演方法。采用轨迹密度矩阵的奇值分解对反演结果的可靠性进行了评估。应用的反演技术可三维重建具有高分辨率的间隙速度。三维间隙流速分布清楚地表明,运移主要受控于基质,而断裂显示对运移反应无明显影响。

Resumo

Foi levado a cabo um estudo laboratorial de tomografia de um traçador, no espaço tridimensional, numa rocha consolidada fraturada. A amostra estudada, arenito fraturado, foi caraterizada por permeabilidade matricial significativa. Os testes laboratoriais de transporte foram realizados utilizando técnicas de transporte de fluxo de gás e de gás traçador que permitiram a geração de vários padrões de campo de fluxo através de condições de fronteira ajustáveis dentro de um período de tempo experimental curto. No total foram realizados 72 testes de gás traçador (hélio), alterando sistematicamente o esquema de injeção e de monitorização após cada teste. Para inversão das curvas de chegada do traçador foi aplicado um esquema de inversão, a partir da transformação da equação de transporte numa forma da equação iconal. A confiança dos resultados da inversão foi avaliada pela decomposição em valores singulares da matriz densidade da trajetória. A técnica de inversão aplicada permitiu a reconstrução tridimensional da velocidade intersticial com uma alta resolução. A distribuição da velocidade intersticial tridimensional mostra claramente que o transporte é dominado pela matriz, enquanto as fraturas não mostram, aparentemente, ter influência nas respostas de transporte.

Appendices

Appendix 1: SIRT algorithm

The description of the SIRT algorithm with regard to the inversion of tracer breakthrough curves, given in the following, is partly based on the work proposed by Dines and Lytle (1979) and by Hu et al. (2011) in context of the inversion of seismic signals and hydraulic transient pressure signals, respectively.

The interstitial or mean velocity v _m of a conservative tracer can be expressed by the following line integral, which relates the peak arrival time of a conservative tracer, assuming a pulse injection, to the ratio between porosity and hydraulic conductivity K multiplied by the potential head Φ (Vasco and Datta-Gupta, 1999):

$$ {t}_{\mathrm{t}}\left({x}_2\right)={\displaystyle \underset{x_1}{\overset{x_2}{\int }}\frac{n(s)}{K(s)\phi (s)}ds={\displaystyle \underset{x_1}{\overset{x_2}{\int }}\frac{ds}{v_{\mathrm{m}}(s)},}} $$

(7)

where t _t is the travel time of the peak of a tracer signal between point x₁ (source) and the observation point x₂ (receiver) along the propagation path (s).

The integral (Eq. 7) can be transformed into a discrete inverse problem by approximating the integral with a summation

$$ {t}_{\mathrm{t},\mathrm{i}}={\displaystyle \sum_{j=1}^n\frac{1}{v_{\mathrm{m},\mathrm{j}}}{d}_{\mathrm{ij}}} $$

(8)

where d _ij is the distance along trajectory i in cell j and v _m,j is the average interstitial velocity within cell j. The applied SIRT algorithm for solving Eq. (8) is based on an iterative technique, where one equation (one trajectory) is analysed at a time. The algorithm requires an initial estimate of the average interstitial velocity. The interstitial velocity is calculated along each trajectory as follows:

$$ {t}_{\mathrm{t},\mathrm{i}}={\displaystyle \sum_{j=1}^n\frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}{d}_{\mathrm{ij}},} $$

(9)

where v ^q_m,j indicates the estimated interstitial velocity after the qth iteration. The applied SIRT algorithm minimizes for each source-receiver configuration the least squares residual between the predicted and the measured peak tracer arrival time,

$$ \varDelta {t}_{\mathrm{t},\mathrm{i}}=\left[{t}_{\mathrm{t},\mathrm{i}}^{\mathrm{m}}-{t}_{\mathrm{t},\mathrm{i}}^{\mathrm{e}}\right]={\displaystyle \sum_{j=1}^n\left(\varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}{d}_{\mathrm{ij}}\right)} $$

(10)

with t ^m_t,i the measured travel time, t ^e_t,i the estimated travel time, Δt _t,i the difference of the ith measured and estimated peak tracer arrival time and ${t}_{\mathrm{t},\mathrm{i}}^{\mathrm{m}}$ describes the applied changes to cell j of the qth iteration. Following Dines and Lytle (1979) an additional minimum criterion can be defined for the determination of the distribution of $\frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}$:

$$ f\left(\varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}\right)={\displaystyle \sum_{j=1}^n{\left(\varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}\right)}^2\to \min .} $$

(11)

Minimizing the function f subject to Eq. (10) yields an update of the interstitial velocity for each cell sampled by trajectory i. For the minimization the Lagrange multipliers method can be utilized. Thereby the new function K is minimized:

$$ K={\displaystyle \sum_{j=1}^n\left[{\left(\varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}\right)}^2-\lambda \varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}{d}_{\mathrm{i},\mathrm{j}}\right]+\lambda \varDelta {t}_{\mathrm{t},\mathrm{i}}} $$

(12)

The first derivative becomes zero if:

$$ \varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}=\frac{\lambda {d}_{\mathrm{i},\mathrm{j}}}{2} $$

(13)

Substituting Eq. (13) into Eq. (10) λ can be expressed as follows:

$$ \lambda =\frac{2\varDelta {t}_{\mathrm{t},\mathrm{i}}}{{\displaystyle \sum_{j=1}^n{d}_{\mathrm{i},\mathrm{j}}^2}} $$

(14)

The final solution is achived by substituting Eq. (14) into Eq. (13). The Lagrange multipliers method leads for a particular trajectory i, in the qth iteration to the incremental correction:

$$ \varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}=\frac{\varDelta {t}_{\mathrm{t},\mathrm{i}}{d}_{\mathrm{ij}}}{{\displaystyle \sum_{j=1}^n{d}_{\mathrm{ij}}^2}} $$

(15)

The corrections are applied by averaging the incremental corrections of each single cell after all N trajectories have been analysed:

$$ \varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}=\frac{1}{M_{\mathrm{j}}}{\displaystyle \sum_{i=1}^N\varDelta \frac{1}{v_{\mathrm{m},\mathrm{j}}^{\mathrm{q}}}} $$

(16)

where M _j is the number of trajectories passing through cell j.

Appendix 2: minimum time principle

Figure 7 displays two points A and B that are located in two half-spaces. The two half-spaces are characterized by their velocities v ₁ and v ₂. In order to minimize the travel time of an induced signal traveling between point A and B, a third point P located on the interface between the two half-spaces is introduced. The travel time between A and B can be expressed as follows:

$$ f(x)=\frac{\sqrt{a^2+{x}^2}}{v_1}+\frac{\sqrt{a{b}^2+{\left(c-x\right)}^2}}{v_2} $$

(17)

The minimum travel time can be found by setting the first derivative to zero:

$$ f\prime (x)=\left[{b}^2+{\left(c-x\right)}^2\right]{x}^2-\frac{\left[{v}_1^2{\left(c-x\right)}^2\left({a}^2+{x}^2\right)\right]}{v_2^2}=0 $$

(18)

The value x can be determined with the bisection method.

Fig. 7

Schematic illustrating the minimum-velocity algorithm